On a class of differential inclusions in the frame of generalized Hilfer fractional derivative

<abstract><p>In the present paper, we extend and develop a qualitative analysis for a class of nonlinear fractional inclusion problems subjected to nonlocal integral boundary conditions (nonlocal IBC) under the $ \varphi $-Hilfer operator. Both claims of convex valued and nonconvex valued right-hand sides are investigated. The obtained existence results of the proposed problem are new in the frame of a $ \varphi $-Hilfer fractional derivative with nonlocal IBC, which are derived via the fixed point theorems (FPT's) for set-valued analysis. Eventually, we give some illustrative examples for the acquired results.</p></abstract>

Uniqueness of continuous solution to $q$-Hilfer fractional hybrid integro-difference equation of variable order

In this paper, the authors introduced a novel definition based on Hilfer fractional derivative, which name $q$-Hilfer fractional derivative of variable order. And the uniqueness of solution to $q$-Hilfer fractional hybrid integro-difference equation of variable order of the form \eqref{eq:varorderfrac} with $0 < \alpha(t) < 1$, $0 \leq \beta \leq 1$, and $0 < q < 1$ is studied. Moreover, an example is provided to demonstrate the result.

Existence Results for Hilfer Fractional Differential Equations with Variable Coefficient

The aim of this paper is to establish the existence and uniqueness results for differential equations of Hilfer-type fractional order with variable coefficient. Firstly, we establish the equivalent Volterra integral equation to an initial value problem for a class of nonlinear fractional differential equations involving Hilfer fractional derivative. Secondly, we obtain the existence and uniqueness results for a class of Hilfer fractional differential equations with variable coefficient. We verify our results by providing two examples.

Integral transforms of the κ-Hilfer fractional derivative

In this paper, some important properties concerning the κ-Hilfer fractional derivative are discussed. Integral transforms for these operators are derived as particular cases of the Jafari transform. These integral transforms are used to derive a fractional version of the fundamental theorem of calculus. Keywords: Integral transforms, Jafari transform, κ-gamma function, κ-beta function, κ-Hilfer fractional derivative, κ-Riesz fractional derivative, κ-fractional operators.

Mild Solutions for Impulsive Integro-Differential Equations Involving Hilfer Fractional Derivative with almost Sectorial Operators

In this manuscript, we establish the mild solutions for Hilfer fractional derivative integro-differential equations involving jump conditions and almost sectorial operator. For this purpose, we identify the suitable definition of a mild solution for this evolution equations and obtain the existence results. In addition, an application is also considered.

ON A CLASS OF FRACTIONAL HARDY-TYPE INEQUALITIES

In this paper, we study a new class of Hardy-type inequalities involving fractional calculus operators. We derive the Hardy-type inequalities for the variant of Riemann–Liouville fractional calculus operators and [Formula: see text]-Hilfer fractional derivative operator. The obtained inequalities involving fractional operators are more general as compared to some existing results in the literature.

Estimates of Mean-Type Fractional Inequalities For Differentiable Functions

In this article, we established a wide range of fractional mean-type integral inequalities for notable Hilfer fractional derivative using twice differentiable convex and $s$-convex functions for $s\in(0,1]$ with related identities. Also the results for Caputo fractional derivatives are derived as a special case of our general results.