Integrodifferential Equations In Hilbert Space Research Articles

Representations of Solutions for Volterra Integro-Differential Equations in Hilbert Spaces

Representations of Solutions for Volterra Integro-Differential Equations in Hilbert Spaces

Well-Posed Solvability of Volterra Integro-Differential Equations in Hilbert Spaces

Well-Posed Solvability of Volterra Integro-Differential Equations in Hilbert Spaces

Controllability of impulsive stochastic functional integrodifferential equations driven by Rosenblatt process and Lévy noise

In this paper, we develop controllability findings for impulsive neutral stochastic delay partial integrodifferential equations in Hilbert spaces driven by Rosenblatt process and Lévy noise. A novel set of adequate requirements is obtained by utilizing a fixed point method without imposing a stringent compactness constraint on the semigroup. The observed results represent a generalization and continuation of previous findings on this topic. Finally, an example is given to demonstrate how the acquired findings may be used.

Exponential stability of abstract Coleman–Gurtin equations in Hilbert spaces

Abstract We study a class of integro-differential equations in Hilbert spaces which fits the Coleman–Gurtin model of heat conduction with memory. Well-posedness of the equation is established and sufficient conditions for exponential stability of the equation are given. This is done by using the semigroup approach. An illustrative example is given and the obtained theoretical results are verified by numerical simulations.

Effective numerical technique for nonlinear Caputo-Fabrizio systems of fractional Volterra integro-differential equations in Hilbert space

The point of this paper is to analyze and investigate the analytic-approximate solutions for fractional system of Volterra integro-differential equations in framework of Caputo-Fabrizio operator. The methodology relies on creating the reproducing kernel functions to gain analytical solutions in a uniform form of a rapidly convergent series in the Hilbert space. Using the Gram-Schmidt orthonomalization process, the orthonormal basis system is constructed in a dense compact domain to encompass the Fourier series expansion in view of reproducing kernel properties. Besides, convergence and error analysis of the proposed technique are discussed. For this purpose, several numerical examples are tested to demonstrate the great feasibility and efficiency of the present method and to support theoretical aspect as well. From a numerical point of view, the acquired solutions simulation indicates that the methodology used is sound, straightforward, and appropriate to deal with many physical issues in light of Caputo-Fabrizio derivatives.

Uniform polynomial stability of second order integro-differential equations in Hilbert spaces with positive definite kernels

<p style='text-indent:20px;'>We are concerned with the polynomial stability and the integrability of the energy for second order integro-differential equations in Hilbert spaces with positive definite kernels, where the memory can be oscillating or sign-varying or not locally absolutely continuous (without any control conditions on the derivative of the kernel). For this stability problem, tools from the theory of existing positive definite kernels can not be applied. In order to solve the problem, we introduce and study a new mathematical concept – <i>generalized positive definite kernel</i> (GPDK). With the help of GPDK and its properties, we obtain an efficient criterion of the polynomial stability for evolution equations with such a general but more complicated and useful memory. Moreover, in contrast to existing positive definite kernels, GPDK allows us to directly express the decay rate of the related kernel.</p>

Spectral Analysis of Integrodifferential Equations in Hilbert Spaces

We investigate the well-posedness of initial-value problems for abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space and provide the spectral analysis of operator-functions describing symbols of such equations. These equations are an abstract form of linear partial integrodifferential equations arising in the viscoelasticity theory and other important applications. We establish the localization and the spectrum structure of operator-functions describing symbols of these equations.

BSDE representation and randomized dynamic programming principle for stochastic control problems of infinite-dimensional jump-diffusions

We consider a general class of stochastic optimal control problems, where the state process lives in a real separable Hilbert space and is driven by a cylindrical Brownian motion and a Poisson random measure; no special structure is imposed on the coefficients, which are also allowed to be path-dependent; in addition, the diffusion coefficient can be degenerate. For such a class of stochastic control problems, we prove, by means of purely probabilistic techniques based on the so-called randomization method, that the value of the control problem admits a probabilistic representation formula (known as non-linear Feynman-Kac formula) in terms of a suitable backward stochastic differential equation. This probabilistic representation considerably extends current results in the literature on the infinite-dimensional case, and it is also relevant in finite dimension. Such a representation allows to show, in the non-path-dependent (or Markovian) case, that the value function satisfies the so-called randomized dynamic programming principle. As a consequence, we are able to prove that the value function is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation, which turns out to be a second-order fully non-linear integro-differential equation in Hilbert space.

Comparison theorem for neutral stochastic integro-differential equations in Hilbert space

In the present paper, we will discuss a comparison result for solutions to the Cauchy problems for two stochastic differential equations with delay. On this subject number of authors have obtained their comparison results. We deal with the Cauchy problems for two neutral stochastic integro-differential equations. Except transient- (or drift-) and diffusion-coefficients, our equations include also one integro-differential term. Basic difference of our case from the case of all earlier investigated problems is presence of this term. We intoduce a concept of solutions to our problems and prove the comparison theorem for them. According to our result, under certain assumptions on coefficients of equations under consideration, their solutions depend on the transient-coefficients in a monotone way.

Stability of fractional neutral stochastic partial integro-differential equations

Abstract In this paper, we are concerned with a class of fractional partial neutral stochastic integro-differential equations in Hilbert spaces. We assume that the linear part of this equation generates an α-resolvent operator and transform it into an integral equation. By the stochastic analysis and fractional calculus technique, and combining some integral inequalities, we obtain some sufficient conditions ensuring the exponential p-stability of the mild solution of the considered equations are obtained. Subsequently, by the weak convergence approach, we have a try to deal with the stability conditions in distribution of the segment process of mild solutions to the stochastic systems under investigation. Last, an example is presented to illustrate our theory in the work.

The Solvability and Optimal Controls for Impulsive Fractional Stochastic Integro-Differential Equations via Resolvent Operators

In this manuscript, we investigate the solvability and optimal controls for impulsive fractional stochastic integro-differential equations in Hilbert space. Sufficient conditions are obtained for the existence of mild solution of the considered system by using analytic resolvent operators, the uniform continuity of the resolvent, and Leray---Schauder fixed point theorem. Then, the existence of optimal controls is discussed for the considered system. Finally, the obtained theoretical result is validated through an example.

Correct solvability of Volterra integrodifferential equations in Hilbert space

Correct solvability of Volterra integrodifferential equations in Hilbert space

Existence of Square-Mean Almost Automorphic Solutions to Stochastic Functional Integrodifferential Equations in Hilbert Spaces

The existence and uniqueness of square-mean almost automorphic mild solution to a stochastic functional integrodifferential equation is studied. Under some appropriate assumptions, the existence and uniqueness of square-mean almost automorphic mild solution is obtained by Banach’s fixed point theorem. Particularly, based on Schauder’s fixed point theorem, the existence of square-mean almost automorphic mild solution is obtained by using the condition which is weaker than Lipschitz conditions. Finally, an example illustrating our main result is given.

Long-time behavior of nonlinear integro-differential evolution equations

We study the long-time behavior as time tends to infinity of globally bounded strong solutions to certain integro-differential equations in Hilbert spaces. Based on an appropriate new Lyapunov function and the Łojasiewicz–Simon inequality, we prove that any globally bounded strong solution converges to a steady state in a real Hilbert space.

Controllability of a class of Volterra equations in Hilbert spaces with completely monotone kernel

We consider a parabolic Volterra integro-differential equation in Hilbert space with a completely monotone convolution kernel and a sectorial operator. Utilizing a state space setting, we show that for a large class of kernels the state cannot be controlled exactly to zero. On the other hand, equations of our type are always approximately controllable, provided the control operator has dense range.

Weighted [formula omitted] Paley–Wiener Theorem, with applications to stability of the linear multi-step methods for Volterra equations in Hilbert spaces

We study the weighted l1 Paley–Wiener Theorem, with applications to stability of the linear multi-step time discretization for a linear Volterra equation of scalar type in a Hilbert space. The results extend to weighted l1 stability some recent l1 stability theorem due to the author (Numer. Math. 109 (2008) 571–595). Our theoretical results are applied to investigate the asymptotic behavior as tn→∞ of the numerical solutions of certain integro-differential equations in Hilbert space.

Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces

In this paper, we study the existence results of mild solutions for a class of stochastic integro-differential equations with nonlocal conditions and stochastic impulsive integro-differential equations with nonlocal conditions in Hilbert spaces. Sufficient conditions for the existence of mild solutions are derived by means of Leray–Schauder nonlinear alternative. An example is provided to illustrate the theory.

Well-defined solvability and spectral analysis of abstract hyperbolic integrodifferential equations

The authors study integrodifferential equations in Hilbert space. The coefficients of the equations are unbounded and the principal part is an abstract hyperbolic equation perturbed by terms with Volterra integral operators. Such equations can be regarded as an abstract generalization of the Gurtin–Pipkin integrodifferential equation that describes heat transfer in materials with memory and has a number of other applications. Well-defined solvability of initial boundary value problems for such equations is established in weighted Sobolev spaces on the positive semi-axis. The authors examine spectral problems for operator-valued functions representing the symbols of the said equations and study the spectrum of the abstract Gurtin–Pipkin integrodifferential equation.

Stability of solutions to integro-differential equations in Hilbert spaces

In this paper, we investigate the uniform exponential stability of solutions to abstract integro-differential equations in Hilbert spaces by the theory of operator semigroups and Laplace transforms of vector-valued functions. New criterions are given based on the growth property of associated vector-valued functions on the right half plane. Examples are presented to illustrate our results.

Stochastic integrodifferential equations in Hilbert spaces with applications in electromagnetics

In this work we present some results on deterministic and stochastic integrodifferential equations in Hilbert spaces, motivated from and applied to problems arising from the mathematical modeling of electromagnetics fields in complex random media. We examine the mild, strong and classical well posedness for the Cauchy problem of the integrodifferential equation which describes Maxwell’s equations complemented with the general (and therefore nonlocal in time) linear constitutive relations describing such media, with either additive or multiplicative noise.