New Generalizations of Set Valued Interpolative Hardy-Rogers Type Contractions in b-Metric Spaces

Abstract

Debnath and De La Sen introduced the notion of set valued interpolative Hardy-Rogers type contraction mappings on b-metric spaces and proved that on a complete b-metric space, whose all closed and bounded subsets are compact, the set valued interpolative Hardy-Rogers type contraction mapping has a fixed point. This article presents generalizations of above results by omitting the assumption that all closed and bounded subsets are compact.

New Results in Controlled Metric Type Spaces

“…e concept of extended b-metric spaces was initiated by Kamran et al [8] in 2017, and their work generalized many results in the literature (see, for example, [12][13][14][15][16]). Definition 1 (see [8]).…”

“…We finally prove that g is a fixed point for ξ. e triangle inequality implies that ℘(g, ξg) ≤ ](g, ξg) ℘ g, g n+1 + ℘ g n+1 , ξg � ](g, ξg)℘ g, g n+1 + ](g, ξg)℘ g n+1 , ξg . (16) Note that the first term of the inequality, ](g, ξg)℘(g, g n+1 ), converges to 0, since g n converges to 0. Also, the second term will be ](g, ξg)℘ g n+1 , ξg ≤ ](g, ξg)℘ ξg n , ξg α g n , g ≤ ](g, ξg)ψ ℘ g n , g ≤ ](g, ξg)℘ g n , g , (17) which converges to 0. erefore, ℘(g, ξg) ≤ 0, and hence g is a fixed point for ξ.…”

On interpolative F-contractions with shrink map

“…Altun and Tasdemir [9] presented the study of best proximity points using interpolative proximal contraction inequalities. Along with the aforementioned studies, many other interesting studies on fixed point theory are available in [10][11][12][13][14][15][16][17][18][19][20][21][22][23]; they help readers to verify the existence of fixed points for self-mappings and best proximity points for nonself mappings. Jleli et al [23] introduced the concept of E-fixed point (also called φ-fixed point), which states that, for maps V : K → K and E : K → [0, ∞), a point k ∈ K is called E-fixed point of V : K → K if V (k) = k and E(k) = 0, and proved the existence of such points by using a single inequality involving both maps V and E. It is important to note that Jleli et al [23] used the lower semicontinuity of E. This use of the lower semicontinuity of E by Jleli et al [23] arises the question whether the condition of lower semicontinuity of E can be left and some other technique be adopted.…”

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.…”