Fractional Integro-differential Inclusions Research Articles

Existence of a mild solution and approximate controllability for fractional random integro-differential inclusions with non-instantaneous impulses

This paper investigates the existence and approximate controllability (ACA) of fractional neutral-type stochastic differential inclusions (NTSDIs) characterized by non-instantaneous impulses within a separable Hilbert space (HS) framework. Employing the Atangana–Baleanu–Caputo (ABC) derivative, we transform the system into an equivalent fixed-point (FP) problem through an integral operator. Subsequently, the Bohnenblust–Karlin FP theorem is leveraged to establish existence results. By assuming ACA of the corresponding linear system, we derive sufficient conditions for the ACA of the nonlinear stochastic impulsive control system. Our analysis relies on concepts from stochastic analysis, fractional calculus, FP theory, semigroup theory, and the theory of multivalued maps (MVMs). The theoretical findings are illustrated through a concrete example.

On controllability for Sobolev-type fuzzy Hilfer fractional integro-differential inclusions with Clarke subdifferential

In this paper, our main purpose is to search and obtain the controllability for Sobolev-type fuzzy Hilfer fractional integro-differential inclusions with Clarke subdifferential. Some sufficient conditions for the controllability results of this inclusion problem are proposed by using related techniques of fuzzy set theory, Sobolev-type, fractional calculus and Clarke subdifferential. The theorem of the controllability results is proved by Bohnenblust–Karlin fixed point theorem. In addition, we show an example to explain the controllability results of this inclusion problem.

A study of nonlocal fractional neutral stochastic integrodifferential inclusions of order 1

This paper considers a class of nonlocal fractional neutral stochastic integrodifferential inclusions of order 1 < α < 2 with impulses in a Hilbert space. We study the existence of the mild solution for the cases when the multi-valued map has convex and non-convex values. The results are obtained by combining fixed-point theorems with the fractional order cosine family, semigroup theory, and stochastic techniques. A new set of sufficient conditions is developed to demonstrate the approximate controllability of the system. Finally, an example is given to illustrate the obtained results.

New exploration on approximate controllability of fractional neutral‐type delay stochastic differential inclusions with non‐instantaneous impulse

This paper aims to derive a new set of sufficient conditions for the existence and approximate controllability of neutral‐type fractional stochastic integrodifferential inclusions with infinite delay and non‐instantaneous impulse in a separable Hilbert space using the Atangana–Baleanu Caputo fractional derivative. We investigate the existence of a mild solution for the Atangana–Baleanu Caputo fractional neutral‐type delay integrodifferential stochastic system while taking into account the non‐instantaneous impulses. For this purpose, the Atangana–Baleanu Caputo fractional neutral‐type impulsive delay stochastic system is transferred into an equivalent fixed point problem via an integral operator, and then, the Bohnenblust–Karlin fixed point approach is applied. Further, the approximate controllability results of the proposed nonlinear stochastic impulsive control system are established under the consideration that the corresponding linear system is approximately controllable. The set of sufficient conditions is established by using the concepts of stochastic analysis, fractional calculus, fixed point technique, semigroup theory of bounded linear operators, and the theory of multivalued maps. To illustrate the abstract results, we provide an example at the end of the paper.

Weak solutions of a fractional orders quadratic functional integro-differential inclusion in a reflexive Banach space

Weak solutions of a fractional orders quadratic functional integro-differential inclusion in a reflexive Banach space

On a Hilfer generalized proportional fractional integro-differential inclusion

Existence of solutions for a class of boundary value problems associated to a Hilfer generalized proportional fractional integro-differential inclusion is studied. A new result is obtained when the right hand side has nonconvex values.

Existence Results on Mittag-Leffler Kernel of Fractional Integro-Differential Inclusion Problem Under Boundary Conditions

Existence Results on Mittag-Leffler Kernel of Fractional Integro-Differential Inclusion Problem Under Boundary Conditions

On a Hilfer type fractional integro-differential inclusion

A boundary value problem associated to a Hilfer generalized proportional fractional integro-differential inclusion is studied. The existence of solutions is established in the case when the set-valued map has nonconvex values.

Sobolev‐type existence results for impulsive nonlocal fractional stochastic integrodifferential inclusions of order ϱ∈(1,2) with infinite delay via sectorial operator

The impulsive effect on the fractional stochastic integrodifferential system of order with infinite delay via sectorial operator is investigated in this work. The primary findings of this study are tested by referring to multivalued functions, sectorial operators, fractional derivatives, and the fixed point approach. Initially, the presence of a mild solution for the systems class was introduced. The existence outcomes of semilinear control problems were then established. Eventually, an application is given for deriving the theory of the key findings.

A study on the approximate controllability results of fractional stochastic integro-differential inclusion systems via sectorial operators

The study deals with the findings of the outcome of the approximate controllability results of inclusion type fractional stochastic system in Banach space with the order of the fractional system varrho in (1,2). At first, we implement Bohnenblust-Karlin's fixed point technique to deduce the required conditions on which the fractional system with inital conditions is approximately controllable, and there by, we postulate the sufficient conditions for extending the obtained results to the system with nonlocal conditions.

Approximate controllability results for the Sobolev type fractional delay impulsive integrodifferential inclusions of order $${r} \in (1,2)$$ via sectorial operator

In this study, we investigate nonlocal problems for a class of the Sobolev type fractional delay impulsive integrodifferential inclusions of order $${r} \in (1,2)$$ via sectorial operator in Hilbert space. The results are obtained by engaging of mixed Volterra-Fredholm integrodifferential systems, sectorial operator of type $$(P,\kappa ,{r},\gamma )$$ and Martelli’s fixed point theorem under the assumption that the corresponding linear system is approximately controllable. Finally, an example is given to illustrate the effectiveness of the results obtained.

Existence results of Atangana-Baleanu fractional integro-differential inclusions of Sobolev type

In this article, we scrutinise the study of the existence of Sobolev type Atangana-Baleanu fractional integro-differential inclusions in Banach space. The results are gained by using Martelli’s fixed point theorem and ρ-resolvent operators. An example is given for theoretical result.

Discussion on the Approximate Controllability of Hilfer Fractional Neutral Integro-Differential Inclusions via Almost Sectorial Operators

This paper focuses on the approximate controllability of Hilfer fractional neutral Volterra integro-differential inclusions via almost sectorial operators. Almost sectorial operators, fractional differential, Leray-Schauder fixed point theorem and multivalued maps are used to prove the result. We start by emphasizing the existence of a mild solution and demonstrate the approximate controllability of the fractional system. In addition, an example is presented to demonstrate the principle.

A Note on the Existence and Controllability Results for Fractional Integrodifferential Inclusions of Order $$r \in (1,2]$$ with Impulses

A Note on the Existence and Controllability Results for Fractional Integrodifferential Inclusions of Order $$r \in (1,2]$$ with Impulses

Existence of Mild Solutions for Hilfer Fractional Neutral Integro-Differential Inclusions via Almost Sectorial Operators

This manuscript focuses on the existence of a mild solution Hilfer fractional neutral integro-differential inclusion with almost sectorial operator. By applying the facts related to fractional calculus, semigroup, and Martelli’s fixed point theorem, we prove the primary results. In addition, the application is provided to demonstrate how the major results might be applied.

"On certain boundary value problems associated to some fractional integro-differential inclusions"

"Two classes of fractional integro-differential inclusions with certain boundary conditions are studied. The existence of solutions is established in the case when the set-valued map has nonconvex values."

Hilfer Fractional Neutral Stochastic Volterra Integro-Differential Inclusions via Almost Sectorial Operators

In our paper, we mainly concentrate on the existence of Hilfer fractional neutral stochastic Volterra integro-differential inclusions with almost sectorial operators. The facts related to fractional calculus, stochastic analysis theory, and the fixed point theorem for multivalued maps are used to prove the result. In addition, an illustration of the principle is provided.

On the approximate controllability results for fractional integrodifferential systems of order [formula omitted] with sectorial operators

In this paper, we investigate the approximate controllability for fractional integrodifferential inclusions of order r∈(1,2) in Banach space with sectorial operators. In particular, we obtain a new set of sufficient conditions for the approximate controllability of nonlinear fractional integrodifferential inclusions of order r∈(1,2) under the assumption that the corresponding linear system is approximately controllable. In addition, we establish the approximate controllability results for the Sobolev type fractional control system with nonlocal conditions. The results are obtained with the help of fractional derivatives, sectorial operators of type (P,η,r,γ), multivalued functions, and Bohnenblust-fixed Karlin’s point theorem. Moreover, an example is also provided to illustrate the effectiveness of the results.

Approximate controllability of neutral delay integro-differential inclusion of order $ \alpha\in (1, 2) $ with non-instantaneous impulses

&lt;p style='text-indent:20px;'&gt;This paper aims to establish the approximate controllability results for fractional neutral integro-differential inclusions with non-instantaneous impulse and infinite delay. Sufficient conditions for approximate controllability have been established for the proposed control problem. The tools for study include the fixed point theorem for discontinuous multi-valued operators with the &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \alpha- $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;resolvent operator. Finally, the proposed results are illustrated with the help of an example.&lt;/p&gt;

A note on approximate controllability for nonlocal fractional evolution stochastic integrodifferential inclusions of order [formula omitted] with delay

In this paper, we formulate a new set of sufficient conditions for the approximate controllability of fractional evolution stochastic integrodifferential delay inclusions of order r∈(1,2) with nonlocal conditions in Hilbert space. Martelli’s fixed point theorem, multivalued functions, cosine and sine families, fractional calculus and operator semigroups are used to establish the results under the assumption that the corresponding linear system is approximately controllable. Finally, an example is provided to illustrate the applicability of the obtained theoretical results.