Abstract
In this paper, we present a new type of set-valued mappings called partial q-set-valued quasi-contraction mappings and give results as regards fixed points for such mappings in b-metric spaces. By providing some examples, we show that our results are real generalizations of the main results of Aydi et al. (Fixed Point Theory Appl. 2012:88, 2012) and many results in the literature. We also consider fixed point results for single-valued mapping, fixed point results for set-valued mapping in b-metric space endowed with an arbitrary binary relation, and fixed point results in a b-metric space endowed with a graph. By using our result, we establish the existence of solution for the following an integral equations: x(c )= φ(c )+ � b
Highlights
The Banach contraction principle is a very popular tool of mathematics in solving many problems in several branches of mathematics since it can be observed and comfortably
In, Czerwik [ ] introduced the concept of b-metric spaces and presented the fixed point theorem for contraction mappings in b-metric spaces, that is, we have a generalization of the Banach contraction principle in metric spaces
In, Aydi et al [ ] extended the concept of q-set-valued quasi-contraction mappings in metric spaces due to Amini-Harandi [ ] to b-metric spaces. They established the fixed point results for q-set-valued quasi-contraction mappings in b-metric spaces
Summary
The Banach contraction principle is a very popular tool of mathematics in solving many problems in several branches of mathematics since it can be observed and comfortably. In a b-metric space (X, d) the following assertions hold: ( ) a convergent sequence has a unique limit; ( ) each convergent sequence is Cauchy; ( ) in general a functional b-metric d : X × X → R+ for coefficient s > is not jointly continuous in all its variables. We say that t is α-admissible if for x, y ∈ X for which α(x, y) ≥ ⇒ α(tx, ty) ≥ They proved the fixed point results for single-valued mapping as regards this concept and showed that these results can be utilized to derive fixed point theorems in partially ordered spaces. As an application, they obtain the existence of solutions for ordinary differential equations. For all n ∈ N, d(xn, xn+ ) ≤ β max d(xn– , xn), d(xn– , Txn– ), d(xn, Txn), d(xn– , Txn), d(xn, Txn– ) ≤ β max d(xn– , xn), d(xn– , xn), d(xn, xn+ ), d(xn– , xn+ ), d(xn, xn) ≤ β max d(xn– , xn), d(xn, xn+ ), s d(xn– , xn) + d(xn, xn+ ) ≤ βs d(xn– , xn) + d(xn, xn+ )
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