Approximating Fixed Points of Nonself Contractive Type Mappings in Banach Spaces Endowed with a Graph

Abstract

Let K be a non-empty closed subset of a Banach space X endowed with a graph G. We obtain fixed point theorems for nonself G-contractions of Chatterjea type. Our new results complement and extend recent related results [Berinde, V., Păcurar, M., The contraction principle for nonself mappings on Banach spaces endowed with a graph, J. Nonlinear Convex Anal. 16 (2015), no. 9, 1925-1936; Balog, L., Berinde, V., Fixed point theorems for nonself Kannan type contractions in Banach spaces endowed with a graph, Carpathian J. Math. 32 (2016), no. 3 (in press)] and thus provide more general and flexible tools for studying nonlinear functional equations.

A Brief Note Concerning Non-Self Contractions in Banach Spaces Endowed with a Graph

“…Our theorems enhance and generalize fixed point theorems for some non-self contractions on Banach spaces involving a graph in Berinde & Pȃcurar (2015), Balog & Berinde (2016) and Balog et al (2016). Within the future scope of the idea, reader might point out the existence and uniqueness of the fixed point point theorem for a Hardy-Rogers multi-valued contractions on a complete metric space via the g−graph preserving condition.…”

“…On the other hand, Berinde and Pȃcurar (2015) [11] defined the notion of G H −contractions in Banach space with a graph and subsequently presented some fixed point results for such classes of non-self contractions. Balog and Berinde (2016) [12] introduced the concept of G H −Kannan contractions on Banach space endowed with a graph and their results extended and generalized the authors [11]. In the sequel, Balog et al [13] showed some fixed point theorems for G H −Catterjea contractions in Banach space involving a graph.…”

“…Balog and Berinde (2016) [12] introduced the concept of G H −Kannan contractions on Banach space endowed with a graph and their results extended and generalized the authors [11]. In the sequel, Balog et al [13] showed some fixed point theorems for G H −Catterjea contractions in Banach space involving a graph. They also have generalized the above-mentioned papers.…”

“…As in [11,12], we relocated the property (L) of the tripled (M, Proof. Verbatim of the proof of Theorem 2, we conclude that the iterative sequence {x n } is Cauchy and converges to z ∈ M .…”

“…In other words, z = gz. Suppose that t is another fixed point of g. Running a similar method like in Theorem 2, we have z = t. (iii) In Theorem 2 and Theorem 2, if we take a = b = 0, we obtain Theorem 2 and Theorem 3 in [13], respectively. (iv) In Theorem 2 and Theorem 2, if we take c = 0, we obtain the existence and uniqueness results for non-self Reich G−contractions in a Banach space with a graph.…”