Showing papers on "C0-semigroup published in 1984"
255 citations
167 citations
145 citations
01 Jan 1984
TL;DR: In this article, the authors give necessary and sufficient conditions for exponential dichotomy of strongly continuous semigroups of operators defined on a Banach space and obtain a Datko theorem for exponentialstability of a semigroup of class Cdefined on a graph.
Abstract: In this paper we give necessary and sufficient conditions forexponential dichotomy of a general class of strongly continuoussemigroups of operators defined on a Banach space. As aparticular case we obtain a Datko theorem for exponentialstability of a strongly continuous semigroup of class Cdefined on a Banach space.
48 citations
TL;DR: In this paper, the existence and properties of resolvent operators for linear Volterra integrodifferential equations in Banach space were investigated and the regularity of weak solutions given by the variation of parameters formula was examined.
Abstract: This paper is concerned with the existence and properties of resolvent operators for linear Volterra integrodifferential equations in Banach space. Regularity of weak solutions given by the variation of parameters formula is also examined.
47 citations
TL;DR: In this article, the Perron-Frobenius spectral theory for positive semigroups on Banach lattices is surveyed and applied to stability theory of retarded differential equations and quasi-periodic flows.
Abstract: In this paper we survey the Perron-Frobenius spectral theory for positive semigroups on Banach lattices and indicate its applications to stability theory of retarded differential equations and quasi-periodic flows.
42 citations
TL;DR: In this article, nonlinear evolution operators in Banach spaces have been discussed and a fundamental result on the construction of an evolution operator in general Banach space has been established by Crandall-Pazy.
Abstract: Publisher Summary This chapter discusses nonlinear evolution operators in Banach spaces. It also presents the generation of evolution operators that provide the generalized solutions. A fundamental result on the construction of an evolution operators in general Banach spaces has been established by Crandall–Pazy. The chapter also presents the extension of the notion of integral solution to the time-dependent case.
41 citations
34 citations
TL;DR: In this article, the authors studied the error between the solution manifold of (1) and its discretizations for arbitrary (finite) dimension of?, and showed that these estimates are applicable to rather general nonlinear boundaryvalue problems for partial differential equations.
Abstract: The paper concerns solution manifolds of nonlinear parameterdependent equations (1)F(u, ?)=y0 involving a Fredholm operatorF between (infinite-dimensional) Banach spacesX=Z×? andY, and a finitedimensional parameter space ?. Differntial-geometric ideas are used to discuss the connection between augmented equations and certain onedimensional submanifolds produced by numerical path-tracing procedures. Then, for arbitrary (finite) dimension of ?, estimates of the error between the solution manifold of (1) and its discretizations are developed. These estimates are shown to be applicable to rather general nonlinear boundaryvalue problems for partial differential equations.
32 citations
TL;DR: In this paper, a method of lines is introduced and developed for the abstract functional problem Lipschitzian-like function in its two variables, and conditions are given for the convergence of the method to a strong solution of (E).
Abstract: Let X be a real Banach space with X* uniformly convex. A method of lines is introduced and developed for the abstract functional problem Lipschitzian-like function in its two variables. Further conditions are given for the convergence of the method to a strong solution of (E). Recent results for perturbed abstract ordinary equations are substantially improved. The method applies also to large classes of functional parabolic problems as well as problems of integral perturbations. The method is straightforward because it avoids the introduction of the operators A( t) and the corresponding use of nonlinear evolution operator theory. 1. Introduction-Preliminaries. Let X be a real Banach space with norm 11 Let
28 citations
01 Jan 1984
TL;DR: In this paper, the authors investigated the probability of large deviations for sums of i.i.d. values of Banach space valued random variables when the laws of the random variables converge weakly and a uniform exponential integrability condition is satisfied.
Abstract: Probabilities of large deviations for sums of i.i.d. Banach space valued random variables are investigated when the laws of the random variables converge weakly and a uniform exponential integrability condition is satisfied. Furthermore, a discussion of possible improvements of the estimates is given, when the probability is estimated that the sum lies in a convex set.
TL;DR: Lie series are used to calculate both closed form and approximate solutions for elementary nonlinear ordinary differential equations as discussed by the authors, and they can be used to compute both closed-form and approximate solution for nonlinear ODEs.
TL;DR: In this article, a positive operator in a Banach space is constructed, the operators and their products and superpositions are investigated and the moment inequality as well as other estimates are proved.
Abstract: For functions , 0$ SRC=http://ej.iop.org/images/0025-5734/47/1/A02/tex_sm_2628_img2.gif/>, nondecreasing, and a positive operator in a Banach space, the operators and are constructed, their products and superpositions are investigated and the moment inequality as well as other estimates are proved. The results are generalized to the case when does not exist or is not bounded.Bibliography: 21 titles.
TL;DR: In this article, the authors prove the existence of a bounded solution of the differential equation y′ = A(t)y + f(t, y) in a Banach space.
Abstract: We prove the existence of bounded solution of the differential equation y′ = A(t)y + f(t, y) in a Banach space. The method used here is based on the concept of “admissibility” due to Massera and Schaffer when f satisfies the Caratheodory conditions and some regularity condition expressed in terms of the measure of noncompactness α.
TL;DR: In this article, it was shown that for strongly monotone operators, the truncation errors are bounded uniformly in time and with error O(Δt)2, ifϑ = 1/2−|O( Δt)|, where Δt is the Lipschitz constant.
Abstract: Classical discretization error estimates for systems of ordinary differential equations contain a factor exp (Lt), whereL is the Lipschitz constant. For strongly monotone operators, however, one may prove that for aϑ-method, 0<ϑ<1/2, the errors are bounded uniformly in time and with errorO(Δt)2, ifϑ=1/2−|O(Δt)|. This was done by this author (1977), for an operator in a reflexive Banach space and includes the case of systems of differential equations as a special case. In the present paper we restate this result as it may have been overlooked and consider also the monotone (inclusive of the conservative) and unbounded cases. We also discuss cases where the truncation errors are bounded by a constant independent of the stiffness of the problem. This extends previous results in [6] and [7]. Finally we discuss a boundary value technique in the context above.
TL;DR: In this paper, a generalization of Marti's method for least square solutions of the linear equation is presented, where T is a bounded pperator from one Hilbert space into another, and the convergence to a solution minimizing the seminorm is proved.
Abstract: An algorithm is presented to compute least squares solutions of the linear equation $Tx = y$, where T is a bounded pperator from one Hilbert space into another. The algorithm is a generalization of Marti’s method. The convergence to a solution minimizing the seminorm $\|Lx\|$ is proved, where the operator L is a closed densely defined linear operator with closed range. The method is applied to integral equations, and three examples are treated numerically.
TL;DR: In this paper, Lie transformations are used to represent solutions of initial value problems for systems of nonlinear ordinary differential equations as exponentials of first order linear partial differential operators, which are then expanded using an analog of the usual exponential identities.
Abstract: Lie transformations are used to represent solutions of initial value problems for systems of nonlinear ordinary differential equations as exponentials of first order linear partial differential operators. These exponentials are then expanded using an analog of the usual exponential identities. This expansion is called the factored product expansion. Such expansions have been found useful in the study of magnetic and optical lenses.
TL;DR: In this paper, exponential estimates on the fundamental matrix, uniform on the perturbation parameter, are obtained for singularly perturbed systems of linear retarded functional differential equations, under the assumption that the eigenvalues of a certain coefficient matrix in the system have negative real parts.
TL;DR: In this article, the authors give sufficient conditions for the existence of a function x: J-+E which has absolutely continuous derivative and satisfies almost everywhere on J the differential equation X = f(t,X,X’) with the boundary conditions ca (ii) there exists an integrable function m: J+ R, such that Ilf( t, x7 Y) II 6 40 forall t EI and x,yEE.
Abstract: LET J = [a, b] BE a compact interval in R, and let E be a real Banach space. In this paper we give sufficient conditions for the existence of a function x: J-+ E which has absolutely continuous derivative and satisfies almost everywhere on J the differential equation X”=f(t,X,X’) (1) with the boundary conditions ca (ii) there exists an integrable function m: J+ R, such that Ilf(t, x7 Y) II 6 40 forall t EI and x,yEE. (2”) WI, w2 E E and the conditions (2) are normed according to di E (0, l}, di = 0 j Ci = 1 for i = 1,2, and there exists a Green’s function G for the problem x” = g(t), CiX(U) diX’(U) = 0, crx(b) + d*x’(b) = 0, where g is a given integrable function from J into E. (3”)h is an integrable function from J into R+ and P, Q are positive numbers such that
TL;DR: In this paper, it was shown that a norm order continuous injective Banach lattice is order isomorphic to an (AL)-space, where the lattice can be expressed as a norm bounded linear operator.
Abstract: LetE be a 1-injective Banach lattice,X any Banach space andT: E ← X a norm bounded linear operator. Then eitherT is an isomorphism on some copy ofl
∞ inE or for all σ > 0 there is φ ∈E
+
′
such that ‖Tu‖≦φ (|u|)+σ ‖u‖ for allu ∈E. We deduce the theorem that: A norm order continuous injective Banach lattice is order isomorphic to an (AL)-space.
TL;DR: In this paper, it was shown that reflexivity is also a necessary and sufficient condition for the bounded weak topology of a Banach space to be locally convex under certain conditions.
Abstract: This paper is devoted to the question "Under what conditions on a Banach space E is it true that the bounded weak topology is locally convex?". The theorem of Banach and Dieudonne shows that reflexivity is a sufficient condition. The author proves that reflexivity is also a necessary condition
TL;DR: In this article, the existence and linear stability of the equilibrium solutions of the dynamical system ドラゴンK dudt = ƒ(u, α)====== which are close to the origin in U×R are studied.
TL;DR: In this article, basic techniques from the theory of interpolation spaces equivalence theorems are established for the intermediate spaces between a given Banach space A and the domain D(Λr of the r-th power of the infinitesimal generator Λ of a strongly continuous cosine operator function C. The results are applied to the study of second order evolution equations including regularity, order reduction and approximation by finite difference methods.
Abstract: Using basic techniques from the theory of interpolation spaces equivalence theorems are established for the intermediate spaces between a given Banach space A and the domain D(Λr) of the r-th power of the infinitesimal generator Λ of a strongly continuous cosine operator function C. The results are applied to the study of second order evolution equations including regularity, order reduction and approximation by finite difference methods.