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.

We prove that the solution to a pair of nonlinear integral equations arising in the thermodynamic Bethe Ansatz can be expressed in terms of the resolvent kernel of the linear integral operator with kernel p-(u(θ)+u(θ'}}

We characterize the well-posedness for second order discrete evolution equations in UMD spaces by means of Fourier multipliers and R-boundedness properties of the resolvent operator which defines the equation. Applications to semilinear problems are given.

Sharp estimates are established for strong solutions of systems of differential-difference equations of both neutral and retarded type. The approach is based on the study of the resolvent corresponding to the generator of the semigroup of shifts along the trajectories of a dynamical system. In the case of neutral type equations, the Riesz basis property of the subsystem of exponential solutions...

This work is concerned with the approximation to the solutions of the stochastic Stokes equations by the splitting up method. We apply the resolvent operator to evaluate the solution of the deterministic equations at the endpoints of every small interval, and the error is estimated.

In the paper stochastic Volterra equations of nonscalar type in Hilbert space are studied. The aim of the paper is to provide some results on stochastic convolution and mild solutions to those Volterra equations. The motivation of the paper comes from a model of aging viscoelastic materials. The pseudo-resolvent approach is used.

The resolvent operator approach is applied to address a system of generalized ordered variational inclusions with ⊕ operator in real ordered Banach space. With the help of the resolvent operator technique, Li et al. (J. Inequal. Appl. 2013:514, 2013; Fixed Point Theory Appl. 2014:122, 2014; Fixed Point Theory Appl. 2014:146, 2014; Appl. Math. Lett. 25:1384-1388, 2012; Fixed Point Theory Appl. 2...

This paper address a new vision for the generalized Mittag-Leffler stability of the fractional differential equations. We mainly focus on a new method, consisting of decomposing a given fractional differential equation into a cascade of many sub-fractional differential equations. And we propose a procedure for analyzing the generalized Mittag-Leffler stability for the given fractional different...