Solving nonlinear and dynamic programming equations on extended b-metric spaces with the fixed-point technique

Abstract

Abstract In this article, we present an approach to solve a wide range of nonlinear equations formulated in extended b-metric spaces based on a new fixed-point theorem on these spaces. This research effort was motivated by challenges arising in solving pattern problems efficiently that can not be addressed by using standard metric spaces. Our approach relies on a novel common fixed-point theorem for Ćirić-type operators on extended b-metric spaces requiring only very weak assumptions that we present and derive in this article. The proposed approach is illustrated by applications asserting the existence and uniqueness of the solutions to Bellman equations, Volterra integral equations, and fractional differential equations formulated in extended b-metric spaces. Moreover, the obtained results provide general constructive recursive procedures to solve the above types of nonlinear equations.

Keywords

• Single-valued mappings
• Extended b-metric spaces
• Common fixed point
• Nonlinear integral equations
• Fractional differential equations system
• Bellman’s equation