 Research
 Open Access
Coincidence point theorems for generalized contractions with application to integral equations
 Nawab Hussain^{1},
 Jamshaid Ahmad^{2},
 Ljubomir Ćirić^{3}Email author and
 Akbar Azam^{2}
https://doi.org/10.1186/s1366301503314
© Hussain et al. 2015
 Received: 21 November 2014
 Accepted: 17 May 2015
 Published: 5 June 2015
Abstract
In this article, we introduce a new type of contraction and prove certain coincidence point theorems which generalize some known results in this area. As an application, we derive some new fixed point theorems for Fcontractions. The article also includes an example which shows the validity of our main result and an application in which we prove an existence and uniqueness of a solution for a general class of Fredholm integral equations of the second kind.
Keywords
 coincidence point
 Fcontractions
 integral equations
MSC
 46S40
 47H10
 54H25
1 Introduction and preliminaries
The Banach contraction principle [1] is one of the earliest and most important results in fixed point theory. Because of its application in many disciplines such as computer science, chemistry, biology, physics, and many branches of mathematics, a lot of authors have improved, generalized, and extended this classical result in nonlinear analysis; see, e.g., [2–10] and the references therein. In 2012, Azam [3] obtained the existence of a coincidence point of a mapping and a relation under a contractive condition in the context of metric space. For coincidence point results see also [11]. Consistent with Azam, we begin with some basic known definitions and results which will be used in the sequel. Throughout this article, \(\mathbb{N}\), \(\mathbb{R}^{+}\), \(\mathbb{R}\) denote the set of all natural numbers, the set of all positive real numbers, and the set of all real numbers, respectively.
Let A and B be arbitrary nonempty sets. A relation R from A to B is a subset of \(A\times B\) and is denoted \(R:A\rightsquigarrow B\). The statement \(( x,y ) \in R\) is read ‘x is Rrelated to y’, and is denoted \(xRy\). A relation \(R:A\rightsquigarrow B\) is called lefttotal if for all \(x\in A\) there exists a \(y\in B\) such that \(xRy\), that is, R is a multivalued function. A relation \(R:A\rightsquigarrow B\) is called righttotal if for all \(y\in B\) there exists an \(x\in A\) such that \(xRy\). A relation \(R:A\rightsquigarrow B \) is known as functional, if \(xRy\), \(xRz\) implies that \(y=z\), for \(x\in A \) and \(y,z\in B\). A mapping \(T:A\rightarrow B\) is a relation from A to B which is both functional and lefttotal.
Wardowski [12] introduced and studied a new contraction called an Fcontraction to prove a fixed point result as a generalization of the Banach contraction principle.
Definition 1
 (F_{1}):

F is strictly increasing;
 (F_{2}):

for all sequence \(\{ \alpha_{n}\} \subseteq R^{+}\), \(\lim_{n\to\infty}\alpha_{n}=0\) if and only if \(\lim_{n\to \infty}F(\alpha_{n})=\infty\);
 (F_{3}):

there exists \(0< k<1\) such that \(\lim_{n\rightarrow 0^{+}}\alpha^{k}F(\alpha)=0\).
Consistent with Wardowski [12], we denote by Ϝ the set of all functions \(F:\mathbb{R}^{+}\rightarrow\mathbb{R}\) satisfying the conditions (F_{1})(F_{3}).
Definition 2
[12]
Theorem 3
[12]
Abbas et al. [13] further generalized the concept of an Fcontraction and proved certain fixed and common fixed point results. Hussain and Salimi [14] introduced some new type of contractions called αGFcontractions and established SuzukiWardowski type fixed point theorems for such contractions. For more details on Fcontractions, we refer the reader to [11, 13–20].
In this paper, we obtain coincidence points of mappings and relations under a new type of contractive condition in a metric space. Moreover, we discuss an illustrative example to highlight the realized improvements.
2 Main results
Now we state and prove the main results of this section.
Theorem 4
Proof
Example 5
From Theorem 4 we deduce the following result immediately.
Theorem 6
Proof
Remark 7
If in the above theorem we choose \(X=Y\), \(R=I\) (the identity mapping on X), we obtain Theorem 3, which is Theorem 3.1 of Wardowski [12].
Corollary 8
Proof
Consider the mapping \(F(t)=\ln(t)\), for \(t>0\). Then obviously F satisfies (F_{1})(F_{3}). From Theorem 4, we obtain the desired conclusion. □
Corollary 9
Proof
Consider the mapping \(F(t)=\ln(t)\), for \(t>0\). Then obviously F satisfies (F_{1})(F_{3}). From Theorem 6, we obtain the desired conclusion. □
Remark 10
If in the above corollary we choose \(X=Y\) and \(R=I\) (the identity mapping on X), we obtain the Banach contraction theorem.
Theorem 11
Proof
3 Applications
Theorem 12
 (i)
\(K:[0,\Theta]\times[0,\Theta]\times \mathbb{R} \rightarrow \mathbb{R} \), \(g:[0,\Theta]\rightarrow \mathbb{R} \) and \(f:\mathbb{R} \rightarrow \mathbb{R} \) are continuous,
 (ii)
\(\int_{0}^{t}K(t,s,\cdot):\mathbb{R} \rightarrow \mathbb{R} \) is increasing, for all \(t,s\in[0,\Theta]\),
 (iii)there exists \(\tau\in(0,+\infty)\) such thatfor all \(t,s\in[0,\Theta]\) and \(hx,hy\in \mathbb{R} \),$$ \biglK\bigl(t,s,hx(s)\bigr)K\bigl(t,s,hy(s)\bigr)\bigr \leq\tau\biglhx(s)hy(s)\bigr $$
 (iv)if f is injective, then for \(\tau>0\) there exists \(e^{\tau}\in \mathbb{R} ^{+}\) such that for all \(x,y\in \mathbb{R} \);$$ \vert hxhy\vert \leq e^{\tau} \vert fxfy\vert $$
Proof
Declarations
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, first author acknowledges with thanks DSR, KAU for financial support.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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