Compact and Noncompact Solutions to Generalized Sturm–Liouville and Langevin Equation with Caputo–Hadamard Fractional Derivative

In this work, through using the Caputo–Hadamard fractional derivative operator with three nonlocal Hadamard fractional integral boundary conditions, a new type of the fractional-order Sturm–Liouville and Langevin problem is introduced. The existence of solutions for this nonlinear boundary value problem is theoretically investigated based on the Krasnoselskii in the compact case and Darbo fixed point theorems in the noncompact case with aiding the Kuratowski’s measure of noncompactness. To demonstrate the applicability and validity of the main gained findings, some numerical examples are included.

Lyapunov-type inequalities for differential equation with Caputo–Hadamard fractional derivative under multipoint boundary conditions

In this work, we establish Lyapunov-type inequalities for the fractional boundary value problems with Caputo–Hadamard fractional derivative subject to multipoint and integral boundary conditions. As far as we know, there is no literature that has studied these problems.

L 1-solutions of the initial value problems for implicit differential equations with Hadamard fractional derivative

In this paper, we study the existence of integrable solutions for initial value problems for fractional order implicit differential equations with Hadamard fractional derivative. Our results are based on Schauder’s fixed point theorem and the Banach contraction principle fixed point theorem.

Chaos Detection of the Chen System with Caputo–Hadamard Fractional Derivative

In this paper, we investigate the chaotic behaviors of the Chen system with Caputo–Hadamard derivative. First, we construct some practical numerical schemes for the Chen system with Caputo–Hadamard derivative. Then, by means of the variational equation, we estimate the bounds of the Lyapunov exponents for the considered system. Furthermore, we analyze the dynamical evolution of the Chen system with Caputo–Hadamard derivative based on the Lyapunov exponents, and we found that chaos does exist in the considered system. Some phase diagrams and Lyapunov exponent spectra are displayed to verify our analysis.

On an extension of Hadamard fractional derivative

In this paper, we introduce a new approach to the fractional derivation which generalizes the classical Hadamard fractional derivative. We prove some properties of this new approach and also establish some results by addressing some standard functions.

A study of fractional integro-differential equations via Hilfer-Hadamard fractional derivative

In this paper, we investigate the existence of solution of integro-differential equations (IDEs) with Hilfer-Hadamard fractional derivative. The main results are obtained by using Schaefer’s fixed point theorem. Some Ulam stability results are presented.