A fixed point theorem for F-Khan-contractions on complete metric spaces and application to integral equations

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1 Available online at J. Nonlinear Sci. Appl., (27), Research Article Journal Homepage: A fixed point theorem for F-Khan-contractions on complete metric spaces and application to integral equations Hossein Piri a,, Samira Rahrovi a, Hamidreza Marasi b, Poom Kumam c,e,d, a Department of mathematics, University of Bonab, Bonab , Iran. b Department of applied mathematics, Faculty of mathematical sciences, University of Tabriz, Tabriz, Iran. c KMUTT Fixed Point Research Laboratory, Department of Mathematics, Room SCL 82 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut s University of Technology Thonburi (KMUTT), 26 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 4, Thailand. d KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkuts University of Technology Thonburi (KMUTT), 26 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 4, Thailand. e Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 442, Taiwan. Communicated by M. Imdad Abstract In this article, we introduce a new concept of contraction called F-Khan-contractions and prove a fixed point theorem concerning this contraction which generalizes the results announced by Khan [M. S. Khan, Rend. Inst. Math. Univ. Trieste., 8 (976), 69 72], Fisher [B. Fisher, Riv. Math. Univ. Parma., 4 (978), 35 37], and Piri et al. [H. Piri, S. Rahrovi, P. Kumam, J. Math. Computer Sci., 7 (27), 76 83]. An example and application for the solution of certain integral equations are given to illustrate the usability of the obtained results. c 27 All rights reserved. Keywords: Fixed point, metric space, integral equations. 2 MSC: 74H, 54H25.. Introduction The Banach contraction principle is one of the most fundamental and important results in modern mathematics which is widely applied in many other branches of science and applied science. The Banach contraction principle provides a constructive method of finding a unique solution for models involving various types of differential and integral equations. This principle is generalized by several authors in various directions (see [, 6 8] and references therein). In recent years an interesting but different generalization of Banach-contraction theorem has been given by Wardowski [9]. He introduced a new contraction called F-contraction and established a fixed Corresponding authors addresses: h.piri@bonabu.ac.ir (Hossein Piri), s.rahrovi@bonabu.ac.ir (Samira Rahrovi), Hamidreza.marasi@gmail.com (Hamidreza Marasi), poom.kumam@mail.kmutt.ac.th (Poom Kumam) doi:.22436/jnsa..9.2 Received

2 H. Piri, S. Rahrovi, H. Marasi, P. Kumam, J. Nonlinear Sci. Appl., (27), point result as a generalization of the Banach contraction principle in a dissimilar way than in the other acknowledged results from the literature. On the other hand, some generalizations of Banach contraction principle are obtained by contraction conditions containing rational expressions. In this direction, in 973, Geraghty [5] introduced a contraction in which the contraction constant was replaced by a function having some specific properties. Since then, several papers which dealt with fixed point theory for rational Geraghty contractive mappings have appeared (see, e.g., [2, ] and references therein). One of the well-known works in this direction was established by Khan [6] and revised by Fisher [4] as follows. Theorem. ([4]). Let (X, d) be a complete metric space and T : X X satisfy { d(x,t x)d(x,t y)+d(y,t y)d(y,t x) k d(tx, Ty) d(x,t y)+d(t x,y), if d(x, Ty) + d(tx, y),, if d(x, Ty) + d(tx, y) =, (.) where k [, ) and x, y X. Then T has a unique fixed point x X. Moreover, for all x X, the sequence {T n x} n N converges to x. Recently, Piri et al., [8] extended the results of Khan [6] and Fisher [4] by introducing a new general contractive condition with rational expressions as follows: Theorem.2 ([8]). Let (X, d) be a complete metric space and T : X X satisfy { d(x,t x)d(x,t y)+d(y,t y)d(y,t x) k d(tx, Ty) max{d(x,t y),d(t x,y)}, if max{d(x, T y), d(t x, y)},, if max{d(x, Ty), d(tx, y)} = (.2) for some k [, ) and x, y X. Then T has a unique fixed point x X. Moreover, for all x X, the sequence {T n x} n N converges to x. Also by providing some examples, Piri et al. [8] showed that their results are a proper generalization of Fisher [4] and Khan [6]. Following this direction of research, in the present article, we will present some fixed point results of F-Khan-type self-mappings on complete metric spaces. Moreover, an example and application for the solution of certain integral equations are given to illustrate the usability of the obtained results. 2. Preliminaries Definition 2. ([9]). Let F be the family of all functions F: (, ) R such that (F) F is strictly increasing, i.e. for all x, y (, ) such that x < y, F(x) < F(y); (F2) for each sequence {α n } n= of positive numbers, lim α n = if and only if lim F(α n ) = ; (F3) there exists k (, ) such that lim α + α k F(α) =. Definition 2.2 ([9]). Let (X, d) be a metric space. A mapping T : X X is said to be an F-contraction on (X, d), if there exist F F and τ (, ) such that x, y X, [d(tx, Ty) > τ + F(d(Tx, Ty)) F(d(x, y))]. Wardowski [9] stated a modified version of Banach contraction principle as follows. Theorem 2.3 ([9]). Let (X, d) be a complete metric space and T : X X an F-contraction. Then T has a unique fixed point x X and for every x X the sequence {T n x} n N converges to x.

3 H. Piri, S. Rahrovi, H. Marasi, P. Kumam, J. Nonlinear Sci. Appl., (27), Definition 2.4. Let F K be the family of all increasing functions F: (, ) R, i.e. for all x, y (, ), if x < y, then F(x) F(y). Definition 2.5. Let (X, d) be a metric space. A mapping T : X X is said to be an F-Khan-contraction if there exists τ (, ) and F F K such that for all x, y X if max{d(x, Ty), d(tx, y)}, then Tx Ty and d(x, Tx)d(x, Ty) + d(y, Ty)d(y, Tx) τ + F(d(Tx, Ty)) F, (2.) max{d(x, Ty), d(tx, y)} and if max{d(x, Ty), d(tx, y)} =, then Tx = Ty. Example 2.6. Let F (α) = ln(α), α >. Obviously F F K and for F -Khan-contraction T, for all x, y X such that max{d(x, Ty), d(tx, y)}, the following condition holds: τ d(x, Tx)d(x, Ty) + d(y, Ty)d(y, Tx) d(tx, Ty) e. (2.2) max{d(x, Ty), d(tx, y)} Example 2.7. Let F 2 (α) = α, α >. Obviously F 2 F K and for F 2 -Khan-contraction T, for all x, y X such that max{d(x, Ty), d(tx, y)}, the following condition holds: d(tx, Ty) d(x, Tx)d(x, Ty) + d(y, Ty)d(y, Tx) τ [d(x, Tx)d(x, Ty) + d(y, Ty)d(y, Tx)] + max{d(x, Ty), d(tx, y)}. (2.3) Example 2.8. Let F 3 : (, ) R be given by the formula F 3 (α) = [α], where [α] denotes the integer part of α. Obviously F 3 F K and for F 3 -Khan-contraction T, for all x, y X such that max{d(x, Ty), d(tx, y)}, the following condition holds: τ + d(tx, Ty) + d(x, Tx)d(x, Ty) + d(y, Ty)d(y, Tx). (2.4) max{d(x, Ty), d(tx, y)} Example 2.9. Let X = {,, 2, 3} and d(x, y) = x y for all x, y X. Then (X, d) is a complete metric space. Let T : X X be defined by Now we consider the following cases: Case. Let x = and y =, then Case 2. Let x = and y = 2, then Case 3. Let x = and y = 3, then T(2) = T() =, T() = 2, T(3) =. d(tx, Ty) = d(2, ) =, d(x, Tx)d(x, Ty) + d(y, Ty)d(y, Tx) = d(, 2)d(, ) + d(, )d(, 2) = 2, max{d(x, Ty), d(tx, y)} = max{d(, ), d(2, )} =. d(tx, Ty) = d(2, ) =, d(x, Tx)d(x, Ty) + d(y, Ty)d(y, Tx) = d(, 2)d(, ) + d(2, )d(2, 2) = 2, max{d(x, Ty), d(tx, y)} = max{d(, ), d(2, 2)} =. d(tx, Ty) = d(2, ) = 2, d(x, Tx)d(x, Ty) + d(y, Ty)d(y, Tx) = d(, 2)d(, ) + d(3, )d(3, 2) = 3, max{d(x, Ty), d(tx, y)} = max{d(, ), d(2, 3)} =.

4 H. Piri, S. Rahrovi, H. Marasi, P. Kumam, J. Nonlinear Sci. Appl., (27), Case 4. Let x = and y = 2, then d(tx, Ty) = d(, ) =, d(x, Tx)d(x, Ty) + d(y, Ty)d(y, Tx) = d(, )d(, ) + d(2, )d(2, ) =, max{d(x, Ty), d(tx, y)} = max{d(, ), d(2, )} =. Case 5. Let x = and y = 3, then d(tx, Ty) = d(, ) =, d(x, Tx)d(x, Ty) + d(y, Ty)d(y, Tx) = d(, )d(, ) + d(3, )d(3, ) = 6, max{d(x, Ty), d(tx, y)} = max{d(, ), d(, 3)} = 2. Case 6. Let x = 2 and y = 3, then d(tx, Ty) = d(, ) =, d(x, Tx)d(x, Ty) + d(y, Ty)d(y, Tx) = d(2, )d(2, ) + d(3, )d(3, 2) = 5, max{d(x, Ty), d(tx, y)} = max{d(2, ), d(, 3)} = 2. For τ (, ln 3 2 ], we have 3e τ 2 and 2e τ. Therefore, T satisfies in condition (2.2). For τ (, 6 ], we have 2 3 2τ+ and 3τ+ 2. Therefore, T satisfies in condition (2.3). For τ (, 2], T satisfies in condition (2.4). 3. Main results Our main theorem is essentially inspired by Khan [6], Fisher [4], Wardowski [9], and Piri et al. [8]. More precisely, we state and prove the following result. Theorem 3.. Let (X, d) be a complete metric space and T : X X be an F-Khan-contraction. Then, T has a unique fixed point x X and for every x X the sequence {T n x} n N converges to x. Proof. Let x = x X. Put x n+ = Tx n = T n+ x for all n =,, 2,. If there exists n N such that x n = x n, then x n is a fixed point of T. This completes the proof. Therefore, we suppose x n x n for all n N. We shall divide the proof into two cases. Cases. Assume that d(x n, Tx n ), for all n N. Then, from (2.) we have F(d(x n, Tx n )) < τ + F(d(Tx n, Tx n )) d(xn, Tx n )d(x n, Tx n ) + d(x n, Tx n )d(x n, Tx n ) F max{d(x n, Tx n ), d(tx n, x n )} = F(d(x n, Tx n )). (3.) Since F F K, so from (3.) we have d(x n, Tx n ) < d(x n, Tx n ), n N. Therefore {d(x n, Tx n )} n N is a strictly decreasing sequence of nonnegative real numbers, and hence lim d(x n, Tx n ) = γ. Since {d(x n, Tx n )} n N is a nonnegative strictly decreasing sequence, so for every n N, we have d(x n, Tx n ) γ. (3.2)

5 H. Piri, S. Rahrovi, H. Marasi, P. Kumam, J. Nonlinear Sci. Appl., (27), Now, we claim that γ =. Arguing by contradiction, we assume that γ >. From (3.2) and F F K, we get F(γ) F(d(x n, Tx n )) F(d(x n, Tx n )) τ F(d(x n 2, Tx n 2 )) 2τ F(d(x, Tx )) nτ (3.3) for all n N. Since F(γ) R and lim [F(d(x, Tx )) nτ] =, so there exists n N such that It follows from (3.3) and (3.4) that It is a contradiction. Therefore, we have F(d(x, Tx )) nτ < F(γ), n > n. (3.4) F(γ) F(d(x, Tx )) nτ < F(γ), n > n. lim d(x n, Tx n ) =. (3.5) Now, we claim that, {x n } n= is a Cauchy sequence. Arguing by contradiction, we assume that there exist ɛ >, the sequences {p(n)} n= and {q(n)} n= of natural numbers such that p(n) > q(n) > n, d(x p(n), x q(n) ) ɛ, d(x p(n), x q(n) ) < ɛ, n N. (3.6) By triangular inequality, we have It follows from (3.5) and (3.6) that d(x p(n), x q(n) ) d(x p(n), Tx q(n) ) + d(tx q(n), x q(n) ). ɛ lim inf d(x p(n), Tx q(n) ). So, there exists n 2 N such that for all n n 2, d(x p(n), Tx q(n) ) > ɛ 2. Therefore Again by triangular inequality, we have > ɛ 2, n n 2. (3.7) d(x p(n), x q(n) ) d(x p(n), Tx p(n) ) + d(tx p(n), Tx q(n) ) + d(tx q(n), x q(n) ). It follows from (3.5) and (3.6) that So, there exists n 3 N such that for all n n 3, ɛ lim inf d(tx p(n), Tx q(n) ). d(tx p(n), Tx q(n) ) > ɛ 2. (3.8) Since F F K, so from (2.), (3.7), and (3.8), for all n max{n 2, n 3 }, we have τ + F( ɛ 2 ) τ + F(d(Tx p(n), Tx q(n) )) d(xp(n), Tx p(n) )d(x p(n), Tx q(n) ) + d(x q(n), Tx q(n) )d(x q(n), Tx p(n) ) F. (3.9)

6 H. Piri, S. Rahrovi, H. Marasi, P. Kumam, J. Nonlinear Sci. Appl., (27), From (3.7), for n n 2, we have d(x p(n), Tx p(n) )d(x p(n), Tx q(n) ) + d(x q(n), Tx q(n) )d(x q(n), Tx p(n) ) = d(x p(n), Tx p(n) )d(x p(n), Tx q(n) ) + d(x q(n), Tx q(n) )d(x q(n), Tx p(n) ) d(x p(n), Tx p(n) )d(x p(n), Tx q(n) ) d(x p(n), Tx q(n) ) = d(x p(n), Tx p(n) ) + d(x q(n), Tx q(n) ). + d(x q(n), Tx q(n) )d(x q(n), Tx p(n) ) d(tx p(n), x q(n) ) (3.) It follows from (3.5), (3.), and sandwich theorem that d(x p(n), Tx p(n) )d(x p(n), Tx q(n) ) + d(x q(n), Tx q(n) )d(x q(n), Tx p(n) ) lim =. So there exists n 4 N such that for all n > n 4, d(x p(n), Tx p(n) )d(x p(n), Tx q(n) ) + d(x q(n), Tx q(n) )d(x q(n), Tx p(n) ) < ɛ 2. Since F F K, so for all n > n 4, we have d(xp(n), Tx p(n) )d(x p(n), Tx q(n) ) + d(x q(n), Tx q(n) )d(x q(n), Tx p(n) ) F From (3.9) and (3.), for all n max{n 2, n 3, n 4 }, we obtain that F( ɛ ). (3.) 2 τ + F( ɛ d(xp(n) 2 ) F, Tx p(n) )d(x p(n), Tx q(n) ) + d(x q(n), Tx q(n) )d(x q(n), Tx p(n) ) F( ɛ 2 ). This contradiction shows that {x n } is a Cauchy sequence. By Completeness of (X, d), {x n } converges to some point x in X. Therefore lim d(x n, x ) = and lim d(tx n, Tx ) = d(x, Tx ). (3.2) Now, we claim that d(x, Tx ) =. By contradiction, we assume that d(x, Tx ) >. We only have the following two cases (I) n N, i n N, i n > i n, i = and x in + = Tx ; (II) N N, n N, d(x n, Tx ) >. In the first case, from (3.2) we have In the second case, for all n N, we have x = lim x i n+ = Tx. max{d(x n, Tx ), d(tx n, x )} >. So from (2.), we get ( τ + F(d(Tx n, Tx d(xn, Tx n )d(x n, Tx ) + d(x, Tx )d(x ), Tx n ) )) F max{d(x n, Tx ), d(tx n, x. (3.3) )}

7 H. Piri, S. Rahrovi, H. Marasi, P. Kumam, J. Nonlinear Sci. Appl., (27), On the other hand, from (3.5) and (3.2) we have d(x n, Tx n )d(x n, Tx ) + d(x, Tx )d(x, Tx n ) lim max{d(x n, Tx ), d(tx n, x =. )} Since d(x, Tx ) >, so there exists n 5 N such that for all n n 5, d(x n, Tx n )d(x n, Tx ) + d(x, Tx )d(x, Tx n ) max{d(x n, Tx ), d(tx n, x )} < 2 d(x, Tx ). Therefore ( d(xn, Tx n )d(x n, Tx ) + d(x, Tx )d(x ), Tx n ) F max{d(x n, Tx ), d(tx n, x )} F 2 d(x, Tx ), n n 5. (3.4) It follows from (3.3) and (3.4) that τ + F(d(Tx n, Tx )) F 2 d(x, Tx ), n n 5. So, we get and from (3.2), we obtain d(tx n, Tx ) 2 d(x, Tx ), n n 5, d(x, Tx ) 2 d(x, Tx ). This is contradiction. So, we have x = Tx. Now, we show that T has a unique fixed point. For this, we assume that y is another fixed point of T in X such that d(x, y ) >. Therefore max{d(x, Ty ), d(tx, y )} >. So from (2.), we get ( d(x F(d(x, y )) = F(d(Tx, Ty )) < τ + F(d(Tx, Ty, Tx )d(x, Ty ) + d(y, Ty )d(y, Tx ) ) )) F max{d(x, Ty ), d(tx, y. )} Since d(x, Tx )d(x, Ty ) + d(y, Ty )d(y, Tx ) max{d(x, Ty ), d(tx, y )} this leads to a contradiction and hence x = y. This completes the proof. Cases 2. Assume that there exists m N such that d(x m, Tx m ) =. By assumption of theorem, we have d(tx m, Tx m ) = and hence x m = Tx m. This completes the proof of the existence of a fixed point of T. The uniqueness follows as in Case. Remark 3.2. Theorem 3. is the generalization of Theorem.2, since for the mapping F of the form in Example 2.6, an F-Khan contraction mapping becomes the contraction explained in Theorem.2. Theorem 3.3. Let (X, d) be a complete metric space and T n : X X be an F-Khan-contraction for some n N. Then, T has a unique fixed point x X and for every x X, the sequence {T n x } n N converges to x. Proof. By Theorem 3., T n has a unique fixed point x X. Then, we have T n (Tx ) = T(T n x ) = Tx and so Tx is a fixed point of T n. Therefore, by uniqueness of the fixed point of T n it must be that Tx = x. =,

8 H. Piri, S. Rahrovi, H. Marasi, P. Kumam, J. Nonlinear Sci. Appl., (27), Example 3.4. Consider the sequence {S n } n N as follows: S = 2, S 2 = ,, S n = n(n + ) = n(n + )(n + 2),. 3 Let X = {S n : n N} and d : X X [, ) be defined by d(x, y) = max{x, y}, if x y and d(x, y) =, if x = y. Then (X, d) is complete metric space. Define the mapping T : X X by T(S ) = S and T(S n ) = S n for every n >. Since lim d(ts n, TS ) d(s n,t S n )d(s n,t S )+d(s,t S )d(s,t S n ) max{d(s n,t S ),d(s,t S n )} = lim = lim d(s n, S ) d(s n,s n )d(s n,s )+d(s,s )d(s,s n ) max{d(s n,s ),d(s,s n )} S n S n S n + S n n = lim n + 2 =. Therefore T is not satisfies in assumption of Theorem.2 and since (.) implies (.2), so T is not satisfies in assumption of Theorem.. On the other hand, taking F 2 : (, ) R, F 2 (α) = ln(α) + α, we obtain that T is an F 2 -Khan-contraction with τ = ln 3. calculation. First observe that To see this, let us consider the following For every m N, m > 2, we have max{d(x n, Tx m ), d(tx n, x m )} [(m > 2 n = ) (m > n > )]. d(ts m, TS ) d(s m,t S m )d(s m,t S )+d(s,t S )d(s,t S m ) max{d(s m,t S ),d(s,t S m )} = S m S m e S m S m = m m + 2 e 3 (3m 2 +3m) < e 3. For every m, n N, m > n >, we have d(ts m, TS n ) d(s m,t S m )d(s m,t S n )+d(s n,t S n )d(s n,ts m ) max{d(s m,t S n ),d(s n,t S m )} = 4. Application S m S m e S SnS m +SmSm m Sm S n S m + S m S m S m S m e S SnS m +SmSm m Sm (S n + S m )S m e d(t S m,t S ) d(sm,t Sm)d(Sm,T S )+d(s,t S )d(s,t Sm) max{d(sm,t S ),d(s,t Sm)} e d(t (S d(sm,t Sm)d(Sm,T Sn)+d(Sn,T Sn)d(Sn,T Sm) m)),t (S n )) max{d(sm,t Sn),d(Sn,T Sm)} S m e SnS m Sm < e 3. S n + S m In this section, we present an application where Theorem 3. can be applied. This application is inspired by [3]. Let X = C([, ]) be the set of all real continuous functions on [, ]. Clearly, (X, d) is a complete metric space, where d is defined by d(f, g) = f g = max f(t) g(t), f, g X. t [,] Let Y = {f X : f(t) 8, t [, ] or f(t) =, t [, ] },

9 H. Piri, S. Rahrovi, H. Marasi, P. Kumam, J. Nonlinear Sci. Appl., (27), and G : [, ] [, ] Y X be defined by { G(t, s, f(r)) = 2, f(r) 8, 4, f(r) = for all r, s, t [, ] and f Y. Obviously Y is complete metric space and G(s, r, f(r)) is integrable with respect to r on [, ] for all s [, ]. Let T be defined on Y by Tf(s) = G(s, r, f(r))dr for all s [, ]. We have { G(s, r, f(r))dr = Tf(s) = G(s, r, f(r))dr = 4 dr = 4 2 dr = 2, f(r) 8,, f(r) =. This proves that Tf Y for all f Y. For all r, s [, ] and f, g Y, we have and {, f(r) = g(r) = or f(r), g(r) G(s, r, f(r)) G(s, r, g(r)) = 8, 4, otherwise, (4.) { f(r) f(r) Tf(r) = 2, f(r) 8, 4 = 3 4, f(r) =, and f(r) Tg(r) = f(r) 2, f(r), g(r) 8, f(r) 4, f(r) 8, g(r) = 2 = 2, f(r) =, g(r) 8 4 = 3 4, f(r) = g(r) =. According symmetry the above relations are established for g(r) T g(r) and g(r) T f(r). Obviously f(r) 8 implies that 3 8 f(r) 2 2 and 8 f(r) 4 4. Therefore Observe that 8 4 f(r) Tf(r) f(r) Tg(r) + g(r) Tg(r) g(r) Tf(r) 4[ f(r) Tf(r) f(r) Tg(r) + g(r) Tg(r) g(r) Tf(r) ] 4[ ] = 3 8. (4.2) From (4.) and (4.2) for all k [ 2 3, ), we have G(s, r, f(r)) G(s, r, g(r)) k Now, we prove that the integral equation has a unique solution f Y. f(r) Tf(r) f(r) Tg(r) + g(r) Tg(r) g(r) Tf(r). (4.3) f(s) = G(s, r, f(r))dr (4.4)

10 H. Piri, S. Rahrovi, H. Marasi, P. Kumam, J. Nonlinear Sci. Appl., (27), For all f, g Y and s [, ] from (4.3), we have Tf(s) Tg(s) = G(s, r, f(r))dr So, for all f, g Y, we have = k G(s, r, g(r))dr G(s, r, f(r))dr G(s, r, g(r)) dr k f(r) Tf(r) f(r) Tg(r) + g(r) Tg(r) g(r) Tf(r) dr f Tf f Tg + g Tg g Tf k dr f Tf f Tg + g Tg g Tf. Tf Tg k Consequently, by passing to logarithms, one can obtain ln k + ln d(tf, Tg) ln f Tf f Tg + g Tg g Tf. d(f, Tf)d(f, Tg) + d(g, Tg)d(g, Tf). max{d(f, Tg), d(g, Tf)} Consequently, all the conditions of Theorem 3. are satisfied by operator T with the function F defined as in Example 2.6 and τ = ln k. Therefore T has a fixed point which is the solution of the integral equation (4.4). Acknowledgment The authors are very thankful to the anonymous reviewers for their careful reading of the manuscript and for their constructive reports, which have been very useful to improve the paper. This project was supported by the Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Cluster (CLASSIC), Faculty of Science, KMUTT. References [] M. Cosentino, P. Vetro, Fixed point results for F-contractive mappings of Hardy-Rogers-type, Filomat, 28 (24), [2] D. Dukić, Z. Kadelburg, S. Radenović, Fixed points of Geraghty-type mappings in various generalized metric spaces., Abstr. Appl. Anal., 2 (2), 3 pages. [3] N. V. Dung, V. T. L. Hang, A Fixed Point Theorem for Generalized F-Contractions on Complete Metric Spaces, Vietnam J. Math,. 43 (25), [4] B. Fisher, On a theorem of Khan, Riv. Math. Univ. Parma., 4 (978), ,.,,, 3 [5] M. A. Geraghty, On contractive maps., Proc. of Amer. Math. Soc., 4 (973), [6] M. S. Khan, A fixed point theorem for metric spaces, Rend. Inst. Math. Univ. Trieste., 8 (976), ,,, 3 [7] P. Kumam, D. Gopal, L. Budhia, A new fixed point theorem under Suzuki type Z-contraction mappings, J. Math. Anal., 8 (27), 3 9. [8] H. Piri, S. Rahrovi, P. Kumam, Generalization of Khan fixed point theorem, J. Math. Computer Sci., 7 (27), ,,.2,, 3 [9] D. Wardowski, Fixed point theory of a new type of contractive mappings in complete metric spaces, Fixed PoinTheory Appl., 22 (22), 6 pages., 2., 2.2, 2, 2.3, 3 [] F. Zabihi, A. Razani, Fixed point theorems for hybrid rational Geraghty contractive mappings in ordered b-metric spaces, J. Math. Appl., 24 (24), 9 pages.