Hilfer Fractional Differential Equations Research Articles

On a Hilfer fractional differential equation with nonlocal Erdélyi-Kober fractional integral boundary conditions

We consider a Hilfer fractional differential equation with nonlocal Erd?lyi-Kober fractional integral boundary conditions. The existence, uniqueness and Ulam-Hyers stability results are investigated by means of the Krasnoselskii?s fixed point theorem and Banach?s fixed point theorem. An example is given to illustrate the main results.

On the theory of fractional terminal value problem with <i>ψ</i>-Hilfer fractional derivative

In this paper, we prove the existence and uniqueness of solutions of a new class of boundary value problems of terminal type for <i>ψ</i>-Hilfer fractional differential equations. The technique used in the analysis relies on the Banach contraction principle and Krasnosleskii fixed point theorem. Moreover, we use generalized Gronwall inequality with singularity to establish uniqueness and continuous dependence of the <i>δ</i>-approximate solution. Finally, we demonstrate some examples to illustrate our main results.

Finite approximate controllability of Hilfer fractional semilinear differential equations

Finite approximate controllability of Hilfer fractional semilinear differential equations

Existence and Attractivity Results for Hilfer Fractional Differential Equations

We present some results on the existence of attracting solutions of some fractional differential equations of the Hilfer type. The results on the existence of solutions are applied to the Schauder fixed-point theorem. It is proved that all solutions are uniformly locally attracting.

Approximate controllability of Hilfer fractional differential equations

In this paper, the approximate controllability for a class of Hilfer fractional differential equations (FDEs) of order 1&lt;α&lt;2 and type 0 ≤ β ≤ 1 is considered. The existence and uniqueness of mild solutions for these equations are established by applying the Banach contraction principle. Further, we obtain a set of sufficient conditions for the approximate controllability of these equations. Finally, an example is presented to illustrate the obtained results.

On the Novel Ulam–Hyers Stability for a Class of Nonlinear $$\psi $$-Hilfer Fractional Differential Equation with Time-Varying Delays

In this paper, we present some alternative results concerning the uniqueness and Ulam–Hyers stability of solutions for a kind of $$\psi $$ -Hilfer fractional differential equations with time-varying delays. Under some updated criteria along with the generalized Gronwall inequality, the new constructive results have been established in the literature. The derived analysis has the ability to generalize and improve some other results from the literature. As an application, two typical examples are delineated to demonstrate the effectiveness of our theoretical results.

Continuous dependence solutions for Hilfer fractional differential equations with nonlocal conditions

Continuous dependence solutions for Hilfer fractional differential equations with nonlocal conditions

On the nonlinear $$\varvec{\varPsi }$$ Ψ -Hilfer fractional differential equations

We consider the nonlinear Cauchy problem for $ \Psi $- Hilfer fractional differential equations and investigate the existence, interval of existence and uniqueness of solution in the weighted space of functions. The continuous dependence of solutions on initial conditions is proved via Weissinger fixed point theorem. Picard's successive approximation method has been developed to solve nonlinear Cauchy problem for differential equations with $ \Psi $- Hilfer fractional derivative and an estimation have been obtained for the error bound. Further, by Picard's successive approximation, we derive the representation formula for the solution of linear Cauchy problem for $ \Psi $-Hilfer fractional differential equation with constant coefficient and variable coefficient in terms of Mittag-Leffler function and Generalized (Kilbas-Saigo) Mittag-Leffler function.

A class of Hilfer fractional stochastic differential equations and optimal controls

In this paper, we investigate a class of Hilfer fractional stochastic differential equations with nonlocal conditions. We first study the existence of mild solutions of these equations by means of stochastic analysis theory, fractional calculations, and operator semigroup theory. Further, the existence of optimal pairs for the corresponding Lagrange control systems is investigated. Finally, an example is presented to illustrate our obtained results.

An Arbitrary Order Differential Equations on Times Scale

Here existence and stability results of $\psi$-Hilfer fractional differential equations on time scales is obtained. Here sufficient condition for existence and uniqueness of solution by using Schauder's fixed point theorem (FPT) and Banach FPT is produced. In addition, generalized Ulam stability of the proposed problem is also discussed. problem.

Impulsive Hilfer fractional differential equations

Existence and controllability results for nonlinear Hilfer fractional differential equations are studied. Sufficient conditions for existence and approximate controllability for Sobolev-type impulsive fractional differential equations are established, where the time fractional derivative is the Hilfer derivative. An example for Sobolev-type Hilfer fractional delay partial differential equation with impulsive condition is considered.

On $\psi$-Hilfer fractional differential equation with complex order

The objectives of this paper is to investigate some adequate results for the existence of solution to a $\psi$-Hilfer fractional derivatives (HFDEs) involving complex order. Appropriate conditions for the existence of at least one solution are developed by using Schauder fixed point theorem (SFPT) to the consider problem. Moreover, we also investigate the Ulam-Hyers stability for the proposed problem.

A survey on Hadamard and Hilfer fractional differential equations: Analysis and stability

In the present survey paper, we prove some results concerning the existence of solutions and weak solutions for some classes of Hadamard and Hilfer fractional differential equations. Some attractivity and Ulam stability results are presented. Our results are provided by applying the fixed point theory, and the technique of strong and weak measures of noncompactness.

Null controllability of nonlocal Hilfer fractional stochastic differential equations

Null controllability of nonlocal Hilfer fractional stochastic differential equations

Existence results for BVP of a class of Hilfer fractional differential equations

In this paper, we consider the existence and uniqueness of solutions to the nonlocal boundary value problem for semi-linear differential equations involving Hilfer fractional derivative. With the help of properties of Hilfer fractional calculus, Mittag-Leffler functions, and fixed point methods, we derive existence and uniqueness results. Finally, examples are given to illustrate our theoretical results.

Fractional Brownian Motions, Fractional Noises and Applications

Fractional Brownian Motions, Fractional Noises and Applications