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Hussain et al Fixed Point Theory and Applications (2015) 2015:78 DOI 10.1186/s13663-015-0331-4 RESEARCH Open Access Coincidence point theorems for generalized contractions with application to integral equations ´ c3* and Akbar Azam2 Nawab Hussain1 , Jamshaid Ahmad2 , Ljubomir Ciri´ * Correspondence: lciric@rcub.bg.ac.rs Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, Belgrade, 11 000, Serbia Full list of author information is available at the end of the article 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 F-contractions 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 MSC: 46S40; 47H10; 54H25 Keywords: coincidence point; F-contractions; integral equations Introduction and preliminaries The Banach contraction principle [] 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., [–] and the references therein In , Azam [] 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 [] Consistent with Azam, we begin with some basic known definitions and results which will be used in the sequel Throughout this article, N, R+ , 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 × B and is denoted R : A B The statement (x, y) ∈ R is read ‘x is R-related to y’, and is denoted xRy A relation R : A B is called left-total if for all x ∈ A there exists a y ∈ B such that xRy, that is, R is a multivalued function A relation R : A B is called right-total if for all y ∈ B there exists an x ∈ A such that xRy A relation R : A B is known as functional, if xRy, xRz implies that y = z, for x ∈ A and y, z ∈ B A mapping T : A → B is a relation from A to B which is both functional and left-total For R : A B, E ⊂ A we define R(E) = {y ∈ B : xRy for some x ∈ E}, © 2015 Hussain et al 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 Hussain et al Fixed Point Theory and Applications (2015) 2015:78 Page of 13 dom(R) = x ∈ A : R {x} = φ , Range(R) = y ∈ B : y ∈ R {x} for some x ∈ dom(R) For convenience, we denote R({x}) by R{x} The class of relations from A to B is denoted by R(A, B) Thus the collection M(A, B) of all mappings from A to B is a proper subcollection of R(A, B) An element w ∈ A is called a coincidence point of T : A → B and R : A B if Tw ∈ R{w} In the following we always suppose that X is a nonempty set and (Y , d) is a metric space For R : X Y and u, v ∈ dom(R), we define D R{u}, R{v} = inf d(x, y) uRx,vRy Wardowski [] introduced and studied a new contraction called an F-contraction to prove a fixed point result as a generalization of the Banach contraction principle Definition Let F : R+ → R be a mapping satisfying the following conditions: (F ) F is strictly increasing; (F ) for all sequence {αn } ⊆ R+ , limn→∞ αn = if and only if limn→∞ F(αn ) = –∞; (F ) there exists < k < such that limn→+ α k F(α) = Consistent with Wardowski [], we denote by satisfying the conditions (F )-(F ) the set of all functions F : R+ → R Definition [] Let (X, d) be a metric space A self-mapping T on X is called an Fcontraction if there exists τ > such that for x, y ∈ X d(Tx, Ty) > ⇒ τ + F d(Tx, Ty) ≤ F d(x, y) , where F ∈ Theorem [] Let (X, d) be a complete metric space and T:X → X be a self-mapping If there exists τ > such that for all x, y ∈ X: d(Tx, Ty) > implies τ + F d(Tx, Ty) ≤ F d(x, y) , where F ∈ , then T has a unique fixed point Abbas et al [] further generalized the concept of an F-contraction and proved certain fixed and common fixed point results Hussain and Salimi [] introduced some new type of contractions called α-GF-contractions and established Suzuki-Wardowski type fixed point theorems for such contractions For more details on F-contractions, we refer the reader to [, –] 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 Hussain et al Fixed Point Theory and Applications (2015) 2015:78 Page of 13 Main results Now we state and prove the main results of this section Theorem Let X be a nonempty set and (Y , d) be a metric space Let T : X → Y , R : X Y be such that R is left-total, Range(T) ⊆ Range(R) and Range(T) or Range(R) is complete If there exist a mapping F : R+ → R and τ > such that ⇒ d(Tx, Ty) > τ + F d(Tx, Ty) ≤ F D R{x}, R{y} (.) for all x, y ∈ X, then there exists w ∈ X such that Tw ∈ R{w} Proof Let x ∈ X be an arbitrary but fixed element We define the sequences {xn } ⊂ X and {yn } ⊂ Range(R) Let y = Tx , Range(T) ⊆ Range(R) We may choose x ∈ X such that x Ry , since R is left-total Let y = Tx , since Range(T) ⊆ Range(R) If Tx = Tx , then we have x Ry This implies that x is the required point that is Tx ∈ R{x } So we assume that Tx = Tx , then from (.) we get τ + F d(y , y ) = τ + F d(Tx , Tx ) ≤ F D R{x }, R{x } (.) We may choose x ∈ X such that x Ry , since R is left-total Let y = Tx , since Range(T) ⊆ Range(R) If Tx = Tx , then we have x Ry This implies that Tx ∈ R{x } and x is the coincidence point So Tx = Tx , then from (.), we get τ + F d(y , y ) = τ + F d(Tx , Tx ) ≤ F D R{x }, R{x } (.) By induction, we can construct sequences {xn } ⊂ X and {yn } ⊂ Range(R) such that yn = Txn– and xn Ryn (.) for all n ∈ N If there exists n ∈ N for which Txn – = Txn Then xn Ryn + Thus Txn ∈ R{xn } and the proof is finished So we suppose now that Txn– = Txn for every n ∈ N Then from (.), (.), and (.), we deduce that τ + F d(yn , yn+ ) = τ + F d(Txn– , Txn ) ≤ F D R{xn– }, R{xn } (.) for all n ∈ N Since xn Ryn and xn+ Ryn+ , by the definition of D, we get D(R{xn– }, R{xn }) ≤ d(yn– , yn ) Thus from (.), we have τ + F d(yn , yn+ ) ≤ F d(yn– , yn ) , (.) which further implies that F d(yn , yn+ ) ≤ F d(yn– , yn ) – τ ≤ F d(yn– , yn– ) – τ ≤ · · · ≤ F d(y , y ) – nτ (.) Hussain et al Fixed Point Theory and Applications (2015) 2015:78 Page of 13 From (.), we obtain lim F d(yn , yn+ ) = –∞ (.) n→∞ Then from (F ), we get lim d(yn , yn+ ) = (.) n→∞ Now from (F ), there exists < k < such that k lim d(yn , yn+ ) F d(yn , yn+ ) = n→∞ (.) By (.), we have d(yn , yn+ )k F d(yn , yn+ ) – d(yn , yn+ )k F d(y , y ) ≤ d(yn , yn+ )k F d(y , y ) – nτ – F d(y , y ) = –nτ d(yn , yn+ ) k ≤ (.) By taking the limit as n → ∞ in (.) and applying (.) and (.), we have lim n d(yn , yn+ ) n→∞ k = (.) It follows from (.) that there exists n ∈ N such that n d(yn , yn+ ) k ≤ (.) for all n > n This implies d(yn , yn+ ) ≤ n/k (.) for all n > n Now we prove that {yn } is a Cauchy sequence For m > n > n we have m– d(yn , ym ) ≤ m– d(yi , yi+ ) ≤ i=n i=n i/k (.) Since < k < , ∞ i= i/k converges Therefore, d(yn , ym ) → as m, n → ∞ Thus we proved that {yn } is a Cauchy sequence in Range(R) Completeness of Range(R) ensures that there exists z ∈ Range(R) such that yn → z as n → ∞ Now since R is left-total, wRz for some w ∈ X Now F d(yn , Tw) = F d(Txn– , Tw) ≤ F D R{xn– }, R{w} – τ < F d(yn– , z) – τ Since limn→∞ d(yn– , z) = , by (F ), we have limn→∞ F(d(yn– , z)) = –∞ This implies that limn→∞ F(d(yn , Tw)) = –∞, which further implies that limn→∞ d(yn , Tw) = Hence Hussain et al Fixed Point Theory and Applications (2015) 2015:78 Page of 13 d(z, Tw) = It follows that z = Tw Hence Tw ∈ R{w} In the case when Range(T) is complete Since Range(T) ⊆ Range(R), there exists an element z∗ ∈ Range(R) such that yn → z∗ The remaining part of the proof is the same as in previous case Example Let X = Y = R, d(x, y) = |x – y| Define T : R → R, R : R Tx = R as follows: if x ∈ Q, if x ∈ Q , R = Q×[, ] ∪ Q ×[, ] Then Range(T) = {, } ⊂ Range(R) = [, ] ∪ [, ] Let F(t) = ln(t) and τ = For x ∈ Q, y ∈ Q or y ∈ Q, x ∈ Q , d(Tx, Ty) > implies that τ + F d(Tx, Ty) ≤ F D R{x}, R{y} Thus all conditions of the above theorem are satisfied and is the coincidence point of T and R From Theorem we deduce the following result immediately Theorem Let X be a nonempty set and (Y , d) be a metric space Let T, R : X → Y be two mappings such that Range(T) ⊆ Range(R) and Range(T) or Range(R) is complete If there exist a mapping F : R+ → R and τ > such that τ + F d(Tx, Ty) ≤ F d(Rx, Ry) for all x, y ∈ X, then T and R have a coincidence point in X Moreover, if either T or R is injective, then R and T have a unique coincidence point in X Proof By Theorem , we see that there exists w ∈ X such that Tw = Rw, where Rw = lim Rxn = lim Txn– , n→∞ n→∞ x ∈ X For uniqueness, assume that w , w ∈ X, w = w , Tw = Rw , and Tw = Rw Then τ + F(d(Tw , Tw )) ≤ F(d(Rw , Rw )) for some τ > If R or T is injective, then d(Rw , Rw ) > and τ + F d(Rw , Rw ) = τ + F d(Tw , Tw ) ≤ F d(Rw , Rw ) , a contradiction to the fact that τ > Remark If in the above theorem we choose X = Y , R = I (the identity mapping on X), we obtain Theorem , which is Theorem . of Wardowski [] Hussain et al Fixed Point Theory and Applications (2015) 2015:78 Page of 13 Corollary Let T : X → Y , R : X Y be such that R is left-total, Range(T) ⊆ Range(R) and Range(T) or Range(R) is complete If there exists λ ∈ [, ) such that for all x, y ∈ X d(Tx, Ty) ≤ λD R{x}, R{y} , then there exists w ∈ X such that Tw ∈ R{w} Proof Consider the mapping F(t) = ln(t), for t > Then obviously F satisfies (F )-(F ) From Theorem , we obtain the desired conclusion Corollary Let X be nonempty set and (Y , d) be a metric space T, R : X → Y be two mappings such that Range(T) ⊆ Range(R) and Range(T) or Range(R) is complete If there exists a λ ∈ [, ) such that for all x, y ∈ X d(Tx, Ty) ≤ λd(Rx, Ry), then R and T have a coincidence point in X Moreover, if either T or R is injective, then R and T have a unique coincidence point in X Proof Consider the mapping F(t) = ln(t), for t > Then obviously F satisfies (F )-(F ) From Theorem , we obtain the desired conclusion Remark If in the above corollary we choose X = Y and R = I (the identity mapping on X), we obtain the Banach contraction theorem In this way, we recall the concept of F-contractions for multivalued mappings and proved Suzuki-type fixed point theorem for such contractions Nadler [] invented the concept of a Hausdorff metric H induced by metric d on X as follows: H(A, B) = max sup d(x, B), sup d(y, A) y∈B x∈A for every A, B ∈ CB(X) He extended the Banach contraction principle to multivalued mappings Since then many authors have studied fixed points for multivalued mappings Very recently, Sgroi and Vetro extended the concept of the F-contraction for multivalued mappings (see also []) Theorem Let (X, d) be a metric space and let T : X → CB(X) Assume that there exist a function F ∈ that is continuous from the right and τ ∈ R+ such that d(x, Tx) ≤ d(x, y) ⇒ τ + F H(Tx, Ty) ≤ F d(x, y) (.) for all x, y ∈ X Then T has a fixed point Proof Let x ∈ X be an arbitrary point of X and choose x ∈ Tx If x ∈ Tx , then x is a fixed point of T and the proof is completed Assume that x ∈/ Tx , then Tx = Tx Now d(x , Tx ) ≤ d(x , x ) < d(x , x ) Hussain et al Fixed Point Theory and Applications (2015) 2015:78 Page of 13 From the assumption, we have τ + F H(Tx , Tx ) ≤ F d(x , x ) Since F is continuous from the right, there exists a real number h > such that F hH(Tx , Tx ) ≤ F H(Tx , Tx ) + τ Now, from d(x , Tx ) ≤ H(Tx , Tx ) < hH(Tx , Tx ), we deduce that there exists x ∈ Tx such that d(x , x ) ≤ hH(Tx , Tx ) Consequently, we get F d(x , x ) ≤ F hH(Tx , Tx ) < F H(Tx , Tx ) + τ , which implies that τ + F d(x , x ) ≤ τ + F H(Tx , Tx ) + τ ≤ F d(x , x ) + τ Thus τ + F d(x , x ) ≤ F d(x , x ) Continuing in this manner, we can define a sequence {xn } ⊂ X such that xn ∈/ Txn , xn+ ∈ Txn and τ + F d(xn , xn+ ) ≤ F d(xn– , xn ) for all n ∈ N ∪ {} Therefore F d(xn , xn+ ) ≤ F d(xn– , xn ) – τ ≤ F d(xn– , xn– ) – τ ≤ · · · ≤ F d(x , x ) – nτ (.) for all n ∈ N Since F ∈ , by taking the limit as n → ∞ in (.) we have lim F d(xn , xn+ ) = –∞ n→∞ ⇐⇒ lim d(xn , xn+ ) = n→∞ (.) Now from (F ), there exists < k < such that k lim d(xn , xn+ ) F d(xn , xn+ ) = n→∞ (.) Hussain et al Fixed Point Theory and Applications (2015) 2015:78 Page of 13 By (.), we have d(xn , xn+ )k F d(xn , xn+ ) – d(xn , xn+ )k F d(x , x ) ≤ d(xn , xn+ )k F d(x , x ) – nτ – F d(x , x ) = –nτ d(xn , xn+ ) k ≤ (.) By taking the limit as n → ∞ in (.) and applying (.) and (.), we have lim n d(xn , xn+ ) n→∞ k = (.) It follows from (.) that there exists n ∈ N such that n d(xn , xn+ ) k ≤ (.) for all n > n This implies d(xn , xn+ ) ≤ n/k (.) for all n > n Now we prove that {xn } is a Cauchy sequence For m > n > n we have m– d(xn , xm ) ≤ m– d(xi , xi+ ) ≤ i=n i=n i/k (.) Since < k < , ∞ i= i/k converges Therefore, d(xn , xm ) → as m, n → ∞ Thus {xn } is a Cauchy sequence Since X is a complete metric space, there exists z ∈ X such that such that xn → z as n → +∞ Now, we prove that z is a fixed point of T If there exists an increasing sequence {nk } ⊂ N such that xnk ∈ Tz for all k ∈ N Since Tz is closed and xn → z as n → +∞, we get z ∈ Tz and the proof is completed So we can assume that there exists n ∈ N such that xn ∈/ Tz for all n ∈ N with n ≥ n This implies that Txn– = Tz for all n ≥ n We first show that d(z, Tx) ≤ d(z, x) for all x ∈ X\{z} Since xn → z, there exists n ∈ N such that d(z, xn ) ≤ d(z, x) for all n ∈ N with n ≥ n Then we have d(xn , Txn ) < d(xn , Txn ) ≤ d(xn , xn+ ) ≤ d(xn , z) + d(z, xn+ ) d(x, z) = d(x, z) – d(x, z) ≤ d(x, z) – d(z, xn ) ≤ d(x, xn ) ≤ Hussain et al Fixed Point Theory and Applications (2015) 2015:78 Page of 13 Thus, by assumption, we get τ + F H(Txn , Tx) ≤ F d(xn , x) (.) Since F is continuous from the right, there exists a real number h > such that F hH(Txn , Tx) < F H(Txn , Tx) + τ Now, from d(xn+ , Tx) ≤ H(Txn , Tx) < hH(Txn , Tx), we obtain F d(xn+ , Tx) ≤ F hH(Txn , Tx) < F H(Txn , Tx) + τ Thus we have τ + F d(xn+ , Tx) ≤ τ + F H(Txn , Tx) + τ ≤ F d(xn , x) + τ Since F is strictly increasing, we have d(xn+ , Tx) < d(xn , x) Letting n tend to +∞, we obtain d(z, Tx) ≤ d(z, x) for all x ∈ X\{z} We next prove that τ + F H(Tz, Tx) ≤ F d(z, x) for all x ∈ X Since F ∈ that , we take x = z Then for every n ∈ N, there exists yn ∈ Tx such d(z, yn ) ≤ d(z, Tx) + d(z, x) n So we have d(x, Tx) ≤ d(x, yn ) ≤ d(x, z) + d(z, yn ) ≤ d(x, z) + d(z, Tx) + d(z, x) n ≤ d(x, z) + d(x, z) + d(z, x) n d(x, z) = + n Hussain et al Fixed Point Theory and Applications (2015) 2015:78 Page 10 of 13 for all n ∈ N and hence d(x, Tx) ≤ d(x, z) Thus by assumption, we get τ + F H(Tz, Tx) ≤ F d(z, x) Thus τ + F d(xn+ , Tz) ≤ τ + F H(Txn , Tz) ≤ F d(xn , z) Since F is strictly increasing, we have d(xn+ , Tz) < d(xn , z) Letting n → ∞, we get d(z, Tz) ≤ Since Tz is closed, we obtain z ∈ Tz Thus z is fixed point of T Applications Fixed point theorems for contractive operators in metric spaces are widely investigated and have found various applications in differential and integral equations (see [, , , ] and references therein) In this section we discuss the existence and uniqueness of solution of a general class of the following Volterra type integral equations under various assumptions on the functions involved Let C[, ] denote the space of all continuous functions on [, ], where > and for an arbitrary x τ = supt∈[, ] {|x(t)|e–τ t }, where τ > is taken arbitrary Note that · τ is a norm equivalent to the supremum norm, and (C([, ], R), · τ ) endowed with the metric dτ defined by dτ (x, y) = sup x(t) – y(t) e–τ t t∈[, ] for all x, y ∈ C([, ], R) is a Banach space; see also [] Consider the integral equation t K t, s, hx(s) ds + g(t), (fy)(t) = (.) where x : [, ] → R is unknown, g : [, ] → R, and h, f : R → R are given functions The kernel K of the integral equation is defined on [, ] × [, ] Theorem Assume that the following conditions are satisfied: (i) K : [, ] × [, ] × R → R, g : [, ] → R and f : R → R are continuous, t (ii) K(t, s, ·) : R → R is increasing, for all t, s ∈ [, ], (iii) there exists τ ∈ (, +∞) such that K t, s, hx(s) – K t, s, hy(s) ≤ τ hx(s) – hy(s) for all t, s ∈ [, ] and hx, hy ∈ R, (iv) if f is injective, then for τ > there exists e–τ ∈ R+ such that for all x, y ∈ R; |hx – hy| ≤ e–τ |fx – fy| Hussain et al Fixed Point Theory and Applications (2015) 2015:78 Page 11 of 13 and {fx : x ∈ C([, ], R)} is complete Then there exists w ∈ C([, ], R) such that for x ∈ C([, ], R) and xn (t) = fxn– (t) t fw(t) = lim fxn (t) = lim g(t) + n→∞ n→∞ K t, s, hxn– (s) ds and w is the unique solution of (.) Proof Let X = Y = C([, ], R) and dτ (x, y) = sup x(t) – y(t) e–τ t t∈[, ] for all x, y ∈ X Let T, R : X → X be defined as follows: t K t, s, hx(s) ds and Rx = fx (Tx)(t) = g(t) + Then by assumptions RX = {Rx : x ∈ X} is complete Let x∗ ∈ TX, then x∗ = Tx for x ∈ X and x∗ (t) = Tx(t) By the assumptions there exists y ∈ X such that Tx(t) = fy(t), hence RX ⊆ TX Since t (Tx)(t) – (Ty)(t) = t K t, s, hx(s) ds – K t, s, hy(s) ds t ≤ K t, s, hx(s) – K t, s, hy(s) ds t ≤ τ hx(s) – hy(s) ds t ≤ τ e–τ fx(s) – fy(s) ds t τ e–τ (Rx)(s) – (Ry)(s) e–τ s eτ s ds = ≤ τ e–τ Rx – Ry t eτ s ds τ ≤ τ e–τ Rx – Ry τ eτ t τ = e–τ Rx – Ry τ eτ t This implies that (Tx)(t) – (Ty)(t) eτ t ≤ e–τ Rx – Ry τ , or equivalently dτ (Tx, Ty) ≤ e–τ dτ (Rx, Ry) Taking logarithms, we have ln dτ (Tx, Ty) ≤ ln e–τ dτ (Rx, Ry) Hussain et al Fixed Point Theory and Applications (2015) 2015:78 Page 12 of 13 After routine calculations, one can easily get τ + ln dτ (Tx, Ty) ≤ ln dτ (Rx, Ry) Now, we observe that the function F : R+ → R defined by F(t) = ln(t) for each t ∈ C([, ], R) and τ > is in Thus all conditions of Theorem are satisfied Hence, there exists a unique w ∈ X such that fw(t) = lim Rxn (t) = lim Txn– (t) = T(w)(t), n→∞ n→∞ x ∈ X, for all t, which is the unique solution of (.) Competing interests The authors declare that they have no competing interests Authors’ contributions All authors contributed equally and significantly in writing this paper All authors read and approved the final paper Author details Department of Mathematics, King Abdulaziz University, P.O Box 80203, Jeddah, 21589, Saudi Arabia Department of Mathematics, COMSATS Institute of Information Technology, Chack Shahzad, Islamabad, 44000, Pakistan Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, Belgrade, 11 000, Serbia 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 Received: 21 November 2014 Accepted: 17 May 2015 References Banach, S: Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales Fundam Math 3, 133-181 (1922) Boyd, DW, Wong, JSW: On nonlinear contractions Proc Am Math Soc 20, 458-464 (1969) Azam, A: Coincidence points of mappings and relations with applications Fixed Point Theory Appl 2012, Article ID 50 (2012) Suzuki, T: A generalized Banach contraction principle that characterizes metric completeness Proc Am Math Soc 136, 1861-1869 (2008) Kirk, WA: Fixed point theory for non-expansive mappings II Contemp Math 18, 121-140 (1983) Murthy, PP: Important tools and possible applications of metric fixed point theory Nonlinear Anal 47, 3479-3490 (2001) Rhoades, BE: A comparison of various definitions of contractive mappings Trans Am Math Soc 26, 257-290 (1977) Rhoades, BE: Contractive definitions and continuity Contemp Math 12, 233-245 (1988) Agarwal, RP, Hussain, N, Taoudi, MA: Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations Abstr Appl Anal 2012, Article ID 245872 (2012) 10 Nadler, J: Multivalued contraction mappings Pac J Math 30, 475-478 (1969) 11 Ansari, QH, Idzik, A, Yao, JC: Coincidence and fixed point theorems with applications Topol Methods Nonlinear Anal 15(1), 191-202 (2000) 12 Wardowski, D: Fixed points of a new type of contractive mappings in complete metric spaces Fixed Point Theory Appl 2012, Article ID 94 (2012) 13 Abbas, M, Ali, B, Romaguera, S: Fixed and periodic points of generalized contractions in metric spaces Fixed Point Theory Appl 2013, Article ID 243 (2013) 14 Hussain, N, Salimi, P: Suzuki-Wardowski type fixed point theorems for α-GF-contractions Taiwan J Math 18, 1879-1895 (2014) 15 Hussain, N, Aziz-Taoudi, M: Krasnosel’skii-type fixed point theorems with applications to Volterra integral equations Fixed Point Theory Appl 2013, Article ID 196 (2013) 16 Acar, Ö, Altun, I: A fixed point theorem for multivalued mappings with δ -distance Abstr Appl Anal 2014, Article ID 497092 (2014) 17 Suzuki, T: A new type of fixed point theorem in metric spaces Nonlinear Anal 71(11), 5313-5317 (2009) 18 Piri, H, Kumam, P: Some fixed point theorems concerning F-contraction in complete metric spaces Fixed Point Theory Appl 2014, Article ID 210 (2014) 19 Sgroi, M, Vetro, C: Multi-valued F-contractions and the solution of certain functional and integral equations Filomat 27(7), 1259-1268 (2013) 20 Secelean, NA: Iterated function systems consisting of F-contractions Fixed Point Theory Appl 2013, Article ID 277 (2013) doi:10.1186/1687-1812-2013-277 Hussain et al Fixed Point Theory and Applications (2015) 2015:78 Page 13 of 13 21 Klim, D, Wardowski, D: Fixed points of dynamic processes of set-valued F-contractions and application to functional equations Fixed Point Theory Appl 2015, Article ID 22 (2015) 22 Hussain, N, Khan, AR, Agarwal, RP: Krasnosel’skii and Ky Fan type fixed point theorems in ordered Banach spaces J Nonlinear Convex Anal 11(3), 475-489 (2010) 23 Hussain, N, Kutbi, MA, Salimi, P: Fixed point theory in α -complete metric spaces with applications Abstr Appl Anal 2014, Article ID 280817 (2014) ... fixed point theorems for α-GF -contractions Taiwan J Math 18, 1879-1895 (2014) 15 Hussain, N, Aziz-Taoudi, M: Krasnosel’skii-type fixed point theorems with applications to Volterra integral equations. .. is fixed point of T Applications Fixed point theorems for contractive operators in metric spaces are widely investigated and have found various applications in differential and integral equations. .. α-GF -contractions and established Suzuki-Wardowski type fixed point theorems for such contractions For more details on F -contractions, we refer the reader to [, –] In this paper, we obtain coincidence
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