On Orthogonal Partial $b$-Metric Spaces with an Application

Abstract

In this paper, we initiate the concept of orthogonal partial b -metric spaces. We ensure the existence of a unique fixed point for some orthogonal contractive type mappings. Some useful examples are given, and an application is also provided in support of the obtained results.

Unique Fixed-Point Results in Fuzzy Metric Spaces with an Application to Fredholm Integral Equations

“…x, tÞ: ð0,∞Þ ⟶ ½0, 1 is continuous for all w, x, z ∈ W and t, s > 0 Definition 3 (see [28,29]). Let ðW, M F , * Þ be a FM-space, w ∈ W, and fw i g be a sequence in W. Then,…”

“…Definition 7 (see [29]). Let ðW, M F , * Þ be a FM-space and G : W ⟶ W. Then, G is known as a fuzzy contraction, if there is 0 < h < 1 so that…”

“…George and Veeramani [28] modified the concept of FM-spaces with the help of continuous t-norms and proved some basic properties in this direction. In 2002, Gregori and Sapena [29] proved some contractive-type fixed-point theorems in complete FM-spaces in the sense of Kramosil and Michalek [26] and in the sense of George and Veeramani [28]. Rana et al [30] established some fixed-point theorems in FM-spaces by using implicit relations.…”

The Existence of Fixed Points for a Different Type of Contractions on Partial b-Metric Spaces

“…Since the inception of this principle, many authors studied fixed point theory vividly and enriched this field with different ideas. is classical result was generalized in different spaces, and different structures were attained using this topic and one may recall the existing notions, partial b-metric spaces [3], R-partial b-metric spaces [4], fuzzy cone b-metric spaces [5], G b -metric spaces [6], orthogonal partial b-metric spaces [7], orthogonal m-metric spaces [8], and several others. More details can be found in [9][10][11][12][13].…”

Results on Complex Partial b-Metric Space with an Application

“…Many authors have studied related interesting metric such as structures along with some applications. And, in this line, significant results have been obtained and can be read in [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. In this paper, under new contraction condition, we prove a fixed point theorem in complex partial b-metric space.…”