Controlled S-Metric-Type Spaces and Applications to Fractional Integrals

Abstract

In this paper, we introduce controlled S-metric-type spaces and give some of their properties and examples. Moreover, we prove the Banach fixed point theorem and a more general fixed point theorem in this new space. Finally, using the new results, we give two applications on Riemann–Liouville fractional integrals and Atangana–Baleanu fractional integrals.

Bipolar b-Metric Spaces in Graph Setting and Related Fixed Points

“…(ii) A bipolar pseudo-metric (BPSM) is one where d is a bipolar pseudo-semi metric meeting (4). (iii) A bipolar metric is a BPSM d satisfying (2).…”

“…and the set of edges given by the following: (1,3), (1,4), (1,5), (1,7), (1,13), (1,15), (3,2), (2,4), (5,2), (2, 7), (13,2), (2,15), (3,4), (3,5), (3,7), (3,13), (3,15), (5,4), (4,7), (13,4), (4,15), (5,7), (5,13), (5,15), (7,13), (7,15), (13,15).…”

“…Many scholars have been drawn to metric spaces (MSs) because of their axiomatic clarity. Several writers have generalized the BCP by using various forms of contraction mappings in different MSs (for example, [1,2]). Mutlu and Gürdal [3] introduced the concept of bipolar MS in 2016, which is a type of partial distance that connects MSs and bipolar MSs.…”

Some Fixed-Point Results in Extended S-Metric Space of Type (α,β)

“…Some researchers generalize the Banach principle by changing the type of contraction and adding conditions that satisfy the existence of a fixed point, like Karman et al [6], who established the notion of extended b-metric space and proved some fixed-point theorems in extended b-metric space. In recent years, there have been several important fixed-point results for some contractions in extended S-metric spaces and S b -metric spaces; see [7][8][9][10][11]. This paper aims to investigate a generalization of S-metric space, called extended S-metric space of type (α, β).…”