Abstract
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.
Highlights
Introduction and preliminariesAll 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
We introduce a new class of cyclic generalized (F, ψ, L)-contractive mappings, and investigate the existence and uniqueness of fixed points for such mappings
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
Summary
Passing to the limit as k → ∞, using ( ) and ( ), we obtain lim d(xn(k), xm(k)–j(k)+ ) = ε It follows from ( )-( ) and the continuity of φ that lim (xm(k)–j(k), xn(k)) = max{ε, } = ε (xm(k)–j(k), xn(k)) = min{ , , ε, ε} =. Passing to the limit as n → ∞ in the above inequality and using ( ), we obtain that lim n→∞. We get immediately the following fixed point theorem. 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. Extends and generalizes many existing fixed point theorems in the literature [ , – ].
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