Integro-differential Inclusions Research Articles

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.

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

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M3">\begin{document}$ \alpha- $\end{document}</tex-math></inline-formula>resolvent operator. Finally, the proposed results are illustrated with the help of an example.</p>

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.

A Study of a More General Class of Nonlocal Integro-Multipoint Boundary-Value Problems for Fractional Integrodifferential Inclusions

A Study of a More General Class of Nonlocal Integro-Multipoint Boundary-Value Problems for Fractional Integrodifferential Inclusions

Solutions of semi-linear stochastic evolution integro-differential inclusions with Poisson jumps and non-local initial conditions

In this paper, we mainly investigate properties on solution sets of a semi-linear stochastic evolution integro-differential inclusion with Poisson jumps and non-local initial conditions. We firstly establish the existence of solutions to the addressed equation via weak topology technique. Through establishing the estimate on noncompactness measure of integrals for Poisson jumps, we further show that the solution set is a compact -set when the resolvent operator is compact or non-compact, respectively.

On a new class of Atangana-Baleanu fractional Volterra-Fredholm integro-differential inclusions with non-instantaneous impulses

• The existence of piecewise-continuous mild solution of Atangana-Baleanu fractional Volterra-Fredholm integro-differential inclusions with non-instantaneous impulses in Banach space is analyzed. • The Martelli’s fixed point theorem and r-resolvent operators are used. • An example is given to support the theoretical results. • The topic allows the deep theoretical investigation AB-derivatives. This manuscripts main objective is to examine the existence of piecewise-continuous mild solution of Atangana-Baleanu fractional Volterra-Fredholm integro-differential inclusions (ABFVFIDI) with non-instantaneous impulses (NII) in Banach space. Based on Martelli’s fixed point theorem and ρ -resolvent operators, we develop the main results. An example is given to support the validation of the theoretical results achieved.

A study of a more general class of nonlocal integro-multipoint boundary-value problems of fractional integro-differential inclusions

UDC 517.9We develop the existence theory for a more general class of nonlocal integro-multipoint boundary value problems ofCaputo type fractional integro-differential inclusions. Our results include the convex and non-convex cases for the givenproblem and rely on standard fixed point theorems for multivalued maps. The obtained results are illustrated with the aidof examples. The paper concludes with some interesting observations.

On a class of partial fractional integro-differential inclusions

On a class of partial fractional integro-differential inclusions

Integrated solutions of non-densely defined semilinear integro-differential inclusions: existence, topology and applications

Given a linear closed but not necessarily densely defined operator on a Banach space with nonempty resolvent set and a multivalued map with weakly sequentially closed graph, we consider the integro-differential inclusion We focus on the case when generates an integrated semigroup and obtain existence of integrated solutions if is weakly compactly generated and satisfies where and denotes the De Blasi measure of noncompactness. When is separable, we are able to show that the set of all integrated solutions is a compact -subset of the space endowed with the weak topology. We use this result to investigate a nonlocal Cauchy problem described by means of a nonconvex-valued boundary condition operator. We also include some applications to partial differential equations with multivalued terms are. Bibliography: 26 titles.

On some evolution inclusions in non separable Banach spaces

We study a Cauchy problem of a class of nonconvex second-order integro-differential inclusions and a boundary value problem associated to a semilinear evolution inclusion defined by nonlocal conditions in non-separable Banach spaces. The existence of mild solutions is established under Filippov type assumptions.

Almost periodic type functions and densities

<p style='text-indent:20px;'>In this paper, we introduce and analyze the notions of <inline-formula><tex-math id="M1">\begin{document}$ \odot_{g} $\end{document}</tex-math></inline-formula>-almost periodicity and Stepanov <inline-formula><tex-math id="M2">\begin{document}$ \odot_{g} $\end{document}</tex-math></inline-formula>-almost periodicity for functions with values in complex Banach spaces. In order to do that, we use the recently introduced notions of lower and upper (Banach) <inline-formula><tex-math id="M3">\begin{document}$ g $\end{document}</tex-math></inline-formula>-densities. We also analyze uniformly recurrent functions, generalized almost automorphic functions and apply our results in the qualitative analysis of solutions of inhomogeneous abstract integro-differential inclusions. We present plenty of illustrative examples, results of independent interest, questions and unsolved problems.</p>

On solvability of the impulsive Cauchy problem for integro-differential inclusions with non-densely defined operators.

We prove the existence of at least one integrated solution to an impulsive Cauchy problem for an integro-differential inclusion in a Banach space with a non-densely defined operator. Since we look for integrated solution we do not need to assume that A is a Hille Yosida operator. We exploit a technique based on the measure of weak non-compactness which allows us to avoid any hypotheses of compactness both on the semigroup generated by the linear part and on the nonlinear term. As the main tool in the proof of our existence result, we are using the Glicksberg-Ky Fan theorem on a fixed point for a multivalued map on a compact convex subset of a locally convex topological vector space. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.

Integrated solutions of non-densely defined semilinear integro-differential inclusions: existence, topology and applications

Пусть заданы линейный замкнутый, но не обязательно плотно определенный оператор $A$ в банаховом пространстве $E$ с непустым резольвентным множеством и многозначное отображение $F\colon I\times E\multimap E$ со слабо секвенциально замкнутым графиком. Рассматривается интегро-дифференциальное включение $$ \dot{u}\in Au+F(t,\int u) \quadна I, \qquad u(0)=x_0. $$ Основное внимание уделяется случаю, когда $A$ порождает интегрированную полугруппу: доказывается существование так называемых интегрированных решений, если пространство $E$ слабо компактно порождено и $F$ удовлетворяет условию $$ \beta(F(t,\Omega))\le \eta(t)\beta(\Omega) \quadдля всех ограниченных множеств \Omega\subset E, $$ где $\eta\in L^1(I)$, а $\beta$ обозначает меру некомпактности Де Блази. В случае, когда $E$ сепарабельно, показано, что множество всех интегрированных решений является компактным $R_\delta$-подмножеством пространства $C(I,E)$ со слабой топологией. Этот результат используется для исследования нелокальной задачи Коши, задаваемой с помощью граничного оператора с невыпуклыми значениями. Приводятся также некоторые приложения к уравнениям в частных производных с многозначными членами. Библиография: 26 названий.

Existence results for Riemann-Liouville fractional integro-differential inclusions with fractional nonlocal integral boundary conditions

We introduce a new class of problems consisting of Riemann-Liouville fractional integro-differential inclusions supplemented with fractional nonlocal multi-point boundary conditions. The existence results for the given problem are derived in the weighted space with the aid of appropriate fixed point theorems for multi-valued maps. Numerical examples are constructed for the illustration of the obtained results.

On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay

In our manuscript, we organize a group of sufficient conditions of neutral integro-differential inclusions of Sobolev-type with infinite delay via resolvent operators. By applying Bohnenblust-Karlin's fixed point theorem for multivalued maps, we proved our results. Lastly, we present an application to support the validity of the study.

Generalized Almost Periodicity in Lebesgue Spaces with Variable Exponents

In this paper, we introduce and analyze Stepanov uniformly recurrent functions, Doss uniformly recurrent functions and Doss almost-periodic functions in Lebesgue spaces with variable exponents. We investigate the invariance of these types of generalized almost-periodicity in Lebesgue spaces with variable exponents under the actions of convolution products, providing also some illustrative applications to the abstract semilinear integro-differential inclusions in Banach spaces.

On a fractional hybrid multi-term integro-differential inclusion with four-point sum and integral boundary conditions

We investigate the existence of solutions for a fractional hybrid multi-term integro-differential inclusion with four-point sum and integral boundary value conditions. By using Dhage’s fixed point results, we prove our main existence result. Finally, we give an example to illustrate our main result.

On fractional hybrid and non-hybrid multi-term integro-differential inclusions with three-point integral hybrid boundary conditions

In this paper, we investigate the existence of solutions for two nonlinear fractional multi-term integro-differential inclusions in two hybrid and non-hybrid versions. The boundary value conditions are in the form of three-point integral hybrid conditions. In this way, we define a new operator based on the integral solution of the given boundary value inclusion problem and then we use assumptions of a Dhage’s fixed point result for this fractional operator in the hybrid case. Also, the approximate endpoint property is applied for the corresponding set-valued maps in the non-hybrid case. Finally, we provide two examples to illustrate our main results.

Controllability of coupled systems for impulsive φ-Hilfer fractional integro-differential inclusions

ABSTRACT This paper studies the controllability of a coupled system for a class of impulsive fractional integro-differential inclusions involving φ-Hilfer fractional derivative and subject to coupled nonlocal integral initial conditions in the case of convex set-valued maps. Some auxiliary conditions are introduced in order to apply a fixed point theorem due to Bohnenblust–Karlin. An illustrative example is provided to exemplify our theoretical results.

Existence of integrable solutions for integro-differential inclusions of fractional order; coupled system approach

Existence of integrable solutions for integro-differential inclusions of fractional order; coupled system approach