On a nonlocal implicit problem under Atangana–Baleanu–Caputo fractional derivative

In this paper, we study a class of initial value problems for a nonlinear implicit fractional differential equation with nonlocal conditions involving the Atangana–Baleanu–Caputo fractional derivative. The applied fractional operator is based on a nonsingular and nonlocal kernel. Then we derive a formula for the solution through the equivalent fractional functional integral equations to the proposed problem. The existence and uniqueness are obtained by means of Schauder’s and Banach’s fixed point theorems. Moreover, two types of the continuous dependence of solutions to such equations are discussed. Finally, the paper includes two examples to substantiate the validity of the main results.

An Equivalent Linear Programming Form of General Linear Fractional Programming: A Duality Approach

Linear fractional programming has been an important planning tool for the past four decades. The main contribution of this study is to show, under some assumptions, for a linear programming problem, that there are two different dual problems (one linear programming and one linear fractional functional programming) that are equivalent. In other words, we formulate a linear programming problem that is equivalent to the general linear fractional functional programming problem. These equivalent models have some interesting properties which help us to prove the related duality theorems in an easy manner. A traditional data envelopment analysis (DEA) model is taken, as an instance, to illustrate the applicability of the proposed approach.

The method of upper and lower solutions for a class of fractional differential coupled systems

In this paper, we investigate a class of nonlocal boundary value problems of nonlinear fractional functional differential coupled systems with state dependent delays. The method of upper and lower solutions is established and some new results for the multiplicity of solutions of the boundary value problem are obtained. An example is also presented to illustrate our main results.

On a study of Sobolev type fractional functional evolution equations

Sobolev type fractional functional evolution equations have many applications in the modeling of many physical processes. Therefore, we investigate fractional-order time-delay evolution equation of Sobolev type with multi-orders in a Banach space and introduce an analytical representation of a mild solution via a new delayed Mittag-Leffler type function which is generated by linear bounded operators. Furthermore, we derive an exact analytical representation of solutions for multi-dimensional fractional functional dynamical systems with nonpermutable and permutable matrices. We also study stability analysis of the given time-delay system in Ulam-Hyers sense with the help of Laplace transform.

Faedo-Galerkin approximation of mild solutions of fractional functional differential equations

In the paper, we discuss the existence and uniqueness of mild solutions of a class of fractional functional differential equations in Hilbert space separable using the Banach fixed point theorem technique. In this sense, Faedo-Galerkin approximation to the solution is studied and demonstrated some convergence results.