A brief survey on the development and applications of Goebel’s coincidence point theorem in differential and integral equations

Abstract

Abstract Goebel’s coincidence theorem, a remarkably simple extension of Banach’s contraction principle widely applied in analysis, has found significant utility in the theory of differential and integral equations. Over the past fifty years, researchers have endeavored to generalize the definition of a metric space, thereby extending the scope of Goebel’s coincidence theorem to diverse settings. This survey paper overviews valuable insights into Goebel’s coincidence theorem’s historical background, its relevance in the field of fixed point theory, and its practical implications in solving problems related to differential and integral equations.