On fractional boundary value problems involving fractional derivatives with Mittag-Leffler kernel and nonlinear integral conditions

Abstract

In this paper, we consider two classes of boundary value problems for nonlinear implicit differential equations with nonlinear integral conditions involving Atangana–Baleanu–Caputo fractional derivatives of orders $0<\vartheta \leq 1$ 0 < ϑ ≤ 1 and $1<\vartheta \leq 2$ 1 < ϑ ≤ 2 . We structure the equivalent fractional integral equations of the proposed problems. Further, the existence and uniqueness theorems are proved with the aid of fixed point theorems of Krasnoselskii and Banach. Lastly, the paper includes pertinent examples to justify the validity of the results.

On boundary value problems of Caputo fractional differential equation of variable order via Kuratowski MNC technique

“…Author details 1 Laboratory ACEDP, Djillali Liabes University, Sidi Bel Abbès, Algeria. 2 Department of Economic Sciences, University of Tiaret, Tiaret, Algeria. 3 Department of Mathematics, Çankaya University, 06790, Etimesgut, Ankara, Turkey.…”

Right Fractional Sobolev Space via Riemann–Liouville Derivatives on Time Scales and an Application to Fractional Boundary Value Problem on Time Scales

“…Although many excellent results have been obtained based on the existence of solutions for fractional boundary value problems [34][35][36][37][38][39][40] and the second-order Hamiltonian systems on time scale T [41][42][43][44][45], it seems that no similar results have been obtained in the literature for FBVP (26) on time scales. The present section seeks to show that the critical point theory is an effective approach to deal with the existence of solutions for FBVP Theorem 26 on time scales.…”

Analysis of a fractional boundary value problem involving Riesz-Caputo fractional derivative

“…For the development of FC, there are sundry common denitions of fractional derivatives and integrals, such as Rimann-Liouville type, Caputo type, Hadamard type, Hilfer type, ψ-Caputo, ψ-Hilfer type, Caputo-Fabrizio type, Atangana-Baleanu type, conformable type, and Erdelyi-Kober type, etc, (see [11,15,16,25,32,33,37,3]). Some recent contributions have been investigated the existence and uniqueness of solutions for dierent kinds of nonlinear fractional dierential equations (FDEs) and inclusion (FDIs) by using various types of xed point theorems, which can be found in [13,6,17,7,39,38,8,19,20,21,4,5,1,2,18], and the references cited therein. The study of FDEs or FDIs with anti-periodic boundary conditions, that are applied in numerous dierent elds, like chemical engineering, physics, economics, dynamics, etc., has received much attention recently, (see [23,27,40,10,26]) and the papers mentioned therein.…”