Fixed point theorems in modular vector spaces

Abstract

In this work, we initiate the metric fixed point theory in modular vector spaces under Nakano formulation. In particular, we establish an analogue to Banach contraction principle, Browder and Göhde fixed point theorems for nonexpansive mappings in the modular sense. Then we finish by proving a common fixed point result of a commutative family of nonexpansive mappings in the modular sense.

On Suzuki Mappings in Modular Spaces

“…Moreover, ρ has the (UUC1)-property if, for each s ≥ 0 and each > 0, one may find η 1 (s, ) > 0 such that δ 1 (r, ) > η 1 (s, ) > 0, for r > s. Definition 5. Given a sequence {x } in X ρ and a nonempty subset S ⊂ X ρ , the following elements may be defined in connection with them (see [10]):…”

“…Let S be a nonempty ρ-closed convex subset of X ρ and {x } be a sequence in X ρ with a finite asymptotic radius relative to S (i.e., r(S) < ∞). If ρ satisfies the (UUC1)-condition, then all the minimizing sequences of τ are modular-convergent, having the same ρ-limit [10].…”

“…In time, the notion was reactivated in the expanded framework of vector spaces and one important direction was settled by Khamsi [6] in connection with the fixed point theory. Nowadays, this approach is fructified in several works: Okeke et al [7], Khan [8], Abbas et al [9], Abdou and Khamsi [10], Alfuraidan et al [11] and the papers referenced there.…”

Feng-Liu Type Fixed Point Results for Multivalued Mappings in GMMS

“…The second, a generalization of a result of Edelstein, is a fixed point theorem for compact set-valued local contractions. Nadler's study is applied through other metric spaces, such as in [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”

Common Fixed Points Results on Non-Archimedean Metric Modular Spaces

“…Lately, various modular structures, viewed as alternatives to classical normed or metric spaces, have been intensely studied in connection with the fixed point theory. Many modular related research papers adopted the setting of a modular vector space (see [1][2][3][4][5]), while others used the more general framework of a metric modular space (see [6][7][8][9][10][11]). The notion of a metric modular, together with its stronger convex version, was firstly introduced and studied by Chistyakov in [6][7][8][9].…”