Differential Equations In Banach Spaces Research Articles

Almost Periodicity for a Nonautonomous Discrete Dispersive Population Model

ABSTRACTIn this article, study the almost periodic profile of solutions for nonautonomous difference equations in Banach spaces. We apply our results in population dynamics.

Integral Inequalities and Pseudo Almost Automorphic Solutions to Semilinear Differential Equations in Banach Spaces

This work aims to investigate the existence and uniqueness of pseudo almost automorphic solutions to semilinear differential equations in Banach space. By generalizing the Bihari type inequality and using Krasnoselski–Schaefer fixed point theorem, the existence results are deduced. Moreover, a new estimate is also built to ensure the existence and uniqueness of solutions. An example is given to illustrate our conclusions.

Pseudo almost automorphy of semilinear fractional differential equations in Banach Spaces

Abstract In this paper, we investigate the existence and uniqueness of (μ,ν)-pseudo almost automorphic mild solutions to semilinear fractional differential equation with Riemann-Liouville derivative in Banach space, where the nonlinear perturbation is of (μ,ν)-pseudo almost automorphic type or Stepanov-like (μ,ν)-pseudo almost automorphic type. As application, we explore the same topic for a fractional relaxation-oscillation equation.

Intertwined evolution operators

We construct a convolution algebra of admissible homomorphisms defined on a ‘test space’ to demonstrate the fundamental role of convolution in the study of intertwined evolution operators of linear ordinary differential equations in Banach spaces and probability theory. The choice of test space makes the framework we present quite versatile. The applications include semigroups of linear operators, empathy, integrated semigroups and empathies and the convolution semigroups of probability theory.

Decay mild solutions for two-term time fractional differential equations in Banach spaces

In this paper, we deal with the nonlocal Cauchy problem for a class of two-term time fractional differential equations in Banach spaces. By constructing a suitable measure of noncompactness on the space of solutions, we prove the existence of a compact set containing decay mild solutions to the mentioned problem.

Nonlinear equations with degenerate operator at fractional Caputo derivative

At first, the existence of a unique solution for the Cauchy problem to nondegenerate fractional differential equation was proved. These results were used for research of the unique solvability for the initial Cauchy and Showalter–Sidorov problems to differential equations in Banach spaces with degenerate operator at fractional Caputo derivative in linear and nonlinear cases. Abstract results are applied to the research of an initial boundary value problem for time‐fractional order Oskolkov system of equations. Copyright © 2016 John Wiley & Sons, Ltd.

Existence and Boundedness of Solutions for Nonlinear Volterra Difference Equations in Banach Spaces

We consider a class of nonlinear discrete-time Volterra equations in Banach spaces. Estimates for the norm of operator-valued functions and the resolvents of quasi-nilpotent operators are used to find sufficient conditions that all solutions of such equations are elements of an appropriate Banach space. These estimates give us explicit boundedness conditions. The boundedness of solutions to Volterra equations with infinite delay is also investigated.

On periodic solutions to nonlinear differential equations in Banach spaces

Let A denote the generator of a strongly continuous periodic one-parameter group of bounded linear operators in a complex Banach space H. In this work, an analog of the resolvent operator which is called quasi-resolvent operator and denoted by R? is defined for points of the spectrum, some equivalent conditions for compactness of the quasi-resolvent operators R? are given. Then using these, some theorems on existence of periodic solutions to the non-linear equations ?(A)x = f (x) are given, where ?(A) is a polynomial of A with complex coefficients and f is a continuous mapping of H into itself.

New Conditions for the Exponential Stability of Pseudolinear Difference Equations in Banach Spaces

We study the local exponential stability of evolution difference systems with slowly varying coefficients and nonlinear perturbations. We establish the robustness of the exponential stability in infinite-dimensional Banach spaces, in the sense that the exponential stability for a given pseudolinear equation persists under sufficiently small perturbations. The main methodology is based on a combined use of new norm estimates for operator-valued functions with the “freezing” method.

Abstract linear partial differential equations related to size-structured population models with diffusion

We study abstract linear partial differential equations in Banach spaces and/or Banach lattices related to size-structured population models with spatial diffusion and their dual problems. We introduce mild solutions through semigroup theory and characteristic method and investigate differentiability of mild solutions. Existence of a unique mild solution is shown. Also, a comparison result is obtained as well as the boundedness of mild solutions is investigated in the Banach lattice setting. Furthermore, we consider the dual problems, and then we introduce weak solutions and establish their uniqueness.

Distributionally chaotic properties of abstract fractional differential equations

In the paper under review, we analyze a class of abstract distributionally chaotic (multi-term) fractional differential equations in Banach spaces, associated with use of the Caputo fractional derivatives.

Bounded solutions of finite lifetime to differential equations in Banach spaces

Consider a smooth vector field $f\colon \mathbb{R}^n\to\mathbb{R}^n$ and a maximal solution $\gamma\colon \,]a,b[\,\to \mathbb{R}^n$ to the ordinary differential equation $x'=f(x)$. It is a well-known fact that, if $\gamma$ is bounded, then $\gamma$ is a global solution, i.e., $\,]a,b[\,=\mathbb{R}$. We show by example that this conclusion becomes invalid if $\mathbb{R}^n$ is replaced with an infinite-dimensional Banach space.

Global existence and stability results for partial fractional random differential equations

Abstract We investigate some existence and stability results for the Darboux problem of partial fractional random differential equations in Banach spaces. Our existence results are based upon some fixed point theorems.

Distributed control problems for a class of degenerate semilinear evolution equations

The unique solvability in the sense of strong solution of initial problems for a class of semilinear first order differential equations in Banach spaces with degenerate operator at the derivative is studied. It allows to prove the existence of a solution for the optimal control problem to systems, described by these initial problems. Abstract results are illustrated by the examples of degenerate systems of partial differential equations not solvable with respect to the time derivatives and of optimal control problems for them.

Nonlocal Cauchy problems close to an asymptotically stable equilibrium point

In this work we investigate the existence of solutions for semilinear Cauchy problems with nonlocal initial conditions in the neighborhood of an asymptotically stable equilibrium point of the evolution equation. Using Granas' continuation principle for contractive maps and the qualitative theory of differential equations in Banach spaces, under mild assumptions, we prove the existence of a unique solution. We also show that the main abstract result can be applied to nonlocal initial boundary value problems for reaction–diffusion equations with non-convex nonlinearities.

On the Existence of Solutions of Ordinary Differential Equations in Banach Spaces

Abstract In this paper we prove an existence theorem for ordinary differential equations in Banach spaces. The main assumptions in our results, formulated in terms of the Kuratowski measure of noncompactness, are motivated by the paper [CONSTANTIN, A.: On Nagumo’s theorem, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), 41-44].

Existence and uniqueness of solutions for nonlinear general fractional differential equations in Banach spaces

This paper is concerned with nonlinear general fractional differential equations. The existence and uniqueness of the solutions of above equations are obtained by using a new norm and Banach contraction principle. The results are the generalization of Theorem 1.1 of the paper (Jankowski, 2013).

μ-Pseudo Almost Periodic and Automorphic Solutions in the α-Norm for Some Partial Functional Differential Equations

In this work, we prove the existence and uniqueness of μ-pseudo almost periodic and μ-pseudo almost automorphic solutions in the α −norm for some classes of partial differential equations in Banach spaces. For illustration, we study some models arising in physical systems.

Dissipativity of the backward Euler method for nonlinear Volterra functional differential equations in Banach space

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.

Measure of noncompactness and semilinear nonlocal functional differential equations in Banach spaces

This paper is concerned with the measure of noncompactness in the spaces of continuous functions and semilinear functional differential equations with nonlocal conditions in Banach spaces. The relationship between the Hausdorff measure of noncompactness of intersections and the modulus of equicontinuity is studied for some subsets related to the semigroup of linear operators in Banach spaces. The existence of mild solutions is obtained for a class of nonlocal semilinear functional differential equations without the assumption of compactness or equicontinuity on the associated semigroups of linear operators.