On the solvability and stability of nonlinear $$\varphi $$-Caputo fractional differential equations in Banach spaces

This review paper is devoted to the numerical analysis of abstract differential equations in Banach spaces. Most of the finite difference, finite element, and projection methods can be considered from the point of view of general approximation schemes (see, e.g., [207,210,211] for such a representation). Results obtained for general approximation schemes make the formulation of concrete numerical methods easier and give an overview of methods which are suitable for different classes of problems. The qualitative theory of differential equations in Banach spaces is presented in many brilliant papers and books. We can refer to the bibliography [218], which contains about 3000 references. Unfortunately, no books or reviews on general approximation theory appear for differential equations in abstract spaces during last 20 years. Any information on the subject can be found in the original papers only. It seems that such a review is the first step towards describing a complete picture of discretization methods for abstract differential equations in Banach spaces. In Sec. 2 we describe the general approximation scheme, different types of convergence of operators, and the relation between the convergence and the approximation of spectra. Also, such a convergence analysis can be used if one considers elliptic problems, i.e., the problems which do not depend on time. Section 3 contains a complete picture of the theory of discretization of semigroups on Banach spaces. It summarizes Trotter–Kato and Lax–Richtmyer theorems from the general and common point of view and related problems. The approximation of ill-posed problems is considered in Sec. 4, which is based on the theory of approximation of local C-semigroups. Since the backward Cauchy problem is very important in applications and admits a stochastic noise, we also consider approximation using a stochastic regularization. Such an approach was never considered in the literature before to the best of our knowledge. In Sec. 5, we present discrete coercive inequalities for abstract parabolic equations in Cτn([0, T ];En), C τn([0, T ];En), L p τn([0, T ];En), and Bτn([0, T ];C (Ωh)) spaces.

Hausdorff and fractal dimensions of attractors for functional differential equations in Banach spaces

On the existence of weak solutions of differential equations in nonreflexive banach spaces

We prove a center-stable manifold theorem for a class of differential equations in (infinite-dimensional) Banach spaces.

A series of stability, contractivity and asymptotic stability results of the solutions to nonlinear stiff Volterra functional differential equations (VFDEs) in Banach spaces is obtained, which provides the unified theoretical foundation for the stability analysis of solutions to nonlinear stiff problems in ordinary differential equations(ODEs), delay differential equations(DDEs), integro-differential equations(IDEs) and VFDEs of other type which appear in practice.

A series of contractivity and exponential stability results for the solutions to nonlinear neutral functional differential equations (NFDEs) in Banach spaces are obtained, which provide unified theoretical foundation for the contractivity analysis of solutions to nonlinear problems in functional differential equations (FDEs), neutral delay differential equations (NDDEs) and NFDEs of other types which appear in practice.

This paper is devoted to second order abstract linear differential equations in a Banach space. For such equations the Cauchy problem is stated, and the behavior of its solutions as \[t\rightarrow +\infty\] is examined. The aim of the paper is to study ergodicity and asymptotic behavior of the solutions of the strongly correct Cauchy problem. For this purpose the theory of complete second order linear differential equations in Banach spaces, developed by Fattorini, is used. As shown in the paper, for a wide class of equations the solutions are either ergodic or unbounded, depending on the initial values. For the solutions to be ergodic, conditions on the linear operators-coefficients of the differential equation and the initial values of the Cauchy problem are obtained. In case of ergodic solutions, exact values of ergodic limits are given. In case of unbounded solutions, asymptotic behavior of solutions is described. Results obtained in this paper are a generalization of the previously known results concerning ergodic properties of the solutions for the Cauchy problem for the incomplete second order equations.

A uniqueness theorem for the Cauchy problem for ordinary differential equations in complex Banach spaces is given. This paper generalizes and extends a number of known results.

Stochastic differential equations in a Banach space driven by the cylindrical Wiener process

<abstract> <p>The aim of the reported results in this manuscript is to handle the existence, uniqueness, extremal solutions, and Ulam-Hyers stability of solutions for a class of $ \Psi $-Caputo fractional relaxation differential equations and a coupled system of $ \Psi $-Caputo fractional relaxation differential equations in Banach spaces. The obtained results are derived by different methods of nonlinear analysis like the method of upper and lower solutions along with monotone iterative technique, Banach contraction principle, and Mönch's fixed point theorem concerted with the measures of noncompactness. Furthermore, the Ulam-Hyers stability of the proposed system is studied. Finally, two examples are presented to illustrate our theoretical findings. Our acquired results are recent in the frame of a $ \Psi $-Caputo derivative with initial conditions in Banach spaces via the monotone iterative technique. As a results, we aim to fill this gap in the literature and contribute to enriching this academic area.</p> </abstract>

A uniqueness theorem for the Cauchy problem for ordinary differential equations in complex Banach spaces is given. This paper generalizes and extends a number of known results.

Existence results for differential delay equations in Banach spaces

This paper concerns the dissipativity of nonlinear Volterra functional differential equations (VFDEs) in Banach space and their numerical discretization. We derive sufficient conditions for the dissipativity of nonlinear VFDEs. The general results provide a unified theoretical treatment for dissipativity analysis to ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs of other type appearing in practice. Then the dissipativity property of the backward Euler method for VFDEs is investigated. It is shown that the method can inherit the dissipativity of the underlying system. The close relationship between the absorbing set of the numerically discrete system generated by the backward Euler method and that of the underlying system is revealed.

This paper is devoted to studying the existence and stability of implicit Volterra difference equations in Banach spaces. The proofs of our results are carried out by using an appropriate extension of the freezing method to Volterra difference equations in Banach spaces. Besides, sharp explicit stability conditions are derived.

We formulate initial value problems for delay difference equations in Banach spaces as fixed-point problems in sequence spaces. By choosing appropriate sequence spaces various types of attractivity can be described. This allows us to establish global attractivity by means of fixed-point results. Finally, we provide an application to delay integrodifference equations in the space of continuous functions over a compact domain.