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.

Existence and Uniqueness of the Mild Solution of an Abstract Semilinear Fractional Differential Equation with State Dependent Nonlocal Condition

The purpose of this paper is to investigate the existence and uniqueness of mild solutions to a semilinear Cauchy problem for an abstract fractional differential equation with state dependent nonlocal condition. Continuous dependence of solutions on initial conditions and local ????-approximate mild solution of the considered problem will be discussed.

On solutions of nonlinear BVPs with general boundary conditions by using a generalized Riesz–Caputo operator

In this work, we study the existence, uniqueness, and continuous dependence of solutions for a class of fractional differential equations by using a generalized Riesz fractional operator. One can view the results of this work as a refinement for the existence theory of fractional differential equations with Riemann–Liouville, Caputo, and classical Riesz derivative. Some special cases can be derived to obtain corresponding existence results for fractional differential equations. We provide an illustrated example for the unique solution of our main result.

Criteria for existence of solutions for a Liouville–Caputo boundary value problem via generalized Gronwall’s inequality

In this research, we first investigate the existence of solutions for a new fractional boundary value problem in the Liouville–Caputo setting with mixed integro-derivative boundary conditions. To do this, Kuratowski’s measure of noncompactness and Sadovskii’s fixed point theorem are our tools to reach this aim. In the sequel, we discuss the continuous dependence of solutions on parameters by means of the generalized Gronwall inequality. Moreover, we consider an inclusion version of the given boundary problem in which we study its existence results by means of the endpoint theory. Finally, we prepare two simulative numerical examples to confirm the validity of the analytical findings.

The mixed problem in the theory of strain gradient thermoelasticity approached with the Lagrange identity

In our paper we address the thermoelasticity theory of the strain gradient. First, we define the mixed problem with initial and boundary data in this context. Then, with the help of an identity of Lagrange type, we prove some uniqueness theorems with regards to the solution of this problem and two theorems with regards to the continuous dependence of solutions on loads and on initial data. We want to highlight that the use of the approach proposed in this work enables obtaining results without recourse to any boundedness assumptions on the coefficients or to any laws of conservation of energy. Also, we do not impose restrictions on thermoelastic coefficients regarding their positive definition.

Stability in Kelvin–Voigt poroelasticity

Hölder continuous dependence of solutions upon the initial data is established for the linear theory of Kelvin–Voigt poroelasticity requiring only symmetry conditions upon the elastic coefficients. A novel functional is introduced to which a logarithmic convexity technique is employed.

Existence and Compactness Results for a System of Fractional Differential Equations

The existence and uniqueness, boundedness, and continuous dependence of solutions for fractional differential equations with Caputo fractional derivative is proven by Perov’s fixed point theorem in vector Banach spaces. We study the existence and compactness of solution sets and the u.s.c. of operator solutions.