Solution of Volterra integral inclusion in b-metric spaces via new fixed point theorem

Abstract

An existence theorem for Volterra-type integral inclusion is establish in b-metric spaces. We first introduce two new F-contractions of Hardy-Rogers type and then establish fixed point theorems for these contractions in the setting of b-metric spaces. Finally, we apply our fixed point theorem to prove the existence theorem for Volterra-type integral inclusion. We also provide an example to show that our fixed point theorem is a proper generalization of a recent fixed point theorem by Cosentino et al.

New Types of F-Contraction for Multivalued Mappings and Related Fixed Point Results in Abstract Spaces

“…Lemma 7 (see [6]). Let ( , , ) be a b-metric space and let { } be any sequence in for which there exist > 0 and ∈ F such that + ( ( , +1 )) ≤ ( ( −1 , )), ∈ N. Then { } is a Cauchy sequence in .…”

“…Ali et al [6] extended the family of mapping F defined by [21] to the family F of all functions : R + → R such that (F1) is strictly increasing, that is, for all , ∈ R + such that < implies ( ) < ( );…”

“…Czewick [5] initiated the study of fixed point in b-metric spaces. Many authors used the concept of b-metric spaces to prove the existence and the uniqueness of a fixed point for several contraction mappings [6][7][8][9]. Furthermore, dislocated quasimetric spaces [10][11][12][13] generalized abstract spaces such as dislocated metric spaces [14] and quasi-metric spaces [15][16][17].…”

“…Wardowski [21] generalized many fixed point results in a beautiful way by introducing −contraction (see also [6,[22][23][24][25][26][27][28][29][30]). Nadler [31] extended Banach's contraction mapping principle to a fundamental fixed point theorem for multivalued mappings.…”

On stability of fixed point inclusion for multivalued type contraction mappings in dislocatedb‐metric spaces with application

“…2,3 One of interesting generalized Banach's contractions is an ℑ-contraction introduced by Wardowski 4 in 2012. After this work of Wardowski, there were other works that studied some fixed point theorems for this contraction (see previous studies [5][6][7][8][9][10][11][12] ). The concept of an -admissible mapping is another one for generalized Banach's contractions.…”