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Best proximity point results with their consequences and applications

Abstract

In the commenced work, we establish some best proximity point results for multivalued generalized contractions on partially ordered complete metric spaces along with the tactic of altering distance function. Furthermore, we deliver some examples to elaborate and explain the usability of the attained results. To arouse further interest in the subject and to show its efficacy, we devote this work to recent applications of fractional calculus and also invoke our findings to the equation of motion modeling to differential equations.

Introduction and preliminaries

Estimating the solution of fixed point problems is well thought-out as one of the main problems in the metric fixed point theory. This forces the researchers to use the contractive conditions on underlying functions, to guarantee the existence of the fixed point. However, this issue becomes more interesting and challenging when mappings involved are non-self. This evolves the concept of best proximity point and related theorems. In fact a best proximity point theorem is principally dedicated to global minimization of the real-valued function \(y \rightarrow \sigma (y,Sy)\), which measures the error involved for an approximate solution of the equation \(Sy = y\) (fixed point problem). In other words, a best proximity point theorem expounds sufficient conditions for the existence of an element y such that the error \(\sigma (y,Sy)\) is minimum. The more general version of best proximity point theorems having more than one non-self-mapping is known as common best proximity point theorems. In 2010, Basha [3] found a best proximity point with the help of the Banach contraction principle. Basha et al. [4] gave the existence of common best proximity points for pairs of non-self-mappings in metric spaces. Karapinar and Erhan [7] also studied best proximity for different types of contractions. Interestingly, these best proximity point theorems also serve as a natural generalization of fixed point theorems. If the mapping under consideration is a self-mapping, then a best proximity point becomes a fixed point. Note that one can convert optimization problems to the problem of finding the best proximity points. Hence, the existence of the best proximity points develops the theory of optimization. Through this theory, one can guarantee that a solution of the multi-objective global minimization problem proposed by a common best proximity point theorem, in turn, becomes a common approximate solution to the system of equations with the least probable error. The theory of fixed point for multivalued mappings plays a key role in the theory of integral inclusion which confirms the existence of solutions. Nadler [13] introduced the study of fixed point theory for multivalued mappings. To this end, the researcher can see notable works in [2, 5, 6, 11, 12, 14, 18]. In the setting of metric spaces, strict contractive conditions for self-mappings and multivalued mappings do not ensure the existence of the fixed point, one can refer to [1]. Recently, as a generalized contraction, Wardwoski [19] introduced the concept of F-contraction. Klim et al. [10] discussed F-contractions for dynamic process and proved fixed point theorems involving F-contractions. To address the above issues, our motivation to define a new concept of multivalued F-contraction on partially ordered complete metric spaces with the notion of altering distance function is to ensure the existence of best proximity point through best proximity point theorems.

Let W be a nonempty set and \((W,\sigma ,\le )\) be a partially ordered metric space. Let L and M be nonempty subsets of the metric space \((W,\sigma )\). Let us assume that \(CB(W)\) is the family of closed and bounded nonempty subsets of metric space W and \(B(W)\) is the family of nonempty bounded subsets of W. The subsequent symbols used in our results are as follows:

$$\begin{aligned}& D (u, M )=\inf \bigl\{ \sigma (u,v ):v\in M \text{ for all } u\in W \bigr\} , \\& \delta (L, M )=\sup \bigl\{ \sigma (u,v ):u\in L \text{ and } v\in M \bigr\} , \\& \sigma (L, M )=\inf \bigl\{ \sigma (u,v ):u\in L \text{ and } v\in M \bigr\} , \\& L_{0}= \bigl\{ u\in L: \sigma (u,v )=\sigma (L,M ) \text{ for some } v\in M \bigr\} , \\& M_{0}=\bigl\{ v\in M: \sigma (u,v )=\sigma (L,M ) \text{ for some } u \in L\bigr\} . \end{aligned}$$

Definition 1.1

([1])

Let L and M be nonempty subsets of a metric space \((W, \sigma )\) and \(S:L\to 2^{M}\) be a multivalued mapping. Then a point \(u\in W\) is called a best proximity point for S if

$$ D (u, Su )=\sigma (L, M). $$

With the idea presented by Khan et al. [8], we use the following version of altering distance function.

Definition 1.2

([8])

A function \(\varphi : (0, +\infty )\to (0, +\infty )\) is said to be an altering distance function if it satisfies the following conditions:

  1. 1.

    φ is continuous,

  2. 2.

    φ is monotonically increasing,

  3. 3.

    \(\varphi (v )> 0 \) for all \(v > 0\).

In 2012, the concept of F-contraction was introduced by Wardowski [19].

Definition 1.3

([19])

Let \(F_{i}\) be the family of all functions \(F: (0, +\infty )\to R\) such that

\((F_{1})\):

F is strictly increasing, that is, for all \(x_{1}, y_{1} \in (0, +\infty )\), so that \(x_{1}< y_{1} \Rightarrow F (x_{1} )< F(y_{1})\);

\((F_{2})\):

For each sequence \(\{{\gamma }_{n}\}\) of positive numbers, \(\lim_{n\to \infty } {\gamma }_{n}=0 \Leftrightarrow \lim_{n\to \infty } F ({ \gamma }_{n} )= -\infty \);

\((F_{3})\):

There exists \(c\in (0, 1)\) such that \(\lim_{\gamma \to 0} {\gamma }^{c}F(\gamma ) =0\).

Let \((W, \sigma )\) be a metric space. A mapping \(S:W\to W\) is said to be an F-contraction on \((W, \sigma )\) if there exist \(F\in F_{i}\) and \(\tau \in (0,+\infty )\) such that, for all \(u,v\in W\),

$$ \sigma (Su, Sv )>0\quad \Rightarrow \quad \tau +F \bigl( \sigma (Su, Sv ) \bigr) \le F\bigl(\sigma (u, v )\bigr). $$

Example 1.1

Let \(F: (0, +\infty )\to R\) be a function defined by \(F (u )= \log u + u\) and a mapping \(S:R\to R\) be defined on a complete metric space R with the metric \(\sigma (u,v )=|u-v|\) by \(S (u )=\frac{u}{3}\). Then it is certified easily that S is an F-contraction.

Raj [17] firstly introduced the idea of P-property as follows.

Definition 1.4

([17])

Let \((L,M)\) be a pair of nonempty subsets of a metric space W so that \(L_{0}\) is nonempty. Then the pair \((L,M)\) is said to have the P-property if and only if

$$ \left . \begin{aligned} \sigma (u_{1}, v_{1} )=\sigma (L, M ) \\ \sigma (u_{2}, v_{2} )=\sigma (L, M) \end{aligned} \right \}\quad \Rightarrow\quad \sigma (u_{1}, u_{2} )=\sigma ( v_{1}, v_{2} ), $$

where \(u_{1}, u_{2}\in L_{0}\) and \(v_{1}, v_{2}\in M_{0}\).

In 2016, Pragadeeswarar et al. [16] established some results on best proximity point for multivalued mappings defined on a partially ordered metric space. Using the concept of F-contraction, in this paper we define a new concept of multivalued F-contractions on a partially ordered complete metric space with the notion of altering distance function and set the results for best proximity point garnished by examples with applications to differential equations and fractional calculus.

Main results

The multivalued F-contraction with altering distance function is defined as follows:

Let L and M be nonempty closed subsets of a metric space \((X, \sigma ) \) and \(T:L\to CB(M)\) be a multivalued mapping, φ is an altering distance function such that \(Tu_{0}\) is included in \(M_{0}\) for all \(u_{0}\in L_{0}\) and

$$ \tau +F\bigl(\varphi \bigl(\delta (Tu, Tv ) \bigr)\bigr)\le F \bigl( \varphi \bigl(N(u, v)\bigr) - \varphi \bigl(\sigma (L,M )\bigr)\bigr) \quad \text{for all } u\le v \text{ in } L, $$
(2.1)

where \(N (u,v )=\max \{ \sigma (u,v ), D (u, Tu ),D (v,Tv ), \frac{D (u,Tv )+D (v,Tu )}{2} \} \) and φ satisfies \(\varphi (u+v )\le \varphi (u )+\varphi (v )\) for all \(u,v\in [0, +\infty )\).

In this condition if we take \(F (x )=\log x \), then we get

$$\begin{aligned}& \tau + \log\bigl(\varphi \bigl(\delta (Tu, Tv ) \bigr)\bigr)\le \log\bigl( \varphi \bigl(N (u, v)\bigr) - \varphi \bigl(\sigma (L,M )\bigr)\bigr) \\& \quad \Leftrightarrow\quad \log e^{\tau }+ \log\bigl(\varphi \bigl( \delta (Tu, Tv ) \bigr) \bigr)\le \log \bigl(\varphi \bigl(N(u, v)\bigr) - \varphi \bigl(\sigma (L,M )\bigr)\bigr) \\& \quad \Leftrightarrow\quad \log \varphi \bigl(\delta (Tu, Tv)\bigr)\leq \log \biggl\{ \frac{\varphi (N(u, v))-\varphi (\sigma (u, v))}{e^{\tau}}\biggr\} \\& \quad \Leftrightarrow\quad \varphi \bigl(\delta (Tu, Tv ) \bigr) \le \frac{1}{e^{\tau }} \bigl( \varphi \bigl(N (u, v ) \bigr)- \varphi \bigl(\sigma (L,M ) \bigr) \bigr) \\& \quad \Leftrightarrow \quad \varphi \bigl(\delta (Tu, Tv ) \bigr) \le k \bigl( \varphi \bigl(N (u, v ) \bigr)- \varphi \bigl(\sigma (L,M ) \bigr) \bigr), \end{aligned}$$

which is a contraction using by Pragadeeswarar et al. [16].

Therefore, our newly defined contraction is a more generalized form of Pragadeeswarar et al. [16].

Theorem 2.1

Let \((X, \le , \sigma )\) be a partially ordered complete metric space. Let L and M be nonempty closed subsets of the metric space \((X, \sigma )\) such that \(L_{0}\) is nonempty and \((L,M)\) satisfies the P-property. Let \(T:L\to CB(M)\) be a multivalued F-contraction with altering distance function φ such that the following conditions are satisfied:

  1. 1.

    There exist two elements \(u_{0}\), \(u_{1}\) in \(L_{0}\) and \(v_{0}\in Tu_{0}\) such that \(\sigma (u_{1}, v_{0} )=\sigma (L, M )\) and \(u_{0}\le u_{1}\);

  2. 2.

    For all \(u,v\in L_{0}\), \(u\le v\) implies \(Tu \subseteq Tv\);

  3. 3.

    If \(\{u_{n}\}\) is a nondecreasing sequence in L such that \(u_{n}\to u\), then \(u_{n}\le u\) for all n.

Then there exists an element u in L such that \(D (u,Tu )= \sigma (L, M)\).

Proof

As in given condition there exist two elements \(u_{0}\), \(u_{1}\) in \(L_{0}\) and \(v_{0}\in Tu_{0}\) such that \(\sigma (u_{1} , v_{0} )=\sigma (L, M )\) and \(u_{0}\le u_{1}\).

Condition (2) implies \(Tu_{0} \subset Tu_{1} \), so there exists \(v_{1}\in Tu_{1}\) with \(\sigma (u_{2} , v_{1} )=\sigma (L, M )\) such that \(u_{1}\le u_{2}\). In general, for each \(n\in N\), there exist \(u_{n+1}\in L_{0}\) and \(v_{n}\in Tu_{n}\) such that \(\sigma (u_{n+1} , v_{n} )=\sigma (L, M ) \), hence we obtain

$$ \sigma (u_{n+1} , v_{n} )=D (u_{n+1}, Tu_{n} )=\sigma (L, M ) \quad \text{for all } n\in N, $$
(2.2)

where \(u_{0}\le u_{1}\le u_{2}\le u_{3}\le\cdots \le u_{n}\le u_{n+1}\le\cdots \) .

If there exists \(n_{0}\) such that \(u_{n_{0}}=u_{n_{0}+1}\), then \(\sigma (u_{n_{0}+1} , {v}_{n_{0}} )=D (u_{n_{0}}, Tu_{n_{0}} )=\sigma (L, M )\). This means that \(u_{n_{0}}\) is the best proximity point of T and the proof is completed. Thus, we suppose that \(u_{n}\neq u_{n+1}\) for all n. Since \(\sigma (u_{n+1} , v_{n} ) = \sigma (L, M )\) and \(\sigma (u_{n} , v_{n-1} )=\sigma (L, M )\) and \((L, M)\) has the P-property

$$ \sigma (u_{n} , u_{n+1} ) = \sigma (v_{n-1} , v_{n} ) \quad \text{for all } n\in N. $$
(2.3)

Since \(u_{n-1}< u_{n}\), so

$$\begin{aligned} F\bigl(\varphi \bigl(\sigma (u_{n}, u_{n+1} ) \bigr) \bigr)&= F\bigl( \varphi \bigl(\sigma (v_{n-1},v_{n} ) \bigr)\bigr) \\ &\le F\bigl(\varphi \bigl(\delta ({Tu}_{n-1}, {Tu}_{n} ) \bigr)\bigr) \\ &\le F\bigl(\varphi \bigl(N (u_{n-1}, u_{n} )\bigr)-\varphi \bigl(\sigma (L,M ) \bigr) \bigr)- \tau , \end{aligned}$$
(2.4)

where

$$\begin{aligned}& N (u_{n-1},u_{n} ) \\& \quad = \max \biggl\{ \sigma (u_{n-1},u_{n} ),D (u_{n-1}, Tu_{n-1} ),D (u_{n},Tu_{n} ), \frac{D (u_{n-1},Tu_{n} )+D (u_{n},Tu_{n-1} )}{2} \biggr\} \\& \quad \le \max \biggl\{ \sigma (u_{n-1},u_{n} ), \sigma (u_{n-1}, v_{n-1} ),\sigma (u_{n},v_{n} ), \frac{\sigma (u_{n-1},v_{n} )+\sigma (u_{n},v_{n-1} )}{2} \biggr\} \\& \quad \le \max \biggl\{ \sigma (u_{n-1},u_{n} ), \sigma (u_{n-1}, v_{n-2} )+\sigma (v_{n-2},v_{n-1} ), \sigma (u_{n},v_{n-1} ) \\& \qquad {}+ \sigma (v_{n-1}, v_{n}), \frac{\sigma (u_{n-1},v_{n-2} )+\sigma (v_{n-2},v_{n-1} ) + \sigma (v_{n-1},v_{n})+ \sigma (u_{n},v_{n-1} )}{2} \biggr\} \\& \quad \le \max \biggl\{ \sigma (u_{n-1},u_{n} ),\sigma (L,M)+ \sigma (u_{n-1}, u_{n} ),\sigma (L,M) \\& \qquad {}+ \sigma (u_{n},u_{n+1} ), \frac{\sigma (L,M )+\sigma (u_{n-1},u_{n} )+ \sigma (u_{n}, u_{n+1} )+\sigma (L,M)}{2} \biggr\} \\& \quad \le \max \bigl\{ \sigma (L,M )+\sigma (u_{n-1}, u_{n} ), \sigma (L,M)+\sigma (u_{n},u_{n+1} ) \bigr\} . \end{aligned}$$

From equation (2.4) and the above inequality we get

$$ \begin{aligned}[b] F\bigl(\varphi \bigl(\sigma (u_{n},u_{n+1} ) \bigr)\bigr) & \le F\bigl(\varphi \bigl({\max \bigl\{ \sigma (L,M )+\sigma (u_{n-1}, u_{n} ), \sigma (L,M)+\sigma (u_{n}, u_{n+1} ) }\bigr\} \bigr) \\ &\quad {}- \varphi \bigl(\sigma (L,M )\bigr) \bigr) - \tau . \end{aligned} $$
(2.5)

If \(\sigma (u_{n} , u_{n+1} )>\sigma (u_{n-1}, u_{n})\), from (2.5) we obtain

$$\begin{aligned}& F\bigl(\varphi \bigl(\sigma (u_{n}, u_{n+1} ) \bigr) \bigr)\le F\bigl( \varphi { \bigl(\sigma (L,M )+\sigma (u_{n}, u_{n+1} ) \bigr) - }\varphi \bigl(\sigma (L,M )\bigr)\bigr)- \tau, \\& F\bigl(\varphi \bigl(\sigma (u_{n}, u_{n+1} ) \bigr) \bigr)\le F\bigl( \varphi { \bigl(\sigma (L,M )\bigr)+\varphi \bigl(\sigma (u_{n}, u_{n+1} ) \bigr) - }\varphi \bigl(\sigma (L,M ) \bigr)\bigr)- \tau, \\& F\bigl(\varphi \bigl(\sigma (u_{n}, u_{n+1} ) \bigr) \bigr)\le F\bigl( \varphi \bigl(\sigma (u_{n}, u_{n+1} ) \bigr)\bigr)- \tau , \end{aligned}$$

which is contradiction, so we have

$$ \sigma (u_{n} , u_{n+1} )\le \sigma (u_{n-1}, u_{n}). $$
(2.6)

Hence, the sequence \(\{\sigma (u_{n} , u_{n+1} )\}\) is monotonic nonincreasing and bounded below. Thus, there exists \(r\ge 0\) such that

$$ \lim_{n\to \infty } \sigma (u_{n} , u_{n+1} ) =r\ge 0. $$
(2.7)

Suppose that \(\lim_{n\to \infty } \sigma (u_{n} , u_{n+1} ) =r>0 \). Using (2.6), inequality (2.5) becomes

$$\begin{aligned}& F\bigl(\varphi \bigl(\sigma (u_{n}, u_{n+1} ) \bigr) \bigr)\le F\bigl( \varphi \bigl(\sigma (u_{n-1}, u_{n} ) \bigr)\bigr)- \tau \\& \quad \Rightarrow\quad F\bigl(\varphi \bigl(\sigma (u_{n}, u_{n+1} ) \bigr)\bigr)\le F\bigl(\varphi \bigl(\sigma (u_{n-2}, u_{n-1} )\bigr)\bigr)-2\tau . \end{aligned}$$

Continuing this process, we get

$$ F\bigl(\varphi \bigl(\sigma (u_{n}, u_{n+1} ) \bigr)\bigr) \le F\bigl(\varphi \bigl(\sigma (u_{0}, u_{1} )\bigr)\bigr) -n\tau . $$
(2.8)

We obtain that

$$ {\lim_{n\to \infty } F\bigl(\varphi \bigl(\sigma (u_{n}, u_{n+1} ) \bigr)\bigr)=- \infty }$$

together with \((F_{2} )\) gives

$$ \lim_{n\to \infty } \varphi \bigl(\sigma (u_{n}, u_{n+1} ) \bigr)=0. $$
(2.9)

In view of \((F_{3}) \) (by the definition of F-contraction) there exists \(k\in (0, 1 )\) such that

$$\begin{aligned}& \lim_{\sigma (u_{n}, u_{n+1} )\to 0} \bigl(\varphi \bigl(\sigma (u_{n}, u_{n+1} ) \bigr)\bigr) ^{k}F \bigl(\varphi \bigl(\sigma (u_{n}, u_{n+1} )\bigr) \bigr)=0, \\& \lim_{n\to \infty } \bigl(\varphi \bigl(\sigma (u_{n}, u_{n+1} )\bigr) \bigr)^{k}F \bigl( \varphi \bigl(\sigma (u_{n}, u_{n+1} )\bigr) \bigr)=0. \end{aligned}$$
(2.10)

By (2.8) again,

$$\begin{aligned}& F\bigl(\varphi \bigl(\sigma (u_{n}, u_{n+1} ) \bigr) \bigr)\le F\bigl( \varphi \bigl(\sigma (u_{0}, u_{1} ) \bigr)\bigr)-n\tau \\& \quad \Rightarrow\quad \bigl(\varphi \bigl(\sigma (u_{n}, u_{n+1} )\bigr) \bigr)^{k} F\bigl(\varphi \bigl(\sigma (u_{n}, u_{n+1} ) \bigr)\bigr)-F\bigl(\varphi \bigl( \sigma (u_{0}, u_{1} ) \bigr)\bigr) \\& \hphantom{\quad \Rightarrow\quad}\quad \le - { \bigl( \varphi \bigl(\sigma (u_{n}, u_{n+1} )\bigr) \bigr)}^{k}n \tau \le 0. \end{aligned}$$

Let \({\beta }_{n}= \varphi (\sigma (u_{n}, u_{n+1} ))\), then \(({\beta }_{n} )^{k} (F ({\beta }_{n} )-F ({\beta }_{0} ) )\le - ({\beta }_{n} )^{k} n\tau \).

Letting \(n\to \infty \) in this, by (2.9) and (2.10), we get

$$\begin{aligned}& \lim_{n\to \infty } { ({\beta }_{n} )}^{k} \bigl(F ({\beta }_{n} )- F ({\beta }_{0} ) \bigr) \le \lim_{n\to \infty } {- ({\beta }_{n} )}^{k} n \tau \le 0 , \\& \lim_{n\to \infty } n{ ({\beta }_{n} )}^{k} = 0. \end{aligned}$$
(2.11)

Now, let us observe that from (2.11) for given \(\epsilon >0 \) there exists \(n_{1}\in N\) such that

$$\begin{aligned}& \bigl\vert n{ ({\beta }_{n} )}^{k}-0 \bigr\vert < \epsilon \quad \text{for all } n\ge n_{1}, \\& \bigl\vert n{ ({\beta }_{n} )}^{k} \bigr\vert < \epsilon , \\& ({\beta }_{n} )< \frac{\epsilon }{n^{\frac{1}{k}}} \quad \text{for all } n\ge n_{1}. \end{aligned}$$

We claim that \(\{u_{n}\}\) is a Cauchy sequence.

Consider \(m, n\in N \) such that \(m>n>n_{1}\). Therefore,

$$\begin{aligned} \varphi \bigl(\sigma (u_{m},u_{n} )\bigr)&\le \varphi \bigl(\sigma (u_{m}, u_{m-1} )\bigr)+ \varphi \bigl(\sigma (u_{m-1},u_{m-2} )\bigr)+ \cdots +\varphi \bigl(\sigma (u_{n+1}, u_{n} )\bigr) \\ &\le {\beta }_{m-1}+{\beta }_{m-2}+{\beta }_{m-3}+ \cdots +{\beta }_{n} \\ &< \sum^{\infty }_{i=n}{\beta }_{i} \le \sum^{\infty }_{i=n} \frac{\epsilon }{i^{\frac{1}{k}}}. \end{aligned}$$

Since \(k\in (0, 1 )\) then \(\frac{1}{k}>1\). Therefore, by the P-series test, the series \(\sum^{\infty }_{i=n}{\frac{1}{i^{\frac{1}{k}}} }\) is convergent for \(\frac{1}{k}>1\). Hence \(\{u_{n} \}\) is a Cauchy sequence in L. Since L is complete, then there exists \(u\in L\) such that

$$ \lim_{n\to \infty } u_{n} =u\quad \text{or}\quad u_{n}\to u . $$

Since \(\sigma (u_{n}, u_{n+1} )=\sigma (v_{n-1} , v_{n} )\). The sequence \(\{v_{n}\}\) in M is Cauchy and then is convergent.

Suppose that \(v_{n}\to v \). By the relation \(\sigma (u_{n+1} , v_{n} )=\sigma (L, M )\) for all n.

We conclude that \(\sigma (u,v) = \sigma ( L,M)\). We now claim that \(v\in Tu\).

Since \(\{u_{n} \}\) is an increasing sequence in L and \(u_{n}\to u\) by hypothesis (3), \(u_{n}\le u \) for all n. Suppose that \(v \notin Tu\). Thus, we consider

$$\begin{aligned}& F\bigl(\varphi \bigl(D (v_{n}, Tu ) \bigr)\bigr) \\& \quad \le F\bigl(\varphi \bigl(\delta (Tu_{n}, Tu ) \bigr)\bigr) \\& \quad \le F \biggl(\varphi \biggl({\max { \biggl\{ \sigma (u_{n},u ),D (u_{n}, Tu_{n} ),D (u,Tu ), \frac{D (u_{n},Tu )+D (u,Tu_{n} )}{2} \biggr\} } } \biggr) \\& \qquad {}- \varphi \bigl(\sigma ( L, M ) \bigr) \biggr)- \tau \\& \quad \le F \biggl(\varphi \biggl({\max { \biggl\{ \sigma (u_{n},u ), \sigma (u_{n}, v_{n} ),D (u,Tu ), \frac{D (u_{n},Tu )+\sigma (u, v_{n} )}{2} \biggr\} } } \biggr) \\& \qquad {}- \varphi \bigl(\sigma ( L, M )\bigr) \biggr)-\tau . \end{aligned}$$

Taking \(n\to \infty \) in the above inequality, using \(u_{n}\to u\), \(v_{n}\to v \) and \(\sigma (u, v )=\sigma (L,M )\), we get

$$\begin{aligned}& \begin{aligned} &F\bigl(\varphi \bigl(D (v,Tu ) \bigr)\bigr) \\ &\quad \le F \biggl( \varphi \biggl({\max { \biggl\{ 0, \sigma (L, M ),D (u,Tu ), \frac{D (u,Tu )+\sigma (L,M)}{2} \biggr\} } } \biggr) - \varphi \bigl(\sigma ( L, M )\bigr) \biggr) - \tau \\ &\quad \le F\bigl(\varphi \bigl(\sigma (L, M )+ D ( v, Tu )\bigr)- \varphi \bigl( \sigma (L,M ) \bigr)\bigr)-\tau \\ &\quad \le F\bigl(\varphi \bigl(\sigma (L,M ) \bigr)+\varphi \bigl(D ( v, Tu ) \bigr)- \varphi \bigl(\sigma (L,M ) \bigr)\bigr)-\tau, \end{aligned} \\& F\bigl(\varphi \bigl(D (v, Tu ) \bigr)\bigr) \le F\bigl(\varphi \bigl(D (v, Tu ) \bigr)\bigr)-\tau , \end{aligned}$$

which is contraction.

This implies that \(v\in Tu\), and hence \(D (u, Tu )=\sigma (L, M )\). That is, u is the best proximity point of T. □

As the consequence of our result, we deduce new best proximity point and fixed point results for multivalued and single valued mappings in the partially ordered metric spaces.

Theorem 2.2

Let \((X, \le ,\sigma )\) be a partially ordered complete metric space. Let L and M be nonempty closed subsets of the metric space \((X, \sigma )\) such that \(L_{0}\) is nonempty and \((L,M)\) satisfies the P-property. Let \(T:L\to CB(M)\) be a multivalued mapping with altering distance function φ such that the following conditions are satisfied:

  1. 1.

    There exist two elements \(u_{0} \), \(u_{1}\) in \(L_{0}\) and \(v_{0}\in Tu_{0}\) such that \(\sigma (u_{1} , v_{0} )=\sigma (L, M )\) and \(u_{0}\le u_{1}\);

  2. 2.

    \(T (L_{0} )\subseteq M_{0}\) and \(F(\varphi (\delta (Tu, Tv ) )) \le F( \varphi (N(u, v))) - \tau \) for all \(u\le v\) in L, where

    $$\begin{aligned} N (u,v ) =& \max \biggl\{ \sigma (u,v ), D (u,Tu )-\sigma (L,M ),D (v,Tv )- \sigma (L,M ), \\ & \frac{D (u,Tv )+D (v,Tu )}{2}-\sigma (L,M ) \biggr\} \end{aligned}$$

    for all \(u,v\in [0, +\infty )\);

  3. 3.

    For all \(u,v\in L_{0}\), \(u\le v\) implies \(Tu\subseteq Tv\);

  4. 4.

    If \(\{u_{n}\}\) is a nondecreasing sequence in L such that \(u_{n}\to u\), then \(u_{n}\le u \) for all n.

Then there exists an element u in L such that \(D (u,Tu )=\sigma (L, M)\).

Proof

The proof is similar to Theorem 2.1. □

As a consequence of Theorem 2.2, we find the following results by using T is a single valued mapping.

Corollary 2.1

Let \((X, \le ,\sigma )\) be a partially ordered complete metric space. Let L and M be nonempty closed subsets of the metric space \((X, \sigma )\) such that \(L_{0}\) is nonempty and \((L,M)\) satisfies the P-property. Let \(T:L\to M\) be a single valued mapping such that the following conditions are satisfied:

  1. 1.

    There exist two elements \(u_{0} \), \(u_{1}\) in \(L_{0}\) and \(v_{0}\in Tu_{0}\) such that \(\sigma (u_{1} , v_{0} )=\sigma (L, M )\) and \(u_{0}\le u_{1}\);

  2. 2.

    \(T (L_{0} )\subseteq M_{0}\) and \(F(\varphi (\sigma (Tu, Tv ) ))\le F( \varphi (N(u, v))) - \tau \) for all \(u\le v\) in L, where

    $$\begin{aligned} N (u,v ) =& \max \biggl\{ \sigma (u,v ), \sigma (u,Tu )-\sigma (L,M ), \sigma (v,Tv )-\sigma (L,M ), \\ &\frac{\sigma (u,Tv )+\sigma (v,Tu )}{2}- \sigma (L,M ) \biggr\} \end{aligned}$$

    and φ is an altering distance function such that \(\varphi (u+v )\le \varphi (u )+\varphi (v ) \) for all \(u,v\in [0, +\infty )\);

  3. 3.

    For all \(u,v\in L_{0}\), \(u\le v\) implies \(Tu\le Tv\);

  4. 4.

    If \(\{u_{n}\}\) is a nondecreasing sequence in L such that \(u_{n}\to u\), then \(u_{n}\le u \) for all n.

Then there exists an element u in L such that \(\sigma (u,Tu )=\sigma (L, M)\).

Proof

It follows from the proof of Theorem 2.1. □

If we take \(L=M\) in Theorem 2.1 and Theorem 2.2, then we get the following results respectively.

Corollary 2.2

Let \((X, \le ,\sigma )\) be a partially ordered complete metric space. Let L be a nonempty closed subset of the metric space X and let \(T:L\to CB(L)\) be a multivalued F-contraction with an altering distance function φ such that the following conditions are satisfied:

  1. 1.

    There exist two elements \(u_{0} \), \(u_{1}\) in L and \(v_{0}\in Tu_{0}\) such that \(\sigma (u_{1} , v_{0} )=0\) and \(u_{0}\le u_{1}=v_{0}\);

  2. 2.

    For all \(u,v\in L\), \(u\le v\) implies \(Tu\subseteq Tv\);

  3. 3.

    If \(\{u_{n}\}\) is a nondecreasing sequence in L such that \(u_{n}\to u\), then \(u_{n}\le u \) for all n.

Then there exists an element u in L such that \(D (u,Tu )=0\) i.e. u is a fixed point of the mapping T.

Corollary 2.3

Let \((X, \le ,\sigma )\) be a partially ordered complete metric space. Let L be a nonempty closed subset of the metric space X, and let \(T:L\to L\) be a single valued mapping and φ be an altering distance function such that the following conditions are satisfied:

  1. 1.

    There exist two elements \(u_{0} \), \(u_{1}\) in L such that \(\sigma (u_{1} , Tu_{0} )=0 \) and \(u_{0}\le u_{1}\);

  2. 2.

    T satisfies \(F(\varphi (\sigma (Tu, Tv ) ))\le F( \varphi (N(u, v))) - \tau \) for all \(u\le v\) in L, where \(N (u,v )={\max \{ \sigma (u,v ), \sigma (u, Tu ),\sigma (v,Tv ), \frac{\sigma (u,Tv )+\sigma (v,Tu )}{2} \} }\) and φ is an altering distance function such that \(\varphi (u+v )\le \varphi (u )+\varphi (v ) \) for all \(u,v\in [0, +\infty )\);

  3. 3.

    For all \(u,v\in L\), \(u\le v\) implies \(Tu\le Tv\);

  4. 4.

    If \(\{u_{n}\}\) is a nondecreasing sequence in L such that \(u_{n}\to u\), then \(u_{n}\le u \) for all n.

Then there exists an element u in L such that \(\sigma (u,Tu )=0\) i.e. u is a fixed point of the mapping T.

Also, if we revenue T is single valued mapping and φ is an identity function, then we realize the next result.

Corollary 2.4

Let \((X, \le ,\sigma )\) be a partially ordered complete metric space. Let L be a nonempty closed subset of the metric space X, and let \(T:L\to L\) be a single valued mapping such that the following conditions are satisfied:

  1. 1.

    There exist two elements \(u_{0} , u_{1} \in L\) such that \(\sigma (u_{1} , Tu_{0} )=0\) and \(u_{0}\le u_{1}\);

  2. 2.

    T satisfies \(F(\sigma (Tu, Tv ))\le F(N(u, v)) - \tau \) for all \(u\le v\) in L, where \(N (u,v )={\max \{ \sigma (u,v ), \sigma (u, Tu ), \sigma (v,Tv ), \frac{\sigma (u,Tv )+\sigma (v,Tu )}{2} \} }\);

  3. 3.

    For all \(u,v\in L\), \(u\le v\) implies \(Tu< Tv\);

  4. 4.

    If \(\{u_{n}\}\) is a nondecreasing sequence in L such that \(u_{n}\to u\), then \(u_{n}\le u \) for all n.

Then there exists an element u in L such that \(\sigma (u,Tu )=0\) i.e. u is a fixed point of the mapping T.

Proof

It follows from the same lines as earlier. □

Example 2.1

Let \(W=R^{2}\) and consider the order \((x,y )\le (z,t )\Longleftrightarrow x\le z\) and \(y\le t\), where ≤ is the usual order in R. Thus, \((W, \le )\) is a partially ordered set. Besides, \((W,\sigma ) \) is a complete metric space where the metric is defined as

$$ \sigma \bigl( (u_{1}, v_{1} ), (u_{2},v_{2} ) \bigr)= \vert u_{1}-u_{2} \vert + \vert v_{1}-v_{2} \vert .$$

Let \(L=\{ (-7,0 ), (0,-7 ), (0,5 )\}\) and \(M=\{ (-2, 0 ), (0,-2 ), (0, 0 ), (-2, 2 ), (2,2 )\}\) be closed subsets of W.

Then \(\sigma (L,M )={\inf \{ \sigma (u,v ):u \in L \text{ and } v\in M\} }=5\) and \(L=L_{0}\) and \(M=M_{0}\). Let \(T:L\to CB(M)\) be defined by

$$ T (u,v )= \textstyle\begin{cases} \{ (0,-2 ), (0,0 ) \} &\text{if } (u,v )= (-7,0 ), \\ \{ (2,2 ), (-2,2 ) \} & \text{if } (u,v)=(0,-7), \\ \{ (-2,2 ), (0,0 ), (0,-2 ), (2,2 ) \} & \text{if } (u,v )=(0,5). \end{cases} $$

Since there exist two elements \((-7,0 ), (0,5 )\in L_{0}\) and \((0,0 )\in T(-7,0)\) such that

$$ \sigma \bigl( (0, 5 ), (0, 0 ) \bigr)=5= \sigma (L, M ) \quad \text{and} \quad (-7,0 )\le (0, 5), $$

so condition (1) is true. Now, we have to prove condition (2.1) of Theorem 2.1. It is easy to prove that \(Tu_{0} \) is included in \(M_{0}\) for all \(u_{0}\in L_{0}\).

Now, by the definition “≤” in L, there are two cases \((-7,0 )\le (0, 5 )\) and \((0, -7 )\le (0, 5)\). In both cases we get

$$ \delta (Tu, Tv )=6, \qquad N (u, v )=12 \quad \text{and}\quad \sigma (L, M )=5.$$

Let F be defined by \(F (\beta )=\log \beta +\beta\) and \(\tau =1 \).

For \(\varphi (t )=2t\), we get \(\varphi (\delta (Tu, Tv ) )=12\) and \(\varphi (N (u, v ) )- \varphi (\sigma (L,M ) )= 14\).

Hence,

$$\begin{aligned}& \frac{ \varphi (\delta (Tu, Tv ) )}{ \varphi (N (u, v ) )- \varphi (\sigma (L,M ) )} e^{ \varphi (\delta (Tu, Tv ) )- ( \varphi (N (u, v ) )- \varphi (\sigma (L,M ) ) )} \\& \quad =\frac{12}{14} \bigl(e^{12-14} \bigr) = \frac{12}{14} e^{-2}< e^{-1}. \end{aligned}$$

Hence, T satisfies condition (2.1).

Also we can prove conditions (2) and (3) easily. Hence, all the hypotheses of Theorem 2.1 are satisfied. Also it can be observed that \((0,5)\) is the best proximity point of the mapping T, that is, \(D ( (0,5 ),T (0,5 ) )=\sigma (L,M )=5\).

Example 2.2

Let \(W=\{ 0, 1, 2, 3, \dots \}\) with the usual order ≤ be a partial order set, and let \(\sigma :W\times W\to R\) be given as

$$ \sigma (u, v )= \textstyle\begin{cases} 0; & u=v, \\ u+v; & u\neq v. \end{cases} $$

Then \((W, \sigma )\) is a complete metric space. Let \(S:W\to W\) be defined as

$$ S (u )= \textstyle\begin{cases} 0 & \text{if } u=0, \\ u-1& \text{if } u \neq 0 .\end{cases} $$

Then we prove that S is an F-contraction with respect to \(F (\beta )= \log \beta +\beta\) and \(\tau =1\).

We discuss the following cases for \(u,v\in W\).

Case I: If \(u>v\) and \(v\neq 0\), then

$$\begin{aligned}& \sigma (Su, Sv )=\sigma (u-1, v-1 )=u+v-2, \\& \begin{aligned} N (u, v )&=\max \biggl\{ \sigma (u,v ), \sigma (u, Tu ),\sigma (v,Tv ), \frac{\sigma (u,Tv )+\sigma (v,Tu )}{2} \biggr\} \\ &=\max \biggl\{ \sigma (u,v ), \sigma (u, u-1 ),\sigma (v, v-1 ), \frac{\sigma (u,v-1 )+\sigma (v,u-1 )}{2} \biggr\} \\ &=\max \{ u+v, 2u-1,2v-1, u+v-1 \} \\ &= 2u-1, \end{aligned} \\& \begin{aligned} \frac{\sigma (Su, Sv )}{N (u, v )} e^{ \sigma (Su, Sv )-N (u, v )}&= \frac{u+v-2}{2u-1} e^{u+v-2- (2u-1 )} \\ & = \frac{u+v-2}{2u-1}e^{-u+v-1} < e^{-1}. \end{aligned} \end{aligned}$$

Case II: If \(v>u\) and \(u\neq 0\), then

$$\begin{aligned}& \sigma (Su, Sv )=\sigma (u-1, v-1 )=u+v-2, \\& \begin{aligned} N (u, v )&=\max \biggl\{ \sigma (u,v ), \sigma (u, Tu ),\sigma (v,Tv ), \frac{\sigma (u,Tv )+\sigma (v,Tu )}{2} \biggr\} \\ &=\max \biggl\{ \sigma (u,v ), \sigma (u, u-1 ),\sigma (v, v-1 ), \frac{\sigma (u,v-1 )+\sigma (v,u-1 )}{2} \biggr\} \\ &=\max \{ u+v, 2u-1,2v-1, u+v-1 \} \\ &= 2v-1, \end{aligned} \\& \begin{aligned} \frac{\sigma (Su, Sv )}{N (u, v )} e^{ \sigma (Su, Sv )-N (u, v )}&= \frac{u+v-2}{2v-1} e^{u+v-2- (2v-1 )} \\ &= \frac{u+v-2}{2v-1}e^{u-v-1} < e^{-1}.\end{aligned} \end{aligned}$$

Case III: If \(u>v\) and \(v=0\), then

$$\begin{aligned}& \sigma (Su, Sv )=\sigma (u-1, 0 )=u-1 , \\& \begin{aligned} N (u, v )&=\max \biggl\{ \sigma (u,v ), \sigma (u, Tu ),\sigma (v,Tv ), \frac{\sigma (u,Tv )+\sigma (v,Tu )}{2} \biggr\} \\ &=\max \biggl\{ \sigma (u,0 ), \sigma (u, u-1 ),\sigma (0, 0 ), \frac{\sigma (u, 0 )+\sigma (0,u-1 )}{2} \biggr\} \\ &=\max \biggl\{ u, 2u-1,0, u-\frac{1}{2} \biggr\} \\ &= 2u-1, \end{aligned} \\& \begin{aligned} \frac{\sigma (Su, Sv )}{N (u, v )} e^{ \sigma (Su, Sv )-N (u, v )}&=\frac{u-1}{2u-1} e^{u-1- (2u-1 )} \\ & = \frac{u-1}{2u-1}e^{-u} < e^{-1}. \end{aligned} \end{aligned}$$

Case IV: If \(v>u\) and \(u=0\), then

$$\begin{aligned}& \sigma (0, Sv )=\sigma (0, v-1 )=v-1, \\& \begin{aligned} N (u, v )&=\max \biggl\{ \sigma (u,v ), \sigma (u, Tu ),\sigma (v,Tv ), \frac{\sigma (u,Tv )+\sigma (v,Tu )}{2} \biggr\} \\ &=\max \biggl\{ \sigma (0,v ), \sigma (0, 0 ),\sigma (v, v-1 ), \frac{\sigma (0,v-1 )+\sigma (v,0 )}{2} \biggr\} \\ &=\max \biggl\{ v, 0,2v-1, v-\frac{1}{2} \biggr\} \\ &= 2v-1, \end{aligned} \\& \begin{aligned} \frac{\sigma (Su, Sv )}{N (u, v )} e^{ \sigma (Su, Sv )-N (u, v )}&=\frac{v-1}{2v-1} e^{v-1- (2v-1 )} \\ &= \frac{v-1}{2v-1}e^{-v} < e^{-1}.\end{aligned} \end{aligned}$$

Case V: If \(v=u\), then

$$\begin{aligned}& \sigma (Su, Su )=\sigma (u-1, u-1 )=0, \\& \begin{aligned} N (u, u )&=\max \biggl\{ \sigma (u,u ), \sigma (u, Tu ),\sigma (u,Tu ), \frac{\sigma (u,Tu )+\sigma (u,Tu )}{2} \biggr\} \\ &=\max \biggl\{ 0, \sigma (u, u-1 ),\sigma (u, u-1 ), \frac{\sigma (u,u-1 )+\sigma (u,u-1 )}{2} \biggr\} \\ &=\max \{ 0, 2u-1,2u-1, 2u-1 \} \\ &= 2u-1, \end{aligned} \\& \frac{\sigma (Su, Sv )}{N (u, v )} e^{ \sigma (Su, Sv )-N (u, v )}=\frac{0}{2u-1} e^{0- (2u-1 )}< e^{-1}. \end{aligned}$$

In all the above cases, the conditions of Corollary 2.4 are satisfied, and it is clear that 0 is a fixed point of T.

Application to equation of motion

Let \(D([0,1])\) be the set of all continuous functions defined on \([0, 1]\) and \(\sigma : D ( [0,1 ] )\times D ( [0,1 ] )\to R\) be the metric defined by \(\sigma (u,v )={ \| u-v \| }_{ \infty }={\max_{t\in [0, 1]} |u(t) }-v(t)|\).

We know that \((D ( [0,1 ] ),\sigma )\) is a complete metric space.

Consider the following problem.

Problem.

A particle of mass m is at rest at \(x=0 \), \(t=0\). A force f starts activity on it in X-direction such that its velocity jumps from 0 to 1 immediately after \(t=0\). Find the position of the particle at time t.

The equation of motion is

$$\begin{aligned}& m\frac{d^{2}x}{dt^{2}}=f\bigl(t, x(t)\bigr)\quad \text{and} \\& x (0 )=0,\qquad x' (0 )=1, \end{aligned}$$
(3.1)

where \(f: [0, 1 ]\times R \to R \) is continuous.

Green’s function associated with (3.1) is defined by

$$ G (t, s )= \textstyle\begin{cases} t & \text{if } 0\le t\le s\le 1 , \\ 2t-s & \text{if } 0\le s \le t\le 1. \end{cases} $$

Let \(\varphi :R\times R\to R \) be a function with the following conditions:

  1. 1.

    \(\vert f (t, a )-f (t, b ) \vert \le \max_{a,b\in R} \vert a-b \vert \) for all \(t \in [0, 1 ] \) and \(a,b\in R\) with \(\varphi (a,b )\ge 0\);

  2. 2.

    There exists \(x_{0} \in D[0,1]\) such that \(\varphi (x_{0} (t),Tx_{0}(t))\geq 0\) for all \(t \in [0,1] \), where \(T:D[0,1]\rightarrow D[0,1]\).

We now prove the existence of a solution of the second order differential equation (3.1).

Theorem 3.1

Under assumptions (1)(2), (3.1) has a solution in \(D^{2}([0,1])\).

Proof

It is well known that a solution of (3.1) is equivalent to \(x\in D ( [0,1 ] )\) a solution of the integral equation

$$ x (t )= \int ^{1}_{0}{G(t, s)f\bigl(s,x(s)\bigr)}\,ds, \quad t \in [0, 1].$$

Let \(T: D ( [0,1 ] )\to D ( [0,1 ] )\) be defined by

$$ T\bigl(x (t )\bigr)= \int ^{1}_{0}{G(t, s)f\bigl(s,x(s)\bigr)}\,ds. $$

Suppose that \(x,y\in D ( [0,1 ] )\) such that \(\varphi (x (t ), y (t ) )\ge 0 \) for all \(t\in [0, 1]\).

Now, consider

$$\begin{aligned} \bigl\vert T \bigl(x (t ) \bigr) -T \bigl(y (t ) \bigr) \bigr\vert &= \biggl\vert \int ^{1}_{0}{G (t, s )f \bigl(s,x (s ) \bigr)} \,ds- \int ^{1}_{0}{G (t, s )f \bigl(s,y (s ) \bigr)} \,ds \biggr\vert \\ &\le \int ^{1}_{0}{G (t, s ) \bigl\vert f \bigl(s,x (s ) \bigr)-f \bigl(s,y (s ) \bigr) \bigr\vert }\,ds \\ &\le \int ^{1}_{0}{G (t, s )\max \bigl\vert x (s )- y (s ) \bigr\vert }\,ds \\ &\le \Vert x-y \Vert _{\infty }\sup_{t\in [0, 1]} \biggl\{ \int ^{1}_{0}{G(t, s)}\,ds \biggr\} . \end{aligned}$$

Since \(\int ^{1}_{0}{G(t, s)}\,ds= -\frac{t^{2}}{2}+\frac{1}{2} \) for all \(t\in [0,1]\), then \(\sup_{t\in [0, 1]} \{\int ^{1}_{0}{G(t, s)}\,ds \}= \frac{1}{2}\).

It follows that \({ \Vert Tx-Ty \Vert }_{\infty }\le \frac{1}{2} { \Vert x-y \Vert }_{\infty }\).

Taking logarithm in both sides

$$\begin{aligned}& {\log \bigl( \Vert Tx-Ty \Vert }_{\infty }\bigr)\le \log \bigl( { \Vert x-y \Vert }_{\infty } \bigr)- \log 2, \\& {\log 2+ \log \bigl( \Vert Tx-Ty \Vert }_{\infty }\bigr)\le \log \bigl( { \Vert x-y \Vert }_{\infty } \bigr). \end{aligned}$$

Now, we consider the function \(F: (0, +\infty )\to R \) defined by \(F (x )= \log x\).

We get \(\log 2+F (\sigma (Tx, Ty ) )\le F(\sigma (x, y))\).

But \(F (\sigma (x, y ) )\le F(N(x, y))\) implies

$$ \log 2+F \bigl(\sigma (Tx, Ty ) \bigr)\le F\bigl(N(x, y)\bigr). $$

Therefore, the mapping T is an F-contraction. It follows from Corollary 2.4 that T has a fixed point in \(D ( [0,1 ] )\), which in turn is the solution of problem. □

Application to fractional calculus

Let (C[0, 1]) be the set of all continuous functions on \([0, 1]\) and \(\sigma :C([0,1])\times C([0,1])\rightarrow R\) be the metric defined by \(\sigma (u,v )={ \| u-v \| }_{ \infty }=\max_{t\in [0, 1]} |u(t) -v(t)|\).

Now, we recall some notation of [9] and [15]. The Caputo derivative of fractional order β for a continuous function \(h:[ 0, +\infty )\rightarrow R\) is defined as

$$ {}^{c}D^{\beta}\bigl(h(t)\bigr)= \frac{1}{\Gamma (m-\beta )} \int _{0}^{t} (t-s)^{m- \beta -1}g^{(m)} (s)\,ds \quad \bigl(m-1< \beta < n,m = [\beta ]+1\bigr), $$

where Γ the gamma function and \([\beta ]\) denotes the integer part of a real number.

In this work, we present the existence of the solution of nonlinear fractional differential equation

$$ {}^{c}D^{\beta}\bigl(u(t)\bigr)+f \bigl(t,u(t)\bigr)= 0 \quad (0 \leq t \leq 1, \beta < 1), $$
(4.1)

with \(u(0)= u(1)=0\) and \(f:[0,1]\times R \rightarrow R\) being a continuous function, and Green’s function associated with Problem (4.1) is given by

$$ G (t, s )= \textstyle\begin{cases} (t(1-s))^{a-1}-(t-s)^{a-1} & \text{if } 0\le t\le s\le 1, \\ \frac{(t(1-s))^{a-1}}{\Gamma (a)} & \text{if } 0\le s\le t\le 1. \end{cases} $$

Assume that the following conditions hold:

  1. 1.

    \(|f(t,u)-f(t,v)|\le e^{-\tau}W(u,v)\) for each \(t\in [0, 1]\) and \(a, b \in R\), where \(W(u, v)= \max\{|u-v|,|u-Tu|,|v-Tv|,\frac{|u-Tv|+|v-Tu|}{2}\}\);

  2. 2.

    There exists \(u_{0} \in C[0,1]\) such that \(\varphi (u_{0} (t),Tu_{0}(t))\geq 0\) for all \(t \in [0,1] \).

Assume that the mapping \(T:C[0,1]\rightarrow C[0,1]\) is defined by

$$ T\bigl(u(t)\bigr)= \int _{0}^{1}G(t,s)f\bigl(s, u(s)\bigr) \,ds.$$

Now, we have to prove the existence of a solution of fractional differential equation (4.1).

Theorem 4.1

Under the assumptions of conditions (1)(2), (4.1) has a solution.

Proof

It is well known that u is a solution of (4.1) is equivalent to \(u\in X\) is a solution of the integral equation

$$ u(t)= \int _{0}^{1}G(t,s)f\bigl(s, u(s)\bigr)\,ds \quad \text{for all } t\in [0,1] .$$

Consider

$$\begin{aligned}& \bigl\vert Tu(x)-Tv(x) \bigr\vert \\& \quad = \biggl\vert \int _{0}^{1}G(x,s)f\bigl(s, u(s)\bigr)\,ds - \int _{0}^{1}G(x,s)f\bigl(s, v(s)\bigr)\,ds \biggr\vert \\& \quad \leq \int _{0}^{1} \bigl\vert G(x,s) \bigl(f\bigl(s, u(s)\bigr)-f\bigl(s, v(s)\bigr)\bigr)\,ds \bigr\vert \\& \quad \leq \int _{0}^{1} \bigl\vert G(x,s) \bigr\vert \bigl\vert \bigl(f\bigl(s, u(s)\bigr)-f\bigl(s, v(s)\bigr)\bigr) \bigr\vert \,ds \\& \quad \leq \int _{0}^{1} \bigl\vert G(x,s) \bigr\vert e^{-\tau}W(u,v)\,ds \\& \quad \leq \int _{0}^{1} \bigl\vert G(x,s) \bigr\vert e^{-\tau}\max\biggl\{ \vert u-v \vert , \vert u-Tu \vert , \vert v-Tv \vert , \frac{ \vert u-Tv \vert + \vert v-Tu \vert }{2}\biggr\} \,ds \\& \quad \leq e^{-\tau}\max\biggl\{ \sigma (u,v), \sigma (u,Tu),\sigma (v,Tv), \frac{\sigma (u,Tv)+\sigma (v,Tu)}{2} \biggr\} \int _{0}^{1}\bigl(G(x,s)\bigr)\,ds \\& \quad \leq e^{-\tau}\max\biggl\{ \sigma (u,v), \sigma (u,Tu),\sigma (v,Tv), \frac{\sigma (u,Tv)+\sigma (v,Tu)}{2} \biggr\} \\& \qquad {}\times \sup_{x\in [0,1]}\biggl( \int _{0}^{1}\bigl(G(x,s)\bigr)\,ds\biggr) \\& \quad \leq e^{-\tau}N(u, v)\sup_{x\in [0, 1]}\biggl( \int _{0}^{1}\bigl(G(x,s)\bigr)\,ds\biggr). \end{aligned}$$

Since \(\int _{0}^{1}(G(x,s))\,ds = \frac{x^{a-1}(1-x)^{a}}{-a \Gamma a} + \frac{x^{a-1}}{a \Gamma a} + \frac{x^{a-1}}{a}+ \frac{x^{a-1} (1-x)^{a}}{a}\) for all \(x\in [0, 1]\), then

$$ \sup_{x\in [0, 1]}\biggl( \int _{0}^{1}\bigl(G(x,s)\bigr)\,ds\biggr) \leq 1 .$$

It follows that

$$ \bigl\vert Tu(x)-Tv(x) \bigr\vert \leq e^{-\tau}N(u, v), $$

where \(N(u,v)= \max\{\sigma (u,v), \sigma (u,Tu),\sigma (v,Tv), \frac{\sigma (u,Tv)+ \sigma (v,Tu)}{2} \}\).

Therefore, for all \(u, v \in X\) and for all \(x \in [0, 1]\), we have

$$ \sigma (Tu, Tv) \leq \bigl(e^{-\tau}N(u, v)\bigr). $$

Taking the logarithm on both sides, we get

$$ \log \bigl(\sigma (Tu, Tv)\bigr)\leq \log \bigl(N(u,v)\bigr)-\tau . $$

Now, we consider the function \(F: (0, +\infty )\to (0,+\infty )\) defined by \(F (x )=\log x\).

We get

$$ \tau +F \bigl(\sigma (Tu, Ty ) \bigr)\le F\bigl(\sigma \bigl(N(u, v)\bigr) \bigr).$$

Therefore, the mapping T is an F-contraction. It follows from Corollary 2.4 that T has a fixed point in \(C ( [0,1 ] )\), which in turn is the solution of Problem (4.1). □

Conclusion

In this paper, we introduced the concept of multivalued F-contraction on partially ordered complete metric spaces with the notion of altering distance function. Results endowed with a partial order have been obtained for aforesaid contractions. All examples agree with the theoretical results of this study. In the end, we applied our main results to provide solutions of differential equations (equation of motion) and also of fractional differential equations. We concluded that the new results are influential and effective in finding the solutions for a wide class of nonlinear equations dealing with science and engineering.

Availability of data and materials

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References

  1. Abkar, A., Gbeleh, M.: The existence of best proximity points for multivalued non-self mappings. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 107, 319–325 (2012)

    MathSciNet  Article  Google Scholar 

  2. Aydi, H., Bota, M.F., Karapinar, E., Mitrovic, S.: Fixed point theorem for set-valued quasicontractions in b-metric spaces. Fixed Point Theory Appl. 2012, 88 (2012)

    Article  Google Scholar 

  3. Basha, S.S.: Extensions of Banach’s contraction principle. Numer. Funct. Anal. Optim. 31, 569–576 (2010)

    MathSciNet  Article  Google Scholar 

  4. Basha, S.S., Shahzad, N., Jeyaraj, R.: Common best proximity points: global optimization of multi-objective functions. Appl. Math. Lett. 24(6), 883–886 (2011)

    MathSciNet  Article  Google Scholar 

  5. Fabiano, N., Parvaneh, V., Mirković, D., Paunović, L., Radenović, S.: On W-contractions of Jungck–Ciric–Wardowski-type in metric spaces. Cogent Math. Stat. 7(1), 1792699 (2020)

    Article  Google Scholar 

  6. Isik, H., Parvaneh, V., Mohammadi, B., Altun, I.: Common fixed point results for generalized Wardowski type contractive multi-valued mappings. Mathematics 7(11), 1130 (2019)

    Article  Google Scholar 

  7. Karapinar, E., Erhan, I.M.: Best proximity point on different type contractions. Appl. Math. Inf. Sci. 5, 342–353 (2011)

    MathSciNet  Google Scholar 

  8. Khan, M.S., Swaleh, M., Sessa, S.: Fixed point theorem by altering distances between the points. Bull. Aust. Math. Soc. 30, 1–9 (1984)

    MathSciNet  Article  Google Scholar 

  9. Kildas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Application to Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

    Google Scholar 

  10. Klim, D., Wardowski, D.: Fixed points of dynamic processes of set-valued F-contractions and application to functional equations. Fixed Point Theory Appl. 2015, 22 (2015)

    MathSciNet  Article  Google Scholar 

  11. Mishra, L.N., Dewangan, V., Mishra, V.N., Karateked, S.: Best proximity points of admissible almost generalized weakly contractive mappings with rational expressions on b-metric spaces. J. Math. Comput. Sci. 22, 97–109 (2021)

    Article  Google Scholar 

  12. Mohammadi, B., Parvaneh, V., Aydi, H.: On extended interpolative Ciric–Reich–Rus type F-contractions and an application. J. Inequal. Appl. 2019, 290 (2019)

    Article  Google Scholar 

  13. Nadler, S.B.: Multi-valued contraction mappings. Pac. J. Math. 30, 475–478 (1969)

    MathSciNet  Article  Google Scholar 

  14. Parvaneh, V., Mohammadi, B., De La Sen, M., Alizadeh, E., Nashine, H.K.: On existence of solutions for some nonlinear fractional differential equations via Wardowski–Mizoguchi–Takahashi type contractions. Int. J. Nonlinear Anal. Appl. 12(1), 893–902 (2021)

    Google Scholar 

  15. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  16. Pragadeeswarar, V., Marudai, M., Kumam, P.: Best proximity point for multivalued mappings for the complete partially ordered metric space. J. Nonlinear Sci. Appl. 9, 1911–1921 (2016)

    MathSciNet  Article  Google Scholar 

  17. Raj, V.S.: A best proximity point theorem for weakly contractive non-self-mappings. Nonlinear Anal. 74, 4804–4808 (2011)

    MathSciNet  Article  Google Scholar 

  18. Shoaib, A.: Fixed point results for \(\alpha ^{*}\)-ψ multivalued mappings. Bull. Math. Anal. Appl. 8(4), 43–55 (2016)

    MathSciNet  Google Scholar 

  19. Wardowski, D.: Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012, 94 (2012)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The authors are thankful to the editors and anonymous referee(s) for their valuable comments and suggestions that helped to improve this manuscript.

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Contributions

SKJ dealt with conceptualization and writing original draft preparation. GM contributed in proper investigation and methodology. DS dealt with the formal analysis and design. JKM contributed in proper supervision and coordination. Finally, all authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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Correspondence to Gopal Meena.

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Jain, S.K., Meena, G., Singh, D. et al. Best proximity point results with their consequences and applications. J Inequal Appl 2022, 73 (2022). https://doi.org/10.1186/s13660-022-02807-y

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  • DOI: https://doi.org/10.1186/s13660-022-02807-y

MSC

  • 54H10
  • 54H25
  • 47H10

Keywords

  • Best proximity point
  • Partially ordered set
  • F-contraction
  • Metric space