Differential Equations In Banach Spaces Research Articles

Approximate controllability of impulsive Riemann- Liouville fractional equations in Banach spaces

In this paper, we study control systems governed by impulsive Riemann-Liouville fractional differential equations in Banach spaces. Firstly, we introduce $PC_{1-\alpha}$-mild solutions for impulsive Riemann-Liouville fractional differential equations. Then, we make a set of assumptions to guarantee the existence and uniqueness of mild solutions. Finally, approximate controllability of the associated impulsive Riemann-Liouville fractional evolution control systems is also formulated and proved.

Order-preserving nonautonomous discrete dynamics: attractors and entire solutions

The concept of pullback convergence turned out to be a central idea to describe the long-term behavior of nonautonomous dynamical systems. This paper provides a general framework for the existence and structure of pullback attractors capturing the asymptotics of nonautonomous and order-preserving difference equations in Banach spaces. Furthermore we obtain criteria for the convergence to bounded entire solutions and additionally discuss various applications.

Boundary-Value Problems for Differential Equations in Banach Spaces with Unbounded Operator in the Linear Part

We find a criterion for the existence of solutions of boundary-value problems in Banach and Hilbert spaces in the case where the linear part contains an unbounded operator. We also establish conditions for the normal and generalized solvability of these problems.

Conditions for Almost Periodicity of Bounded Solutions of Nonlinear Differential Equations Unsolved with Respect to the Derivative

We establish conditions for the existence of almost periodic solutions of nonlinear almost periodic differential equations in Banach spaces without using the H-classes of these equations.

Solvability of boundary value problems for impulsive fractional differential equations in Banach spaces

Of concern is the existence of solutions for a class of boundary value problems for impulsive fractional differential equations involving the Caputo fractional derivative in a Banach space. Our approach is based upon the techniques of noncompactness measures and the fixed point theory. Two examples are presented to illustrate the results. MSC: 34A08, 34A37, 34K30, 47H08.

Differentiable mappings on products with different degrees of differentiability in the two factors

We develop differential calculus of Cr,s-mappings on products of locally convex spaces and prove exponential laws for such mappings. As an application, we consider differential equations in Banach spaces depending on a parameter in a locally convex space. Under suitable assumptions, the associated flows are mappings of class Cr,s.

Exponential behavior in Banach spaces: robustness of trichotomies in discrete time

For difference equations in Banach spaces, we consider a generalization of the notion of exponential dichotomy, usually called trichotomy in the literature, for which the behaviors in $$\mathbb {Z}^+$$ and $$\mathbb {Z}^-$$ are still exponential but need not agree at the origin. Our main aim is to show that this exponential behavior is robust, in the sense that it persists under sufficiently small linear perturbations.

Decay integral solutions for neutral fractional differential equations with infinite delays

Our aim in this work is to find decay integral solutions for a class of neutral fractional differential equations in Banach spaces involving unbounded delays. By constructing a suitable measure of noncompactness on the space of solutions and establishing new estimates for fractional resolvent operators, we prove the existence of a compact set of decay integral solutions to the mentioned problem. Copyright © 2014 John Wiley & Sons, Ltd.

Asymptotic stability of solution to nonlinear neutral and Volterra functional differential equations in Banach spaces

This paper is concerned with the asymptotic stability properties of the solution to nonlinear neutral and Volterra functional differential equations. Some sufficient conditions for the asymptotic stability of the systems are given. As an illustration of the applications of these investigations, the contractivity and asymptotic stability results of the solution to Volterra partial functional differential equations and delay integro-differential equations of “Hale’s” form are obtained respectively. These results form the basis for obtaining insight into the analogous properties of their numerical solutions.

Existence of Bounded Solutions of Nonlinear Difference Equations in Banach Spaces

We establish sufficient conditions for the existence of bounded solutions of the nonlinear difference equations $$ x\left( {n+1} \right)=F\left( {n,x(n)} \right)x(n)+f(n),\;n\in \mathbb{Z} $$ , in a Banach space.

Linear measure functional differential equations with infinite delay

We use the theory of generalized linear ordinary differential equations in Banach spaces to study linear measure functional differential equations with infinite delay. We obtain new results concerning the existence, uniqueness, and continuous dependence of solutions. Even for equations with a finite delay, our results are stronger than the existing ones. Finally, we present an application to functional differential equations with impulses.

On the initial value problem of fractional evolution equations with noncompact semigroup

In this paper, we study the initial value problem of fractional semilinear evolution equations with noncompact semigroup in Banach spaces. The existence of saturated mild solutions and global mild solutions is obtained under weaker conditions. Particularly, we find that the results obtained are also valid for the initial value problem of fractional ordinary differential equations in Banach spaces. The theorems proved in this paper improve and extend some related conclusions on this topic. An example is given to illustrate that our results are valuable.

Generalized Uniqueness Theorem for Ordinary Differential Equations in Banach Spaces

We consider nonlinear ordinary differential equations in Banach spaces. Uniqueness criterion for the Cauchy problem is given when any of the standard dissipative-type conditions does apply. A similar scalar result has been studied by Majorana (1991). Useful examples of reflexive Banach spaces whose positive cones have empty interior has been given as well.

Existence Results for a Coupled System of Nonlinear Fractional Differential Equations in Banach Spaces

We investigate boundary value problems for a coupled system of nonlinear fractional differential equations involving Caputo derivative in Banach spaces. A generalized singular type coupled Gronwall inequality system is given to obtain an important a priori bound. Existence results are obtained by using fixed point theorems and an example is given to illustrate the results.

Eigenvalues of a Class of Singular Boundary Value Problems of Impulsive Differential Equations in Banach Spaces

This paper is devoted to investigating the eigenvalue problems of a class of nonlinear impulsive singular boundary value problem in Banach spaces:μx′′+f(t,x)=0,t∈(0,1),t≠ti;Δx|t=ti=αix(ti-0),i=1,2,…,k;ax(0)-bx′(0)=θ;cx(1)+dx′(1)=θ,whereθdenotes the zero element of Banach space,Δx|t=ti=x(ti+0)-x(ti-0),αi>-1,a,b,c,d∈R+,γ=ac+ad+bc>0,μis a parameter, andf(t,x)may be singular att=0,1andx=θ. The arguments are mainly based upon the theory of fixed point index, measure of noncompactness, and the special cone, which is constructed to overcome the singularity.

Existence of Almost Periodic Solutions for Impulsive Neutral Functional Differential Equations

The existence of piecewise almost periodic solutions for impulsive neutral functional differential equations in Banach space is investigated. Our results are based on Krasnoselskii’s fixed-point theorem combined with an exponentially stable strongly continuous operator semigroup. An example is given to illustrate the theory.

Inverse problems for singular differential-operator equations with higher order polar singularities

In this paper we study an inverse problem for strongly degenerate differential equations in Banach spaces. Projection method on suitable subspaces will be used to solve the given problem. A number of concrete applications to ordinary and partial differential equations is described.

Decay integral solutions for a class of impulsive fractional differential equations in Banach spaces

Abstract The aim of this paper is to establish the existence of decay integral solutions to a class of retarded fractional differential equations involving impulsive effects. The results are obtained by using the fixed point approach and fractional calculus tools in Banach spaces. Applications to both ordinary and partial differential equations are presented.

Generalized Nonuniform Dichotomies and Local Stable Manifolds

We establish the existence of local stable manifolds for semiflows generated by nonlinear perturbations of nonautonomous ordinary linear differential equations in Banach spaces, assuming the existence of a general type of nonuniform dichotomy for the evolution operator that contains the nonuniform exponential and polynomial dichotomies as a very particular case. The family of dichotomies considered allow situations for which the classical Lyapunov exponents are zero. Additionally, we give new examples of application of our stable manifold theorem and study the behavior of the dynamics under perturbations.

Application of measure of noncompactness to a Cauchy problem for fractional differential equations in Banach spaces

Abstract This paper is devoted to study the existence of solutions of a Cauchy type problem for a nonlinear fractional differential equation, via the techniques of measure of noncompactness. The investigation is based on a new fixed point result which is a generalization of the well known Darbo’s fixed point theorem. The main result is less restrictive than those given in the literature. Some illustrative examples are given.