Notes on Krasnoselskii-type fixed-point theorems and their application to fractional hybrid differential problems

Abstract

Fixed-point theory has experienced quick improvement over the most recent quite a few years. The development has been firmly advanced by the vast number of utilizations in the existence theory of functional, fractional, differential, partial differential, and integral equations. Two fundamental theorems concerning fixed points are those of Schauder and of Banach. The Schauder’s fixed point theorem, involving a compactness condition, may be stated as ”if S is a closed convex and bounded subset of a Banach space X, then every completely continuous operator A : S → S has at least one fixed point”. Note that an operator A on a Banach space X is called completely continuous if it is continuous and A(D) is totally bounded for any bounded subset D of X. Banach’s fixed point theorem, involving a metric assumption on the mapping, states that ”if X is complete metric space and if A is a contraction on X, then it has a unique fixed point, i.e., there is a unique point x∗ ∈ X such that Ax∗ = x∗. Moreover, the sequence Ax converges to x∗ for every x ∈ X,”. The idea of the hybrid fixed point theorems, that is, a blend of the nonlinear contraction principle and Schauder’s fixed-point theorem goes back to 1964, with Krasnoselskii [14], who still maintains an interest in the subject. He gave intriguing applications to differential equations by finding the existence of solutions under some hybrid conditions. Burton [4] extended Krasnoselskii’s result for a wide class of operators in 1998. In 2013, Dhage [6] and Dhage and Lakshmikantham [7] proposed an important Krasnoselskii-type fixed-point