Integro-differential Inclusions Research Articles

On S-asymptotically $$\omega$$-periodic mild solutions of some integrodifferential inclusions of Volterra-type

On S-asymptotically $$\omega$$-periodic mild solutions of some integrodifferential inclusions of Volterra-type

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.

A new conversation on the existence of Hilfer fractional stochastic Volterra–Fredholm integro-differential inclusions via almost sectorial operators

The existence of Hilfer fractional stochastic Volterra–Fredholm integro-differential inclusions via almost sectorial operators is the topic of our paper. The researchers used fractional calculus, stochastic analysis theory, and Bohnenblust–Karlin’s fixed point theorem for multivalued maps to support their findings. To begin with, we must establish the existence of a mild solution. In addition, to show the principle, an application is presented.

Almost Periodic Solutions of Abstract Impulsive Volterra Integro-Differential Inclusions

In this paper, we introduce and systematically analyze the classes of (pre-)(B,ρ,(tk))-piecewise continuous almost periodic functions and (pre-)(B,ρ,(tk))-piecewise continuous uniformly recurrent functions with values in complex Banach spaces. We weaken substantially, or remove completely, the assumption that the sequence (tk) of possible first kind discontinuities of the function under consideration is a Wexler sequence (in order to achieve these aims, we use certain results about Stepanov almost periodic type functions). We provide many applications in the analysis of the existence and uniqueness of almost periodic type solutions for various classes of the abstract impulsive Volterra integro-differential inclusions.

Abstract Impulsive Volterra Integro-Differential Inclusions

In this work, we provide several applications of (a,k)-regularized C-resolvent families to the abstract impulsive Volterra integro-differential inclusions. The resolvent operator families under our consideration are subgenerated by multivalued linear operators, which can degenerate in the time variable. The use of regularizing operator C seems to be completely new within the theory of the abstract impulsive Volterra integro-differential equations.

A computational method for investigating a quantum integrodifferential inclusion with simulations and heatmaps

<abstract><p>We aim to investigate an integro-differential inclusion using a novel computational approach in this research. The use of quantum calculus, and consequently the creation of discrete space, allows the computer and computational algorithms to solve our desired problem. Furthermore, to guarantee the existence of the solution, we use the endpoint property based on fixed point methods, which is one of the most recent techniques in fixed point theory. The above will show the novelty of our work, because most researchers use classical fixed point techniques in continuous space. Moreover, the sensitivity of the parameters involved in controlling the existence of the solution can be recognized from the heatmaps. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables and some figures in our examples that are presented at the end of the work.</p></abstract>

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.

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

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.

Analysis of a hybrid integro-differential inclusion

Our main objectives in this paper are to investigate the existence of the solutions for an integro-differential inclusion of second order with hybrid nonlocal boundary value conditions. The sufficient condition for the uniqueness of the solution will be given and the continuous dependence of the solution on the set of selections and on other functions will be proved. As an application, the nonlocal problem of the Chandrasekhar hybrid second-order functional integrodifferential inclusion and some particular cases will be presented. Also, we provide some examples to illustrate our results.

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.

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.

"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."

Solvability of a State–Dependence Functional Integro-Differential Inclusion with Delay Nonlocal Condition

There is great focus on phenomena that depend on their past history or their past state. The mathematical models of these phenomena can be described by differential equations of a self-referred type. This paper is devoted to studying the solvability of a state-dependent integro-differential inclusion. The existence and uniqueness of solutions to a state-dependent functional integro-differential inclusion with delay nonlocal condition is studied. We, moreover, conclude the existence of solutions to the problem with the integral condition and the infinite-point boundary one. Some properties of the solutions are given. Finally, two examples illustrating the main result are presented.

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.

Optimal control of nonconvex integro-differential sweeping processes

This paper is devoted to the study, for the first time in the literature, of optimal control problems for sweeping processes governed by integro-differential inclusions of the Volterra type with different classes of control functions acting in nonconvex moving sets, external dynamic perturbations, and integral parts of the sweeping dynamics. We establish the existence of optimal solutions and then obtain necessary optimality conditions for a broad class of local minimizers in such problems. Our approach to deriving necessary optimality conditions is based on the method of discrete approximations married to basic constructions and calculus rules of first-order and second-order variational analysis and generalized differentiation. The obtained necessary optimality conditions are expressed entirely in terms of the problem data and are illustrated by nontrivial examples that include applications to optimal control models of non-regular electrical circuits.

Existence results for Caputo fractional mixed Volterra-Fredholm-type integrodifferential inclusions of order r ∈ (1, 2) with sectorial operators

Fractional differential equations are used in many areas, such as the fields of science. In this paper, we study existence results for a class of fractional mixed Volterra-Fredholm-type integrodifferential inclusions of order 1 <r < 2 with the sectorial operator in Banach space. Firstly, a more appropriate concept for mild solutions is introduced. Secondly, the existence of mild solution is investigated by using the fractional calculus, multivalued map, and fixed point theorem with the sectorial operator of type (P,η,r,γ). Further, the result is extended to obtain the conditions for the existence of fractional mixed Volterra-Fredholm-type integrodifferential inclusions with nonlocal conditions. Finally, an application is developed to illustrate the theory of the obtained result.

Solvability for a Class of Integro-Differential Inclusions Subject to Impulses on the Half-Line

In this paper, we study a semilinear integro-differential inclusion in Banach spaces, under the action of infinitely many impulses. We provide the existence of mild solutions on a half-line by means of the so-called extension-with-memory technique, which consists of breaking down the problem in an iterate sequence of non-impulsive Cauchy problems, each of them originated by a solution of the previous one. The key that allows us to employ this method is the definition of suitable auxiliary set-valued functions that imitate the original set-valued nonlinearity at any step of the problem’s iteration. As an example of application, we deduce the controllability of a population dynamics process with distributed delay and impulses. That is, we ensure the existence of a pair trajectory-control, meaning a possible evolution of a population and of a feedback control for a system that undergoes sudden changes caused by external forces and depends on its past with fading memory.