Fixed Point Theorems for Geraghty Contraction Type Mappings in b-Metric Spaces and Applications

Abstract

In this paper, some new results are given on fixed and common fixed points of Geraghty type contractive mappings defined in b-complete b-metric spaces. Moreover, two examples are represented to show the compatibility of our results. Some applications for nonlinear integral equations are also given.

Some fixed point results of generalized $$(\phi , \psi )$$-contractive mappings in ordered b-metric spaces

“…This means that for . Hence, we obtain from ( 6 ) that Since, then the sequence is a Cauchy sequence by [ 3 , 5 , 8 , 10 ]. Since P is complete, then there exists some such that .…”

Solution of fractional differential equations in quasi-b-metric and b-metric-like spaces

“…(1) by α-ψ-contractions. Similar ideas have also been considered by some authors; see, for example, [2,3,8,9,[14][15][16][18][19][20], and the references therein.…”

“…where M = sup η∈J 1 0 G(η, ζ ) dζ , also conditions (ii)-(iv) from Theorem 2.3 are satisfied, where G(η, ζ ) is given in (9). Then problem (8) has at least one solution.…”

The Existence and Uniqueness of the Solution of a Nonlinear Fredholm–Volterra Integral Equation with Modified Argument via Geraghty Contractions

“…As a generalization of the metric space, Bakhtin [1] introduced the notion of b-metric space and since then, several papers have been published on the fixed point theory in such spaces. Subsequently, many results in b-metric spaces have been obtained and published and some of them can be found in ( [2][3][4][5][6][7][8][9]).…”