In this paper, we establish results concerning, existence, uniqueness, global continuation, and regularity of integral solutions to some partial neutral functional differential equations with infinite delay. These equations find their origin in the description of heat flow models, viscoelastic and thermoviscoelastic materials, and lossless transmission lines models; see for example [15] and [38].

For a class of linear autonomous delay differential equations with parameter α we give upper bounds for the integral ∞ 0 |X (t, α)| dt of the fundamental solution X (·, α). The asymptotic estimations are sharp at a critical value α0 where x = 0 loses stability. We use these results to study the stability properties of perturbed equations.

In this paper, we prove the existence, uniqueness, and continuous dependence of the mild solutions for a class of fractional abstract differential equations with infinite delay. The results are obtained by using the Krasnoselskii’s fixed point theorem and the theory of resolvent operators for integral equations.

in this work, a new iterative method is proposed for obtaining the approximate solution of a class of hammerstein type integral equations system. the main structure of this method is based on the richardson iterative method for solving an algebraic linear system of equations. some conditions for existence and unique solution of this type equations are imposed. convergence analysis and error bou...

The main goal of this article is to study an averaging principle for a class two-time-scale stochastic differential delay equations in which the slow-varying process includes multiplicative fractional Brownian noise with Hurst parameter H ∈ ( 1 2 , ) and fast-varying rapidly-changing diffusion. We would like emphasize that approach proposed paper based on fact integral respect motion can be def...

In this paper, we introduce hybrid of block-pulse functions and Bernstein polynomials and derive operational matrices of integration, dual, differentiation, product and delay of these hybrid functions by a general procedure that can be used for other polynomials or orthogonal functions. Then, we utilize them to solve delay differential equations and time-delay system. The method is based upon e...

A Legendre wavelet method is presented for numerical solutions of stochastic Volterra-Fredholm integral equations. The main characteristic of the proposed method is that it reduces stochastic Volterra-Fredholm integral equations into a linear system of equations. Convergence and error analysis of the Legendre wavelets basis are investigated. The efficiency and accuracy of the proposed method wa...