Abstract
AbstractIn this paper, we consider the set-valued contractions defined on product spaces when the underlying space is a complete metric space endowed with a graph. Some fixed point results for the so-called set-valued G-Prešić operators are established. Our theorems extend and generalize some known results in product spaces of the recent literature. As an application of our main result, fixed point results for various types of set-valued contractions on product spaces are derived, and a sufficient condition for the existence of a weakly asymptotically stable and global attractor equilibrium point of a kth order nonlinear difference inclusion is established.
Highlights
In, Prešić [, ] extended the famous Banach contraction principle to the product spaces and obtained some convergence results for some particular sequences
Luong and Thuan [ ] and Shukla and Radenović [ ] considered the Prešić type mappings in partially ordered sets and proved the ordered version of Prešić theorem. These results generalize the result of Ran and Reurings [ ] in product spaces
Shukla and Shahzad [ ] extended, generalized and unified the result of Jachymski [ ], Prešić [, ], Luong and Thuan [ ] by proving fixed point results for G-Prešić type operators in the spaces endowed with a graph
Summary
In , Prešić [ , ] extended the famous Banach contraction principle to the product spaces and obtained some convergence results for some particular sequences. (Prešić) Let (X, d) be a complete metric space, k be a positive integer and T : Xk → X be a mapping satisfying the following contractive type condition:. Luong and Thuan [ ] and Shukla and Radenović [ ] considered the Prešić type mappings in partially ordered sets and proved the ordered version of Prešić theorem These results generalize the result of Ran and Reurings [ ] in product spaces. Shukla and Shahzad [ ] extended, generalized and unified the result of Jachymski [ ], Prešić [ , ], Luong and Thuan [ ] by proving fixed point results for G-Prešić type operators in the spaces endowed with a graph. Let (X, d) be a complete metric space, k be a positive integer and T : Xk → CB(X) be a set-valued G-Prešić operator. There exists a termwise connected sequence {xn} in X such that xn+k ∈ T(xn, xn+ , . . . , xn+k– ) for all n ∈ N and {xn} converges to a fixed point of T
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