Fixed point theorems in some types of metric-like spaces

Abstract

It is well known that a normed space X is uniformly convex (smooth) if and only if its dual X∗ is uniformly smooth (convex). We extend some of these geometric properties to the so-called 2-normed spaces and also we introduce definition of uniformly smooth 2-normed spaces. We get some fundamental links between Lindenstrauss duality formu- las. Besides, a duality property between uniform convexity and uniform smoothness of 2-normed space is also given. Moreover, we introduced a definition of Metric projection in a 2-normed space and also theorem of the approximation of fixed points in 2-Hilbert spaces.

1. In chapter 1: we give a summary of quotient spaces and state the Hahn-Banach Theorem; which is the one of the fundamental theorems of functional analysis. We study some of the geometric properties in linear normed spaces and we give a brief history of Metric projection existence and uniqueness in different spaces. 2. In chapter 2: we discuss some spaces like 2-metric spaces and linear 2-normed spaces. We define a Cauchy sequence and a con- vergent sequence in both spaces and mention the relation between both concepts and also the notions of bounded bilinear function. 3. In chapter 3: we study the completion of linear 2-normed spaces. 4. In chapter 4: we study some geometric properties for linear 2- normed spaces like strictly convex, strictly 2-convex, uniformly convex and uniformly 2-convex 2-Banach spaces. We give new definitions and prove new results. We prove an important result which state that: “a 2-normed space X is uniformly convex if and only if all dual spaces (X/V (c))∗ or X∗, for all c ̸= 0 in