Some common fixed point and invariant approximation results for nonexpansive mappings in convex metric space

Abstract

In this work, we introduce a new class of self-maps which satisfy the (E.A.) property with respect to some q ∈ M, where M is q-starshaped subset of a convex metric space and common fixed point results are established for this new class of self-maps. After that we obtain some invariant approximation results as an application. Our results represent a very strong variant of the several recent results existing in the literature. We also provide some illustrative examples in the support of proved results. MSC: 46T99; 47H10; 54H25

Existence of common fixed point and best proximity point for generalized nonexpansive type maps in convex metric space

“…in the context of convex metric space and extend the results of Kumar and Rathee (2014) to four self-maps by utilizing this newly introduced concept. Further, we ensure the existence of common best proximity point for generalized non-expansive type maps.…”

“…Then after, Beg and Azam (1987), Fu and Huang (1991), Ciric (1993), and many others have obtained fixed point theorems in convex metric spaces. Very recently, Kumar and Rathee (2014) defined the concept of (E.A.) property in the setup of convex metric space and ensure the existence of common fixed point for a pair of maps satisfying this property by omitting the assumption that the range of one map is contained in other.…”

“…A mapping is said to beaffine (Al-Thagafi and Shahzad 2006; Huang and Li 1996), if M is convex and for all and . q -affine (Al-Thagafi and Shahzad 2006; Kumar and Rathee 2014), if M is q -starshaped and for all and . …”

“…q -affine (Al-Thagafi and Shahzad 2006; Kumar and Rathee 2014), if M is q -starshaped and for all and .…”

“…property in a convex metric space defined by Kumar and Rathee (Fixed Point Theory Appl 1–14, 2014) by introducing a new class of self-maps which satisfies the common property (E.A.) in the context of convex metric space and ensure the existence of common fixed point for this newly introduced class of self-maps.…”