Fixed point theorems for set-valued G -contractions in a graphical convex metric space with applications
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Journal of Fixed Point Theory and Applications
Fixed point theorems for set-valued G-contractions in a graphical convex metric space with applications Lili Chen, Ni Yang, Yanfeng Zhao and Zhenhua Ma Abstract. In this paper, we first introduce the concept of graphical convex metric spaces and some basic properties of the underlying spaces. Different from related literature, we generalize Mann iterative scheme and Agrawal iterative scheme for set-valued mappings to above spaces by introducing the concepts of T -Mann sequences and T -Agrawal sequences. Furthermore, by using the iterative techniques and graph theory, we investigate the existence and uniqueness of fixed points for setvalued G-contractions in a graphical convex metric space. Moreover, we present some notions of well-posedness and G-Mann stability of the fixed point problems in the above space. Additionally, as an application of our main results, we discuss the well-posedness and G-Mann stability of the fixed point problems for set-valued G-contractions in a graphical convex metric space. Mathematics Subject Classification. Primary 47H09, Secondary 47H10. Keywords. Graphical convex metric spaces, Mann iterative scheme, Agrawal iterative scheme, set-valued mappings, fixed point.
1. Introduction In recent decades, Banach contraction principle which is one of the most widely applied fixed point theorems in all branches of Mathematics has been extensively investigated [1–3]. In 2004, Ran and Reuring [4] extended Banach contraction principle in the context of partially ordered set. In [5], Jachymski generalized these spaces and introduced the graphical metric spaces, by replacing the previous partially ordered structure with the graph structure. Afterwards, many researchers extended and generalized various types of fixed point theorems to the graphical metric spaces [6–11]. 0123456789().: V,-vol
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In 1969, Nadler [12] extended Banach contraction principle to contractive set-valued mappings. In 1972, Reich [13] studied the fixed points of contractive functions. In 1974, Lim obtained a fixed point theorem for nonexpansive set-valued mappings in uniformly convex Banach spaces by means of Edelstein’s method of asymptotic centers. In 1990, Kirk and Massa obtained an important result for non-expansive set-valued mappings in Banach spaces. In 2011, Nicolae, O’Regan, and Petru¸sel [8] extended the notion of setvalued contraction on a graphical metric space. In 2013, Dinevari and Frigon [7] proposed a more general definition of set-valued contraction on a graphical metric space. In 2014, Sultana and Vetrivel [14] extended Mizoguchi– Takahashi’s fixed point theorem for set-valued mappings on a metric space endowed with a graph. For more details on set-valued mappings we refer the readers to [15–19] and the abundant list of references. In 1970, Takahashi [20] introduced the concepts of the convex structure and the convex metric space, he also obtained some fixed point theorems for non-expansive mappings in the convex metric spaces. In addition, Goe
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