Infinite Delay Research Articles

Existence and Asymptotic Periodicity of Solutions for Neutral Integro-Differential Evolution Equations with Infinite Delay

Abstract In this work, making use of the theory of resolvent operators and Banach fixed point theorem, we first discuss the existence and regularity of mild solutions for neutral partial functional integro-differential equations with infinite delay. We assume that the linear part of the considered equation generates a resolvent operator and the nonlinear function satisfies Lipschitz conditions. Then we investigate the asymptotic periodicity of mild solutions under asymptotic periodic assumption on the nonlinear function. The obtained results extend somewhat the related conclusions in literature. In the end, an example is presented to illustrate the obtained results.

Mixed Neutral Caputo Fractional Stochastic Evolution Equations with Infinite Delay: Existence, Uniqueness and Averaging Principle

The aim of this article is to consider a class of neutral Caputo fractional stochastic evolution equations with infinite delay (INFSEEs) driven by fractional Brownian motion (fBm) and Poisson jumps in Hilbert space. First, we establish the local and global existence and uniqueness theorems of mild solutions for the aforementioned neutral fractional stochastic system under local and global Carathéodory conditions by using the successive approximations, stochastic analysis, fractional calculus, and stopping time techniques. The obtained existence result in this article is new in the sense that it generalizes some of the existing results in the literature. Furthermore, we discuss the averaging principle for the proposed neutral fractional stochastic system in view of the convergence in mean square between the solution of the standard INFSEEs and that of the simplified equation. Finally, the obtained averaging theory is validated with an example.

An investigation on the approximate controllability of impulsive neutral delay differential inclusions of second order

This paper mainly focuses on the approximate controllability for second‐order impulsive neutral differential inclusions with infinite delay. By using the results related to cosine and sine function of operators, Dhage's fixed point theorem, and their outcome when combined with the properties of differential inclusions, our primary discussions are proved. Finally, we provided an example for the illustration of the theory obtained.

Optimal controls problems for some impulsive stochastic integro-differential equations with state-dependent delay

ABSTRACT In this paper, optimal control problems for a class of stochastic functional integral-differential equations in Hilbert spaces are investigated. First, the existence of mild solutions is investigated using stochastic analysis theory, fixed point theorems, and Grimmer's resolvent operator theory. Following that, the existence requirements of optimal pairs of systems governed by stochastic partial integro-differential equations with infinite delay are discussed. The results are achieved by the use of a combination of Lipschitz and Carathéodory conditions. At the end of the paper, an illustration is supplied to help highlight the key findings.

Optimal control and approximate controllability for fractional integrodifferential evolution equations with infinite delay of order r∈(1,2)

AbstractThis article investigates the issue of optimal control and approximate controllability results for fractional integrodifferential evolution equations with infinite delay of in Banach space. In the beginning, we analyze approximate controllability results for fractional integrodifferential evolution equations using the fractional calculations, cosine families, and Banach fixed point theorem. After, we developed the continuous dependence of the fractional integrodifferential evolution equations by using the Henry–Gronwall inequality. Furthermore, we tested the existence of optimal controls for the Lagrange problem. Lastly, an application is presented to illustrate the theory of the main results.

Solution of differential equations with infinite delay via coupled fixed point

This manuscript aims to prove exciting results that unify and generalize several fixed point results for metric spaces endowed with graphs. As an application, we apply our own results to give and introduce sufficient conditions to guarantee existence solutions of such differential equations with infinite delay.

Controllability of neutral impulsive stochastic functional integrodifferential equations driven by a fractional Brownian motion with infinite delay via resolvent operator

This paper is concerned with the controllability results of neutral impulsive stochastic functional integrodifferential equations driven by a fractional Brownian motion with infinite delay in a real separable Hilbert space. The controllability results are obtained using stochastic analysis, the theory of resolvent operator in the sense of Grimmer and Krasnoselskii fixed point theorem. An example is provided to illustrate the obtained theory.

New criteria for exponential stability in mean square of stochastic functional differential equations with infinite delay

<p style='text-indent:20px;'>Stochastic functional differential equations with infinite delay are considered. A novel approach to exponential stability of such equations is proposed. New criteria for the mean square exponential stability of general stochastic functional differential equations with infinite delay are presented. Illustrative examples are given.</p>

Multi-valued random dynamics of stochastic wave equations with infinite delays

<p style='text-indent:20px;'>This paper is devoted to the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with infinite delays and additive white noise. The nonlinear terms of the equation are not expected to be Lipschitz continuous, but only satisfy continuity assumptions along with growth conditions, under which the uniqueness of the solutions may not hold. Using the theory of multi-valued non-autonomous random dynamical systems, we prove the existence and measurability of a compact global pullback attractor.</p>

Local existence and blowing up phenomena for a class of non-autonomous partial functional differential equations with infinite delay

Abstract In this work, we study a class of abstract non-autonomous partial functional differential equations with infinite delay. Our main results concern the local existence of the mild solution which can blow up at the finite time. The unbounded operators associated to the non-autonomous system are assumed to be stable family which generates C 0-semigroups while the nonlinear part is supposed to be continuous. Under Lipschitz condition on the nonlinear term of the equation, we prove the existence and uniqueness of the mild solution. For illustration, we provide an example for some reaction-diffusion non-autonomous partial functional differential equations involving infinite delay.

Global existence for neutral partial functional integrodifferential equations with infinite delay

The aim of this work is to prove some results about the existence and regularity of solutions for some neutral partial functional integrodifferential equations with infinite delay in Banach spaces. The results are based on semigroup theory and on Banach’s fixed point theorems. The method used treats the equations in its subspace D(A). One example is given to illustrate the theory.

Periodic Solution for some Class of Linear Partial Differential Equation with infinite Delay using Semi-Fredholm perturbations

Abstract In this work, we study the existence of periodic solutions for a class of linear partial functional differential equations with infinite delay. Inspiring by an existing study, by applying the perturbation theory of semi-Fredholm operators, we introduce a suitable a priori estimate on the norm of the operator L to establish the periodicity of solutions in the case where the linear part is nondensely defined and satisfies the Hille-Yosida condition and without considering the exponential stability condition on the semigroup generated by the part of this operator on the closure of it’s domain. Moreover, in the special case where the linear part generates a strongly continuous semigroup and perturbed by a compact linear operator, we give some sufficient conditions to derive periodic solution from bounded ones. Finally, our theoretical results are illustrated by applications in both densely and nondensely cases.

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>

Optimal Controls for Neutral Stochastic Integrodifferential Equations with Infinite Time Delay and Deviated Argument Driven by Rosenblatt Process

Optimal Controls for Neutral Stochastic Integrodifferential Equations with Infinite Time Delay and Deviated Argument Driven by Rosenblatt Process

‐stability of hybrid neutral stochastic differential equations with infinite delay

AbstractIn this article, we study the ‐stability in qth moment (.s.q.m) of hybrid neutral stochastic differential equations with infinite delay (HNSDEID) using the Lyapunov techniques and the method of M‐matrix. Finally, we apply the main result to some examples.

Lyapunov Functional Method in the Stability Problem of Volterra Integro-Differential Equations with Infinite Delay

Lyapunov Functional Method in the Stability Problem of Volterra Integro-Differential Equations with Infinite Delay

Controllability of impulsive neutral stochastic integro-differential systems driven by fractional Brownian motion with delay and Poisson jumps

In this paper the controllability of a class of impulsive neutral stochastic integro-differential systems driven by fractional Brownian motion and Poisson process in a separable Hilbert space with infinite delay is studied. The controllability result is obtained by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result.

New results on stability of discrete-time systems with infinite delays

This paper is concerned with problems of stability of discrete-time systems with infinite delays. Several new theorems for stability analysis of the concerned delayed systems are developed. Specifically, uniform stability, uniform asymptotic stability, global uniform asymptotic stability, exponential stability and global exponential stability are included. Our results include some existing works on bounded delays as special cases. Furthermore, an example is furnished to demonstrate the effectiveness of the proposed theorems.

Robust non‐fragile Mittag‐Leffler synchronization of fractional order non‐linear complex dynamical networks with constant and infinite distributed delays

The present article is an attempt to investigate the robust non‐fragile Mittag‐Leffler synchronization of fractional order non‐linear complex dynamical networks with uncertain parameters and mixed delays. First, we develop a generalized fractional order complex dynamical network (GFOCDN) and then derive its synchronization criteria by using certain Lemmas based on the theory of Lyapunov stability and, second, by adding parameters of uncertainty to the GFOCDN and derive the synchronization analysis by including the appropriate non‐fragile controller into the system model to achieve a non‐fragile Mittag‐Leffler synchronization. Third, we incorporate the parameter uncertainty into the inner couplings and derive as a special case to achieve its synchronization. Finally, we present two numerical simulations to show the reliability of our proposed work.

Limit theorems for additive functionals of stochastic functional differential equations with infinite delay

This paper examines limit theorems for a class of stochastic functional differential equations with infinite delay. Under non-Lipschitz conditions, the strong law of large numbers and the central limit theorem for additive functionals of the segment processes are established by using limit theorems of uniform mixing Markovian processes. Under some dissipative assumptions, the law of iterated logarithm is also derived for additive functionals of such equations via a martingale difference sequence of square integrable martingales.