Fixed Point Theorems in Generalized Banach Algebras and an Application to Infinite Systems in 𝒞([0, 1], c₀) × 𝒞([0, 1], c₀)

Abstract

The purpose of this paper is to extend Boyd and Wong’s fixed point theorem in complete generalized metric spaces and to apply this result under the so-called G-weak topology in generalized Banach algebras. Some tools will be introduced and involved in our work as the generalized measures of weak non-compactness and the sequential condition $({\mathcal{P}}_G).$ Our results will extend many known theorems in the literature. Also, we will give an application for an infinite system of nonlinear integral equations defined on the generalized Banach algebra $(\mathcal{C}([0,1],c_0)\times \mathcal{C}([0,1],c_0)),$ where c₀ is the space of all real sequences converging to zero.

Keywords:

• $\varphi$-contractive mappings
• condition $({\mathcal{P}}_G)$
• generalized Banach space
• generalized measures of weak non-compactness
• integral equations system

Mathematics Subject Classification:

• 47H10
• 45G15