Non-autonomous Integro-differential Equations Research Articles

New discussion on global existence and attractivity of mild solutions for nonautonomous integrodifferential equations with state-dependent delay

New discussion on global existence and attractivity of mild solutions for nonautonomous integrodifferential equations with state-dependent delay

Stability and robust stability of non-autonomous linear differential equations with infinite delay

By the idea of comparison principle, we first introduce novel explicit criteria for the $ \varrho $-stability of general linear non-autonomous functional differential equations with infinite delay. Some characterizations for the time-independent exponential stability are also investigated. We then present a sufficient condition for the exponential stability of the equations subject to time-varying structured perturbations. These results lead to applications to the linear non-autonomous differential system with unbounded delay and the linear non-autonomous integro-differential equations with delay. Some examples are given to illustrate the obtained results.

A Note on Existence of Mild Solutions for Second-Order Neutral Integro-Differential Evolution Equations with State-Dependent Delay

This article is mainly devoted to the study of the existence of solutions for second-order abstract non-autonomous integro-differential evolution equations with infinite state-dependent delay. In the first part, we are concerned with second-order abstract non-autonomous integro-differential retarded functional differential equations with infinite state-dependent delay. In the second part, we extend our results to study the second-order abstract neutral integro-differential evolution equations with state-dependent delay. Our results are established using properties of the resolvent operator corresponding to the second-order abstract non-autonomous integro-differential equation and fixed point theorems. Finally, an application is presented to illustrate the theory obtained.

Existence and uniqueness of extremal mild solutions for non-autonomous nonlocal integro-differential equations via monotone iterative technique

In this work, we will discuss the existence and uniqueness of extremal mild solutions for non-autonomous integro-differential equations having nonlocal condition via monotone iterative method with upper and lower solutions in an ordered Banach space X, using evolution system and measure of noncompactness.

Existence of solutions of abstract non-autonomous second order integro-differential equations

In this paper the existence of solutions of a non-autonomous abstract integro-differential equation of second order is considered. Assuming the existence of an evolution operator corresponding to the associate abstract non-autonomous Cauchy problem of second order, we establish the existence of a resolvent operator for the homogeneous integro-differential equation and the existence of mild solutions to the inhomogeneous integro-differential equation. Furthermore, we study the existence of classical solutions of the integro-differential equation. Finally, we apply our results to the study of the existence of solutions of a non-autonomous wave equation.

Pseudo Asymptotic Behavior of Mild Solution for Nonautonomous Integrodifferential Equations with Nondense Domain

By the weighted ergodic function based on the measure theory, we study pseudo asymptotic behavior of mild solution for nonautonomous integrodifferential equations with nondense domain. The existence and uniqueness ofμ-pseudo antiperiodic (μ-pseudo periodic,μ-pseudo almost periodic, andμ-pseudo automorphic) solution are investigated. Some interesting examples are presented to illustrate the main findings.

𝐶^{𝛼}-classical solutions for abstract non-autonomous integro-differential equations

In this paper we study the existence of strict and classical solutions for a class of abstract non-autonomous integro-differential equations in Banach spaces. Some applications of the abstract results are considered.

Global attractor for a non-autonomous integro-differential equation in materials with memory

Abstract The long-time behavior of an integro-differential parabolic equation of diffusion type with memory terms, expressed by convolution integrals involving infinite delays and by a forcing term with bounded delay, is investigated in this paper. The assumptions imposed on the coefficients are weak in the sense that uniqueness of solutions of the corresponding initial value problems cannot be guaranteed. Then, it is proved that the model generates a multivalued non-autonomous dynamical system which possesses a pullback attractor. First, the analysis is carried out with an abstract parabolic equation. Then, the theory is applied to the particular integro-differential equation which is the objective of this paper. The general results obtained in the paper are also valid for other types of parabolic equations with memory.

Solutions to non-autonomous integrodifferential equations with infinite delay

The conclusions of Da Prato and Sinestrari concerning the non-autonomous evolution operator of hyperbolic type for the equation u ′ ( t ) = A ( t ) u ( t ) + f ( t ) , t ∈ [ 0 , T ] , u ( 0 ) = z are applied to find the uniqueness and existence of solution to u ′ ( t ) = A ( t ) u ( t ) + ∫ 0 t K ( t , θ , u ( θ ) ) d θ + L ( t ) u t + f ( t ) , t ∈ [ 0 , T ] , u 0 = φ , where A ( ⋅ ) maybe nondensely defined. Moreover, some relative problems are concerned.

Non-Autonomous Integrodifferential Equations with Non-Local Conditions

Non-Autonomous Integrodifferential Equations with Non-Local Conditions

Nonautonomous integro-differential equations of hyperbolic type

This paper is devoted to two problems for the nonautonomous integrodifferential equation in a Banach space $X$ of hyperbolic type $$ \text{$u'(t) = A(t)u(t)+ \int_0^t B(t,s)u(s) ds + f(t)$ for $t\in[0,T]$, and $u(0) = u_0$.} $$ One is the problem of existence and uniqueness of classical solutions without assuming that the common domain of $A(t)$ is dense in $X$, and the other is the regularity problem in the case where the common domain of $A(t)$ is dense in $X$. The regularity result will play a role in developing an abstract theory which can be applied to the second-order integrodifferential equations with the third kind boundary conditions.

Existence and sharp regularity results for linear parabolic non-autonomous integro-differential equations

We study existence, uniqueness and maximal regularity of the strict solutionu∈C 1([0,T],E) of the integro-differential equation\(u'(t) - A(t)u(t) - \int {_0^1 } B(t,s)u(s)ds = f(t),t \in [0,T],\) with the initial datumu(0)=x, in a Banach spaceE, {itA(itt)}f∈|0,1| is a family of generators of analytic semigroups whose domainsD A(t) are not constant int as well as (possibly) not dense inE, whereas {itB(itt)}0≦11≦T is a family of closed linear operators withD B(t,s) ⊇D A(s) ∨t∈[s, T]. We prove necessary and sufficient conditions for existence of the strict solution and for Holder continuity of its derivative; well-posedness of the problem with respect to the Holder norms is also shown.