Multiple nontrivial solutions of semilinear strongly resonant elliptic equations

Abstract

Fixed point theory is an important and actual topic of nonlinear analysis. Moreover, it’s well known that the contraction mapping principle, formulated and proved in the PhD dissertation of Banach in 1920 which was published in 1922 is one of the most important theorems in classical functional analysis. During the last four decades, this theorem has undergone various generalizations either by relaxing the condition on contractivity or withdrawing the requirement of completeness or sometimes even both. Recently, a very interesting generalization was obtained in [1] by changing the structure of the space itself. In fact, Branciari [1] introduced a concept of generalized metric space by replacing the triangle inequality by a more general inequality. As such, any metric space is a generalized metric space but the converse is not true [1]. He proved the Banach’s fixed point theorem in such a space. For more, the reader can refer to [2–11]. It is also known that common fixed point theorems are generalizations of fixed point theorems. Thus, over the past few decades, there have been many researchers who have interested in generalizing fixed point theorems to coincidence point theorems and common fixed point theorems. In this paper, we prove some common fixed point theorems for a larger class of α-ψφ-contractions in generalized metric spaces and improve the results obtained by Lakzian and Samet [12] and Di Bari and Vetro [13].