Existence, uniqueness, and relaxation results in initial value type problems for nonlinear fractional differential equations

Abstract

This paper is concerned with the nonlinear fractional differential equations with initial values. Applying some new contraction conditions to a new-found Banach space $$L_{p,\alpha ;\psi }$$ , we give a refinement of some important inequalities in the theorems appeared in the literature. Our main work, in this paper, is in two themes. First, the contraction constant which so far was strictly less than one is extended so that it covers the case when it equals one. Second, the continuity condition of the nonlinear term is relaxed. The former is a new approach, while the latter has been done in our previous works and we extend those methods, here. This method may constitute a step forward in helping us to overcome the obstacles, in some practical aims. We state two examples, in this regard, in each direction. We also express two real problems of physics applications which can be converted to the problems to which we could apply our method.