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Solution of fractional integral equations via fixed point results

Abstract

In this paper, we introduce two new concepts of F-contraction, called dual \(F^{*}\)-weak contraction and triple \(F^{*}\)-weak contraction, which generalize the existing contractions in the sense of Wardowski, Jleli and Samet as well as Skof. These new generalizations embed their roots in the aim devoted to extending the generalized Banach contraction conjuncture to the class of F-contraction type mappings with the use of multiple F-type functions. Furthermore, we establish the existence of a unique fixed point for such contractions under certain conditions. Fractional calculus can be used to precisely change or control the fractal dimension of any random or deterministic fractal with coordinates that can be expressed as functions of one independent variable. We apply our main result to weaken certain conditions on the fractional integral equations. Finally, we discuss the significance of our obtained results in comparison with certain renowned ones in the literature.

1 Introduction and preliminaries

In a plethora of real and theoretical world problems, the existence of the solution to a problem is equivalent to the existence of a fixed point. Therefore, fixed points are of phenomenal importance in different areas of science and have become the subject of scientific research. Metric fixed point theory was initiated by Banach [1] with a principle known as Banach contraction principle. Banach established a remarkable fixed point theorem for a contraction \(\mathfrak{F}:X\to X\) in a metric space \((X,d)\) by introducing the following contraction condition:

$$ d(\mathfrak{F}x,\mathfrak{F}y) \le k d (x,y) \quad \text{for all } k \in [0,1), x,y\in X. $$
(1)

Over the years, the Banach contraction principle has been generalized in numerous directions. In most of the generalizations, either the topology is weakened or the contractive nature of the mapping is weakened (for example, see [215] etc.).

The aim of this research is to establish some interesting results concerning new generalizations or extensions of the Banach contraction principle. The idea behind this research has its roots in some extensions of the generalized Banach contraction conjuncture by introducing three auxiliary functions in the main contraction condition and by weakening some constraints on the fractional integral equations. At first, we review the literature and observe some interesting generalizations and extensions of the Banach contraction condition.

Boyd and Wong [4] generalized the Banach contraction condition by replacing the constant k with a control function \(\varphi : [0, \infty ] \to [0, \infty ]\) such that

$$ d(\mathfrak{F}x,\mathfrak{F}y)\le \varphi \bigl(d(x,y)\bigr) \quad \text{for all } x,y\in X,$$
(2)

where, for each \(s>0\), \(s > \varphi (s)\).

Geraghty [5] presented a mapping \(\mathfrak{F}:X\to X\) in a metric space \((X,d)\) with the following contraction condition:

$$ d(\mathfrak{F}x,\mathfrak{F}y)>0 \quad \text{implies} \quad d( \mathfrak{F}x, \mathfrak{F}y)\le \zeta \bigl(d(x,y)\bigr)d(x,y),\quad x,y\in X,$$
(3)

where the function \(\zeta :(0,\infty )\to (0,1)\).

Skof [6] established a fixed point theorem for a mapping \(\mathfrak{F}:X\to X\) in a metric space \((X,d)\) using the following contraction condition:

$$ \varphi \bigl(d(\mathfrak{F}x,\mathfrak{F}y)\bigr)\le \alpha \varphi \bigl(d(x,y)\bigr)+ \beta \varphi \bigl(d(x,\mathfrak{F}x)\bigr)+\gamma \varphi (d(y,\mathfrak{F}y) $$
(4)

for all \(x,y\in X\), and \(\varphi :[0,\infty )\to [0,\infty )\), \(\alpha ,\beta ,\gamma \in [0,1)\) with \(0\le \alpha +\beta +\gamma <1\).

In 2001, James Merryfield [15] generalized the Banach contraction principle by introducing a conjecture named generalized Banach contraction conjecture. Merryfield established fixed point results using a minimum of a set of powers for the operator rather than requiring a single operator. He proposed a mapping \(\mathfrak{F}:X\to X\) in a metric space \((X,d)\) such that

$$ \min \bigl\{ d \bigl( {{\mathfrak{F}}^{i}}x,{{\mathfrak{F}}^{i}}y \bigr):i=1,\ldots p \bigr\} \le kd (x,y) \quad \text{for all } x,y \in X. $$
(5)

In 2012, [16] Wardowski generalized Banach contraction in an interesting way by applying a control function F on the mapping involved. The use of this method brings a wide range of potential applications in distinct disciplines, including physics, economics, engineering, computer science, and so forth. Wardowski [16] introduced a mapping \(\mathfrak{F}:X\to X\) in a metric space \((X,d)\) using the following contraction condition:

$$ d ( \mathfrak{F}x,\mathfrak{F}y )>0 \quad \text{implies} \quad \tau +F \bigl( d ( \mathfrak{F}x,\mathfrak{F}y ) \bigr)\le F \bigl( d ( x,y ) \bigr) $$
(6)

for all \(x,y\in X \) and \(F:{{\mathbb{R}}_{+}}\mapsto \mathbb{R}\), \(\tau >0\). Furthermore, Sumati Kumari [10] generalized the notion of F-contraction in the view of d-metric spaces and investigated the uniqueness of fixed point and coincidence point of such mappings. Recently, Proinov [7] introduced two auxiliary functions in his main contraction condition as follows:

$$ \psi \bigl( d ( \mathfrak{F}x,\mathfrak{F}y ) \bigr) \le \varphi \bigl( d (x,y ) \bigr) \quad \text{with } d( \mathfrak{F}x, \mathfrak{F}y)>0 $$
(7)

for all \(x,y\in X, \psi \), \(\varphi : ( 0,\infty )\to \mathbb{R}\).

We shall focus on the most interesting extensions of the Banach contraction principle following the approach inspired by Wardowski [16] via the idea presented by James Marryfield [15]. We extend the idea of generalized Banach contraction conjecture to the class of F-contraction mappings. That is, following Wardowski’s idea of F-contraction and the conjecture presented by James Merryfield [15], we generalize F-contraction by introducing two types: a dual \(F^{*}\)-weak contraction and a triple \(F^{*}\)-weak contraction. We also establish existence and uniqueness theorems for these new kinds of contractions.

In recent years, fractional analysis has been augmented as a powerful tool on the strength of its effective applications in various scientific research areas like mathematical biology, applied mathematics, dynamics, chaos theory, statistics, control theory, and optimization. The development of new differential and integral operators in the field of fractional analysis along with the applications to fractional-order differential and integral equations has attracted the consideration of researchers worldwide (for details, see [2, 8, 1722]). The nonlinear fractional differential equations have a valuable role in various fields of science, such as engineering, biology, fluid mechanics, physics, chemistry, and bio-physics.

In the literature, the contractive type conditions and their generalizations have been used for establishing the fixed point results and applied for developing the existence theorems of real and theoretical word problems whose mathematical models are differential and integral equations involving fractional operators. In 2016, Gopal and Abbas [23] presented an application to nonlinear fractional differential equation and proved the existence of solutions for the nonlinear fractional differential problem

$$\begin{aligned}& ^{C}{{D}^{\beta }}x(t)=f\bigl(t,x(t)\bigr)\quad (0< t< 1, 1< \beta \le 2)\\& \text{via}\quad x(0)=0,\qquad x(1)= \int _{0}^{\eta }{x(s)\,\mathrm{d}s\quad (0< \eta < 1).} \end{aligned}$$
(8)

Gopal and Abbas [23] established the existence theorem by assuming that there exists a function \(\xi :\mathbb{R}\times \mathbb{R}\to \mathbb{R}\) and \(\tau >0\) such that

$$ \bigl\vert f(t,a)-f(t,b) \bigr\vert \le \frac{\Gamma (\beta +1)}{5}{{e}^{- \tau }} \vert a-b \vert $$
(9)

for all \(t\in [0,1]\) and \(a,b\in \mathbb{R}\) with \(\xi (a,b)>0\). By using the conditions of the existence theorem along with the defined mapping in a certain space they proved that the mapping is restricted by the following F-contraction condition:

$$ \tau +F\bigl(d(\mathfrak{F}x,\mathfrak{F}y)\bigr)\le F\bigl(d(x,y) \bigr).$$
(10)

In 2020, Hanadi Zahed, based on the proved fixed point theorem, provided some new sufficient conditions for the existence of solutions of an integral boundary value problem for the following scalar nonlinear Caputo fractional differential equations with fractional order (1, 2):

$$ {}_{a}^{C}D{}_{t}^{q}\bigl(x(t) \bigr)=f\bigl(t,x(t)\bigr)\quad \text{for } t \in (a,b),$$
(11)

with

$$x(a)=0,\qquad x(b)= \int _{a}^{\lambda }{x(s)\,ds}\quad(a< \lambda < b). $$

Hanadi Zahed [24] established the existence theorem by applying certain conditions on an unknown function along with the concerned operator. That is, the function \(f\in C([a,b]\times \mathbb{R},\mathbb{R})\) and there exists a constant K such that

$$ \Lambda = \frac{K{{(b-a)}^{q}}}{\Gamma (1+q)} \biggl( 1+ \frac{2K(b-a)}{(2(b-a)-{{(\lambda -a)}^{2}})} \biggl( 1+ \frac{\lambda -a}{1+q} \biggr) \biggr)\in (0,1)$$

and a number \(p\in (0,1]\) such that

$$ \bigl\vert f(t,x)-f(t,y) \bigr\vert \le K{{ \vert x-y \vert }^{p}},\quad x,y\in \mathbb{R}, t\in [a,b].$$
(12)

Zahed used these conditions of the existence theorem along with the defined mapping in a certain space and proved that the mapping is restricted by the following F-contraction condition:

$$ \ln {{ \biggl( \frac{1}{\Lambda } \biggr)}^{\frac{1}{p}}}+\frac{1}{p} \ln \bigl(d \bigl(\mathfrak{F}(x),\mathfrak{F}(y)\bigr)\bigr)\le \ln \bigl(d (x,y) \bigr).$$
(13)

We observed from the literature that the auxiliary function or the control function used in Wardowski’s F-contraction condition has the ability to weaken some stronger conditions on the existence theorems concerning certain integral boundary value problems (see [2, 8, 18, 2023, 25]). After establishing the fixed point theorems for certain generalizations of F-contraction involving two or three auxiliary functions, we provide some new sufficient weaker conditions for the existence of the solutions of an integral boundary value problem for a scalar nonlinear Caputo fractional differential equation in comparison with an integral boundary value problem for the Caputo fractional derivative considered in [22, 23]. Furthermore, we use our obtained results to find a solution to an engineering problem in which the transformed mathematical model of a problem, representing a damping force in the case of a vertical spring moving through a fluid, is a boundary value problem for a second-order differential equation.

Throughout the following discussions, \(\mathbb{N}\), \(\mathbb{N}_{0}\), \(\mathbb{R}\), and \(\mathbb{R}\mathbbm{_{+}}\) denote the set of natural numbers, \(\mathbb{N}\cup \{0\}\), real numbers, and positive real numbers, respectively. Throughout this article, every set X taken into account is nonempty.

Definition 1.1

([16])

Let \(\mathcal{H}\) be a collection of functions \(F:\mathbb{R}\mathbbm{_{+}}\mapsto \mathbb{R}\) satisfying the following conditions:

\((F_{1})\):

F is strictly increasing; i.e., if \(x< y\), then \(F(x)< F(y)\);

\((F_{2})\):

\(\lim_{n\rightarrow +\infty} \alpha _{n}=0\) if and only if \(\lim_{n\rightarrow +\infty} F(\alpha _{n})=-\infty \), where \(\{\alpha _{n}\}_{n=1}^{+\infty }\) is any sequence of positive numbers;

\((F_{3})\):

\(\lim_{\alpha \rightarrow 0^{+}} \alpha ^{k}F(\alpha )=0\), \(k\in (0,1)\).

A mapping \(\mathfrak{F}:X\mapsto X\) is said to be an F-contraction in a metric space \((X,d)\) if there exist a function \(F:\mathbb{R}\mathbbm{_{+}}\mapsto \mathbb{R}\) and a real number \(\tau >0\) such that, for all \(x,y\in X\),

$$\begin{aligned} \bigl[d(\mathfrak{F}x,\mathfrak{F}y)>0 \text{ implies } \tau +F(d( \mathfrak{F}x,\mathfrak{F}y)\leq F\bigl(d(x,y)\bigr)\bigr], \end{aligned}$$

where \(F \in \mathcal{H}\).

Example 1.1

([16])

The real-valued functions \(f(t)=\ln t\), \(g(t)=\ln (t^{2}+t)\), and \(h(t)=t+\ln t\) belong to \(\mathcal{H}\).

Remark 1.1

([17])

Consider a real-valued function \(\mathfrak{S}(t)=\frac{-1}{\sqrt[p]{\alpha }}\), where \(p>1\), \(\alpha > 0\). Then \(\mathfrak{S} \in \mathcal{H}\).

The following Wardowski’s theorem [16] represents a significant generalization of the Banach contraction principle.

Theorem 1.1

([16])

An F-contraction mapping \(\mathfrak{F}\) has a unique fixed point \(x^{*}\) on a complete metric space \((X,d)\), and for any \(x \in X\), the sequence \(\{ {{\mathfrak{F}}^{n}}x \}_{n=1}^{\infty }\) converges to \(x^{*}\).

Inspired by Wardowski’s result of F-contraction, there are continuous efforts by many authors to improve and extend this idea by modifying or eliminating certain conditions on control function or generalizing the definition space. In this regard, Secelean [26] and Piri and Kumam [27] have shown that condition \((F3)\) can be removed from Theorem 1.1 by considering the continuity of F. Vetro [28] replaced the constant τ with a function to generalize F-contraction. Secelean and Wardowski [29] presented \(\psi F-\) contraction that modifies F-contraction by weakening condition \((F1)\). Lukács and Kajántó [30] relaxed condition \((F2)\) from the definition of F-contraction. Focusing on the definition space, Sumati Kumari [9] introduced the new classes of Hardy–Rogers type cyclic contractions and proved pertinent fixed point theorems for these Hardy–Rogers type contractions in the generating space of a b-dislocated metric family.

We need the following definitions, results, and remarks for our main discussion.

Definition 1.2

([31])

A mapping \(\mathfrak{F}\) on a metric space \((X,d)\) is said to be orbitally continuous if, for any sequence \(\{y_{n}\}\) in \(O_{x}(\mathfrak{F})\), \(y_{n} \rightarrow u\) implies \(\mathfrak{F}y_{n} \rightarrow \mathfrak{F}u\) as \(n\rightarrow +\infty \), where \(O_{x}(\mathfrak{F})=\{\mathfrak{F}^{n}x: n \geq 0\}\) is the orbit of \(\mathfrak{F}\) at x.

Definition 1.3

([32])

A self-mapping \(\mathfrak{F}\) of a metric space \((X,d)\) is called k-continuous, \(k=1,2,3, \ldots \) , if \(\mathfrak{F}^{k} x_{n}\rightarrow \mathfrak{F}t\) whenever \(\{x_{n}\}\) is a sequence in X such that \(\mathfrak{F}^{k-1}x_{n} \rightarrow t\), \(t \in X\).

It is noteworthy to mention that for \(k>1\), the k-continuity of \(\mathfrak{F}\) does not depend on the continuity of \(\mathfrak{F}^{k}\) and continuity implies 2-continuity implies 3-continuity implies … , but the converse is not true. Moreover, one can easily observe that continuity is equivalent to 1-continuity.

Definition 1.4

([33])

Let \((X,d)\) be a metric space. A mapping \(\mathfrak{F}:X\mapsto \mathbb{R}\) is said to be a \(\mathfrak{F}\)-orbitally lower semi-continuous at \(z\in X\) if \(\{x_{n}\}\) is a sequence in \(O_{x}(\mathfrak{F})\) for some \(x\in X\), \(\lim_{n\rightarrow \infty} x_{n}=z\) implies \(\mathfrak{F}(z)\leq \lim_{n\rightarrow \infty} \inf \mathfrak{F}(x_{n})\).

Proposition 1.1

([34])

Let \((X,d)\) be a metric space, \(\mathfrak{F}:X\mapsto X\) and \(z\in X\). If \(\mathfrak{F}\) is orbitally continuous at z or \(\mathfrak{F}\) is k-continuous at z for some \(k\neq 1\), then the function \(\mathfrak{F}(x)=d(x,\mathfrak{F}x)\) is \(\mathfrak{F}\)-orbitally lower semi-continuous at z.

Lemma 1.1

([35])

Let \((X,d)\) be a metric space and \(\{x_{n}\}\) be a sequence in X which is not Cauchy and \(\lim_{n\rightarrow \infty} d(x_{n},x_{n+1})=0\). Then there exists \(\varepsilon >0\) and two subsequences \(\{x_{n_{k}}\}\) and \(\{x_{m_{k}}\}\) of \(\{x_{n}\}\) such that

$$\begin{aligned} \lim_{k\rightarrow \infty} d(x_{n_{k}},x_{m_{k}})= \varepsilon \quad \textit{and}\quad \lim_{k\rightarrow \infty} d(x_{n_{k}+1},x_{m_{k}+1})= \varepsilon _{+}. \end{aligned}$$

2 Main results

In 2020, Alfaqih et al. [25] provided \(F^{*}\)-weak contractions by introducing a new class of auxiliary functions that eliminate conditions \((F1)\), \((F3)\) and only satisfy one-way implication of condition \((F2)\).

Firstly, we suppose that \(\mathcal{F}\) is the set of all functions \(F:(0,\infty )\rightarrow \mathbb{R}\) satisfying the following condition:

\((C_{1})\):

\(\inf_{t>\varepsilon}F(t)>-\infty \) for any \(\varepsilon >0\).

It follows from Lemma 2.3 [7] that condition \((C_{1})\) is equivalent to \((C_{1}')\) and \((C_{1}'')\) stated as follows:

\((C_{1}')\):

\(\lim_{n\rightarrow \infty} F(t_{n})=-\infty \) implies \(\lim_{n \rightarrow \infty} t_{n}=0\);

\((C_{1}'')\):

\(\liminf_{t\rightarrow \varepsilon +} F(t)>-\infty \) for any \(\varepsilon >0\).

Apparently, \(\mathcal{H}\subset \mathcal{F}\). Next, we will introduce the concepts of the dual \(F^{*}-\) weak contraction and triple \(F^{*}-\) weak contraction, which can be regarded as a generalization of F-contraction.

Definition 2.1

Let \((X,d)\) be a metric space. We say that a mapping \(\mathfrak{F}:X\mapsto X\) is a dual \(F^{*}\)-weak contraction of type-I if there exists a real number \(\tau >0\) and \(F_{1}, F_{2}\in \mathcal{F}\) such that for all \(x,y\in X\), we have

$$\begin{aligned} &d\bigl(\mathfrak{F}^{2}x,\mathfrak{F}^{2}y \bigr)>0 \\ &\quad \text{implies} \\ &\qquad \tau +\min \bigl\{ F_{2}\bigl(d\bigl( \mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr) \bigr),F_{1}\bigl(d(\mathfrak{F}x,\mathfrak{F}y)\bigr)\bigr\} \leq \alpha _{2}F_{2}\bigl(d(x,y)\bigr)+\alpha _{1}F_{1}\bigl(d(x,y)\bigr), \end{aligned}$$
(14)

where

$$\textstyle\begin{cases} \alpha _{1}=0,\alpha _{2}=1, & \text{if } F_{2}(d( \mathfrak{F}^{2}x,\mathfrak{F}^{2}y))\leq F_{1}(d(\mathfrak{F}x, \mathfrak{F}y)), \\ \alpha _{1}=1,\alpha _{2}=0, & \text{if } F_{2}(d( \mathfrak{F}^{2}x,\mathfrak{F}^{2}y))> F_{1}(d(\mathfrak{F}x, \mathfrak{F}y)). \end{cases} $$

Remark 2.1

For some \(x,y\in X\), the conditions of Definition 2.1 yield either

$$ \tau +F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr)\bigr)\leq F_{2}\bigl(d(x,y)\bigr) \quad (L1) $$

or

$$ \tau +F_{1}\bigl(d(\mathfrak{F}x,\mathfrak{F}y)\bigr) \leq F_{1}\bigl(d(x,y)\bigr).\quad (L2) $$

It is worth noting that Definition 2.1 can be viewed as a combination of \((L1)\) and \((L2)\), which could reduce to the form of Wardowski’s contraction while it includes inequality \((L2)\) only.

Definition 2.2

Let \((X,d)\) be a metric space. We say that a mapping \(\mathfrak{F}:X\mapsto X\) is a dual \(F^{*}\)-weak contraction of type-II if there exists a real number \(\tau >0\) and \(F_{1}, F_{2}\in \mathcal{F}\) such that for all \(x,y\in X\), we have

$$\begin{aligned} &d\bigl(\mathfrak{F}^{2}x,\mathfrak{F}^{2}y \bigr)>0 \\ &\quad \text{implies} \\ & \qquad \tau +\min \bigl\{ F_{2}\bigl(d\bigl( \mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr) \bigr),F_{1}\bigl(d(\mathfrak{F}x,\mathfrak{F}y)\bigr)\bigr\} \leq \alpha _{2}F_{2}\bigl(d(x,y)\bigr)+\alpha _{1}F_{1}\bigl(d(x,y)\bigr), \end{aligned}$$
(15)

where either \(\alpha _{1}=0\) or \(\alpha _{2}=0 \) and \(\alpha _{1}+\alpha _{2}=1\).

Remark 2.2

For all \(x,y\in X\), the dual \(F^{*}\)-weak contraction of type-II deals with one of the following two cases \((R1)\), \((R2)\):

$$\begin{aligned}& \tau +\min \bigl\{ F_{2}\bigl(d\bigl( \mathfrak{F}^{2}x,\mathfrak{F}^{2}y\bigr) \bigr),F_{1}\bigl(d( \mathfrak{F}x,\mathfrak{F}y)\bigr)\bigr\} \leq F_{2}\bigl(d(x,y)\bigr), \quad (R1) \\& \tau +\min \bigl\{ F_{2}\bigl(d\bigl( \mathfrak{F}^{2}x,\mathfrak{F}^{2}y\bigr) \bigr),F_{1}\bigl(d( \mathfrak{F}x,\mathfrak{F}y)\bigr)\bigr\} \leq F_{1}\bigl(d(x,y)\bigr). \quad (R2) \end{aligned}$$

For some \(x,y\in X, (R1)\) yields either

$$ \tau +F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr)\bigr)\leq F_{2}\bigl(d(x,y)\bigr) \quad (a) $$

or

$$ \tau +F_{1}\bigl(d(\mathfrak{F}x,\mathfrak{F}y)\bigr) \leq F_{2}\bigl(d(x,y)\bigr). \quad (d) $$

Similarly, for some \(x,y\in X, (R2)\) yields either

$$ \tau +F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr)\bigr)\leq F_{1}\bigl(d(x,y)\bigr) \quad (c) $$

or

$$ \tau +F_{1}\bigl(d(\mathfrak{F}x,\mathfrak{F}y)\bigr) \leq F_{1}\bigl(d(x,y)\bigr). \quad (d) $$

Apparently, inequality \((R1)\) takes the form as a combination of subcases \((a)\) and \((d)\). If the inequality \((R1)\) yields subcase \((d)\) only, we have \(\tau +F_{1}(d(\mathfrak{F}x,\mathfrak{F}y))\leq F_{2}(d(x,y))\), which is equivalent to the form of Wardowski’s contraction by taking \(F_{2}=F_{1}\).

Definition 2.3

Let \((X,d)\) be a metric space. We say that a mapping \(\mathfrak{F}:X\mapsto X\) is a triple \(F^{*}\)-weak contraction if there exists a real number \(\tau >0\) and \(F, F_{1}, F_{2}\in \mathcal{F}\) such that for all \(x,y\in X\) we have

$$\begin{aligned} &d\bigl(\mathfrak{F}^{2}x,\mathfrak{F}^{2}y \bigr)>0 \\ &\quad \text{implies}\quad \tau +\min \bigl\{ F_{2}\bigl(d\bigl( \mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr) \bigr),F_{1}\bigl(d(\mathfrak{F}x,\mathfrak{F}y)\bigr)\bigr\} \leq F \bigl(d(x,y)\bigr). \end{aligned}$$
(16)

Remark 2.3

One can easily observe that if \(\min \{{{F}_{2}}(d({{\mathfrak{F}}^{2}}x,{{\mathfrak{F}}^{2}}y)),{{F}_{1}}(d( \mathfrak{F}x,\mathfrak{F}y))\}={{F}_{1}}(d(\mathfrak{F}x, \mathfrak{F}y))\) and \(F (t )=F_{1} (t )\) for all \(x,y \in X\), \(t\in (0,\infty )\), then inequality (16) becomes the form of Wardowski’s contraction.

Example 2.1

Let \(F_{1}, F_{2}\in \mathcal{F}\) be given by \(F_{1}(\alpha )=\ln \alpha \), \(F_{2}(\alpha )=\ln k\alpha \), where \(\alpha , k>0\).

The dual \(F^{*}\)-weak contraction of type-I will take the form

$$\begin{aligned} \tau +\min \bigl\{ \ln \bigl(kd\bigl(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr)\bigr),\ln \bigl(d( \mathfrak{F}x,\mathfrak{F}y) \bigr)\bigr\} \leq \alpha _{2}\ln \bigl(kd(x,y)\bigr)+\alpha _{1} \ln \bigl(d(x,y)\bigr), \end{aligned}$$

which can also be rewritten as

$$\begin{aligned} \tau +\ln \bigl(\min \bigl\{ kd\bigl(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr),d( \mathfrak{F}x,\mathfrak{F}y)\bigr\} \bigr) \leq \ln \bigl(k^{\alpha _{2}}d(x,y)\bigr)^{{ \alpha _{1}}+{\alpha _{2}}}. \end{aligned}$$

From the definition of dual \(F^{*}\)-weak contraction of type-I, we have \(\alpha _{1}+\alpha _{2}=1\) and

$$\begin{aligned} \min \bigl\{ kd\bigl(\mathfrak{F}^{2}x,\mathfrak{F}^{2}y \bigr),d(\mathfrak{F}x, \mathfrak{F}y)\bigr\} \leq e^{-\tau}k^{\alpha _{2}}d(x,y). \end{aligned}$$

It is worth noting that \(\mathfrak{F}\) may be a contraction provided by \(d(\mathfrak{F}x,\mathfrak{F}y)< kd(\mathfrak{F}^{2}x,\mathfrak{F}^{2}y)\) for all \(x,y\in X\), that is, \(d(\mathfrak{F}x,\mathfrak{F}y)\leq e^{-\tau}d(x,y)\) with \(\alpha _{2}=0\) or be neither a contraction nor an expansion provided by \(d(\mathfrak{F}x,\mathfrak{F}y)>kd(\mathfrak{F}^{2}x,\mathfrak{F}^{2}y)\) for all \(x,y\in X\), that is, \(d(\mathfrak{F}^{2}x,\mathfrak{F}^{2}y)\leq e^{-\tau}kd(x,y)\).

Example 2.2

Let \(F_{1}, F_{2}\in \mathcal{F}\) be given by \(F_{1}(\alpha )=\ln \alpha \), \(F_{2}(\alpha )=\ln k\alpha \), where \(\alpha , k>0\).

Then, by the definition of dual \(F^{*}\)-weak contraction of type-II, for all \(x,y\in X\), we have

$$\begin{aligned} \tau +\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x, \mathfrak{F}y)\bigr)\bigr\} \leq \alpha _{2}F_{2}\bigl(d(x,y) \bigr)+\alpha _{1}F_{1}\bigl(d(x,y)\bigr). \end{aligned}$$

From \((R1)\), \((R2)\), we have

$$\begin{aligned}& \tau +\min \bigl\{ F_{2}\bigl(d\bigl( \mathfrak{F}^{2}x,\mathfrak{F}^{2}y\bigr) \bigr),F_{1}\bigl(d( \mathfrak{F}x,\mathfrak{F}y)\bigr)\bigr\} \leq F_{1}\bigl(d(x,y)\bigr), \end{aligned}$$
(17)
$$\begin{aligned}& \tau +\min \bigl\{ F_{2}\bigl(d\bigl( \mathfrak{F}^{2}x,\mathfrak{F}^{2}y\bigr) \bigr),F_{1}\bigl(d( \mathfrak{F}x,\mathfrak{F}y)\bigr)\bigr\} \leq F_{2}\bigl(d(x,y)\bigr). \end{aligned}$$
(18)

Together with the definition of \(F_{1}\), inequality (17) can be rewritten as

$$\begin{aligned} \tau +\ln \bigl(\min \bigl\{ kd\bigl(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr),d( \mathfrak{F}x,\mathfrak{F}y)\bigr\} \bigr) \leq \ln \bigl(d(x,y)\bigr). \end{aligned}$$

Then we have

$$\begin{aligned} \min \bigl\{ kd\bigl(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr),d(\mathfrak{F}x, \mathfrak{F}y)\bigr\} \leq e^{-\tau}d(x,y). \end{aligned}$$
(19)

Similarly, inequality (18) can be rewritten as

$$\begin{aligned} \min \bigl\{ kd\bigl(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr),d(\mathfrak{F}x, \mathfrak{F}y)\bigr\} \leq e^{-\tau}kd(x,y). \end{aligned}$$
(20)

It is noted that inequalities (19) and (20) can imply that \(\mathfrak{F}\) is a Lipschitzian mapping provided by \(kd(\mathfrak{F}^{2}x,\mathfrak{F}^{2}y)>d(\mathfrak{F}x,\mathfrak{F}y)\) for all \(x,y\in X\) or a contraction provided by \(kd(\mathfrak{F}^{2}x,\mathfrak{F}^{2}y)< d(\mathfrak{F}x,\mathfrak{F}y)\) for all \(x,y\in X\).

Example 2.3

Define \(F, F_{1}, F_{2}\in \mathcal{F}\) by \(F(\alpha )=\ln k\alpha \), \(F_{1}(\alpha )=\ln k_{1}\alpha \), \(F_{2}(\alpha )=\ln k_{2}\alpha \), where \(k, k_{1}, k_{2}>0\).

Then, by the definition of triple \(F^{*}\)-weak contraction, for all \(x,y\in X\), we have

$$\begin{aligned} \tau +\min \bigl\{ \ln \bigl(k_{2}d\bigl(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr)\bigr),\ln \bigl(k_{1}b( \mathfrak{F}x,\mathfrak{F}y)\bigr)\bigr\} \leq \ln \bigl(kd(x,y)\bigr), \end{aligned}$$

which can also be rewritten as

$$\begin{aligned} \min \bigl\{ k_{2}d\bigl(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr),k_{1}b( \mathfrak{F}x,\mathfrak{F}y) \bigr\} \leq e^{-\tau}kd(x,y). \end{aligned}$$

That is, either

$$\begin{aligned} k_{2}d\bigl(\mathfrak{F}^{2}x,\mathfrak{F}^{2}y \bigr)\leq e^{-\tau}kd(x,y) \end{aligned}$$

or

$$\begin{aligned} k_{1}b(\mathfrak{F}x,\mathfrak{F}y)\leq e^{-\tau}kd(x,y). \end{aligned}$$

We can define \(k_{2}=2\alpha _{2}\), \(k_{1}=2\alpha _{1}\), \(k_{1}+k_{2}=2\), where \(\alpha _{1}+\alpha _{2}=1\). Then we have

$$\begin{aligned} \alpha _{2}d\bigl(\mathfrak{F}^{2}x,\mathfrak{F}^{2}y \bigr)\leq \frac{k}{2}e^{- \tau}d(x,y) \end{aligned}$$

or

$$\begin{aligned} \alpha _{1}d(\mathfrak{F}x,\mathfrak{F}y)\leq \frac{k}{2} e^{-\tau}d(x,y). \end{aligned}$$

Therefore, we have

$$\begin{aligned} \alpha _{2}\bigl(d\bigl(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr)\bigr)+\alpha _{1}\bigl(d( \mathfrak{F}x,\mathfrak{F}y)\bigr)\leq ke^{-\tau}\bigl(d(x,y)\bigr), \end{aligned}$$

which shows that \(\mathfrak{F}\) is a mean Lipschitzian mapping [36].

Next, we present the existence of a unique fixed point for dual \(F^{*}\)-weak contractions and triple \(F^{*}\)-weak contraction as follows.

Theorem 2.1

Suppose that a mapping \(\mathfrak{F}:X\mapsto X\) is a dual \(F^{*}\)-weak contraction of type-I in a complete metric space \((X,d)\). If \(\mathfrak{F}\) is orbitally continuous or k-continuous for some \(k \in \mathbb{N}\) and for all \(t_{1}, t_{2}\in \mathbb{R}\mathbbm{_{+}}\), there exist \(\upsilon >0\), \(\tau >2\upsilon \) such that

$$\begin{aligned} &F_{2}(t_{2})< F_{1}(t_{1})\leq F_{2}(t_{2})+\upsilon \end{aligned}$$
(A)

or

$$\begin{aligned} &F_{1}(t_{1})< F_{2}(t_{2})\leq F_{1}(t_{1})+\upsilon . \end{aligned}$$
(B)

Then, for every \(x_{0}\in X\), the sequence \(\{\mathfrak{F}^{m}x_{0}\}_{m=1}^{+\infty}\) converges to the unique fixed point of \(\mathfrak{F}\).

Proof

Consider a sequence \(\{x_{m}\}\subseteq X\) such that, for all \(m\in \mathbb{N}_{0}\), \(x_{m+1}=\mathfrak{F}x_{m}=\mathfrak{F}^{m}x_{0}\), where \(x_{0}\) is an arbitrary point in X.

If there exists \(m\in \mathbb{N}_{0}\) such that \(d(x_{m},\mathfrak{F}x_{m})=0\), then \(\mathfrak{F}\) admits a fixed point. So, we assume that \(d(x_{m},\mathfrak{F}x_{m})=d(\mathfrak{F}x_{m-1},\mathfrak{F}x_{m})>0\) for all \(m\in \mathbb{N}\).

We will prove that \(\lim_{m\rightarrow +\infty} d(x_{m},\mathfrak{F}x_{m})=0\).

From the definition of dual F-contraction of type I, we have

$$\begin{aligned} &\tau +\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad \leq \alpha _{2}F_{2}\bigl(d(x_{m-1},x_{m}) \bigr)+\alpha _{1}F_{1}\bigl(d(x_{m-1},x_{m}) \bigr), \end{aligned}$$

so

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad \leq \alpha _{2}F_{2}\bigl(d(x_{m-1},x_{m}) \bigr)+\alpha _{1}F_{1}\bigl(d(x_{m-1},x_{m}) \bigr)- \tau . \end{aligned}$$
(23)

Now we will discuss the following two possible cases \((\mathit{I})\), \((\mathit{II})\):

$$\begin{aligned} &F_{1}\bigl(d(\mathfrak{F}x_{m-1},\mathfrak{F}x_{m}) \bigr)=\min \bigl\{ F_{2}\bigl(d\bigl( \mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr)\bigr\} , \quad (I) \\ &F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr)=\min \bigl\{ F_{2}\bigl(d\bigl( \mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr)\bigr\} . \quad (\mathit{II}) \end{aligned}$$

If \((I)\) holds, then inequality (21) will take the form

$$\begin{aligned} \min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \leq F_{1}\bigl(d(x_{m-1},x_{m})\bigr)- \tau . \end{aligned}$$
(24)

Moreover, we also have either

$$\begin{aligned} F_{1}\bigl(d(x_{m-1},x_{m})\bigr)= \min \bigl\{ F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr),F_{1}\bigl(d(x_{m-1},x_{m}) \bigr)\bigr\} \end{aligned}$$
(25)

or

$$\begin{aligned} F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr)=\min \bigl\{ F_{2}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr),F_{1} \bigl(d(x_{m-1},x_{m})\bigr)\bigr\} . \end{aligned}$$
(26)

If (23) holds, then (22) yields

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad \leq \min \bigl\{ F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr),F_{1}\bigl(d(x_{m-1},x_{m}) \bigr) \bigr\} -\tau . \end{aligned}$$
(27)

If (24) holds, we can write

$$\begin{aligned} F_{2}\bigl(d(\mathfrak{F}x_{m-1},\mathfrak{F}x_{m}) \bigr)< F_{1}\bigl(d(x_{m-1},x_{m})\bigr). \end{aligned}$$

It follows from condition \((A)\) that

$$\begin{aligned} F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr)< F_{1}\bigl(d(x_{m-1},x_{m}) \bigr) \leq F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr)+\upsilon . \end{aligned}$$
(28)

Applying (26) in (22), we have

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad \leq F_{2}\bigl(d(\mathfrak{F}x_{m-1},\mathfrak{F}x_{m}) \bigr)+\upsilon -\tau . \end{aligned}$$

So, it follows from (24) that

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad \leq \min \bigl\{ F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr),F_{1}\bigl(d(x_{m-1},x_{m}) \bigr) \bigr\} +\upsilon -\tau . \end{aligned}$$
(29)

Combining the both possible cases of (25) and (27), we have

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad \leq \min \bigl\{ F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr),F_{1}\bigl(d(x_{m-1},x_{m}) \bigr) \bigr\} +\delta _{m}\upsilon -\tau , \end{aligned}$$
(30)

where

$$\delta _{m}= \textstyle\begin{cases} 0, & \text{if } F_{2}(t_{2})>F_{1}(t_{1}), \\ 1, & \text{if } F_{2}(t_{2})< F_{1}(t_{1}), \end{cases} $$

\(t_{1}, t_{2}\in \mathbb{R}\mathbbm{_{+}}\), \(t_{1}\neq t_{2}\).

Therefore, \((\mathit{I})\) implies (28).

Similarly, with the existence of \((\mathit{II})\), (21) takes the following form:

$$\begin{aligned} \min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \leq F_{2}\bigl(d(x_{m-1},x_{m})\bigr)- \tau . \end{aligned}$$
(31)

If \(F_{2}(d(x_{m-1},x_{m}))\leq F_{1}(d(x_{m-1},x_{m}))\), then (29) can be rewritten as

$$\begin{aligned} \min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \leq F_{1}\bigl(d(x_{m-1},x_{m})\bigr)- \tau . \end{aligned}$$
(32)

If \(F_{2}(d(x_{m-1},x_{m}))>F_{1}(d(x_{m-1},x_{m}))\), from condition \((B)\), we have

$$\begin{aligned} \min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \leq F_{1}\bigl(d(x_{m-1},x_{m})\bigr)+ \upsilon -\tau . \end{aligned}$$
(33)

Combining inequalities (30), (31), we have

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad \leq F_{1}\bigl(d(x_{m-1},x_{m})\bigr)+ \eta _{m}\upsilon -\tau , \end{aligned}$$
(34)

where

$$\eta _{m}= \textstyle\begin{cases} 1, & \text{if } F_{2}(t)>F_{1}(t), \\ 0, & \text{if } F_{2}(t)< F_{1}(t), \end{cases} $$

\(t\in \mathbb{R}\mathbbm{_{+}}\).

Together with inequalities (22)–(28), inequality (32) will take the form

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad \leq \min \bigl\{ F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr),F_{1}\bigl(d(x_{m-1},x_{m}) \bigr) \bigr\} +(\delta _{m}+\eta _{m})\upsilon -\tau . \end{aligned}$$
(35)

Therefore, \((\mathit{II})\) implies (33).

The existence of \((\mathit{I})\) and \((\mathit{II})\) implies the existence of (28) and (33) respectively. Combining inequalities (28) and (33), we have

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad\leq \min \bigl\{ F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr),F_{1}\bigl(d(x_{m-1},x_{m}) \bigr) \bigr\} +(\delta _{m}+\varsigma _{m}\eta _{m})\upsilon -\tau \\ &\quad=\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-2}, \mathfrak{F}^{2}x_{m-1}\bigr)\bigr),F_{1} \bigl(d(x_{m-2},x_{m-1})\bigr) \bigr\} +(\delta _{m}+ \varsigma _{m}\eta _{m})\upsilon -\tau , \end{aligned}$$

where \(\varsigma _{m}\) is either 0 or 1.

Repeating this process, we have

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad\leq \min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-3}, \mathfrak{F}^{2}x_{m-2}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-3},\mathfrak{F}x_{m-2})\bigr)\bigr\} \\ &\quad\quad{}+(\delta _{m}+\varsigma _{m}\eta _{m})\upsilon +(\delta _{m-1}+ \varsigma _{m-1}\eta _{m-1})\upsilon -2\tau \\ &\quad\cdots \\ &\quad\leq \min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{1}, \mathfrak{F}^{2}x_{0}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{1},\mathfrak{F}x_{0})\bigr)\bigr\} +\upsilon \sum_{j=1}^{m}( \delta _{j}+ \varsigma _{j}\eta _{j})-m\tau . \end{aligned}$$

Since \(\tau >2\upsilon \) and \(\upsilon \sum_{j=1}^{m}(\delta _{j}+\varsigma _{j}\eta _{j})< m \tau \), we deduce

$$\begin{aligned} \lim_{m\rightarrow +\infty} \upsilon \sum_{j=1}^{m}( \delta _{j}+\varsigma _{j}\eta _{j})-m\tau =- \infty . \end{aligned}$$

Therefore,

$$\begin{aligned} \lim_{m\rightarrow +\infty} \min \bigl\{ F_{2}\bigl(d \bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m} \bigr)\bigr),F_{1}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr) \bigr\} =-\infty . \end{aligned}$$
(36)

Now, equation (34) further has two possible cases:

$$\begin{aligned} &\lim_{m\rightarrow +\infty} F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr)=-\infty , \end{aligned}$$
(E)
$$\begin{aligned} &\lim_{m\rightarrow +\infty} F_{1}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr)=-\infty . \end{aligned}$$
(F)

Condition \((C_{1})\) with case \((E)\) yields

$$\begin{aligned} \lim_{m\rightarrow +\infty} d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)=0 \end{aligned}$$

or equivalently,

$$\begin{aligned} \lim_{m\rightarrow +\infty} d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)=\lim_{m\rightarrow +\infty} d(x_{m+1}, \mathfrak{F}x_{m+1})=\lim_{m\rightarrow +\infty} d(x_{m}, \mathfrak{F}x_{m})=0. \end{aligned}$$

Condition \((C_{1})\) with case \((F)\) yields

$$\begin{aligned} \lim_{m\rightarrow +\infty} d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})=\lim_{m\rightarrow +\infty} d(x_{m}, \mathfrak{F}x_{m})=0. \end{aligned}$$

Therefore, from (34) we get

$$\begin{aligned} \lim_{m\rightarrow +\infty} d(x_{m}, \mathfrak{F}x_{m})=0. \end{aligned}$$
(39)

Now, in order to prove that the sequence \(\{x_{m}\}_{m=1}^{+\infty }\) is a Cauchy sequence, we suppose on the contrary that there exists \(\varepsilon >0\) and two subsequences \(\{ x_{g(m)}\}_{m=1}^{+\infty }\) and \(\{x_{h(m)}\}_{m=1}^{+\infty }\) of \(\{x_{n}\}\),

$$\begin{aligned} \lim_{m\rightarrow \infty} d(x_{g(m)+2},x_{h(m)+2})= \lim_{m\rightarrow \infty} d(x_{g(m)+1},x_{h(m)+1})= \lim _{m\rightarrow \infty} d(x_{g(m)},x_{h(m)})=\varepsilon _{+}. \end{aligned}$$
(40)

Applying (14) by taking \(x=x_{g(m)}\) and \(y=x_{h(m)}\), we get

$$\begin{aligned} &\tau +\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{g(m)} \bigr),\mathfrak{F}^{2}x_{h(m)}\bigr),F_{1}\bigl(d( \mathfrak{F}x_{g(m)},\mathfrak{F}x_{h(m)})\bigr)\bigr\} \\ &\quad \leq \alpha _{2}F_{2}\bigl(d(x_{g(m)},x_{h(m)}) \bigr)+\alpha _{1}F_{1}\bigl(d(x_{g(m)},x_{h(m)}) \bigr), \end{aligned}$$

which concludes the following two cases:

$$\begin{aligned} \tau +F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{g(m)}, \mathfrak{F}^{2}x_{h(m)}\bigr)\bigr) \leq F_{2} \bigl(d(x_{g(m)},x_{h(m)})\bigr) \end{aligned}$$

or

$$\begin{aligned} \tau +F_{1}\bigl(d\bigl(\mathfrak{F}^{2}x_{g(m)}, \mathfrak{F}^{2}x_{h(m)}\bigr)\bigr) \leq F_{1} \bigl(d(x_{g(m)},x_{h(m)})\bigr). \end{aligned}$$

Taking limits in the above two inequalities as \(k\rightarrow \infty \), we have

$$\begin{aligned} \tau +\min \bigl\{ F_{2}(\varepsilon _{+}),F_{1}( \varepsilon _{+})\bigr\} \leq \alpha _{2}F_{2}( \varepsilon _{+})+\alpha _{1}F_{1}(\varepsilon _{+}), \end{aligned}$$

which yields either \(\tau +F_{2}(\varepsilon _{+})\leq F_{2}(\varepsilon _{+})\) or \(\tau +F_{1}(\varepsilon _{+})\leq F_{1}(\varepsilon _{+})\).

Both of the above inequalities are contradictions. Therefore, \(\{x_{m}\}_{m=1}^{+\infty}\) is a Cauchy sequence. The completeness of \((X,d)\) proves that \(\{x_{m}\}_{m=1}^{+\infty}\) converges to some point \(x^{*}\) in X.

Now, if \(\mathfrak{F}\) is orbitally continuous, then we have

$$\begin{aligned} d\bigl(\mathfrak{F}x^{*},x^{*}\bigr)=\lim _{m\rightarrow +\infty} d( \mathfrak{F}x_{m},x_{m})=\lim _{m\rightarrow +\infty} d(x_{m+1},x_{m})=d \bigl(x^{*},x^{*}\bigr)=0. \end{aligned}$$

If \(\mathfrak{F}\) is k-continuous for some \(k\in \mathbb{N}\), we have

$$\begin{aligned} d\bigl(\mathfrak{F}x^{*},x^{*}\bigr)=\lim _{m\rightarrow +\infty} d\bigl( \mathfrak{F}\bigl(\mathfrak{F}^{k-1}x_{m} \bigr),\mathfrak{F}^{k-1}x_{m}\bigr)= \lim _{m\rightarrow +\infty} d(x_{k+m},x_{k+m-1})=d \bigl(x^{*},x^{*}\bigr)=0. \end{aligned}$$

Therefore, \(\mathfrak{F}\) has a fixed point \(x^{*}\).

Furthermore, from Proposition (1.1), it follows that \(x\mapsto d(x,\mathfrak{F}x)\) is \(\mathfrak{F}\)-orbitally lower semi-continuous, then we have

$$\begin{aligned} d\bigl(x^{*},\mathfrak{F}x^{*}\bigr)\leq \liminf _{m\rightarrow \infty} d(x_{m}, \mathfrak{F}x_{m})=0, \end{aligned}$$

which implies that \(\mathfrak{F}x^{*}=x^{*}\) and \(x^{*}\) is a fixed point of \(\mathfrak{F}\).

Now, for the uniqueness, let us suppose that \(\mathfrak{F}\) has more than one fixed point, that is, there exist two distinct \(x,y\in X\) such that \(\mathfrak{F}x=x\neq y=\mathfrak{F}y\).

Therefore, \(d(x,y)=d(\mathfrak{F}x,\mathfrak{F}y)=d(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y)>0\). From (14), we have either

$$\begin{aligned} F_{1}\bigl(d(x,y)\bigr)=F_{1}\bigl(d( \mathfrak{F}x,\mathfrak{F}y )\bigr)< \tau +F_{1}\bigl(d( \mathfrak{F}x, \mathfrak{F}y)\bigr)\leq F_{1}\bigl(d(x,y)\bigr) \end{aligned}$$
(41)

or

$$\begin{aligned} F_{2}\bigl(d(x,y)\bigr)=F_{2}\bigl(d\bigl( \mathfrak{F}^{2}x,\mathfrak{F}^{2}y \bigr)\bigr)< \tau +F_{2}\bigl(d\bigl( \mathfrak{F}^{2}x,\mathfrak{F}^{2}y \bigr)\bigr)\leq F_{2}\bigl(d(x,y)\bigr). \end{aligned}$$
(42)

Both (37), (38) are two contradictions that prove the existence of a unique fixed point. □

Theorem 2.2

Let \((X,d)\) be a complete metric space. Suppose that a mapping \(\mathfrak{F}:X\mapsto X\) is a dual \(F^{*}\)-weak contraction of type-II and is orbitally continuous or k-continuous for some \(k\in \mathbb{N}\). Moreover, for all \(t_{1}, t_{2}\in \mathbb{R}\mathbbm{_{+}}\), there exist \(\upsilon >0\), \(\tau >2\upsilon \) such that

$$\begin{aligned} &F_{2}(t_{2})< F_{1}(t_{1})\leq F_{2}(t_{2})+\upsilon \end{aligned}$$
(A)

or

$$\begin{aligned} &F_{1}(t_{1})< F_{2}(t_{2})\leq F_{1}(t_{1})+\upsilon . \end{aligned}$$
(B)

Then, for every \(x_{0}\in X\), the sequence \(\{\mathfrak{F}^{m}x_{0}\}_{m=1}^{+\infty}\) converges to the unique fixed point of \(\mathfrak{F}\).

Proof

Let \(x_{0}\in X\) be an arbitrary point and define a sequence \(\{x_{m}\}\subseteq X\) by \(x_{m+1}=\mathfrak{F}x_{m}=\mathfrak{F}^{m}x_{0}\) for all \(m\in \mathbb{N}_{0}\).

If there exists some \(m\in \mathbb{N}_{0}\) such that \(d(x_{m},\mathfrak{F}x_{m})=0\), then \(\mathfrak{F}\) admits a fixed point. So, we assume that \(d(x_{m},\mathfrak{F}x_{m})=d(\mathfrak{F}x_{m-1},\mathfrak{F}x_{m})>0\) for all \(m\in \mathbb{N}\).

We first take into account case \((R2)\). Analysis similar to the procedure of obtaining inequalities (22) to (28) in the proof of Theorem 2.1 shows that

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad \leq \min \bigl\{ F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr),F_{1}\bigl(d(x_{m-1},x_{m}) \bigr) \bigr\} +\delta _{m}\upsilon -\tau , \end{aligned}$$

where

$$\delta _{m}= \textstyle\begin{cases} 0 & \text{if } F_{2}(t_{2})>F_{1}(t_{1}), \\ 1 & \text{if } F_{2}(t_{2})< F_{1}(t_{1}), \end{cases} $$

\(t_{1}, t_{2}\in \mathbb{R}\mathbbm{_{+}}\), \(t_{1}\neq t_{2}\).

The above relation takes the following form:

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad \leq \min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-2}, \mathfrak{F}^{2}x_{m-1}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-2},\mathfrak{F}x_{m-1})\bigr)\bigr\} +\delta _{m}\upsilon - \tau . \end{aligned}$$

By continuing this process, we can write

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad\leq \min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-3}, \mathfrak{F}^{2}x_{m-2}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-3},\mathfrak{F}x_{m-2})\bigr)\bigr\} \\ &\quad\quad{}+\delta _{m}\upsilon +\delta _{m-1}\upsilon -2\tau \\ &\quad\leq \cdots \\ &\quad\leq \min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{1}, \mathfrak{F}^{2}x_{0}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{1},\mathfrak{F}x_{0})\bigr)\bigr\} + \sum _{j=1}^{m} \delta _{j}\upsilon -m \tau \end{aligned}$$

or

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad \leq F_{1}\bigl(d(x_{1},x_{0})\bigr)+ \sum _{j=1}^{m}\delta _{j} \upsilon -(m+1)\tau . \end{aligned}$$

Since \(\tau >2\upsilon \) and \(\sum_{j=1}^{m}\delta _{j}<2(m+1)\), therefore

$$\begin{aligned} \lim_{m\rightarrow +\infty} \sum_{j=1}^{m} \delta _{j} \upsilon -(m+1)\tau =-\infty . \end{aligned}$$

Therefore,

$$\begin{aligned} \lim_{m\rightarrow +\infty} \min \bigl\{ F_{2}\bigl(d\bigl( \mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m} \bigr)\bigr),F_{1}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr) \bigr\} =-\infty . \end{aligned}$$

Similar arguments apply to case \((R1)\). According to the procedure of inequalities (29) to (32) in the proof of Theorem 2.1, we have

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad \leq F_{1}\bigl(d(x_{m-1},x_{m})\bigr)+\eta _{m}\upsilon -\tau , \end{aligned}$$

where

$$\eta _{m}= \textstyle\begin{cases} 1 & \text{if } F_{2}(t)>F_{1}(t), \\ 0 & \text{if } F_{2}(t)< F_{1}(t), \end{cases} $$

\(t\in \mathbb{R}\mathbbm{_{+}}\).

The above inequality will take the form

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad \leq \min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-2}, \mathfrak{F}^{2}x_{m-1}\bigr)\bigr),F_{1} \bigl(d(x_{m-2},x_{m-1})\bigr) \bigr\} +(\delta _{m}+ \eta _{m})\upsilon -\tau . \end{aligned}$$

Repeating this process, we have

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad\leq \min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-3}, \mathfrak{F}^{2}x_{m-2}\bigr)\bigr),F_{1} \bigl(d(x_{m-3},x_{m-2})\bigr) \bigr\} \\ &\quad\quad{}+(\delta _{m}+\eta _{m})\upsilon +(\delta _{m-1}+\eta _{m-1}) \upsilon -2\tau \\ &\quad\leq \cdots \\ &\quad\leq \min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{1}, \mathfrak{F}^{2}x_{0}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{1},\mathfrak{F}x_{0})\bigr)\bigr\} + \sum _{j=1}^{m}( \delta _{j}+\eta _{j})\upsilon -m\tau . \end{aligned}$$

Since \(\tau >2\upsilon \) and \(\sum_{j=1}^{m}(\delta _{j}+\eta _{j})\upsilon < m\tau \), therefore

$$\begin{aligned} \lim_{m\rightarrow +\infty} \sum_{j=1}^{m}( \delta _{j}+ \eta _{j})\upsilon -m\tau =-\infty . \end{aligned}$$

So that

$$\begin{aligned} \lim_{m\rightarrow +\infty} \min \bigl\{ F_{2}\bigl(d\bigl( \mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m} \bigr)\bigr),F_{1}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr) \bigr\} =-\infty . \end{aligned}$$

Hence, for both cases \((R1)\), \((R2)\), it follows from condition \((C_{1})\) that

$$\begin{aligned} \lim_{m\rightarrow +\infty} d(x_{m}, \mathfrak{F}x_{m})=0. \end{aligned}$$
(45)

Now, in order to prove that the sequence \(\{x_{m}\}_{m=1}^{+\infty }\) is a Cauchy sequence, we suppose on the contrary that there exist \(\varepsilon >0\) and two subsequences \(\{ x_{g(m)}\}_{m=1}^{+\infty }\) and \(\{x_{h(m)}\}_{m=1}^{+\infty }\) of \(\{x_{n}\}\),

$$\begin{aligned} \lim_{m\rightarrow \infty} d(x_{g(m)+2},x_{h(m)+2})= \lim_{m\rightarrow \infty} d(x_{g(m)+1},x_{h(m)+1})= \lim _{m\rightarrow \infty} d(x_{g(m)},x_{h(m)})=\varepsilon _{+}. \end{aligned}$$
(46)

Applying (15) by taking \(x=x_{g(m)}\) and \(y=x_{h(m)}\), we get

$$\begin{aligned} &\tau +\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{g(m)}, \mathfrak{F}^{2}x_{h(m)}\bigr)\bigr),F_{1}\bigl(d \bigl( \mathfrak{F}^{2}x_{g(m)},\mathfrak{F}^{2}x_{h(m)} \bigr)\bigr)\bigr\} \\ &\quad \leq \alpha _{2}F_{2}\bigl(d(x_{g(m)},x_{h(m)}) \bigr)+\alpha _{1}F_{1}\bigl(d(x_{g(m)},x_{h(m)}) \bigr). \end{aligned}$$

Taking limits in the above inequalities as \(m\rightarrow \infty \), we have

$$\begin{aligned} \tau +\min \bigl\{ F_{2}(\varepsilon _{+}),F_{1}( \varepsilon _{+})\bigr\} \leq \alpha _{2}F_{2}( \varepsilon _{+})+\alpha _{1}F_{1}(\varepsilon _{+}). \end{aligned}$$

For both cases \((R1)\), \((R2)\), the above inequality yields the following possibilities:

$$\begin{aligned}& \tau +F_{2}(\varepsilon _{+})\leq F_{2}(\varepsilon _{+}), \end{aligned}$$
(47)
$$\begin{aligned}& \tau +F_{1}(\varepsilon _{+})\leq F_{1}(\varepsilon _{+}), \end{aligned}$$
(48)
$$\begin{aligned}& \tau +F_{2}(\varepsilon _{+})\leq F_{1}(\varepsilon _{+}), \end{aligned}$$
(49)
$$\begin{aligned}& \tau +F_{1}(\varepsilon _{+})\leq F_{2}(\varepsilon _{+}). \end{aligned}$$
(50)

As \(\tau >0\), inequalities (41) and (42) are the contradictions.

Now we consider inequality (43). Since \(\tau >2\upsilon \), we have

$$\begin{aligned} 2\upsilon +F_{2}(\varepsilon )\leq F_{1}(\varepsilon ). \end{aligned}$$

That is,

$$\begin{aligned} \upsilon +\bigl(\upsilon +F_{2}(\varepsilon )\bigr)< F_{1}( \varepsilon ). \end{aligned}$$

Moreover, from condition \((A)\) we also have

$$\begin{aligned} \upsilon +F_{1}(\varepsilon )< F_{1}(\varepsilon ). \end{aligned}$$

Therefore, inequality (43) yields a contradiction.

Similarly, \(\tau +F_{1}(\varepsilon )\leq F_{2}(\varepsilon )\) implies a contradiction \(\upsilon +F_{2}(\varepsilon )< F_{2}(\varepsilon )\). Contradictions of inequalities (41)–(44) prove that \(\{x_{m}\}_{m=1}^{+\infty }\) is a Cauchy sequence. Since \((X,d)\) is a complete metric space, the sequence \(\{x_{m}\}_{m=1}^{\infty }\) is convergent in X and \(x^{*}\) is the point of convergence. The completeness of \((X,d)\) proves that \(\{x_{m}\}_{m=1}^{\infty }\) converges to some point \(x^{*}\) in X.

Now, if \(\mathfrak{F}\) is orbitally continuous, then we have

$$\begin{aligned} d\bigl(\mathfrak{F}x^{*},x^{*}\bigr)=\lim _{m\rightarrow +\infty} d( \mathfrak{F}x_{m},x_{m})=\lim _{m\rightarrow +\infty} d(x_{m+1},x_{m})=d \bigl(x^{*},x^{*}\bigr)=0. \end{aligned}$$

If \(\mathfrak{F}\) is k-continuous for some \(k\in \mathbb{N}\), we have

$$\begin{aligned} d\bigl(\mathfrak{F}x^{*},x^{*}\bigr)=\lim _{m\rightarrow +\infty} d( \mathfrak{F}\bigl(\mathfrak{F}^{k-1}x_{m}, \mathfrak{F}^{k-1}x_{m}\bigr)= \lim_{m\rightarrow +\infty} d(x_{k+m},x_{k+m-1})=d\bigl(x^{*},x^{*} \bigr)=0. \end{aligned}$$

Therefore, \(\mathfrak{F}\) has a fixed point \(x^{*}\).

Furthermore, from Proposition 1.1 it follows that \(x\mapsto d(x,\mathfrak{F}x)\) is \(\mathfrak{F}\)-orbitally lower semi-continuous, then we have

$$\begin{aligned} d\bigl(x^{*},\mathfrak{F}x^{*}\bigr)\leq \liminf _{m\rightarrow \infty} d(x_{m}, \mathfrak{F}x_{m})=0, \end{aligned}$$

which implies that \(\mathfrak{F}x^{*}=x^{*}\) and \(x^{*}\) is a fixed point of \(\mathfrak{F}\).

Now, in order to prove the uniqueness, we suppose that there exist two distinct \(x,y\in X\) such that \(\mathfrak{F}x=x\neq y=\mathfrak{F}y\). Therefore, \(d(x,y)=d(\mathfrak{F}x,\mathfrak{F}y)=d(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y)>0\). From the assumptions of the theorem, we may have the following four possibilities:

$$\begin{aligned}& F_{1}\bigl(d(x,y)\bigr)=F_{1}\bigl(d( \mathfrak{F}x,\mathfrak{F}y)\bigr)< \tau +F_{1}\bigl(d( \mathfrak{F}x, \mathfrak{F}y)\bigr)\leq F_{1}\bigl(d(x,y)\bigr), \end{aligned}$$
(51)
$$\begin{aligned}& F_{2}\bigl(d(x,y)\bigr)=F_{2}\bigl(d\bigl( \mathfrak{F}^{2}x,\mathfrak{F}^{2}y\bigr)\bigr)< \tau +F_{2}\bigl(d\bigl( \mathfrak{F}^{2}x,\mathfrak{F}^{2}y \bigr)\bigr)\leq F_{2}\bigl(d(x,y)\bigr), \end{aligned}$$
(52)
$$\begin{aligned}& F_{2}\bigl(d(x,y)\bigr)=F_{2}\bigl(d\bigl( \mathfrak{F}^{2}x,\mathfrak{F}^{2}y\bigr)\bigr)< \tau +F_{2}\bigl(d\bigl( \mathfrak{F}^{2}x,\mathfrak{F}^{2}y \bigr)\bigr)\leq F_{1}\bigl(d(x,y)\bigr), \end{aligned}$$
(53)
$$\begin{aligned}& F_{1}\bigl(d(x,y)\bigr)=F_{1}\bigl(d( \mathfrak{F}x,\mathfrak{F}y)\bigr)< \tau +F_{1}\bigl(d( \mathfrak{F}x, \mathfrak{F}y)\bigr)\leq F_{2}\bigl(d(x,y)\bigr). \end{aligned}$$
(54)

Inequalities (45) and (46) are both contradictions. Inequality (47) with condition \((B)\) yields a contradiction as follows:

$$\begin{aligned} \tau +F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr)\bigr)\leq F_{1}\bigl(d(x,y)\bigr). \end{aligned}$$

Since \(\tau >2\upsilon \), we have

$$\begin{aligned} 2\upsilon +F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr)\bigr)\leq F_{1}\bigl(d(x,y)\bigr) \end{aligned}$$

or

$$\begin{aligned} \upsilon +\bigl(\upsilon +F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr)\bigr)\bigr) \leq F_{1}\bigl(d(x,y) \bigr), \end{aligned}$$

or

$$\begin{aligned} \upsilon +F_{1}\bigl(d\bigl(\mathfrak{F}^{2}x, \mathfrak{F}^{2}y\bigr)\bigr))\leq F_{1}\bigl(d(x,y)\bigr). \end{aligned}$$

As \(\mathfrak{F}^{2}x=x\), \(\mathfrak{F}^{2}y=y\), we have

$$\begin{aligned} \upsilon +F_{1}\bigl(d(x,y)\bigr))\leq F_{1}\bigl(d(x,y) \bigr), \end{aligned}$$

which is a contradiction.

Similarly, inequality (48) with condition \((A)\) implies a contradiction as follows:

$$\begin{aligned} \tau +F_{1}\bigl(d(\mathfrak{F}x,\mathfrak{F}y)\bigr)\leq F_{2}\bigl(d(x,y)\bigr). \end{aligned}$$

Since \(\tau >2\upsilon \), we can write

$$\begin{aligned} 2\upsilon +F_{1}\bigl(d(\mathfrak{F}x,\mathfrak{F}y)\bigr)\leq F_{2}\bigl(d(x,y)\bigr) \end{aligned}$$

or

$$\begin{aligned} \upsilon +\bigl(\upsilon +F_{1}\bigl(d(\mathfrak{F}x,\mathfrak{F}y) \bigr)\bigr)\leq F_{2}\bigl(d(x,y)\bigr), \end{aligned}$$

or

$$\begin{aligned} \upsilon +F_{2}\bigl(d(\mathfrak{F}x,\mathfrak{F}y)\bigr))\leq F_{2}\bigl(d(x,y)\bigr). \end{aligned}$$

As \(\mathfrak{F}x=x\), \(\mathfrak{F}y=y\), we have

$$\begin{aligned} \upsilon +F_{2}\bigl(d(x,y)\bigr))\leq F_{2}\bigl(d(x,y) \bigr), \end{aligned}$$

which is a contradiction. That proves the existence of a unique fixed point. □

Theorem 2.3

Let \((X,d)\) be a complete metric space. Suppose that a mapping \(\mathfrak{F}:X\mapsto X\) is a triple \(F^{*}\)-weak contraction and is orbitally continuous or k-continuous for some \(k\in \mathbb{N}\). Moreover, for all \(t, t_{1}, t_{2}\in \mathbb{R}\mathbbm{_{+}}\), there exist \(\upsilon >0\), \(\tau >2\upsilon \) such that

$$\begin{aligned} &F_{2}(t_{2})< F_{1}(t_{1})\leq F_{2}(t_{2})+\upsilon \end{aligned}$$
(A)

or

$$\begin{aligned} &F_{1}(t_{1})< F_{2}(t_{2})\leq F_{1}(t_{1})+\upsilon , \end{aligned}$$
(B)

or

$$\begin{aligned} &{{F}_{i}} ({{t}_{1}} )< F ( {{t}_{2}} )\le {{F}_{i}} ({{t}_{1}} )+\upsilon ,\quad i=1,2. \end{aligned}$$
(C)

Then, for every \(x_{0}\in X\), the sequence \(\{\mathfrak{F}^{m}x_{0}\}_{m=1}^{+\infty}\) converges to the unique fixed point of \(\mathfrak{F}\).

Proof

Consider a sequence \(\{x_{m}\}\subseteq X\) such that, for all \(m\in \mathbb{N}_{0}\), \(x_{m+1}=\mathfrak{F}x_{m}=\mathfrak{F}^{m}x_{0}\), where \(x_{0}\) is an arbitrary point in X. If there exists some \(m\in \mathbb{N}_{0}\) such that \(d(x_{m},\mathfrak{F}x_{m})=0\), then \(\mathfrak{F}\) admits a fixed point. So, we assume that \(d(x_{m},\mathfrak{F}x_{m})=d(\mathfrak{F}x_{m-1},\mathfrak{F}x_{m})>0\) for all \(m\in \mathbb{N}\).

We will show that \(\lim_{m\rightarrow +\infty} d(x_{m},\mathfrak{F}x_{m})=0\). From the definition of triple \(F^{*}\)-weak contraction, we can write

$$\begin{aligned} \min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \leq F \bigl(d(x_{m-1},x_{m})\bigr)- \tau . \end{aligned}$$
(58)

Using condition \((C)\), we can rewrite inequality (49) as follows:

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad \leq F_{1}\bigl(d(x_{m-1},x_{m})\bigr)- \tau . \end{aligned}$$
(59)

Then we have either

$$\begin{aligned} F_{1}\bigl(d(x_{m-1},x_{m})\bigr)= \min \bigl\{ F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr),F_{1}\bigl(d(x_{m-1},x_{m}) \bigr)\bigr\} \end{aligned}$$
(60)

or

$$\begin{aligned} F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr)=\min \bigl\{ F_{2}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr),F_{1} \bigl(d(x_{m-1},x_{m})\bigr)\bigr\} . \end{aligned}$$
(61)

If relation (51) exists, then (50) can be written as

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad \leq \min \bigl\{ F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr),F_{1}\bigl(d(x_{m-1},x_{m}) \bigr) \bigr\} -\tau . \end{aligned}$$
(62)

If inequality (52) holds, we can write \(F_{2}(d(\mathfrak{F}x_{m-1},\mathfrak{F}x_{m}))< F_{1}(d(x_{m-1},x_{m}))\).

Using \((A)\), we can write

$$\begin{aligned} F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr)< F_{1}\bigl(d(x_{m-1},x_{m}) \bigr) \leq F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr)+\upsilon . \end{aligned}$$
(63)

Using inequality (54) in (50), we have

$$\begin{aligned} \min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \leq F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m}) \bigr)+\upsilon -\tau . \end{aligned}$$

Moreover, from (52), we have

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad\leq \min \bigl\{ F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr),F_{1}\bigl(d(x_{m-1},x_{m}) \bigr) \bigr\} +\upsilon -\tau . \end{aligned}$$
(64)

Combining both inequalities (53) and (55), we have

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad\leq\min \bigl\{ F_{2}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr),F_{1}\bigl(d(x_{m-1},x_{m}) \bigr) \bigr\} +\delta _{m}\upsilon -\tau , \end{aligned}$$
(65)

where

$$\delta _{m}= \textstyle\begin{cases} 0 & \text{if } F_{2}(t_{2})>F_{1}(t_{1}), \\ 1 & \text{if } F_{2}(t_{2})\leq F_{1}(t_{1}), \end{cases} $$

\(t_{1}, t_{2}\in \mathbb{R}\mathbbm{_{+}}\), \(t_{1}\neq t_{2}\).

The above relation is equivalent to

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad \leq \min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-2}, \mathfrak{F}^{2}x_{m-1}\bigr)\bigr),F_{1} \bigl(d(x_{m-2},x_{m-1})\bigr) \bigr\} +\delta _{m} \upsilon -\tau . \end{aligned}$$

Repeating this process, we have

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad\leq \min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-3}, \mathfrak{F}^{2}x_{m-2}\bigr)\bigr),F_{1} \bigl(d(x_{m-3},x_{m-2})\bigr) \bigr\} \\ &\quad\quad{}+\delta _{m}\upsilon +\delta _{m-1}\upsilon -2\tau \\ &\quad\leq \cdots \\ &\quad\leq \min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{1}, \mathfrak{F}^{2}x_{0}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{1},\mathfrak{F}x_{0})\bigr)\bigr\} + \sum _{j=1}^{m} \delta _{j}\upsilon -m \tau \end{aligned}$$

or

$$\begin{aligned} &\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr),F_{1}\bigl(d( \mathfrak{F}x_{m-1},\mathfrak{F}x_{m})\bigr)\bigr\} \\ &\quad \leq F_{1}\bigl(d(x_{1},x_{0})\bigr)+ \sum _{j=1}^{m}\delta _{j} \upsilon -(m+1)\tau . \end{aligned}$$

Since \(\tau >2\upsilon \) and \(\sum_{j=1}^{m}\delta _{j}< m+1\), we have

$$\begin{aligned} \lim_{m\rightarrow +\infty} \sum_{j=1}^{m} \delta _{j} \upsilon -(m+1)\tau =-\infty . \end{aligned}$$

So that we have

$$\begin{aligned} \lim_{m\rightarrow +\infty} \min \bigl\{ F_{2}\bigl(d \bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m} \bigr)\bigr),F_{1}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr) \bigr\} =-\infty . \end{aligned}$$
(66)

Now, equation (57) further has two possible cases:

$$\begin{aligned} &\lim_{m\rightarrow +\infty} F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)\bigr)=-\infty , \end{aligned}$$
(G)
$$\begin{aligned} &\lim_{m\rightarrow +\infty} F_{1}\bigl(d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})\bigr)=-\infty . \end{aligned}$$
(H)

Condition \((C_{1})\) with case \((G)\) yields

$$\begin{aligned} \lim_{m\rightarrow +\infty} d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)=0, \end{aligned}$$

or equivalently,

$$\begin{aligned} \lim_{m\rightarrow +\infty} d\bigl(\mathfrak{F}^{2}x_{m-1}, \mathfrak{F}^{2}x_{m}\bigr)=\lim_{m\rightarrow +\infty} d(x_{m+1}, \mathfrak{F}x_{m+1})=\lim_{m\rightarrow +\infty} d(x_{m}, \mathfrak{F}x_{m})=0. \end{aligned}$$

Condition \((C_{1})\) with case \((H)\) yields

$$\begin{aligned} \lim_{m\rightarrow +\infty} d(\mathfrak{F}x_{m-1}, \mathfrak{F}x_{m})=\lim_{m\rightarrow +\infty} d(x_{m}, \mathfrak{F}x_{m})=0. \end{aligned}$$

Therefore, from (57), we get

$$\begin{aligned} \lim_{m\rightarrow +\infty} d(x_{m}, \mathfrak{F}x_{m})=0. \end{aligned}$$
(69)

Now, in order to prove that the sequence \(\{x_{m}\}_{m=1}^{+\infty }\) is a Cauchy sequence, we suppose on the contrary that there exist \(\varepsilon >0\) and two subsequences \(\{ x_{g(m)}\}_{m=1}^{+\infty }\) and \(\{x_{h(m)}\}_{m=1}^{+\infty }\) of \(\{x_{n}\}\),

$$\begin{aligned} \lim_{m\rightarrow \infty} d(x_{g(m)+2},x_{h(m)+2})= \lim_{m\rightarrow \infty} d(x_{g(m)+1},x_{h(m)+1})= \lim _{m\rightarrow \infty} d(x_{g(m)},x_{h(m)})=\varepsilon _{+}. \end{aligned}$$
(70)

Applying (16) by taking \(x=x_{g(m)}\) and \(y=x_{h(m)}\), we get

$$\begin{aligned} &\tau +\min \bigl\{ F_{2}\bigl(d\bigl(\mathfrak{F}^{2}x_{g(m)}, \mathfrak{F}^{2}x_{h(m)}\bigr)\bigr),F_{1}\bigl(d \bigl( \mathfrak{F}^{2}x_{g(m)},\mathfrak{F}^{2}x_{h(m)} \bigr)\bigr)\bigr\} \\ &\quad \leq F\bigl(d(x_{g(m)},x_{h(m)})\bigr). \end{aligned}$$

Taking limits in the above inequalities as \(m\rightarrow \infty \), we have

$$\begin{aligned} \tau +\min \bigl\{ F_{2}(\varepsilon _{+}),F_{1}( \varepsilon _{+})\bigr\} \leq F_{1}( \varepsilon _{+}), \end{aligned}$$

which yields either \(\tau +F_{2}(\varepsilon _{+})\leq F(\varepsilon _{+})\) or \(\tau +F_{1}(\varepsilon _{+})\leq F(\varepsilon _{+})\).

If \(\tau +{{F}_{2}} (\varepsilon _{+} )\le F ( \varepsilon _{+} )\), condition \((C)\) allows us to write \(\tau +{{F}_{2}} (\varepsilon _{+} )\le {{F}_{2}} ( \varepsilon _{+} )+\upsilon \). That yields a contradiction as \(\tau >2\upsilon \). Likewise, \(\tau +{{F}_{1}} (\varepsilon _{+} )\le F ( \varepsilon _{+} )\) gives a contradiction \(\tau +{{F}_{2}} (\varepsilon _{+} )\le {{F}_{2}} ( \varepsilon _{+} )+\upsilon \).

The completeness of \((X,d)\) proves that \(\{x_{m}\}_{m=1}^{+\infty}\) converges to some point \(x^{*}\) in X.

The rest of the proof runs as the proof of Theorem 2.1. For brevity, we omit it. □

Example 2.4

Consider a closed unit ball \(\mathcal{B}\) in the \(\ell _{1}\) space of all absolutely summable sequences, \(u =(u _{1},u_{2},\ldots )\) with a metric inherited from the standard norm \(\|u\|=\sum_{i=1}^{+\infty}|u_{i}|\). Let \(h:[-1,1]\mapsto [-1,1]\) be the function given by

$$h(w)= \textstyle\begin{cases} 1+2w,& \text{if } -1\leq w\leq -\frac{1}{2}, \\ 0,& \text{if } -\frac{1}{2}< w\leq \frac{1}{2}, \\ -1+2w,& \text{if } \frac{1}{2}< w\leq 1. \end{cases} $$

Observe that for all \(s, w\in [-1,1]\), we have \(|h(s)-h(w)|\leq 2|s-w|\) and \(|h(w)|\leq |w|\).

Define a continuous mapping \(\mathfrak{F}:\mathcal{B}\mapsto \mathcal{B}\) by

$$\begin{aligned} \mathfrak{F}u=\mathfrak{F}(u_{1},u_{2},\ldots )=e^{-\kappa}\biggl(h(u_{2}), \frac{2}{3}u_{3},u_{4},u_{5}, \ldots \biggr), \end{aligned}$$

where \(\kappa >0\) is a real number. Then we have

$$\begin{aligned} \mathfrak{F}^{2}u=e^{-\kappa}\biggl(h\biggl( \frac{2}{3}u_{3}\biggr),\frac{2}{3}u_{4},u_{5},u_{6}, \ldots \biggr). \end{aligned}$$

For each \(u= (u_{1},u_{2}, \ldots )\) and \(v=(v_{1},v_{2},\ldots )\) in \(\mathcal{B}\), we have

$$\begin{aligned} e^{\kappa} \Vert \mathfrak{F}u-\mathfrak{F}v \Vert &= \bigl\vert h(u_{2})-h(v_{2}) \bigr\vert + \frac{2}{3} \vert u_{3}-v_{3} \vert +\sum_{k=4}^{+\infty } \vert u_{k}-v_{k} \vert \\ & \leq 2 \vert u_{2}-v_{2} \vert +\frac{2}{3} \vert u_{3}-v_{3} \vert + \sum _{k=4}^{+ \infty} \vert u_{k}-v_{k} \vert \\ & \leq 2 \Vert u- v \Vert . \end{aligned}$$

That is,

$$\begin{aligned} \frac{e^{\kappa}}{2} \Vert \mathfrak{F}u-\mathfrak{F}v \Vert \leq \Vert u-v \Vert . \end{aligned}$$
(71)

The above inequality can be written as

$$\begin{aligned} \kappa +\ln \biggl(\frac{1}{2} \Vert \mathfrak{F}u- \mathfrak{F}v \Vert \biggr)\leq \ln \Vert u-v \Vert . \end{aligned}$$
(72)

Likewise,

$$\begin{aligned} e^{\kappa} \bigl\Vert \mathfrak{F}^{2}u-\mathfrak{F}^{2}v \bigr\Vert &= \biggl\vert h\biggl(\frac{2}{3}u_{3}\biggr)-h \biggl( \frac{2}{3}v_{3}\biggr) \biggr\vert + \frac{2}{3} \vert u_{3}-v_{3} \vert \\ &\quad{}+ \sum_{k=5}^{+\infty} \vert u_{k}-v_{k} \vert \\ & \leq \frac{4}{3} \vert u_{3}-v_{3} \vert + \frac{2}{3} \vert u_{3}-v_{3} \vert +\sum _{k=}^{+\infty} \vert u_{k}-v_{k} \vert \\ & \leq \frac{4}{3} \Vert u-v \Vert . \end{aligned}$$

That is,

$$\begin{aligned} \frac{3e^{\kappa}}{4} \bigl\Vert \mathfrak{F}^{2}u- \mathfrak{F}^{2}v \bigr\Vert \leq \Vert u-v \Vert . \end{aligned}$$

The above inequality can be written as

$$\begin{aligned} \kappa + \ln \biggl(\frac{3}{4} \bigl\Vert \mathfrak{F}^{2}u-\mathfrak{F}^{2}v \bigr\Vert \biggr) \leq \ln \Vert u-v \Vert . \end{aligned}$$
(73)

Now, we define \(F_{1}(t)=\ln (\frac{t}{2})\), \(F_{2}(t)=\ln (\frac{3t}{4})\), \(F(t)=\ln (t)\). So, inequalities (61), (62) will take the form

$$\begin{aligned} \kappa +\min \bigl\{ F_{1}\bigl( \Vert \mathfrak{F}u- \mathfrak{F}v \Vert \bigr),F_{2}\bigl( \bigl\Vert \mathfrak{F}^{2}u-\mathfrak{F}^{2}v \bigr\Vert \bigr)\bigr\} \leq \min \{F\bigl( \Vert u-v \Vert \bigr). \end{aligned}$$
(74)

Further, the definitions of \(F_{1}\), \(F_{2}\), and F yield \(F_{1}(t)< F_{2}(t)\). So that we have \(\ln \frac{t}{2}<\ln \frac{3t}{4}\). Now, we can define \(0<\upsilon \leq \ln \frac{3}{2}\) such that

$$\begin{aligned} \ln \frac{t}{2}< \ln \frac{3t}{4}\leq \ln \frac{t}{2}+ \upsilon . \end{aligned}$$

That is, \(F_{1}(t)< F_{2}(t)\leq F_{1}(t)+\upsilon \).

Therefore, we conclude from inequality (63) that \(\mathfrak{F}:\mathcal{B}\mapsto \mathcal{B}\) represents the triple \(F^{*}\)-weak mapping.

Next we will show that for every \(x_{0} \in \mathcal{B}\), the sequence \(\{\mathfrak{F}^{m}x_{0}\}_{m=1}^{+\infty }\) converges to a unique fixed point \(x^{*}=(0, 0,\ldots )\).

For some fixed \(i\in \mathbb{N}\), consider the absolutely summable sequences \(a=(a_{1},a_{2},\ldots ,a_{i},0,0, \ldots )\in \mathcal{B}\) with a metric inherited from the standard norm. As \(\mathcal{B}\) is a closed unit ball, we have \(\sum_{k=1}^{i}{|a_{k}|\leq 1}\).

Now,

$$\begin{aligned} \mathfrak{F}a=e^{-\kappa}\biggl(\tau (a_{2}), \frac{2}{3}a_{3},\ldots ,a_{i},0,0, \ldots \biggr) \end{aligned}$$

implies that

$$\begin{aligned} \mathfrak{F}^{2}a=e^{-\kappa}\biggl(\tau \biggl( \frac{2}{3}a_{3}\biggr),a_{4},\ldots ,a_{i},0,0, \ldots \biggr). \end{aligned}$$

Then, for all \(m>i\), we have \(\mathfrak{F}^{m}a=(0,0,\ldots ,0)\). That is, \(\{\mathfrak{F}^{m}a\}_{m=1}^{+\infty}=(0,0,\ldots )\) is a unique fixed point.

3 Applications

3.1 Application to Caputo fractional derivative \({}^{C}{{D}^{\chi }}\) of order χ

We apply our obtained result to weaken the condition on the Caputo fractional derivative \({}^{C}{{D}^{\chi }}\) of order χ, considered in [22, 23].

Consider a Caputo fractional derivative \({}^{C}{{D}^{\chi }}\) of order χ and the following problem:

$$ ^{C}{{D}^{\chi }}\bigl({x_{1}}(t)\bigr)=h \bigl(t,{x_{1}}(t)\bigr),\quad (0< t< 1, 1< \chi \le 2), $$
(75)

with the integral boundary conditions

$$ {x_{1}}(0)=0,\qquad {x_{1}}(1)= \int _{0}^{\delta }{{x_{1}}(s)}\,ds\quad (0< \delta < 1).$$

Let \(h:[0,1]\times \mathbb{R}\to \mathbb{R}\) be a continuous function and \((X, \|.\|_{\infty })\) be a Banach space of continuous functions \(C([0,1],\mathbb{R})\) from \([0,1] \) into \(\mathbb{R}\) endowed with supremum norm \({{ \Vert x \Vert }_{\infty }}= \sup_{t\in [ 0,1 ]} \vert x(t) \vert \). Then, for a continuous function \(\mathfrak{F}:(0, \infty )\to \mathbb{R}\), a Caputo derivative of fractional order χ can be defined as follows:

$$ {}^{C}{{D}^{\chi }}\mathfrak{F}(t)=\frac{1}{\Gamma (n-\chi )} \int _{0}^{t}{{{(t-s)}^{n- \chi -1}} {{ \mathfrak{F}}^{n}}(s)\,\mathrm{d}s\quad \bigl(n-1< \chi < n,n=[ \chi ]+1 \bigr)},$$

where \([\chi ] \) denotes the greatest integer not greater than χ. Consider a continuous function \(\mathfrak{F}:{{\mathbb{R}}^{+}}\to \mathbb{R}\), and define the Riemann–Liouville fractional derivatives of order χ as follows:

$$ {{D}^{\chi }}\mathfrak{F}(t)=\frac{1}{\Gamma (n-\chi )} \frac{{{d}^{n}}}{d{{t}^{n}}} \int _{0}^{t}{ \frac{\mathfrak{F}(s)}{{{(t-s)}^{\chi -n+1}}}}\,ds\quad \bigl(n=[ \chi ]+1\bigr),$$

where the function of t on the right-hand side is pointwise defined on \((0,+\infty ) \).

Next, we establish the existence theorem.

Theorem 3.1

Suppose that:

  1. 1.

    A function \(\xi : \mathbb{R}^{2} \to \mathbb{R}\), \(\xi (a,d)>0\) for all \(a,d\in \mathbb{R}\) and a real number \(\tau >0\) such that

    $$ \bigl\vert h(t,a)-h(t,d) \bigr\vert \le \frac{\Gamma (\chi +1)}{5}{{e}^{-\tau }} \vert a-d \vert ^{m} \quad \textit{for all } t\in [0,1] \textit{ and } m \in \mathbb{R} \backslash \{0\}. $$
  2. 2.

    There exists \(\mathfrak{F}:X\to X\) such that

    $$\begin{aligned} \mathfrak{F} {x_{1}}(t)&=\frac{1}{\Gamma (\chi )} \int _{0}^{t}{{(t-s)}^{ \chi -1}}h \bigl(s,{x_{1}}(s)\bigr)\,\mathrm{d}s\\ &\quad {}- \frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{1}{{{(1-s)}^{ \chi -1}}h \bigl(s,{x_{1}}(s)\bigr)\,\mathrm{d}s}\\ &\quad {}+\frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{\delta }{ \biggl( \int _{0}^{s}{{{(s-\kappa )}^{\chi -1}}h\bigl( \kappa ,{x_{1}}( \kappa )\bigr)\,\mathrm{d}\kappa} \biggr)} \,\mathrm{d}s \end{aligned}$$

    and

    $$\begin{aligned} \mathfrak{F}^{2} {x_{1}}(t)&=\frac{\omega}{\Gamma (\chi )} \int _{0}^{t}{{(t-s)}^{ \chi -1}}h\bigl(s, \mathfrak{F} {x_{1}}(s)\bigr)\,\mathrm{d}s\\ &\quad {}- \frac{2t\omega}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{1}{{{(1-s)}^{ \chi -1}}h\bigl(s, \mathfrak{F} {x_{1}}(s)\bigr)\,\mathrm{d}s}\\ &\quad {}+\frac{2t\omega}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{\delta }{ \biggl( \int _{0}^{s}{{{(s-\kappa )}^{\chi -1}}h\bigl( \kappa ,\mathfrak{F} {x_{1}}( \kappa )\bigr)\,\mathrm{d}\kappa} \biggr)} \,\mathrm{d}s, \end{aligned}$$

    with \(\xi ({{x}_{{{1}_{0}}}}(t),\mathfrak{F}{{x}_{{{1}_{0}}}}(t))>0\), \({{x}_{{{1}_{0}}}}\in X\) and \(t\in [0,1]\), \(\omega \in [0, \infty ) \).

  3. 3.

    For each \(t\in [0,1] \) and \({x_{1}},{x_{2}}\in X\), \(\xi ({x_{1}}(t), {x_{2}}(t))>0\) implies \(\xi (\mathfrak{F}{x_{1}}(t),\mathfrak{F}{x_{2}}(t))>0\).

  4. 4.

    If \(\{ {{x}_{{{1}_{n}}}}\}\) is a sequence in X such that \({{x}_{{{1}_{n}}}}\to {x_{1}}\) in X and \(\xi ({{x}_{{{1}_{n}}}},{{x}_{{{1}_{n+1}}}})>0\) for all \(n\in \mathbb{N}\), then \(\xi ({{x}_{{{1}_{n}}}},{x_{1}})>0\) for all \(n\in \mathbb{N}\).

Then problem (64) has at least one solution.

Proof

Function \({x_{1}}\in X\) represents the solution of (64) if and only if it satisfies the following integral equation:

$$\begin{aligned} {x_{1}}(t)&=\frac{1}{\Gamma (\chi )} \int _{0}^{t}{{{(t-s)}^{\chi -1}}h \bigl(s,{x_{1}}(s)\bigr) \,\mathrm{d}s-\frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )}} \int _{0}^{1}{{{(1-s)}^{ \chi -1}}h\bigl(s,\chi (s)\bigr)\,\mathrm{d}s}\\ &\quad {}+\frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{\delta }{ \biggl( \int _{0}^{s}{{{(s-\kappa )}^{\chi -1}}h\bigl( \kappa ,{x_{1}}( \kappa )\bigr)\,\mathrm{d}\kappa} \biggr)}\,\mathrm{d}s, \quad t\in [0,1]. \end{aligned}$$

The solution of problem (64) is equivalent to finding the solution \({x_{1}}^{*}\in X\), which is a fixed point of \(\mathfrak{F}\).

Now, let \({x_{1}},{x_{2}}\in X\) such that \(\xi ({x_{1}}(t),{x_{2}}(t))>0\) for all \(t\in [0,1]\). By (i), we have

$$\begin{aligned} &\bigl\vert {\mathfrak{F}} {x_{1}}(t)-\mathfrak{F} {x_{2}}(t) \bigr\vert \\ &\quad= \biggl\vert \frac{1}{\Gamma (\chi )} \int _{0}^{t}{{{(t-s)}^{\chi -1}}h \bigl(s,{x_{1}}(s)\bigr)\,ds- \frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{1}{{{(1-s)}^{ \chi -1}}h \bigl(s,{x_{1}}(s)\bigr)\,ds}} \\ &\quad\quad {}+\frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{\delta }{ \biggl( \int _{0}^{s}{{{(s-\kappa )}^{\chi -1}}h\bigl( \kappa ,{x_{1}}( \kappa )\bigr)\,d\kappa} \biggr)\,ds}\\ &\quad\quad {}-\frac{1}{\Gamma (\chi )} \int _{0}^{t}{{{(t-s)}^{\chi -1}}h \bigl(s,{x_{2}}(s)\bigr)\,ds+} \frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{1}{{{(1-s)}^{ \chi -1}}h \bigl(s,{x_{2}}(s)\bigr)\,ds}\\ &\quad\quad {}-\frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{ \delta }{ \biggl( \int _{0}^{s}{{{(s-\kappa )}^{\chi -1}}h\bigl( \kappa ,{x_{2}}( \kappa )\bigr)\,d\kappa} \biggr)\,ds} \biggr\vert \\ &\quad \le \frac{1}{\Gamma (\chi )} \int _{0}^{t}{ \vert t-s \vert ^{\chi -1} \bigl\vert h\bigl(s,{x_{1}}(s)\bigr)-h\bigl(s,{x_{2}}(s) \bigr) \bigr\vert \,ds}\\ &\quad\quad {}+\frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{1}{ \bigl\vert {{(1-s)}^{ \chi -1}} \bigr\vert h\bigl(s,{x_{1}}(s)\bigr)-h\bigl(s,{x_{2}}(s) \bigr)|\,ds}\\ &\quad\quad {}+\frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{\delta }{ \biggl\vert \int _{0}^{s}{{{(s-\kappa )}^{\chi -1}}\bigl(h \bigl(\kappa ,{x_{1}}( \kappa )\bigr)-h\bigl(\kappa , {x_{2}}(\kappa )\bigr)\bigr)\,d\kappa} \biggr\vert \,ds }\\ &\quad\le \frac{1}{\Gamma (\chi )} \int _{0}^{t}{ \vert t-s \vert ^{\chi -1} \frac{\Gamma (\chi +1)}{5}} {{e}^{-\tau }} \bigl\vert {x_{1}}(t)-{x_{2}}(t) \bigr\vert ^{m} \,ds\\ &\quad\quad {}+\frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{1}{{{(1-s)}^{ \chi -1}} \frac{\Gamma (\chi +1)}{5}} {{e}^{-\tau }} \bigl\vert {x_{1}}(s)-{x_{2}}(s) \bigr\vert ^{m} \,ds\\ &\quad\quad {}+\frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{\delta }{ \biggl( \int _{0}^{s}{ \vert s-\kappa \vert ^{\chi -1} \frac{\Gamma (\chi +1)}{5}} {{e}^{-\tau }} \bigl\vert {x_{1}}(\kappa )-{x_{2}}( \kappa ) \bigr\vert ^{m} \,d\kappa \biggr)}\,ds\\ &\quad \le \frac{\Gamma (\chi +1)}{5}{{e}^{-\tau }} \Vert {x_{1}}-{x_{2}} \Vert ^{m}_{ \infty }\sup_{t\in (0,1)} \biggl( \frac{1}{\Gamma (\chi )} \int _{0}^{1}{ \vert \mathfrak{t}-s \vert ^{\chi -1}\,ds} \\ &\quad\quad {}+\frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{1}{{{(1-s)}^{ \chi -1}}\,ds+ \frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{ \delta }{ \int _{0}^{s}{ \vert s-\kappa \vert ^{\chi -1}\,d\kappa \,ds}}} \biggr)\\ &\quad\le {{e}^{-\tau }} \Vert {x_{1}}-{x_{2}} \Vert ^{m} _{\infty }. \end{aligned}$$

Thus, for each \({x_{1}},{x_{2}} \in X\), we have

$$ \Vert \mathfrak{F} {x_{1}}-\mathfrak{F} {x_{2}} \Vert _{\infty }\le {{e}^{-\tau }} \Vert {x_{1}}-{x_{2}} \Vert ^{m}_{\infty }. $$
(76)

The above relation can be written as

$$ \ln {{ \bigl\Vert \mathfrak{F} ({x_{1}})-\mathfrak{F} ({x_{2}}) \bigr\Vert }_{\infty }}\le \ln \bigl(e^{-\tau} \bigr) +\ln \Vert {x_{1}}-{x_{2}} \Vert _{\infty }^{m}. $$
(77)

Likewise,

$$\begin{aligned}& \bigl\vert {\mathfrak{F}}^{2} {x_{1}}(t)- \mathfrak{F}^{2} {x_{2}}(t) \bigr\vert \\& \quad = \biggl\vert \frac{\omega}{\Gamma (\chi )} \int _{0}^{t}{{{(t-s)}^{\chi -1}}h\bigl(s, \mathfrak{F} {x_{1}}(s)\bigr)\,ds- \frac{2t\omega}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{1}{{{(1-s)}^{ \chi -1}}h\bigl(s, \mathfrak{F} {x_{1}}(s)\bigr)\,ds}} \\& \qquad {}+\frac{2t\omega}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{\delta }{ \biggl( \int _{0}^{s}{{{(s-\kappa )}^{\chi -1}}h\bigl( \kappa ,\mathfrak{F} {x_{1}}( \kappa )\bigr)\,d\kappa} \biggr)\,ds}\\& \qquad {}-\frac{\omega}{\Gamma (\chi )} \int _{0}^{t}{{{(t-s)}^{\chi -1}}h\bigl(s, \mathfrak{F} {x_{2}}(s)\bigr)\,ds+} \frac{2t\omega}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{1}{{{(1-s)}^{ \chi -1}}h\bigl(s, \mathfrak{F} {x_{2}}(s)\bigr)\,ds}\\& \qquad {}-\frac{2t\omega}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{ \delta }{ \biggl( \int _{0}^{s}{{{(s-\kappa )}^{\chi -1}}h\bigl( \kappa , \mathfrak{F} {x_{2}}(\kappa )\bigr)\,d\kappa} \biggr)\,ds} \biggr\vert \\& \quad \le \frac{\omega}{\Gamma (\chi )} \int _{0}^{t}{ \vert t-s \vert ^{\chi -1} \bigl\vert h\bigl(s, \mathfrak{F} {x_{1}}(s)\bigr)-h\bigl(s, \mathfrak{F} {x_{2}}(s)\bigr) \bigr\vert \,ds}\\& \qquad {}+\frac{2t\omega}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{1}{ \bigl\vert {{(1-s)}^{ \chi -1}} \bigr\vert h\bigl(s,\mathfrak{F} {x_{1}}(s)\bigr)-h\bigl(s, \mathfrak{F} {x_{2}}(s)\bigr)|\,ds}\\& \qquad {}+\frac{2t\omega}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{\delta }{ \biggl\vert \int _{0}^{s}{{{(s-\kappa )}^{\chi -1}}\bigl(h \bigl(\kappa , \mathfrak{F} {x_{1}}(\kappa )\bigr)-h\bigl(\kappa , \mathfrak{F} {x_{2}}(\kappa )\bigr)\bigr)\,d \kappa} \biggr\vert \,ds }\\& \quad \le \frac{\omega}{\Gamma (\chi )} \int _{0}^{t}{ \vert t-s \vert ^{\chi -1} \frac{\Gamma (\chi +1)}{5}} {{e}^{-\tau }} \bigl\vert \mathfrak{F} {x_{1}}(t)- \mathfrak{F} {x_{2}}(t) \bigr\vert ^{m} \,ds\\& \qquad {}+\frac{2t\omega}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{1}{{{(1-s)}^{ \chi -1}} \frac{\Gamma (\chi +1)}{5}} {{e}^{-\tau }} \bigl\vert \mathfrak{F} {x_{1}}(s)- \mathfrak{F} {x_{2}}(s) \bigr\vert ^{m} \,ds\\& \qquad {}+\frac{2t\omega}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{\delta }{ \biggl( \int _{0}^{s}{ \vert s-\kappa \vert ^{\chi -1} \frac{\Gamma (\chi +1)}{5}} {{e}^{-\tau }} \bigl\vert \mathfrak{F} {x_{1}}(\kappa )- \mathfrak{F} {x_{2}}(\kappa ) \bigr\vert ^{m} \,d\kappa \biggr)}\,ds\\& \quad \le \frac{\Gamma (\chi +1)}{5}{\omega{e}^{-\tau }} \Vert \mathfrak{F} {x_{1}}- \mathfrak{F} {x_{2}} \Vert ^{m}_{\infty } \sup_{t\in (0,1)} \biggl( \frac{1}{\Gamma (\chi )} \int _{0}^{1}{ \vert \mathfrak{{F}}-S \vert ^{ \chi -1}\,ds} \\& \qquad {}+\frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{1}{{{(1-S)}^{ \chi -1}}\,ds+ \frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{ \delta }{ \int _{0}^{s}{ \vert S-K \vert ^{\chi -1}\,d \kappa \,ds}}} \biggr)\\& \quad \le \omega{{e}^{-\tau }} \Vert \mathfrak{F} {x_{1}}- \mathfrak{F} {x_{2}} \Vert ^{m} _{\infty }. \end{aligned}$$

Therefore,

$$ \bigl\vert {\mathfrak{F}}^{2} {x_{1}}- \mathfrak{F}^{2} {x_{2}} \bigr\vert \le \omega{{e}^{- \tau }} \Vert \mathfrak{F} {x_{1}}-\mathfrak{F} {x_{2}} \Vert ^{m} _{\infty }. $$
(78)

Further, relation (65) along with (67) yields

$$ \frac{1}{{{\omega }^{\frac{1}{m}}}} \bigl\Vert {{\mathfrak{F}}^{2}} {x_{1}}-{{ \mathfrak{F}}^{2}} {x_{2}} \bigr\Vert _{\infty }^{\frac{1}{m}}\le {{\bigl({{e}^{- \tau }} \bigr)}^{1+\frac{1}{m}}} \Vert {x_{1}}-{x_{2}} \Vert _{\infty }^{m}.$$

The above inequality can be written as

$$ \ln \biggl( \frac{1}{{{\omega }^{\frac{1}{m}}}} \bigl\Vert {{ \mathfrak{F}}^{2}} {x_{1}}-{{\mathfrak{F}}^{2}} {x_{2}} \bigr\Vert _{ \infty }^{\frac{1}{m}} \biggr) \le -\tau \biggl( 1+ \frac{1}{m} \biggr)+\ln \Vert {x_{1}}-{x_{2}} \Vert _{\infty }^{m}. $$
(79)

Now, we define

$$ {{F}_{1}} ( \alpha )=\ln \alpha ,\qquad {{F}_{2}} ( \alpha )=\ln {{ \biggl( \frac{\alpha }{\omega } \biggr)}^{ \frac{1}{m}}} ,\qquad F ( \alpha )=\ln {{ ( \alpha )}^{m}}.$$

So that relations (66) and (68) will take the following forms:

$$\begin{aligned}& \tau + {{F}_{1}} \bigl( d ( \mathfrak{F} {x_{1}}, \mathfrak{F} {x_{2}} ) \bigr)\le F \bigl( d ( {x_{1}},{x_{2}} ) \bigr), \end{aligned}$$
(80)
$$\begin{aligned}& \tau + \frac{\tau}{m}+ {{F}_{2}} \bigl( d \bigl( {{\mathfrak{F} }^{2}} {x_{1}},{{ \mathfrak{F} }^{2}} {x_{2}} \bigr) \bigr)\le F \bigl( d ( {x_{1}},{x_{2}} ) \bigr). \end{aligned}$$
(81)

By combining (69) and (70), we can write

$$ \tau + \frac{\tau}{m} + Min \bigl\{ {{F}_{2}} \bigl( d \bigl( {{ \mathfrak{F} }^{2}} {x_{1}},{{\mathfrak{F} }^{2}} {x_{2}} \bigr) \bigr),{{F}_{1}} \bigl( d \bigl( \mathfrak{F} ({x_{1}}), \mathfrak{F} {x_{2}} \bigr) \bigr) \bigr\} \le F \bigl( d ( {x_{1}},{x_{2}} ) \bigr).$$

Therefore, \(\mathfrak{F} :X \to X \) is a tripe F-contraction, and the operator \(\mathfrak{F}\) has a fixed point in X. □

3.2 Application to second-order differential equation

Next, we apply the triple \(F^{*}\)-weak contraction to find the solution of a second-order differential equation presenting an engineering problem subject to a damping force in the case of a vertical spring moving through a fluid. If an external force affects the motion of spring, then the system for critical damped motion represents the following form of differential equation:

$$ \frac{{{d}^{2}}w}{d{{t}^{2}}}+\frac{c}{m}\frac{dw}{dt}=B^{2} \bigl( t,w(t) \bigr);\quad w(0)=0, \dot{w}(0)=a,$$

where the continuous function \(B^{2}: [ 0,I ]\times {{\mathbb{R}}_{+}}\to \mathbb{R}\), \(I>0\) is the self-composition of \(B: [ 0,J ], J>0\). The equivalent integral equation to the above problem is as follows:

$$ y(s)= \int _{0}^{s}{{B^{2}} \bigl( w,y(w) \bigr) D(s,w)}\,dw, \quad s\in [ 0,I ]. $$
(82)

Here \(D(s,w)\) is the green functions defined by

$$ D(s,w)= \textstyle\begin{cases} (s-w){{e}^{\kappa (s-w)}}&\text{if } 0\le w\le s\le I, \\ 0&\text{if }0\le s\le w\le I, \end{cases} $$
(83)

with a constant \(\kappa (c,m)>0\).

Suppose that the collection of all continuous functions \(v: [ 0,I ]\to {{\mathbb{R}}^{+}}\) is X.

We define

$$ {{ \Vert v \Vert }_{\kappa }}= \sup_{s\in [ 0,I ]} \bigl\{ \bigl\vert v(s) \bigr\vert {{e}^{-2\kappa s}} \bigr\} .$$

Next, we consider a distance function \(d:X\times X\to [ 0,\infty )\) defined by

$$ d(a,b)=\max \bigl\{ {{ \Vert a \Vert }_{\kappa }},{{ \Vert b \Vert }_{\kappa }} \bigr\} \quad \text{for all } a,b\in X.$$

Now, in order to find the existence of solution to boundary value problem (71), we consider a function \(\mathfrak{F}:X \to X\) defined by

$$ \mathfrak{F}^{2} \bigl( y(s) \bigr)= \int _{0}^{s}{B^{2} \bigl( w,y(w) \bigr)D(s,w)}\,dw $$
(84)

for all \(y\in X\) and \(s\in [ 0,I ]\). We will prove that there exists some \(v\in X\) such that \(\mathfrak{F} ( v(s) )=v(s)\). That is, the fixed point of triple \(F^{*}\)-weak contraction will represent the solution of problem (71).

Theorem 3.2

Nonlinear integral equation (71) has a solution if the following conditions hold:

  1. a)

    \(B ( w,y(w) ) \) and \(B^{2} ( w,y(w) )\) are increasing functions;

  2. d)

    There exist \(k, \kappa >0\) such that

    $$ \bigl\vert B^{2}(w,y) \bigr\vert \le k{{\kappa }^{2}} {{e}^{-\kappa }}y,\quad w\in [ 0,I ]\textit{ and }y\in {{\mathbb{R}}_{+}}.$$

Proof

For all \(v,w\in X\), we have

$$ \bigl\vert \mathfrak{F}^{2} \bigl( v(w) \bigr) \bigr\vert = \int _{0}^{s}{ \bigl\vert B^{2} \bigl( w,v(w) \bigr) \bigr\vert } D(s,w)\,dw.$$

Now, using conditions a) and d), we can write

$$\begin{aligned} \bigl\vert \mathfrak{F}^{2} \bigl( v(w) \bigr) \bigr\vert &\le k \int _{0}^{s}{{{\kappa }^{2}} {{e}^{-\kappa }} \bigl\vert v(w) \bigr\vert }D(s,w)\,dw \\ &=k \int _{0}^{s}{{{\kappa }^{2}} {{e}^{-\kappa }} \bigl\vert v(w) \bigr\vert }(s-w){{e}^{\kappa (s-w)}}\,dw \\ &=k \int _{0}^{s}{{{\kappa }^{2}} {{e}^{-\kappa }} {{e}^{2 \kappa w}} {{e}^{-2\kappa w}} \bigl\vert v(w) \bigr\vert }(s-w){{e}^{ \kappa (s-w)}}\,dw \\ &=k \int _{0}^{s}{{{\kappa }^{2}} {{e}^{-\kappa }} {{e}^{2 \kappa w}} {{ \Vert v \Vert }_{\kappa }} }(s-w){{e}^{\kappa (s-w)}}\,dw \\ &=k{{\kappa }^{2}} {{e}^{-\kappa }} {{e}^{s\kappa }} {{ \Vert v \Vert }_{\kappa }} \int _{0}^{s}{{{e}^{2\kappa w}} }(s-w){{e}^{- \kappa w}}\,dw \\ &=k{{\kappa }^{2}} {{e}^{s\kappa -\kappa }} {{ \Vert v \Vert }_{ \kappa }} \int _{0}^{s}{(s-w){{e}^{\kappa w}}}\,dw \\ &=k{{\kappa }^{2}} {{e}^{s\kappa -\kappa }} {{ \Vert v \Vert }_{ \kappa }} \biggl( -\frac{s}{\kappa }+ \frac{{{e}^{\kappa s}}}{{{\kappa }^{2}}}- \frac{1}{{{\kappa }^{2}}} \biggr) \\ &=k{{e}^{-\kappa }} {{ \Vert v \Vert }_{\kappa }} {{e}^{\kappa s}} \bigl( -s\kappa +{{e}^{\kappa s}}-1 \bigr). \end{aligned}$$

Therefore,

$$ {{e}^{-2\kappa s}} \bigl\vert \mathfrak{F}^{2} \bigl( v(w) \bigr) \bigr\vert =k{{e}^{-\kappa }} {{ \Vert v \Vert }_{\kappa }} \bigl( -s \kappa {{e}^{-\kappa s}}+1-{{e}^{-\kappa s}} \bigr).$$

Since \(-s\kappa {{e}^{-\kappa s}}+1-{{e}^{-\kappa s}}\le 1\), we have

$$ {{ \bigl\Vert \mathfrak{F}^{2} \bigl( v(w) \bigr) \bigr\Vert }_{\kappa }} \le k{{e}^{-\kappa }} {{ \Vert v \Vert }_{\kappa }}.$$

Likewise, we can find that

$$ {{ \bigl\Vert \mathfrak{F}^{2} \bigl( z(w) \bigr) \bigr\Vert }_{\kappa }} \le k{{e}^{-\kappa }} {{ \Vert z \Vert }_{\kappa }}.$$

Since

$$\begin{aligned} \max \bigl\{ {{ \bigl\Vert \mathfrak{F}^{2}v \bigr\Vert }_{\kappa }},{{ \bigl\Vert \mathfrak{F}^{2}z \bigr\Vert }_{\kappa }} \bigr\} &\le \max \bigl\{ k{{e}^{-\kappa }} {{ \Vert v \Vert }_{\kappa }},k{{e}^{- \kappa }} {{ \Vert z \Vert }_{\kappa }} \bigr\} , \\ &=k{{e}^{-\kappa }}\max \bigl\{ {{ \Vert v \Vert }_{\kappa }},{{ \Vert z \Vert }_{\kappa }} \bigr\} . \end{aligned}$$

Therefore, for all \(v,z\in X\), we have

$$ d \bigl( \mathfrak{F}^{2}v,\mathfrak{F}^{2}z \bigr)\le k{{e}^{- \kappa }}\max \bigl\{ {{ \Vert v \Vert }_{\kappa }},{{ \Vert z \Vert }_{\kappa }} \bigr\} $$

or

$$ d \bigl( \mathfrak{F}^{2}v,\mathfrak{F}^{2}z \bigr)\le k{{e}^{- \kappa }}d ( v,z ). $$
(85)

For all \(v, z \in X\), the condition of equation (73) along with the assumptions of the theorem shows that the distance \(d ( \mathfrak{F}^{2}v,\mathfrak{F}^{2}z )\) is restricted to attaining a value not greater than \(k{{e}^{-\kappa }}d ( v,z )\). In spite of the fact that the constraints on the expansion or the contraction of the distance \(d ( {{\mathfrak{F}}}v,{{\mathfrak{F}}}z )\) in comparison with \(d ( v,z )\) are unknown, we can write

$$ \min \bigl\{ d \bigl( {{\mathfrak{F}}^{2}}v,{{\mathfrak{F}}^{2}}z \bigr),d ( {{\mathfrak{F}}}v,{{\mathfrak{F}}}z ) \bigr\} \le k{{e}^{-\kappa }}d ( v,z ).$$

So that

$$ \kappa +\min \bigl\{ \ln d \bigl( {{\mathfrak{F}}^{2}}v,{{ \mathfrak{F}}^{2}}z \bigr),\ln d ( {{\mathfrak{F}}}v,{{ \mathfrak{F}}}z ) \bigr\} \le \ln kd ( v,z ).$$

Define

$$ {{F}_{1}} ( \alpha )=\ln \alpha ={{F}_{2}} ( \alpha ), F ( \alpha )=k\ln \alpha . $$

So,

$$ \kappa +\min \bigl\{ {{F}_{2}} \bigl( d \bigl( {{ \mathfrak{F}}^{2}}v,{{ \mathfrak{F}}^{2}}z \bigr) \bigr),{{F}_{1}} \bigl( d ( {{ \mathfrak{F}}}v,{{\mathfrak{F}}}z ) \bigr) \bigr\} \le F \bigl( d ( v,z ) \bigr).$$

Therefore, \(\mathfrak{F}\) is a triple \(F^{*}\)-weak contraction and all the conditions of Theorem 3.2 are satisfied by operator \(\mathfrak{F}\). Consequently, \(\mathfrak{F}\) has a fixed point which is the solution of integral equation (71), and hence the spring mass system has a solution. □

4 Significance of new results

  • In Example 2.2, we have shown that the Lipschitzian mapping is a particular case of dual \(F^{*}\)-weak contraction. Therefore, the existence of a fixed point of dual \(F^{*}\)-weak contraction assures the existence of a fixed point of Lipschitzian mapping under certain conditions. In the literature, there exist certain fixed point theorems concerning the existence of fixed points of Lipschitzian mapping with a strong condition of rotativeness of mapping (for details, see [3740]).

  • Goebel and Japón Pineda [41, 42] proved some fixed point theorems concerning mean nonexpansive mapping by using the fixed point property in a nonempty closed convex and bounded subset of a Banach space (see Theorem 3.5 in [42]). Several research papers in the literature observe the asymptotic behavior of Lipschitz constants for iterates of mean Lipschitzian mappings (for details, see [36, 43]). We found that the mean Lipschitzian mapping is a special case of our developed triple \(F^{*}\)-weak contraction (see Example 2.3). Therefore, based on the proven fixed point Theorem 2.3 along with certain conditions on the control function F, we conclude that the existence of a fixed point of triple \(F^{*}\)-weak contraction guarantees the existence of a fixed point of mean Lipschitzian mapping.

  • Relation (60) considered in Example 2.4 shows that the mapping \(\mathfrak{F}\) is 2-Lipschitzian. The results in the literature require a strong additional condition of n-rotative mapping to prove the existence of its fixed point. We used the triple F-weak contraction to assure the existence of a fixed point of \(\mathfrak{F}\) in a closed unit ball \(\mathcal{B}\) in the \(\ell _{1}\) space of all absolutely summable sequences.

  • Certain applications in the literature apply the established fixed point theorems to find the solution of engineering problems concerning spring-mass system or activation of spring affected by an exterior force (for example, see [8, 44]). Focusing on the unknown function used in the mathematical modeling of certain engineering problems, we modified the differential equation and used the second composition of the unknown function. We applied our obtained fixed point results and proved the existence of a solution to these kinds of problems with a significant modification.

  • In the statement of Theorem 3.1, we have generalized the expression \(\frac{\Gamma (\chi +1)}{5}{{e}^{-\tau }}|a-d|\), used in [23], by replacing it with the expression \(\frac{\Gamma (\chi +1)}{5}{{e}^{-\tau }}|a-d|^{m}\), where \(m \in \mathbb{R}\backslash \{0\}\). Moreover, in Theorem 3.1 (see [23]), Gopal and Abbas proved the existence of solution of problem (64) by assuming

    $$\begin{aligned} \mathfrak{F} {x_{1}}(t)&=\frac{1}{\Gamma (\chi )} \int _{0}^{t}{{{(t-s)}^{ \chi -1}}h \bigl(s,{x_{1}}(s)\bigr)\,\mathrm{d}s- \frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{1}{{{(1-s)}^{ \chi -1}}h \bigl(s,{x_{1}}(s)\bigr)\,\mathrm{d}s}}\\ &\quad {}+\frac{2t}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{\delta }{ \biggl( \int _{0}^{s}{{{(s-\kappa )}^{\chi -1}}h\bigl( \kappa ,{x_{1}}( \kappa )\bigr)\,\mathrm{d}\kappa} \biggr)} \,\mathrm{d}s, \end{aligned}$$

    which was set to be a Lipschitzian type mapping. In our application, we have weakened the condition on \(\mathfrak{F}^{2}\) for given \(\mathfrak{F}\) by assuming

    $$\begin{aligned} \mathfrak{F}^{2} {x_{1}}(t)&=\frac{\omega}{\Gamma (\chi )} \int _{0}^{t}{{(t-s)}^{ \chi -1}}h\bigl(s, \mathfrak{F} {x_{1}}(s)\bigr)\,\mathrm{d}s\\ &\quad {}- \frac{2t\omega}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{1}{{{(1-s)}^{ \chi -1}}h\bigl(s, \mathfrak{F} {x_{1}}(s)\bigr)\,\mathrm{d}s}\\ &\quad {}+\frac{2t\omega}{(2-{{\delta }^{2}})\Gamma (\chi )} \int _{0}^{\delta }{ \biggl( \int _{0}^{s}{{{(s-\kappa )}^{\chi -1}}h\bigl( \kappa ,\mathfrak{F} {x_{1}}( \kappa )\bigr)\,\mathrm{d}\kappa} \biggr)} \,\mathrm{d}s, \end{aligned}$$

    which represents a generalization of Lipschitzian type mappings. In particular, the presented application exhibits the significance of triple \(F^{*}\)-weak contraction as with the use of its multiple control functions, we found a solution to the given problem with the above-mentioned generalized and weakened conditions.

5 Questions

  1. 1.

    There may be a possibility of generalizing F-contraction, F-Suzuki, and F-expanding mappings with \(p > 2\).

  2. 2.

    Instead of using the minimum of a set, one may use the mean of the iterations to generalize F-contraction, F-Suzuki, and F-expanding mappings.

  3. 3.

    One may develop some fixed point theorems concerning generalized F-contraction, F-Suzuki, and F-expanding mappings in a complete G-metric space.

6 Conclusion

In this paper, we introduced two new concepts of F-contraction, called dual \(F^{*}\)-weak contraction and triple \(F^{*}\)-weak contraction, which generalize the existing contractions in the sense of Wardowski, Jleli and Samet as well as Skof. Further, we established the existence of a unique fixed point for such contractions under certain conditions. In 2002, James Marryfield generalized Banach contraction using the minimum of the set of operators rather than requiring a single operator. Whereas in 2012, Wardowski [16] generalized Banach contraction using control function F on the mapping. Amalgamating the ideas presented by James Marryfield [15] and Wardowski [16], we use multiple control functions on the mapping and the minimum of the set of powers of mapping along with fractional calculus to weaken certain conditions on the fractional integral equations. This work can be generalized in different generalized metric spaces.

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Acknowledgements

The authors thank the colleagues for their proofreading and other helpful suggestions.

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Conceptualization, M.Z., S.B.; formal analysis, M.Z., S.B., N.S.; investigation, M.Z., N.S.; writing|original draft preparation, M.Z., S.B., N.S.; writing|review and editing, M.Z., N.S.. All authors have read and agreed to the published version of the manuscript.

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Correspondence to Naeem Saleem.

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Zhou, M., Saleem, N. & Bashir, S. Solution of fractional integral equations via fixed point results. J Inequal Appl 2022, 148 (2022). https://doi.org/10.1186/s13660-022-02887-w

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MSC

  • 47H10
  • 47H19
  • 54H25

Keywords

  • Dual \(F^{*}\)-weak contraction
  • Triple \(F^{*}\)-weak contraction
  • F-contraction
  • Fixed point
  • Fractional integral equations