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Fixed points of admissible almost contractive type mappings on bmetric spaces with an application to quadratic integral equations
Journal of Inequalities and Applications volume 2015, Article number: 32 (2015)
Abstract
Samet et al. in (Nonlinear Anal. 75:21542165, 2012) introduced the concepts of αψcontractive type mappings and αadmissible mappings in metric spaces. The purpose of this paper is to present a new class of almost contractive mappings called almost generalized \((\alpha\mbox{}\psi\mbox{}\varphi\mbox {}\theta)\)contractive mappings and to establish some fixed and common fixed point results for this class of mappings in complete ordered bmetric spaces. Our results improve and generalize several known results from the current literature and its extension. Moreover, an application to integral equations is given here to illustrate the usability of the obtained results.
Introduction
It is well known that the Banach contraction principle has been improved in different directions at different spaces by mathematicians over the years. In 1998, Czerwik [1, 2] introduced the concept of bmetric space. In the sequel, several papers have been published on the fixed point theory of various classes of singlevalued and multivalued operators in bmetric spaces (see, e.g., [1, 3–20]). On the other hand, more recently, Samet et al. in [21] introduced the concepts of αψcontractive type mappings and αadmissible mappings in metric spaces. Then, Karapınar and Samet [22] introduced the concept of generalized αψcontractive type, which was inspired by the notion of αψcontractive mappings. Furthermore, they [22] obtained various fixed point theorems for this generalized class of contractive mappings. Also, it should be noted that the study of common fixed points of mappings satisfying certain contractive conditions has been at the center of rigorous research activity (see [23–27]). In this paper, first, we introduce the concept of almost generalized \((\alpha\mbox{}\psi\mbox{}\varphi\mbox{}\theta) \)contractive mappings, and then we prove some common fixed point and coincidence fixed point theorems for this class of mappings in partially ordered complete bmetric spaces. Finally, as an application of our main results, we prove the existence of a unique solution to a class of nonlinear quadratic integral equations. The results of this paper improve and generalize the obtained results in papers [21, 22, 28].
Definition 1.1
[1]
Let X be a (nonempty) set and \(s\geq1\) be a given real number. A function \(d:X\times X\rightarrow\mathbb{R}^{+}\) is said to be a bmetric space iff for all \(x, y,z\in X\), the following conditions are satisfied:

(i)
\(d(x,y)=0\) iff \(x=y\);

(ii)
\(d(x,y)= d(y,x)\);

(iii)
\(d(x,y)\leq s[d(x,z)+d(z,y)]\).
The pair \((X,d)\) is called a bmetric space with the parameter s.
It is obvious that a bmetric space with base \(s=1\) is a metric space. There are examples of bmetric spaces which are not metric spaces (see, e.g., Singh and Prasad [18]).
The notions of a Cauchy sequence and a convergent sequence in bmetric spaces are defined by Boriceanu [29]. As usual, a bmetric space is said to be complete if and only if each Cauchy sequence in this space is convergent. Note that a bmetric, in the general case, is not continuous [3].
Definition 1.2
[30]
Let X be a nonempty set and \(T,g:X \to X\) be given selfmappings on X. The pair \(\{T,g\}\) is said to be weakly compatible if \(Tgx=gTx\), whenever \(Tx=gx\) for some x in X.
Samet et al. [21] defined the notion of αadmissible mappings as follows.
Definition 1.3
Let \(T:X \to X\) be a map and \(\alpha:X \times X \to\mathbb{R}\) be a function. Then T is said to be αadmissible if
Recently, Rosa and Vetro [31] introduced the following new notions of gαadmissible mapping.
Definition 1.4
Let \(T,g:X \to X\) and \(\alpha:X \times X \to\mathbb{R}\). The mapping T is gαadmissible if, for all \(x,y \in X\) such that \(\alpha (gx,gy)\geq1\), we have \(\alpha(Tx,Ty)\geq1\). If g is the identity mapping, then T is called αadmissible.
Definition 1.5
[32]
An αadmissible map T is said to be triangular αadmissible if
Main results
In this section, we prove some common fixed point results for two selfmappings satisfying an almost generalized \((\alpha\mbox{}\psi\mbox{}\varphi\mbox{}\theta)\)contractive mapping (for the notion of αψcontractive type mappings, see Samet et al. [21]).
Let \((X,d)\) be a bmetric space with a constant s and \(T:X\rightarrow X\) and \(g:X\rightarrow X\) be two mappings. Set
and
Now, we introduce the novel notion of an almost generalized \((\alpha \mbox{}\psi\mbox{}\varphi\mbox{}\theta)\)contractive mapping as follows.
Definition 2.1
Let T and g be two selfmappings on a bmetric space \((X,d)\). We say that T is an almost generalized \((\alpha\mbox{}\psi\mbox{}\varphi\mbox{}\theta )\)contractive mapping with respect to g if there exist \(\alpha:X \times X \to\mathbb{R}\) and some \(L\geq0\) such that for all \(x,y \in X\), we have
where \(\psi,\varphi,\theta:[0,\infty) \to[0,\infty)\) are continuous functions with \(\varphi(t)< \psi(t)\), \(\theta(t)>0\) for each \(t>0\), \(\varphi(0)=\psi(0)=\theta(0)=0\) and ψ is increasing.
Definition 2.2
Let \((X,d)\) be a bmetric space, \(g:X \to X\) and \(\alpha:X \times X \to\mathbb{R}\). X is αregular with respect to g if for every sequence \(\{ x_{n}\}\subseteq X\) such that \(\alpha(gx_{n},gx_{n+1})\geq 1\) for all \(n \in\mathbb{N}\) and \(gx_{n} \to gx \in gX\) as \(n \to\infty \), then there exists a subsequence \(\{gx_{n(k)}\}\) of \(\{gx_{n}\}\) such that for all \(k \in\mathbb{N}\), \(\alpha(gx_{n(k)},gx)\geq1\). If g is the identity mapping, then T is called αregular.
Lemma 2.1
Let \(T,g:X \to X\) and \(\alpha:X \times X \to\mathbb{R}\). Suppose that T is gαadmissible and triangular αadmissible. Assume that there exists \(x_{0}\in X\) such that \(\alpha(gx_{0},Tx_{0})\geq1\). Then
where
Proof
Since there exists \(x_{0}\in X\) such that \(\alpha(gx_{0},Tx_{0})\geq 1\) and T is gαadmissible, we deduce that
By continuing this process, we get
Suppose that \(m< n\). Since \(\alpha(gx_{m},gx_{m+1})\geq1\), \(\alpha (gx_{m+1},gx_{m+2})\geq1\) and T is triangular αadmissible, we have \(\alpha(gx_{m},gx_{m+2})\geq1\). Again, since \(\alpha (gx_{m},gx_{m+2})\geq1\) and \(\alpha(gx_{m+2}, gx_{m+3})\geq1\), we have \(\alpha(gx_{m},gx_{m+3})\geq1\). Continuing this process inductively, we obtain
□
Now, we establish some results for the existence of a common fixed point of mappings satisfying an almost generalized \((\alpha\mbox{}\psi\mbox{}\varphi\mbox{}\theta)\)contractive condition in the setup of bmetric spaces. The main result in this paper is the following common fixed point theorem.
Theorem 2.2
Let \((X,d)\) be a complete bmetric space, \(T,g:X \to X\) be such that \(TX\subseteq gX\) and suppose that gX is closed. Assume that the mapping T is an almost generalized \((\alpha\mbox{}\psi\mbox{}\varphi\mbox{}\theta )\)contractive mapping with respect to g and the following conditions hold:

(i)
T is gαadmissible and triangular αadmissible;

(ii)
there exists \(x_{0}\in X\) such that \(\alpha (gx_{0},Tx_{0})\geq1\);

(iii)
X is αregular with respect to g.
Then T and g have a coincidence point.
Moreover, if the following conditions hold:

(a)
the pair \(\{T,g\}\) is weakly compatible;

(b)
either \(\alpha(u,v)\geq1\) or \(\alpha(v,u)\geq1\) whenever \(Tu=gu\) and \(Tv=gv\).
Then T and g have a unique common fixed point.
Proof
Let \(x_{0}\in X\) be such that \(\alpha(gx_{0},Tx_{0})\geq1\) (using condition (ii)). Since \(TX\subseteq gX\), we can choose a point \(x_{1}\in X\) such that \(Tx_{0}=gx_{1}\). Also, there exists \(x_{2}\in X\) such that \(Tx_{1}=gx_{2}\), this can be done through the reality \(TX\subseteq gX\). Continuing this process having chosen \(x_{1},x_{2},\ldots,x_{n} \in X\), we have \(x_{n+1} \in X\) such that
By Lemma 2.1, we have
If \(Tx_{n_{0}}=Tx_{n_{0}+1}\) for some \(n_{0}\), then by (2.2) we get
that is, T and g have a coincidence point at \(x=x_{n_{0}+1}\), and so the proof is completed. So, we suppose that for all \(n \in\mathbb{N}\), \(Tx_{n} \neq Tx_{n+1}\). Since the mapping T is an almost generalized \((\alpha\mbox{}\psi\mbox{}\varphi\mbox{}\theta)\)contractive mapping with respect to g and using (2.3), we obtain
for all \(n \in\mathbb{N}\), where
and
Since
then we get
If for some \(n \in\mathbb{N}\), \(\max\{ d(gx_{n1},gx_{n}),d(gx_{n},gx_{n+1})\}=d(gx_{n},gx_{n+1})\), then by (2.6) and using the properties of the function φ, we get
which is a contradiction. So
From (2.7), we deduce that \(\{\psi(d(gx_{n},gx_{n+1}))\}\) is a nonnegative nonincreasing sequence. Since ψ is increasing, the sequence \(\{d(gx_{n},gx_{n+1})\}\) is nonincreasing, and consequently there exists \(\delta\geq0\) such that
We claim that \(\delta=0\). On the contrary, assume that
Since ψ and φ are continuous, then from (2.7) and (2.8) we have
and so \(\delta=0\), that is a contradiction. Thus
Now, we claim that
Assume, on the contrary, that there exist \(\epsilon>0\) and subsequences \(\{gx_{m(k)}\}\), \(\{gx_{n(k)}\}\) of \(\{gx_{n}\}\) with \(n(k)>m(k)\geq k\) such that
Additionally, corresponding to \(m(k)\), we may choose \(n(k)\) such that it is the smallest integer satisfying (2.11) and \(n(k)>m(k)\geq k\). Thus,
Using the triangle inequality in a bmetric space and (2.11) and (2.12), we obtain that
Taking the upper limit as \(k\rightarrow\infty\) and using (2.9), we obtain
Also
So, from (2.9) and (2.13), we have
Also
So from (2.9) and (2.13), we get
Also
so from (2.9) and (2.15), we have
Taking (2.9), (2.13), (2.14) and (2.15) into account, we get
So,
Similarly, we have
Now, using inequality (2.1) and Lemma 2.1, we have
which is a contradiction. So, we conclude that \(\{gx_{n}\}\) is a Cauchy sequence in \((X,d)\). By virtue of (2.2) we get \(\{Tx_{n}\}=\{gx_{n+1}\}\subseteq gX\) and gX is closed, there exists \(x \in X\) such that
Now, we claim that x is a coincidence point of T and g. On the contrary, assume that \(d(Tx,gx)>0\). Since X is αregular with respect to g and (2.19), we have
Also by the use of triangle inequality in a bmetric space, we have
In the above inequality, if k tends to infinity, then we have
By property of ψ, (2.20) and (2.21), we have
which is a contradiction. Indeed,
and
When n tends to infinity, we deduce
and
Hence, \(d(gx,Tx)=0\), that is, \(gx=Tx\) and x is a coincidence point of T and g. We claim that if \(Tu=gu\) and \(Tv=gv\), then \(gu=gv\). By hypotheses, \(\alpha (u,v)\geq1\) or \(\alpha(v,u)\geq1\). Suppose that \(\alpha(u,v)\geq1\), then
where
and
So,
which is a contradiction. Thus we deduce that \(gu=gv\). Similarly, if \(\alpha(v,u)\geq1\), we can prove that \(gu=gv\). Now, we show that T and g have a common fixed point. Indeed, if \(w=Tu=gu\), owing to the weak compatibility of T and g, we get \(Tw=T(gu)=g(Tu)=gw\). Thus w is a coincidence point of T and g, then \(gu=gw=w=Tw\). Therefore, w is a common fixed point of T and g. The uniqueness of the common fixed point of T and g is a consequence of conditions (2.1) and (b), and so we omit the details. □
Example 2.1
Let X be the set of Lebesgue measurable functions on \([0, 1]\) such that \(\int_{0}^{1}x(t)\, dt<\infty\). Define \(D:X\times X \rightarrow[0,\infty)\) by
Then D is a bmetric on X, with \(s = 2\).
The operator \(T: X\rightarrow X\) is defined by
and the operator \(g: X\rightarrow X\) is defined by
Now, we prove that T and g have a unique common fixed point. For all \(x,y\in X\), we have
Now, if we define \(\varphi(t)=\ln(1+\sqrt{t})\), \(\psi(t)=\sqrt{t}\), \(\alpha(x, y)=1\) and \(x_{0}=0\). Thus, by using Theorem 2.2, we obtain that T and g have a unique common fixed point.
From Theorem 2.2, if we choose \(g=I_{X}\) the identity mapping on X, we deduce the following corollary.
Corollary 2.3
Let \((X,d)\) be a complete bmetric space and \(T:X \to X\) be a selfmapping on X. Suppose that there exist \(\alpha:X \times X \to\mathbb{R}\) and some \(L\geq0\) such that for all \(x,y \in X\),
where \(\psi,\varphi,\theta:[0,\infty) \to[0,\infty)\) are continuous functions with \(\varphi(t)< \psi(t)\), \(\theta(t)>0\) for each \(t>0\), \(\varphi(0)=\psi(0)=\theta(0)=0\), ψ is increasing,
and
Suppose also that the following conditions hold:

(i)
T is αadmissible and triangular αadmissible;

(ii)
there exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq1\);

(iii)
X is αregular;

(iv)
either \(\alpha(u,v)\geq1\) or \(\alpha(v,u)\geq1\) whenever \(Tu=u\) and \(Tv=v\).
Then T has a unique fixed point.
Example 2.2
Let X be the set of Lebesgue measurable functions on \([0, 1]\) such that \(\int_{0}^{1}x(t)\, dt<\infty\). Define \(D:X\times X \rightarrow[0,\infty)\) by
Then D is a bmetric on X, with \(s = 2\).
The operator \(T: X\rightarrow X\) defined by
Now, we prove that T has a unique fixed point. For all \(x,y\in X\), we have
Now, if we define \(\varphi(t)=\ln(1+\sqrt{t})\), \(\psi(t)=\sqrt{t}\), \(\alpha(x, y)=1\) and \(x_{0}=0\). Thus, by Corollary 2.3 we obtain that T has a unique fixed point.
From Theorem 2.2, if the function \(\alpha:X \times X \to\mathbb {R}\) is such that \(\alpha(x,y)=1\) for all \(x,y \in X\), we deduce the following corollary.
Corollary 2.4
Let \((X,d)\) be a complete bmetric space, \(T,g:X \to X\) be such that \(TX\subseteq gX\). Assume that gX is closed and there exists \(L\geq0\) such that for all \(x,y \in X\),
where \(\psi,\varphi,\theta:[0,\infty) \to[0,\infty)\) are continuous functions with \(\varphi(t)< \psi(t)\), \(\theta(t)>0\) for each \(t>0\), \(\varphi(0)=\psi(0)=\theta(0)=0\), ψ is increasing,
and
Then T and g have a coincidence point. Moreover, if T and g are weakly compatible, then T and g have a unique common fixed point.
From Theorem 2.2, if \(\psi(t)=\psi_{1}(t)\) and \(\varphi(t)=\psi _{1}(t)\varphi_{1}(t)\) for each \(t\in\mathbb{R}_{+}\), where \(\psi_{1}, \varphi_{1}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}\) are continuous functions such that \(\psi_{1}(t)>\varphi_{1}(t)>0\) for \(t>0\), \(\psi _{1}(0)=\varphi_{1}(0)=0\) and \(\psi_{1}\) is increasing, we deduce the following corollary.
Corollary 2.5
Let \((X,d)\) be a complete bmetric space, \(T,g:X \to X\) be such that \(TX\subseteq gX\). Assume that gX is closed and there exist \(\alpha:X \times X \to\mathbb{R}\) and \(L\geq0\) such that for all \(x,y \in X\),
where \(\psi_{1}, \varphi_{1},\theta:\mathbb{R}_{+}\rightarrow\mathbb {R}_{+}\) are continuous functions such that \(\psi_{1}(t)>\varphi _{1}(t)>0\) for \(t>0\), \(\psi_{1}(t)=\varphi_{1}(t)=\theta(t)=0\) if and only if \(t=0\) and \(\psi_{1}\) is increasing,
and
Assume also that the following conditions hold:

(i)
T is gαadmissible and triangular αadmissible;

(ii)
there exists \(x_{0}\in X\) such that \(\alpha (gx_{0},Tx_{0})\geq1\);

(iii)
X is αregular with respect to g.
Then T and g have a coincidence point.
Moreover, the following conditions hold:

(a)
the pair \(\{T,g\}\) is weakly compatible;

(b)
either \(\alpha(u,v)\geq1\) or \(\alpha(v,u)\geq1\) whenever \(Tu=gu\) and \(Tv=gv\).
Then T and g have a unique common fixed point.
From Corollary 2.5, if we choose \(L=0\) and \(g=I_{X}\) the identity mapping on X, we deduce the following corollary.
Corollary 2.6
Let \((X,d)\) be a complete bmetric space, \(T:X \to X\) be a selfmapping on X and \(\alpha:X \times X \to\mathbb{R}\). Assume that the following condition holds:
for all \(x,y \in X\), where \(\psi_{1}, \varphi_{1}:\mathbb {R}_{+}\rightarrow\mathbb{R}_{+}\) are continuous functions such that \(\psi_{1}(t)>\varphi_{1}(t)>0\) for \(t>0\), \(\psi_{1}(0)=\varphi _{1}(0)=0\) and \(\psi_{1}\) is increasing and
Assume also that the following conditions hold:

(i)
T is αadmissible and triangular αadmissible;

(ii)
there exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq1\);

(iii)
X is αregular;

(iv)
either \(\alpha(u,v)\geq1\) or \(\alpha(v,u)\geq1\) whenever \(Tu=u\) and \(Tv=v\).
Then T has a unique fixed point.
From Corollary 2.5, if the function \(\alpha:X \times X \to\mathbb {R}\) is such that \(\alpha(x,y)=1\) for all \(x,y \in X\), we deduce the following corollary.
Corollary 2.7
Let \((X,d)\) be a complete bmetric space, \(T,g:X \to X\) be such that \(TX\subseteq gX\). Assume that gX is closed and there exists \(L\geq0\) such that for all \(x,y \in X\),
where \(\psi_{1}, \varphi_{1},\theta:\mathbb{R}_{+}\rightarrow\mathbb {R}_{+}\) are continuous functions such that \(\psi_{1}(t)>\varphi _{1}(t)>0\) for \(t>0\) and \(\psi_{1}(t)=\varphi_{1}(t)=\theta(t)=0\) if and only if \(t=0\) and \(\psi_{1}\) is increasing,
and
Then T and g have a coincidence point. Moreover, if T and g are weakly compatible, then T and g have a unique common fixed point.
From Corollary 2.7, if \(\psi_{1}(t)=t\), \(g=I\) and \(L=0\), we deduce the following corollary.
Corollary 2.8
Let \((X,d)\) be a complete bmetric space, \(T:X \to X\) be a selfmapping on X. Assume that the following condition holds:
for all \(x,y \in X\), where \(\varphi_{1}:\mathbb{R}_{+}\rightarrow\mathbb {R}_{+}\) is a continuous function such that \(\varphi_{1}(t)< t\) for \(t>0\), \(\varphi_{1}(0)=0\) and
Then T has a unique fixed point.
From Theorem 2.2, if \(\psi(t)=t\) and \(\varphi(t)=\beta(t)t\) which \(\beta\in\mathcal{F}\) (ℱ defined in [28]), we deduce the following corollary.
Corollary 2.9
Let \((X,d)\) be a complete bmetric space, \(T,g:X \to X\) be such that \(TX\subseteq gX\). Assume that gX is closed and there exist \(\alpha:X \times X \to\mathbb{R}\) and \(L\geq0\) such that for all \(x,y \in X\),
where \(\theta:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}\) is a continuous function such that \(\theta(t)=0\) if and only if \(t=0\), \(\beta\in S\) and
and
Assume also that the following conditions hold:

(i)
T is gαadmissible and triangular αadmissible;

(ii)
there exists \(x_{0}\in X\) such that \(\alpha (gx_{0},Tx_{0})\geq1\);

(iii)
X is αregular with respect to g.
Then T and g have a coincidence point.
Moreover, if the following conditions hold:

(a)
the pair \(\{T,g\}\) is weakly compatible;

(b)
either \(\alpha(u,v)\geq1\) or \(\alpha(v,u)\geq1\) whenever \(Tu=gu\) and \(Tv=gv\).
Then T and g have a unique common fixed point.
From Corollary 2.9, if the function \(\alpha:X \times X \to \mathbb{R}\) is such that \(\alpha(x,y)=1\) for all \(x,y \in X\), we deduce the following corollary.
Corollary 2.10
Let \((X,d)\) be a complete bmetric space, \(T,g:X \to X\) be such that \(TX\subseteq gX\). Assume that gX is closed and that the following conditions hold:
for all \(x,y \in X\), \(\beta\in S\) and
Then T and g have a coincidence point. Moreover, if T and g are weakly compatible, then T and g have a unique common fixed point.
Remark 2.1
Since a bmetric space is a metric space when \(s= 1\), so our results can be viewed as the generalization and the extension of several comparable results.
Application to integral equations
Here, in this section, we wish to study the existence of a unique solution for a nonlinear quadratic integral equation, as an application of our fixed point theorem. Consider the nonlinear quadratic integral equation
Let Γ denote the class of those functions \(\gamma:[0,+\infty) \rightarrow[0,+\infty)\) which satisfy the following conditions:

(i)
γ is nondecreasing and \((\gamma(t))^{p}\leq \gamma(t^{p})\) for all \(p\geq1\).

(ii)
There exists \(\varphi:[0,+\infty) \rightarrow [0,+\infty)\) which is a continuous function and \(\varphi(t)< t\) for all \(t>0\) and \(\varphi(0)=0\) such that \(\gamma(t)=t\varphi(t)\) for all \(t\in[0,+\infty)\).
For example, \(\gamma_{1}(t)=kt\), where \(0\leq k<1\) and \(\gamma_{2}(t)=\frac{t}{t+1}\) are in Γ.
We will analyze Eq. (3.1) under the following assumptions:
 (A_{1}):

\(h:I\rightarrow\mathbb{R}\) is a continuous function.
 (A_{2}):

\(f:I\times\mathbb{R}\rightarrow\mathbb{R}\) is a continuous function, \(f(t,x)\geq0\) and there exist constant \(0\leq L<1\) and \(\gamma\in\Gamma\) such that for all \(x,y\in\mathbb{R}\),
$$\bigl\vert f(t,x)f(t,y)\bigr\vert \leq L\gamma\bigl(\vert xy\vert \bigr). $$  (A_{3}):

\(k:I\times I\rightarrow\mathbb{R}\) is continuous at \(t\in I\) for every \(s\in I\) and measurable at \(s\in I\) for all \(t\in I\) such that \(k(t,x)\geq0\) and \(\int_{0}^{1}k(t,s)\, ds\leq K\).
 (A_{4}):

\(\lambda^{p} K^{p}L^{p}\leq\frac{1}{2^{3p3}}\).
Also, consider the space \(X=C(I)\) of continuous functions defined on \(I=[0,1]\) with the standard metric given by
Now, for \(p\geq1\), we define
It is easy to see that \((X,d)\) is a complete bmetric space with \(s=2^{p1}\) [3].
We formulate the main result of this section.
Theorem 3.1
Under assumptions (A_{1})(A_{4}), Eq. (3.1) has a unique solution in \(C(I)\).
Proof
We consider the operator \(T:X\rightarrow X\) defined by
By virtue of our assumptions, T is well defined (this means that if \(x\in X\) then \(Tx\in X\)). Also, for \(x,y \in X\), we have
Since the function γ is nondecreasing, we have
hence
Then we can obtain
This proves that the operator T satisfies the contractive condition (2.25) appearing in Corollary 2.8. So Eq. (3.1) has a unique solution in \(C(I)\) and the proof is complete. □
Example 3.1
Consider the following functional integral equation:
for \(t\in[0,1]\). Observe that this equation is a special case of Eq. (3.1) with
Indeed, by using \(\gamma(t)=\frac{1}{3}t\), we see that \(\gamma\in\Phi\) and \((\gamma(t))^{p}=(\frac{1}{3}t)^{p}=\frac{1}{3^{p}}t^{p}\leq\frac {1}{3}t^{p}= \gamma(t^{p})\) for all \(p\geq1\). Further, for arbitrarily fixed \(x,y\in \mathbb{R}\) such that \(x\geq y\) and for \(t\in[0,1]\), we obtain
Thus, the function f satisfies assumption (A_{2}) with \(L=\frac {1}{6}\). It is also easily seen that h is a continuous function. Further, notice that the function k is continuous in \(t\in I\) for every \(s\in I\) and measurable in \(s\in I\) for all \(t\in I\) and \(k(t,s)\geq0\). Moreover, we have
This shows that assumption (A_{3}) holds. Taking \(L=\frac{1}{6}\), \(K=\frac{1}{2}\) and \(\lambda=\frac{1}{27}\), then inequality \(L^{p}\lambda^{p}K^{p}\leq\frac{1}{2^{3p3}}\) appearing in assumption (A_{4}) has the following form:
It is easily seen that each number \(p\geq1\) satisfies the above inequality. Consequently, all the conditions of Theorem 3.1 are satisfied. Hence the integral equation (3.2) has a unique solution in \(C(I)\).
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Allahyari, R., Arab, R. & Shole Haghighi, A. Fixed points of admissible almost contractive type mappings on bmetric spaces with an application to quadratic integral equations. J Inequal Appl 2015, 32 (2015). https://doi.org/10.1186/s136600150549z
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DOI: https://doi.org/10.1186/s136600150549z
MSC
 47H10
 54H25
Keywords
 common fixed point
 coincidence point
 bmetric space
 gαadmissible mapping
 αregular
 triangular αadmissible
 integral equations
 almost contractive mapping