Fixed point theorems for multivalued G-contractions in Hausdorff b-Gauge spaces

Abstract

In this paper, we extend gauge spaces in the setting of b metric spaces and prove fixed point theorems for multivalued mappings in this new setting endowed with a graph. An example is constructed to substantiate our result. We also discuss possible application of our result for solving integral equations. c 2015 All rights reserved.

Generalized Fixed-Point Results for Almost (α,Fσ)-Contractions with Applications to Fredholm Integral Inclusions

“…Isik [14] used the above family of functions and established multivalued fixed-point theorem in complete metric space. For more details in the direction of multivalued generalization, we refer the reader to (see [15][16][17][18][19][20][21][22] ).…”

Periodic and Fixed Points for Caristi-Type G -Contractions in Extended b -Gauge Spaces

“…In 1973, Reilly [27] studied quasi-gauge spaces and proved that it generalizes topological spaces, quasi uniform spaces, and quasi metric spaces. This notion was extended as b-gauge spaces by Ali et al [28] in 2015. For further facts on gauge spaces, we recommend the reader to [29][30][31][32][33][34][35][36].…”

“…In 2015, Ali et al [28] have defined gauge spaces in the setting of b s -pseudo metrics called b-gauge spaces. In order to introduce extended b-gauge spaces, we start here with the introduction of the notion of extended pseudo-b-metric.…”

“…Let J φ;Ω = fJ β : β ∈ Ωg be the extended J φ;Ω -family on U. A sequence ðu m : m ∈ NÞ is extended J φ;Ω -cauchy sequence in U if, for all β ∈ Ω, lim m⟶∞ sup n>m J β ðu m , u n Þ = 0: We observe that if φ β ðu, vÞ = s for all β ∈ Ω, where s ≥ 1 and J φ;Ω = Q φ;Ω , the above definitions of completeness reduce to the corresponding definitions in b-gauge spaces (see [28])…”

New theorems on extended b-metric spaces under new contractions

“…Bakhtin introduces the notion of b-metric spaces in [11] as a generalization of metric spaces and proved many exciting fixed point theorem. For some interesting fixed point results in b-metric spaces, see [3][4][5][6][7][8][9][19][20][21][23][24][25][26]29]. Kamran et al in [17] introduced the definition of extended b-metric spaces as a generalization of b-metric spaces.…”