Abstract
The objective of this paper is to study the existence of extremal solutions for nonlinear boundary value problems of fractional differential equations involving the ψ−Caputo derivative CDa+σ;ψϱ(t)=V(t,ϱ(t)) under integral boundary conditions ϱ(a)=λIν;ψϱ(η)+δ. Our main results are obtained by applying the monotone iterative technique combined with the method of upper and lower solutions. Further, we consider three cases for ψ*(t) as t, Caputo, 2t, t, and Katugampola (for ρ=0.5) derivatives and examine the validity of the acquired outcomes with the help of two different particular examples.
Highlights
The notion of fractional calculus refers to the last three centuries and it can be described as the generalization of classical calculus to orders of integration and differentiation that are not necessarily integers
We investigated the existence of solutions for a nonlinear fractional differential equation (FDE) in the frame of the ψ−Caputo derivative with integral boundary conditions
We constructed mis that uniformly converged to the extremal solutions of BVP
Summary
The notion of fractional calculus refers to the last three centuries and it can be described as the generalization of classical calculus to orders of integration and differentiation that are not necessarily integers. One of the most recent definitions of a fractional derivative was delivered by Kilbas et al, where the fractional differentiation of a function with respect to another function in the sense of Riemann–Liouville was introduced [5]. They further defined appropriate weighted spaces and studied some of their properties by using the corresponding fractional integral. In [6], Almaida defined the following new fractional derivative and integrals of a function with respect to some other function: Licensee MDPI, Basel, Switzerland
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