Nashine et al Fixed Point Theory and Applications 2012, 2012:217 http://www.fixedpointtheoryandapplications.com/content/2012/1/217 RESEARCH Open Access Cyclic generalized contractions and fixed point results with applications to an integral equation Hemant Kumar Nashine1 , Wutiphol Sintunavarat2 and Poom Kumam2* * Correspondence: poom.kum@kmutt.ac.th Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, 10140, Thailand Full list of author information is available at the end of the article Abstract We set up a new variant of cyclic generalized contractive mappings for a map in a metric space and present existence and uniqueness results of fixed points for such mappings Our results generalize or improve many existing fixed point theorems in the literature To illustrate our results, we give some examples At the same time as applications of the presented theorems, we prove an existence theorem for solutions of a class of nonlinear integral equations MSC: 47H10; 54H25 Keywords: fixed point; cyclic generalized (F, ψ , L)-contraction; integral equation Introduction and preliminaries All the way through this paper, by R+ , we designate the set of all real nonnegative numbers, while N is the set of all natural numbers The celebrated Banach’s [] contraction mapping principle is one of the cornerstones in the development of nonlinear analysis This principle has been extended and improved in many ways over the years (see, e.g., [–]) Fixed point theorems have applications not only in various branches of mathematics but also in economics, chemistry, biology, computer science, engineering, and other fields In particular, such theorems are used to demonstrate the existence and uniqueness of a solution of differential equations, integral equations, functional equations, partial differential equations, and others Owing to the magnitude, generalizations of the Banach fixed point theorem have been explored heavily by many authors This celebrated theorem can be stated as follows Theorem . ([]) Let (X, d) be a complete metric space and T be a mapping of X into itself satisfying d(Tx, Ty) ≤ kd(x, y), ∀x, y ∈ X, () where k is a constant in (, ) Then T has a unique fixed point x* ∈ X Inequality () implies the continuity of T A natural question is whether we can find contractive conditions which will imply the existence of a fixed point in a complete metric space but will not imply continuity © 2012 Nashine et al.; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Nashine et al Fixed Point Theory and Applications 2012, 2012:217 http://www.fixedpointtheoryandapplications.com/content/2012/1/217 Page of 13 On the other hand, cyclic representations and cyclic contractions were introduced by Kirk et al [] A mapping T : A ∪ B → A ∪ B is called cyclic if T(A) ⊆ B and T(B) ⊆ A, where A, B are nonempty subsets of a metric space (X, d) Moreover, T is called a cyclic contraction if there exists k ∈ (, ) such that d(Tx, Ty) ≤ kd(x, y) for all x ∈ A and y ∈ B Notice that although a contraction is continuous, a cyclic contraction need not to be This is one of the important gains of this theorem Definition . (See [, ]) Let (X, d) be a metric space Let p be a positive integer, p A , A , , Ap be nonempty subsets of X, Y = i= Ai , and T : Y → Y Then Y is said to be a cyclic representation of Y with respect to T if (i) Ai , i = , , , p are nonempty closed sets, and (ii) T(A ) ⊆ A , , T(Ap– ) ⊆ Ap , T(Ap ) ⊆ A Following the paper in [], a number of fixed point theorems on a cyclic representation of Y with respect to a self-mapping T have appeared (see, e.g., [, –]) In this paper, we introduce a new class of cyclic generalized (F , ψ, L)-contractive mappings, and then investigate the existence and uniqueness of fixed points for such mappings Our main result generalizes and improves many existing theorems in the literature We supply appropriate examples to make obvious the validity of the propositions of our results To end with, as applications of the presented theorems, we achieve fixed point results for a generalized contraction of integral type and we prove an existence theorem for solutions of a system of integral equations Main results In this section, we introduce two new notions of a cyclic contraction and establish new results for such mappings In the sequel, we fixed the set of functions by F , ψ : [, +∞) → [, +∞) such that (i) F is nondecreasing, continuous, and F () = < F (t) for every t > ; (ii) ψ is nondecreasing, right continuous, and ψ(t) < t for every t > Define F = {F : F satisfies (i)} and = {ψ : ψ satisfies (ii)} We state the notion of a cyclic generalized (F , ψ, L)-contraction as follows Definition . Let (X, d) be a metric space Let p be a positive integer, A , A , , Ap be p nonempty subsets of X and Y = i= Ai An operator T : Y → Y is said to be a cyclic generalized (F , ψ, L)-contraction for some ψ ∈ , F ∈ F , and L ≥ if p (a) Y = i= Ai is a cyclic representation of Y with respect to T; (b) for any (x, y) ∈ Ai × Ai+ , i = , , , p (with Ap+ = A ), F d(Tx, Ty) ≤ ψ F (x, y) + LF (x, y) , where (x, y) = max d(x, y), d(x, Tx), d(y, Ty), d(x, Ty) + d(y, Tx) and (x, y) = d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx) Our first main result is the following Nashine et al Fixed Point Theory and Applications 2012, 2012:217 http://www.fixedpointtheoryandapplications.com/content/2012/1/217 Page of 13 Theorem . Let (X, d) be a complete metric space, p ∈ N, A , A , , Ap be nonempty p closed subsets of X, and Y = i= Ai Suppose T : Y → Y is a cyclic generalized (F , ψ, L)contraction mapping for some ψ ∈ and F ∈ F Then T has a unique fixed point Morep over, the fixed point of T belongs to i= Ai Proof Let x ∈ A (such a point exists since A = ∅) Define the sequence {xn } in X by xn+ = Txn , n = , , , We shall prove that lim d(xn , xn+ ) = () n→∞ If, for some k, we have xk+ = xk , then () follows immediately So, we can suppose that d(xn , xn+ ) > for all n From the condition (a), we observe that for all n, there exists i = i(n) ∈ {, , , p} such that (xn , xn+ ) ∈ Ai × Ai+ Then, from the condition (b), we have F d(xn , xn+ ) ≤ ψ F (xn– , xn ) + LF (xn– , xn ) , n = , , () On the other hand, we have (xn– , xn ) = max d(xn– , xn ), d(xn+ , xn ), d(xn– , xn+ ) = max d(xn– , xn ), d(xn , xn+ ) and (xn– , xn ) = d(xn– , xn ), d(xn , xn+ ), d(xn– , xn+ ), d(xn , xn ) = Suppose that max{d(xk– , xk ), d(xk , xk+ )} = d(xk , xk+ ) for some k ∈ N Then d(xk , xk+ ), so (xk– , xk ) = F d(xk , xk+ ) ≤ ψ F d(xk , xk+ ) < F d(xk , xk+ ) , a contradiction Hence, (xn– , xn ) = d(xn– , xn ), and thus F d(xn , xn+ ) ≤ ψ F d(xn– , xn ) < F d(xn– , xn ) () Similarly, we have F d(xn– , xn ) < F d(xn– , xn– ) () Nashine et al Fixed Point Theory and Applications 2012, 2012:217 http://www.fixedpointtheoryandapplications.com/content/2012/1/217 Page of 13 Thus, from () and (), we get F d(xn+ , xn ) < F d(xn , xn– ) for all n ∈ N Now, from F d(xn+ , xn ) ≤ ψ F d(xn , xn– ) < · · · < ψ n F d(x , x ) and the property of ψ, we obtain limn→∞ F (d(xn+ , xn )) = , and consequently () holds Now, we shall prove that {xn } is a Cauchy sequence in (X, d) Suppose, on the contrary, that {xn } is not a Cauchy sequence Then there exists ε > for which we can find two sequences of positive integers {m(k)} and {n(k)} such that for all positive integers k, m(k) > n(k) ≥ k, d(xm(k) , xn(k) ) ≥ ε () Further, corresponding to n(k), we can choose m(k) in such a way that it is the smallest integer with m(k) > n(k) ≥ k satisfying () Then we have d(xm(k)– , xn(k) ) < ε () Using (), (), and the triangular inequality, we get ε ≤ d(xn(k) , xm(k) ) ≤ d(xn(k) , xm(k)– ) + d(xm(k)– , xm(k) ) < ε + d(xm(k)– , xm(k) ) Thus, we have ε ≤ d(xn(k) , xm(k) ) < ε + d(xm(k)– , xm(k) ) Passing to the limit as k → ∞ in the above inequality and using (), we obtain lim d(xn(k) , xm(k) ) = ε+ () k→∞ On the other hand, for all k, there exists j(k) ∈ {, , p} such that n(k) – m(k) + j(k) ≡ [p] Then xm(k)–j(k) (for k large enough, m(k) > j(k)) and xn(k) lie in different adjacently labeled sets Ai and Ai+ for certain i ∈ {, , p} Using (b), we obtain F d(xm(k)–j(k)+ , xn(k)+ ) ≤ ψ F (xm(k)–j(k) , xn(k) ) + LF (xm(k)–j(k) , xn(k) ) () for all k Now, we have (xm(k)–j(k) , xn(k) ) = max d(xm(k)–j(k) , xn(k) ), d(xm(k)–j(k)+ , xm(k)–j(k) ), d(xn(k)+ , xn(k) ), d(xm(k)–j(k) , xn(k)+ ) + d(xn(k) , xm(k)–j(k)+ ) , () Nashine et al Fixed Point Theory and Applications 2012, 2012:217 http://www.fixedpointtheoryandapplications.com/content/2012/1/217 Page of 13 and (xm(k)–j(k) , xn(k) ) = d(xm(k)–j(k)+ , xm(k)–j(k) ), d(xn(k)+ , xn(k) ), d(xm(k)–j(k) , xn(k)+ ), d(xn(k) , xm(k)–j(k)+ ) () for all k Using the triangular inequality, we get d(xm(k)–j(k) , xn(k) ) – d(xn(k) , xm(k) ) ≤ d(xm(k)–j(k) , xm(k) ) j(k)– ≤ d(xm(k)–j(k)+l , xm(k)–j(k)+l+ ) l= p– d(xm(k)–j(k)+l , xm(k)–j(k)+l+ ) → as k → ∞ from () , ≤ l= which implies from () that lim d(xm(k)–j(k) , xn(k) ) = ε k→∞ () Using (), we have lim d(xm(k)–j(k)+ , xm(k)–j(k) ) = () lim d(xn(k)+ , xn(k) ) = () k→∞ and k→∞ Again, using the triangular inequality, we get d(xm(k)–j(k) , xn(k)+ ) – d(xm(k)–j(k) , xn(k) ) ≤ d(xn(k) , xn(k)+ ) Passing to the limit as k → ∞ in the above inequality, using () and (), we get lim d(xm(k)–j(k) , xn(k)+ ) = ε k→∞ () Similarly, we have d(xn(k) , xm(k)–j(k)+ ) – d(xm(k)–j(k) , xn(k) ) ≤ d(xm(k)–j(k) , xm(k)–j(k)+ ) Passing to the limit as k → ∞, using () and (), we obtain lim d(xn(k) , xm(k)–j(k)+ ) = ε k→∞ () Similarly, we have lim d(xm(k)–j(k)+ , xn(k)+ ) = ε k→∞ () Nashine et al Fixed Point Theory and Applications 2012, 2012:217 http://www.fixedpointtheoryandapplications.com/content/2012/1/217 Page of 13 Now, it follows from ()-() and the continuity of ϕ that lim (xm(k)–j(k) , xn(k) ) = max{ε, } = ε () lim (xm(k)–j(k) , xn(k) ) = min{, , ε, ε} = () k→∞ and k→∞ Passing to the limit as k → ∞ in (), using (), (), (), and the condition (ii), we obtain F (ε) ≤ ψ F (ε) + L · < F (ε), which is a contradiction Thus, we proved that {xn } is a Cauchy sequence in (X, d) Since (X, d) is complete, there exists x* ∈ X such that lim xn = x* () n→∞ We shall prove that p x ∈ * Ai () i= From the condition (a), and since x ∈ A , we have {xnp }n≥ ⊆ A Since A is closed, from (), we get that x* ∈ A Again, from the condition (a), we have {xnp+ }n≥ ⊆ A Since A is closed, from (), we get that x* ∈ A Continuing this process, we obtain () Now, we shall prove that x* is a fixed point of T Indeed, from (), since for all n there exists i(n) ∈ {, , , p} such that xn ∈ Ai(n) , applying (b) with x = x* and y = xn , we obtain F d Tx* , xn+ = F d Tx* , Txn ≤ψ F x* , xn + LF x* , xn () for all n On the other hand, we have x* , xn = max d x* , xn , d x* , Tx* , d(xn , xn+ ), d(x* , xn+ ) + d(xn , Tx* ) and x* , xn = d x* , Tx* , d(xn , xn+ ), d x* , xn+ , d xn , Tx* Passing to the limit as n → ∞ in the above inequality and using (), we obtain that lim n→∞ x* , xn = max d x* , Tx* , d x* , Tx* and lim n→∞ Passing to the limit as n → ∞ in (), using () and (), we get F d x* , Tx* ≤ ψ F max d x* , Tx* , d x* , Tx* x* , xn = () Nashine et al Fixed Point Theory and Applications 2012, 2012:217 http://www.fixedpointtheoryandapplications.com/content/2012/1/217 Page of 13 Suppose that d(x* , Tx* ) > In this case, we have max d x* , Tx* , d(x* , Tx* ) = d x* , Tx* , which implies that F d x* , Tx* ≤ ψ F d x* , Tx* < F d x* , Tx* , a contradiction Then we have d(x* , Tx* ) = , that is, x* is a fixed point of T Finally, we prove that x* is the unique fixed point of T Assume that y* is another fixed p point of T, that is, Ty* = y* From the condition (a), this implies that y* ∈ i= Ai Then we can apply (b) for x = x* and y = y* We obtain F d x* , y* = F d Tx* , Ty* ≤ψ F x* , y* + LF x* , y* Since x* and y* are fixed points of T, we can show easily that * * * * (x , y ) = If d(x , y ) > , we get F d x* , y* = F d Tx* , Ty* = ψ F d x* , y* ≤ψ F (x* , y* ) = d(x* , y* ) and x* , y* < F d x* , y* , a contradiction Then we have d(x* , y* ) = , that is, x* = y* Thus, we proved the uniqueness of the fixed point In the following, we deduce some fixed point theorems from our main result given by Theorem . If we take p = and A = X in Theorem ., then we get immediately the following fixed point theorem Corollary . Let (X, d) be a complete metric space and T : X → X satisfy the following condition: there exist ψ ∈ , F ∈ F , and L ≥ such that F d(Tx, Ty) ≤ ψ F max d(x, y), d(Tx, x), d(y, Ty), d(x, Ty) + d(y, Tx) + LF d(x, y), d(Tx, x), d(y, Ty), d(x, Ty), d(y, Tx) for all x, y ∈ X Then T has a unique fixed point Remark . Corollary . extends and generalizes many existing fixed point theorems in the literature [, –] Corollary . Let (X, d) be a complete metric space, p ∈ N, A , A , , Ap be nonempty p closed subsets of X, Y = i= Ai , and T : Y → Y Suppose that there exist ψ ∈ and F ∈ F such that (a ) Y = p i= Ai is a cyclic representation of Y with respect to T; Nashine et al Fixed Point Theory and Applications 2012, 2012:217 http://www.fixedpointtheoryandapplications.com/content/2012/1/217 Page of 13 (b ) for any (x, y) ∈ Ai × Ai+ , i = , , , p (with Ap+ = A ), F d(Tx, Ty) ≤ ψ F d(x, y) Then T has a unique fixed point Moreover, the fixed point of T belongs to p i= Ai Remark . Corollary . is similar to Theorem . in [] Remark . Taking in Corollary . ψ(t) = kt with k ∈ (, ), we obtain a generalized version of Theorem . in [] Corollary . Let (X, d) be a complete metric space, p ∈ N, A , A , , Ap be nonempty p closed subsets of X, Y = i= Ai , and T : Y → Y Suppose that there exist ψ ∈ and F ∈ F such that p (a ) Y = i= Ai is a cyclic representation of Y with respect to T; (b ) for any (x, y) ∈ Ai × Ai+ , i = , , , p (with Ap+ = A ), F d(Tx, Ty) ≤ ψ F d(x, Ty) + d(y, Tx) Then T has a unique fixed point Moreover, the fixed point of T belongs to p i= Ai Remark . Taking in Corollary . ψ(t) = kt with k ∈ (, ), we obtain a generalized version of Theorem in [] Corollary . Let (X, d) be a complete metric space, p ∈ N, A , A , , Ap be nonempty p closed subsets of X, Y = i= Ai , and T : Y → Y Suppose that there exist ψ ∈ and F ∈ F such that p (a ) Y = i= Ai is a cyclic representation of Y with respect to T; (b ) for any (x, y) ∈ Ai × Ai+ , i = , , , p (with Ap+ = A ), F d(Tx, Ty) ≤ ψ F max d(x, Tx), d(y, Ty) Then T has a unique fixed point Moreover, the fixed point of T belongs to p i= Ai Remark . Taking in Corollary . ψ(t) = kt with k ∈ (, ), we obtain a generalized version of Theorem in [] Corollary . Let (X, d) be a complete metric space, p ∈ N, A , A , , Ap be nonempty p closed subsets of X, Y = i= Ai , and T : Y → Y Suppose that there exist ψ ∈ and F ∈ F such that p (a) Y = i= Ai is a cyclic representation of Y with respect to T; (b) for any (x, y) ∈ Ai × Ai+ , i = , , , p (with Ap+ = A ), F d(Tx, Ty) ≤ ψ F max d(x, y), d(x, Tx), d(y, Ty) Then T has a unique fixed point Moreover, the fixed point of T belongs to p i= Ai Nashine et al Fixed Point Theory and Applications 2012, 2012:217 http://www.fixedpointtheoryandapplications.com/content/2012/1/217 Page of 13 We provide some examples to illustrate our obtained Theorem . Example . Let X = R with the usual metric Suppose A = [–, ] and A = [, ] and for all x ∈ Y It is clear that i= Ai is a Y = i= Ai Define T : Y → Y such that Tx = –x cyclic representation of Y with respect to T Let ψ ∈ be defined by ψ(t) = t and F ∈ F of the form F (t) = kt, k > For all x, y ∈ Y and L ≥ , we have F d(Tx, Ty) = kd(Tx, Ty) k|x – y| k|x – y| ≤ k( (x, y)) ≤ k( (x, y)) ≤ + Lk = =ψ F (x, y) (x, y) + LF (x, y) So, T is a cyclic generalized (F , ψ, L)-contraction for any L ≥ Therefore, all conditions of Theorem . are satisfied (p = ), and so T has a unique fixed point (which is x* = ∈ i= Ai ) Example . Let X = R with the usual metric Suppose A = [–π/, ] and A = [, π/] and Y = i= Ai Define the mapping T : Y → Y by ⎧ ⎨– x| cos(/x)| if x ∈ [–π/, ) ∪ (, π/], Tx = ⎩ if x = Clearly, we have T(A ) ⊆ A and T(A ) ⊆ A Moreover, A and A are nonempty closed subsets of X Therefore, i= Ai is a cyclic representation of Y with respect to T Now, let (x, y) ∈ A × A with x = and y = , we have d(Tx, Ty) = |Tx – Ty| = x cos(/x) + y cos(/y) |x| cos(/x) + |y| cos(/y) ≤ |x| + |y| = On the other hand, we have |x| = –x ≤ –x + x cos(/x) = –x – x cos(/x) ≤ x + x cos(/x) = d(x, Tx) and |y| = y ≤ y + y cos(/y) = y + y cos(/y) = d(y, Ty) Nashine et al Fixed Point Theory and Applications 2012, 2012:217 http://www.fixedpointtheoryandapplications.com/content/2012/1/217 Page 10 of 13 Then we have max d(x, Tx), d(y, Ty) d(x, Ty) + d(y, Tx) ≤ max d(x, y), d(x, Tx), d(y, Ty), d(Tx, Ty) ≤ Define the function ψ : [, ∞) → [, ∞) by ψ(t) = F (t) = kt, k > and L ≥ Then we have F d(Tx, Ty) ≤ ψ F (x, y) + LF t , for all t ≥ and F ∈ F of the form (x, y) () Moreover, we can show that () holds if x = or y = Similarly, we also get () holds for (x, y) ∈ A × A Now, all the conditions of Theorem . are satisfied (with p = ), we deduce that T has a unique fixed point x* ∈ A ∩ A = {} An application to an integral equation In this section, we apply the result given by Theorem . to study the existence and uniqueness of solutions to a class of nonlinear integral equations We consider the nonlinear integral equation T u(t) = G(t, s)f s, u(s) ds for all t ∈ [, T], () where T > , f : [, T] × R → R and G : [, T] × [, T] → [, ∞) are continuous functions Let X = C([, T]) be the set of real continuous functions on [, T] We endow X with the standard metric for all u, v ∈ X d∞ (u, v) = max u(t) – v(t) t∈[,T] It is well known that (X, d∞ ) is a complete metric space Let (α, β) ∈ X , (α , β ) ∈ R such that α ≤ α(t) ≤ β(t) ≤ β for all t ∈ [, T] () We suppose that for all t ∈ [, T], we have T α(t) ≤ G(t, s)f s, β(s) ds () G(t, s)f s, α(s) ds () and T β(t) ≥ We suppose that for all s ∈ [, T], f (s, ·) is a decreasing function, that is, x, y ∈ R, x≥y =⇒ f (s, x) ≤ f (s, y) () Nashine et al Fixed Point Theory and Applications 2012, 2012:217 http://www.fixedpointtheoryandapplications.com/content/2012/1/217 Page 11 of 13 We suppose that T sup G(t, s) ds ≤ () t∈[,T] Finally, we suppose that, for all s ∈ [, T], for all x, y ∈ R with (x ≤ β and y ≥ α ) or (x ≥ α and y ≤ β ), f (s, x) – f (s, y) ≤ ψ max |x – y|, |x – Tx|, |y – Ty|, |x – Ty| + |y – Tx| + L |x – Tx|, |y – Ty|, |x – Ty|, |y – Tx| , where ψ : [, ∞) → [, ∞) is a nondecreasing function that belongs to Now, define the set () and L ≥ C = u ∈ C [, T] : α ≤ u(t) ≤ β for all t ∈ [, T] We have the following result Theorem . Under the assumptions ()-(), problem () has one and only one solution u* ∈ C Proof Define the closed subsets of X, A and A , by A = {u ∈ X : u ≤ β} and A = {u ∈ X : u ≥ α} Define the mapping T : X → X by T Tu(t) = G(t, s)f s, u(s) ds for all t ∈ [, T] We shall prove that T(A ) ⊆ A and T(A ) ⊆ A () Let u ∈ A , that is, u(s) ≤ β(s) for all s ∈ [, T] Using condition (), since G(t, s) ≥ for all t, s ∈ [, T], we obtain that G(t, s)f s, u(s) ≥ G(t, s)f s, β(s) for all t, s ∈ [, T] Nashine et al Fixed Point Theory and Applications 2012, 2012:217 http://www.fixedpointtheoryandapplications.com/content/2012/1/217 Page 12 of 13 The above inequality with condition () implies that T T G(t, s)f s, u(s) ds ≥ G(t, s)f s, β(s) ds ≥ α(t) for all t ∈ [, T] Then we have Tu ∈ A Similarly, let u ∈ A , that is, u(s) ≥ α(s) for all s ∈ [, T] Using condition (), since G(t, s) ≥ for all t, s ∈ [, T], we obtain that G(t, s)f s, u(s) ≤ G(t, s)f s, α(s) for all t, s ∈ [, T] The above inequality with condition () implies that T T G(t, s)f s, u(s) ds ≤ G(t, s)f s, α(s) ds ≤ β(t) for all t ∈ [, T] Then we have Tu ∈ A Finally, we deduce that () holds Now, let (u, v) ∈ A × A , that is, for all t ∈ [, T], u(t) ≤ β(t), v(t) ≥ α(t) This implies, from condition (), that for all t ∈ [, T], v(t) ≥ α u(t) ≤ β , Now, using conditions () and (), we can write that for all t ∈ [, T], we have |Tu – Tv|(t) T ≤ G(t, s) f s, u(s) – f s, v(s) ds T ≤ G(t, s)ψ max u(s) – v(s) , u(s) – Tu(s) , v(s) – Tv(s) , |u(s) – Tv(s)| + |v(s) – Tu(s)| ds T + G(t, s)L u(s) – Tu(s) , v(s) – Tv(s) , u(s) – Tv(s) , v(s) – Tu(s) ds ≤ ψ max d∞ (u, v), d∞ (u, Tu), d∞ (v, Tv), d∞ (u, Tv) + d∞ (v, Tu) T + L d∞ (u, Tu), d∞ (v, Tv), d∞ (u, Tv), d∞ (v, Tu) ≤ ψ max d∞ (u, v), d∞ (u, Tu), d∞ (v, Tv), G(t, s) ds d∞ (u, Tv) + d∞ (v, Tu) + L d∞ (u, Tu), d∞ (v, Tv), d∞ (u, Tv), d∞ (v, Tu) T G(t, s) ds Nashine et al Fixed Point Theory and Applications 2012, 2012:217 http://www.fixedpointtheoryandapplications.com/content/2012/1/217 Page 13 of 13 This implies that F d∞ (Tu, Tv) ≤ ψ F (u, v) + LF (u, v) , where F ∈ F of the form F (t) = t Using the same technique, we can show that the above inequality holds also if we take (u, v) ∈ A × A Now, all the conditions of Theorem . are satisfied (with p = ), we deduce that T has a unique fixed point u* ∈ A ∩ A = C , that is, u* ∈ C is the unique solution to () 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 manuscript Author details Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Mandir Hasaud, Raipur, Chhattisgarh 492101, India Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, 10140, Thailand Acknowledgements The second author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) Moreover, the third author was supported by the Commission on Higher Education (CHE), the Thailand Research Fund (TRF) and the King Mongkut’s University of Technology Thonburi (KMUTT) (Grant No MRG5580213) Received: 13 June 2012 Accepted: 12 November 2012 Published: 28 November 2012 References Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux equations itegrales Fundam Math 3, 133-181 (1922) Chauhan, S, Sintunavarat, W, Kumam, P: Common fixed point theorems for weakly compatible mappings in fuzzy metric spaces using (JCLR) property Appl Math 3(9), 976-982 (2012) Pacurar, M: Fixed point theory for cyclic Berinde operators Fixed Point Theory 12(2), 419-428 (2011) Sintunavarat, W, Cho, YJ, Kumam, P: Common fixed point theorems for c-distance in ordered cone metric spaces Comput Math Appl 62, 1969-1978 (2011) Sintunavarat, W, Kumam, P: Generalized common fixed point theorems in complex valued 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(1971) doi:10.1186/1687-1812-2012-217 Cite this article as: Nashine et al.: Cyclic generalized contractions and fixed point results with applications to an integral equation Fixed Point Theory and Applications 2012 2012:217 ... Cite this article as: Nashine et al.: Cyclic generalized contractions and fixed point results with applications to an integral equation Fixed Point Theory and Applications 2012 2012:217 ... et al Fixed Point Theory and Applications 2012, 2012:217 http://www.fixedpointtheoryandapplications.com/content/2012/1/217 Page of 13 On the other hand, cyclic representations and cyclic contractions. .. T has a unique fixed point Moreover, the fixed point of T belongs to p i= Ai Nashine et al Fixed Point Theory and Applications 2012, 2012:217 http://www.fixedpointtheoryandapplications.com/content/2012/1/217