Showing papers on "C0-semigroup published in 2010"
Book•
29 Apr 2010
TL;DR: The Cauchy Problem in Banach Spaces has been studied in this article. But the authors focus on the existence of nonlinear dissipative dynamics and do not consider nonlinear nonlinear operators.
Abstract: Fundamental Functional Analysis.- Maximal Monotone Operators in Banach Spaces.- Accretive Nonlinear Operators in Banach Spaces.- The Cauchy Problem in Banach Spaces.- Existence Theory of Nonlinear Dissipative Dynamics.
675 citations
TL;DR: In this article, the authors present fixed point theorems in a complete metric space endowed with a partial order by using altering distance functions and present some applications to first and second order ordinary differential equations.
Abstract: The purpose of this paper is to present some fixed point theorems in a complete metric space endowed with a partial order by using altering distance functions We also present some applications to first and second order ordinary differential equations
264 citations
TL;DR: In this article, a class of hyperbolic partial differential equations on a one-dimensional spatial domain with control and observation at the boundary is studied, and it is shown that these systems are well-posed in the sense of Weiss and Salamon if and only if the state operator generates a C0-semigroup.
Abstract: We study a class of hyperbolic partial differential equations on a one dimensional spatial domain with control and observation at the boundary. Using the idea of feedback we show these systems are well-posed in the sense of Weiss and Salamon if and only if the state operator generates a C0-semigroup. Furthermore, we show that the corresponding transfer function is regular, i.e., has a limit for s going to infinity.
92 citations
TL;DR: The results of this paper are new and generalize some known inequalities in the literatures.
85 citations
TL;DR: This work gives a new composition theorem of weighted pseudo-almost periodic functions that improves the known one by making use of a uniform continuity condition instead of the Lipschitz condition.
Abstract: In this work, we give some theorems on (mild) weighted pseudo-almost periodic solutions for some abstract semilinear differential equations with uniform continuity. To facilitate this we give a new composition theorem of weighted pseudo-almost periodic functions. Our composition theorem improves the known one by making use of a uniform continuity condition instead of the Lipschitz condition.
84 citations
TL;DR: In this article, the structure of various classes of spaces of vector-valued functions was studied, ranging between periodic functions and bounded continuous functions, and a general operator theoretical approach was proposed to study a class of semilinear integro-differential equations.
Abstract: In this paper we study the structure of various classes of spaces of vector-valued functions $${\mathcal{M}(\mathbb{R};X)}$$
ranging between periodic functions and bounded continuous functions. Some of these functions are introduced here for the first time. We propose a general operator theoretical approach to study a class of semilinear integro-differential equations. The results obtained are new and they recover, extend or improve variety of recent works.
67 citations
TL;DR: In this paper, several types of periodicity to a class of fractional order differential equations have been studied, and several different types of periods are defined. But none of them are applicable to the present paper.
Abstract: We study several types of periodicity to a class of fractional order differential equations.
64 citations
Book•
01 Jan 2010
TL;DR: In this paper, the authors present an outgrowth of a course which they have given almost periodically over the last eight years, addressed to beginning graduate students of mathematics, engineering, and the physical sciences.
Abstract: i Preface This book is an outgrowth of a course which we have given almost periodically over the last eight years. It is addressed to beginning graduate students of mathematics, engineering, and the physical sciences. Thus, we have attempted to present it while presupposing a minimal background: the reader is assumed to have some prior acquaintance with the concepts of " linear " and " continuous " and also to believe L 2 is complete. An undergraduate mathematics training through Lebesgue integration is an ideal background but we dare not assume it without turning away many of our best students. The formal prerequisite consists of a good advanced calculus course and a motivation to study partial differential equations. A problem is called well-posed if for each set of data there exists exactly one solution and this dependence of the solution on the data is continuous. To make this precise we must indicate the space from which the solution is obtained, the space from which the data may come, and the corresponding notion of continuity. Our goal in this book is to show that various types of problems are well-posed. These include boundary value problems for (stationary) elliptic partial differential equations and initial-boundary value problems for (time-dependent) equations of parabolic, hyperbolic, and pseudo-parabolic types. Also, we consider some nonlinear elliptic boundary value problems, variational or unilateral problems, and some methods of numerical approximation of solutions. We briefly describe the contents of the various chapters. Chapter I presents all the elementary Hilbert space theory that is needed for the book. The first half of Chapter I is presented in a rather brief fashion and is intended both as a review for some readers and as a study guide for others. Non-standard items to note here are the spaces C m (¯ G), V * , and V. The first consists of restrictions to the closure of G of functions on R n and the last two consist of conjugate-linear functionals. Chapter II is an introduction to distributions and Sobolev spaces. The latter are the Hilbert spaces in which we shall show various problems are well-posed. We use a primitive (and non-standard) notion of distribution which is adequate for our purposes. Our distributions are conjugate-linear and have the pedagogical advantage of being independent of any discussion of topological vector space theory. Chapter III is an exposition of the theory of linear elliptic boundary …
50 citations
TL;DR: In this paper, some properties of the generalized f-projection operator are proved in Banach spaces and the strong convergence theorems for relatively nonexpansive mappings are studied inBanach spaces.
Abstract: In this paper, some properties of the generalized f-projection operator are proved in Banach spaces. Using these results, the strong convergence theorems for relatively nonexpansive mappings are studied in Banach spaces. As applications, the strong convergence of general H-monotone mappings in Banach spaces is also given. The results presented in this paper generalize and improve the main results of Matsushita and Takahashi (2005) [9].
45 citations
TL;DR: In this article, the authors present a fixed point theorem for generalized contractions in partially ordered complete metric spaces. And they also present an application to first-order ordinary differential equations, which is a generalization of the generalized contraction theorem presented in this paper.
Abstract: The purpose of this paper is to present a fixed point theorem for generalized contractions in partially ordered complete metric spaces. We also present an application to first-order ordinary differential equations.
45 citations
TL;DR: This paper investigates the existence of solutions for a class of second-order boundary-value problems with integral boundary conditions of nonlinear impulsive integro-differential equations in Banach spaces based upon the fixed point theorem of strict set contraction operators.
TL;DR: The solutions of some evolution equations with fractional orders in a Banach space are considered and conditions are given which ensure the existence of a resolvent operator for an evolution equation in aBanach space.
Abstract: In this paper the solutions of some evolution equations with fractional orders in a Banach space are considered. Conditions are given which ensure the existence of a resolvent operator for an evolution equation in a Banach space.
TL;DR: In this paper, the authors studied the existence and uniqueness of S -asymptotically ω -periodic and asymptotic solutions to a first-order differential equation with linear part dominated by a Hille-Yosida operator with non-dense domain.
Abstract: In this work, we study the existence and uniqueness of S -asymptotically ω -periodic and asymptotically ω -periodic solutions to a first-order differential equation with linear part dominated by a Hille–Yosida operator with non-dense domain. Applications to partial differential equations, fractional integro-differential and neutral differential equations are given.
TL;DR: In this paper, a method of accelerated Landweber type for the iterative regularization of nonlinear ill-posed operator equations in Banach spaces is investigated, based on an auxiliary algorithm with a simplified choice of the step-size parameter.
Abstract: We investigate a method of accelerated Landweber type for the iterative regularization of nonlinear ill-posed operator equations in Banach spaces. Based on an auxiliary algorithm with a simplified choice of the step-size parameter we present a convergence and stability analysis of the algorithm under consideration. We will close our discussion with the presentation of a numerical example.
TL;DR: In this paper, it was shown that there exist infinite dimensional Banach spaces of functions that satisfy properties that apparently should be destroyed by the linear combination of two of them, except for 0.
Abstract: The aim of this paper is to show that there exist infinite dimensional Banach spaces of functions that, except for 0, satisfy properties that apparently should be destroyed by the linear combination of two of them. Three of these spaces are: a Banach space of differentiable functions on R(n) failing the Denjoy-Clarkson property; a Banach space of non Riemann integrable bounded functions, but with antiderivative at each point of an interval; a Banach space of infinitely differentiable functions that vanish at infinity and are not the Fourier transform of any Lebesgue integrable function.
TL;DR: In this paper, the existence and uniqueness of exponentially stable almost periodic solutions for abstract impulsive differential equations in Banach space were investigated by means of the fractional powers of operators, and an example was constructed to illustrate the feasibility of their results.
Abstract: This paper studies the existence and uniqueness of exponentially stable almost periodic solutions for abstract impulsive differential equations in Banach space. The investigations are carried out by means of the fractional powers of operators. We construct an example to illustrate the feasibility of our results.
TL;DR: This work provides necessary and sufficient conditions for switched systems on Banach and Hilbert spaces governed by strongly continuous one-parameter semigroups of linear evolution operators, in terms of the existence of a Lyapunov function common to all modes.
Abstract: We consider switched systems on Banach and Hilbert spaces governed by strongly continuous one-parameter semigroups of linear evolution operators. We provide necessary and sufficient conditions for their global exponential stability, uniform with respect to the switching signal, in terms of the existence of a Lyapunov function common to all modes.
TL;DR: In this article, a class of Banach algebras satisfying certain sequential condition (P) was introduced and fixed point theorems for the sum and the product of nonlinear weakly sequentially continuous operators were proved.
TL;DR: In this paper, the existence of mild solutions for first-order impulsive semilinear neutral functional differential equations with infinite delay in Banach spaces is investigated and conditions in respect of the Hausdorff measure of noncompactness under which the mild solutions exist are derived.
Abstract: In this paper, the existence of mild solutions for first-order impulsive semilinear neutral functional differential equations with infinite delay in Banach spaces is investigated. We derive conditions in respect of the Hausdorff measure of noncompactness under which the mild solutions exist in Banach spaces. Our results improve and generalize some previous results.
TL;DR: In this paper, a hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings was introduced.
Abstract: We introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings, the set of the variational inequality for an α-inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2 uniformly convex and uniformly smooth Banach space. The results presented in this paper improve and extend some recent results.
TL;DR: In this paper, conditions for the existence of solutions vanishing at a singular point for various classes of systems of quasilinear differential equations appearing in the investigation of the asymptotic behavior of solutions of essentially nonlinear nonautonomous differential equations of higher orders were established.
Abstract: We establish conditions for the existence of solutions vanishing at a singular point for various classes of systems of quasilinear differential equations appearing in the investigation of the asymptotic behavior of solutions of essentially nonlinear nonautonomous differential equations of higher orders.
TL;DR: In this paper, a class of nonlinear operator equations x = Ax + x0 on ordered Banach spaces, where A is a monotone generalized concave operator, is studied.
Abstract: In this paper, we study a class of nonlinear operator equations x = Ax + x0 on ordered Banach spaces, where A is a monotone generalized concave operator. Using the properties of cones and monotone iterative technique, we establish the existence and uniqueness of solutions for such equations. In particular, we do not demand the existence of upper-lower solutions and compactness and continuity conditions. As applications, we study first-order initial value problems and two-point boundary value problems with the nonlinear term is required to be monotone in its second argument. In the end, applications to nonlinear systems of equations and to nonlinear matrix equations are also considered.
TL;DR: In this article, the authors considered the control of processes described by oscillating second-order systems of differential equations with a single delay, and developed conditions of relative controllability and constructed a specific control function solving the problem of transferring an initial function to a prescribed point in the phase space.
Abstract: Systems are considered related to the control of processes described by oscillating second-order systems of differential equations with a single delay. An explicit representation of solutions with the aid of special matrix functions called a delayed matrix sine and a delayed matrix cosine is used to develop the conditions of relative controllability and to construct a specific control function solving the relative controllability problem of transferring an initial function to a prescribed point in the phase space.
TL;DR: A boundary for a real Banach space is a subset of the dual unit sphere with the property that each element of the Banach spaces attains its norm on an element of that subset as discussed by the authors.
Abstract: A boundary for a real Banach space is a subset of the dual unit sphere with the property that each element of the Banach space attains its norm on an element of that subset. Trivially, the pointwise convergence with respect to such a boundary is coarser than the weak topology on the Banach space. The boundary problem asks whether nevertheless both topologies have the same norm bounded compact sets.
TL;DR: In this paper, the sum of a weakly-strongly continuous map and a nonexpansive map on a Banach space is shown to be a fixed point theorem.
Abstract: We present some fixed point theorems for the sum of a weakly-strongly continuous map and a nonexpansive map on a Banach space . Our results cover several earlier works by Edmunds, Reinermann, Singh, and others.
01 Jan 2010
TL;DR: In this paper, the existence of mild solutions for impulsive differential equations with nonlocal conditions in Banach spaces is studied under the conditions in respect of the Hausdorff measure of noncompactness.
Abstract: The paper is concerned with the existence of mild solutions for impulsive differential equations with nonlocal conditions in Banach spaces. The results are obtained under the conditions in respect of the Hausdorff measure of noncompactness. Since we do not assume the compactness of semigroup T(t) and f , our theorems extend some existing results in this area.
TL;DR: In this article, the existence and uniqueness of strong solutions for initial functions from a Banach space are proved for a class of non-linear partial differential equations with discrete state-dependent delays.
Abstract: We investigate a class of non-linear partial differential equations with discrete state-dependent delays. The existence and uniqueness of strong solutions for initial functions from a Banach space are proved. To get the well-posed initial value problem we restrict our study to a smaller metric space, construct the dynamical system and prove the existence of a compact global attractor.
TL;DR: In this article, the existence and regularity of solutions for a class of neutral partial functional integrodifferential equations with infinite delay in Banach spaces were investigated and sufficient conditions ensuring the existence of strict solutions were given.
TL;DR: In this paper, the existence and uniqueness of mild solution of fractional differential equations with nonlocal conditions in a Banach space was studied. And the uniqueness theorem was extended to the case of mild solutions with local conditions.
Abstract: This paper is concerned with the existence and uniqueness of mild solution of the fractional differential equations with nonlocal conditions , in a Banach space , where . General existence and uniqueness theorem, which extends many previous results, are given.