Fractional Differential Inclusions Research Articles

Existence of solutions to Caputo fractional differential inclusions of $ 1 < \alpha < 2 $ with initial and impulsive boundary conditions

<abstract><p>This paper is concerned with the existence of solutions to the Caputo fractional differential inclusion of $ 1 < \alpha < 2 $ with initial and impulsive boundary conditions. A novel existence result is presented based on the fixed-point theorem of Dhage for multi-valued operators with some assumptions. Finally, two examples are provided to explicate the applicability of the main result.</p></abstract>

Existence Results for Nonlinear Fractional Differential Inclusions via q-ROF Fixed Point

Fractional Differential inclusions, the multivalued version of fractional differential equations, yellow play a vital role in various fields of applied sciences. In the present article, a class of q-rung orthopair fuzzy (q-ROF) set valued mappings along with q-ROF upper/lower semi-continuity have been introduced. Based on these ideas, existence theorems for a numerical solution of a distinct class of fractional differential inclusions have been achieved with the help of Schaefer type and Banach contraction fixed point theorems. A physical example is also provided to validate the hypothesis of the main results. The notion of q-rung orthopair fuzzy mappings along with the use of fixed point techniques and a new-fangled Caputo type fractional derivative are the principal novelty of this article.

Atangana–Baleanu Semilinear Fractional Differential Inclusions With Infinite Delay: Existence and Approximate Controllability

Abstract The main focus of this paper is centered around approximate controllability results of Atangana–Baleanu fractional differential systems with infinite delay. Using principles and ideas from the theory of multivalued maps, fractional calculus, and Bohnenblust–Karlin fixed point techniques, the key findings are established. We begin by emphasizing the existence of mild solutions, and then demonstrate the approximate controllability of the Atangana–Baleanu fractional control system. We then apply our findings to the theory of the neutral system.

Existence study of semilinear fractional differential equations and inclusions for multi–term problem under Riemann–Liouville operators

In the present paper, we study the existence and uniqueness of the solution for multi–term fractional boundary value problem under Riemann–Liouville fractional operators by using Banach's fixed point theorem. Afterward, we investigate some existence results for a semilinear fractional differential inclusions multi–term problem by using some notions and properties on set‐valued maps together with classical fixed point theorems due to Leray–Schauder. The cases when the set‐valued function has convex as well as nonconvex values are considered. Finally, some examples are given to illustrate our main results.

Analysis and Applications of Sequential Hybrid $$\psi $$-Hilfer Fractional Differential Equations and Inclusions in Banach Algebra

Analysis and Applications of Sequential Hybrid $$\psi $$-Hilfer Fractional Differential Equations and Inclusions in Banach Algebra

Null Controllability of Hilfer Fractional Stochastic Differential Inclusions

This paper gives the null controllability for nonlocal stochastic differential inclusion with the Hilfer fractional derivative and Clarke subdifferential. Sufficient conditions for null controllability of nonlocal Hilfer fractional stochastic differential inclusion are established by using the fixed-point approach with the proof that the corresponding linear system is controllable. Finally, the theoretical results are verified with an example.

A sufficient condition for local controllability of a Caputo type fractional differential inclusion

Abstract We consider a Cauchy problem for a fractional differential inclusion defined by a Caputo type fractional derivative and we obtain a sufficient condition for local controllability along a reference trajectory in terms of a certain fractional variational differential inclusion associated to the initial problem.

Development on a Fractional Hybrid Differential Inclusion with a Nonlinear Nonlocal Fractional-Order Integral Inclusion

In this article, we consider a Riemann–Liouville fractional-order nonlinear hybrid delay differential inclusion with a nonlinear set-valued nonlocal integral condition of fractional order. We prove some existence and uniqueness results in C(I,R). We also study the continuous dependence of the solutions on the two sets of selections of the two set-valued functions, considered in our problem, and on some other parameters. Finally, to validate our results, we present an example and some particular cases.

Existence and controllability of fractional evolution inclusions with impulse and sectorial operator

Many evolutionary operations fromdiverse fields of engineering and physical sciences go through abrupt modifications of state at specific moments of time among periods of non-stop evolution. These operations are more conveniently modeled via impulsive differential equations and inclusions. In this work, firstly we address the existence of mild solutions for nonlocal fractional impulsive semilinear differential inclusions related to Caputo derivative in Banach spaces when the linear part is sectorial. Secondly, we determine the enough, conditions for the controllability of the studied control problem. We apply effectively fixed point theorems, contraction mapping, multivalued analysis and fractional calculus. Moreover, we enhance our results by introducing an illustrative examples.

Observer‐based bipartite Mittag‐Leffler time‐varying formation for fractional multi‐agent systems with discontinuous inherent dynamics

SummaryIn this article, the bipartite Mittag‐Leffler time‐varying formation issue of fractional multi‐agent systems is discussed, where the dynamical nodes of followers are modeled to be discontinuous and nondecreasing. A observer‐based adaptive distributed control scheme, which is dependent on output information, is designed to realized the bipartite formation control goal. Meanwhile, the efficient algorithm with respect to the control protocol also is presented. Under fractional Filippov differential inclusion framework, by constructing Lur'e Postnikov‐type Lyapunov functional and applying Clarke's non‐smooth analysis technique, the bipartite Mittag‐Leffler time‐varying formation conditions are derived in terms of linear matrix inequalities. At last, a simulation example is carried out to verify the effectiveness of the proposed scheme.

Proportional Caputo Fractional Differential Inclusions in Banach Spaces

In this work, we introduce the notion of a (weak) proportional Caputo fractional derivative of order α∈(0,1) for a continuous (locally integrable) function u:[0,∞)→E, where E is a complex Banach space. In our definition, we do not require that the function u(·) is continuously differentiable, which enables us to consider the wellposedness of the corresponding fractional relaxation problems in a much better theoretical way. More precisely, we systematically investigate several new classes of (degenerate) fractional solution operator families connected with the use of this type of fractional derivatives, obeying the multivalued linear approach to the abstract Volterra integro-differential inclusions. The quasi-periodic properties of the proportional fractional integrals as well as the existence and uniqueness of almost periodic-type solutions for various classes of proportional Caputo fractional differential inclusions in Banach spaces are also considered.

Mathematical analysis of impulsive fractional differential inclusion of pantograph type

In this article, we formulate fractional differential inclusion of pantograph type (IFDIP), incorporating impulsive behavior of the solution. The boundary conditions taken into account are nonlocal in nature. We will consider the convex problem and prove the Filippov–Wazewski‐type theorem. Moreover, existence of solution, uniqueness of a solution, and the topological properties of the solution's set will be examined for the problem under consideration. In the second part, the study will be confined to the second‐order impulsive fractional differential equation of pantograph type. For certain geometric characteristics of the solution's set, Aronszajn–Browder–Gupta‐type results will be explored for the newly introduced differential equation. Also, it will prove the existence of solution for the first‐order fractional differential equation of pantograph type having impulsive behavior of the solution.

Discussion on the existence of mild solution for fractional derivative by Mittag–Leffler kernel to fractional stochastic neutral differential inclusions

Fractional calculus is now used to accurately depict a range of real occurrences because it can explain the “long-tail memory” phenomena that have been seen through empirical research. Standard differential equations with integer order derivatives cannot predict this influence since the future state in this case depends on a number of prior states that is equal to the maximum order of derivatives present in the differential equation. This paper is mainly focusing the existence outcomes of Atangana-Baleanu fractional stochastic systems as well as fractional neutral stochastic systems. The essential findings are developed utilizing ideals and principles of stochastic systems, multivalued map theory, fractional derivative, and fixed point approaches. We start by focusing on the existence of mild solutions for the abstract systems and we extend the analysis to the neutral system. Finally, an illustration is presented to define our primary findings.

Nonlocal Boundary Value Problems for (k,ψ)-Hilfer Fractional Differential Equations and Inclusions

In the present research, single and multi-valued (k,ψ)-Hilfer type fractional boundary value problems of order in (1,2] involving nonlocal integral boundary conditions were studied. In the single-valued case, the Banach and Krasnosel’skiĭ fixed point theorems as well as the Leray–Schauder nonlinear alternative were used to establish the existence and uniqueness results. In the multi-valued case, when the right-hand side of the inclusion has convex values, we established an existence result via the Leray–Schauder nonlinear alternative method for multi-valued maps, while the second existence result, dealing with the non-convex valued right-hand side of the inclusion, was obtained by applying Covitz-Nadler fixed point theorem for multi-valued contractions. The obtained theoretical results are well illustrated by the numerical examples provided.

On (k,ψ)-Hilfer Fractional Differential Equations and Inclusions with Mixed (k,ψ)-Derivative and Integral Boundary Conditions

In this paper we study single-valued and multi-valued (k,ψ)-Hilfer-type boundary value problems of fractional order in (1,2], subject to nonlocal boundary conditions involving (k,ψ)-Hilfer-type derivative and integral operators. The results for single-valued case are established by using Banach and Krasnosel’skiĭ fixed point theorems as well as Leray–Schauder nonlinear alternative. In the multi-valued case, we establish an existence result for the convex valued right-hand side of the inclusion via Leray–Schauder nonlinear alternative for multi-valued maps, while the second one when the right-hand side has non-convex values is obtained by applying Covitz–Nadler fixed point theorem for multi-valued contractions. Numerical examples illustrating the obtained theoretical results are also presented.

(k,ψ)-Hilfer Nonlocal Integro-Multi-Point Boundary Value Problems for Fractional Differential Equations and Inclusions

In this paper, we establish existence and uniqueness results for single-valued as well as multi-valued (k,ψ)-Hilfer boundary value problems of order in (1,2], subject to nonlocal integro-multi-point boundary conditions. In the single-valued case, we use Banach and Krasnosel’skiĭ fixed point theorems as well as a Leray–Schauder nonlinear alternative to derive the existence and uniqueness results. For the multi-valued problem, we prove two existence results for the convex and non-convex nature of the multi-valued map involved in a problem by applying a Leray–Schauder nonlinear alternative for multi-valued maps, and a Covitz–Nadler fixed point theorem for multi-valued contractions, respectively. Numerical examples are presented for illustration of all the obtained results.

Analysis of Fractional Differential Inclusion Models for COVID-19 via Fixed Point Results in Metric Space

We examine in this paper some new problems on coincidence point and fixed point theorems for multivalued mappings in metric space. By applying the characterizations of a modified M T ~ -function, under the name D -function, a few novel fixed point results different from the existing fixed point theorems are launched. It is well-known that differential equation of either integer or fractional order is not sufficient to capture ambiguity, since the derivative of a solution to any differential equation inherits all the regularity properties of the mapping involved and of the solution itself. This does not hold in the case of differential inclusions. In particular, fractional-order differential inclusion models are more suitable for describing epidemics. Thus, as a generalization of a newly launched existence result for fractional-order model for COVID-19, using Banach and Shauder fixed point theorems, we investigate solvability criteria of a novel Caputo-type fractional-order differential inclusion model for COVID-19 by applying a standard fixed point theorem of multivalued contraction. Stability analysis of the proposed model in the framework of Ulam-Hyers is also discussed. Nontrivial comparative illustrations are constructed to show that our ideas herein complement, unify and, extend a significant number of existing results in the corresponding literature.

Optimality of Non-instantaneous Impulsive Fractional Stochastic Differential Inclusion with fBm

Optimality of Non-instantaneous Impulsive Fractional Stochastic Differential Inclusion with fBm

Investigating existence results for fractional evolution inclusions with order r ∈ (1, 2) in Banach space

Abstract This manuscript investigates the issue of existence results for fractional differential evolution inclusions of order r ∈ (1, 2) in the Banach space. In the beginning, we analyze the existence results by referring to the fractional calculations, cosine families, multivalued function, and Martelli’s fixed point theorem. The result is also used to investigate the existence of nonlocal fractional evolution inclusions of order r ∈ (1, 2). Finally, a concrete application is given to illustrate our main results.

On a fractional differential inclusion involving a generalized Caputo type derivative with certain fractional integral boundary conditions

We study a class of fractional differential inclusions defined by Caputo-Katugampola fractional derivativeinvolving a nonconvex set-valued map in the presence of certain fractional integral boundary conditions.Using a technique developed by Filippov we establish an existence result for the problem considered underthe hypothesis that the set-valued map is Lipschitz in the state variable. Also, based on a result concerningthe arcwise connectedness of the fixed point set of a class of set-valued contractions, we prove the arcwiseconnectedness of the solution set of the problem considered. The paper is the first in literature which containssuch kind of results in the framework of the problem studied.