Showing papers on "C0-semigroup published in 1973"
TL;DR: In this article, the eigenvalue of minimum modulus of the Frechet derivative of a nonlinear operator is estimated along a bifurcating curve of zeros of the operator.
Abstract: : The eigenvalue of minimum modulus of the Frechet derivative of a nonlinear operator is estimated along a bifurcating curve of zeroes of the operator. This result is applied to the study of a number of differential equations. Parallel results are developed for a class of nonlinear eigenvalue problems of positive type. (Author)
750 citations
370 citations
31 Dec 1973
212 citations
192 citations
TL;DR: In this article, the main theorem states that a solution Y: R+ → B(H, H) is said to be oscillatory if there exists a sequence of points ti ∈ R+ and Y(ti) fails to have a bounded inverse.
Abstract: Let H be a fixed Hilbert space and B(H, H) be the Banach space of bounded linear operators from H to H with the uniform operator topology. Oscillation criteria are obtained for the operator differential equationwhere the coefficients A, C are linear operators from B(H, H) to B(H, H), for each t ≤ 0. A solution Y: R+ → B(H, H) is said to be oscillatory if there exists a sequence of points ti ∈ R+, so that ti → ∞ as i → ∞, and Y(ti) fails to have a bounded inverse. The main theorem states that a solution Y is oscillatory if an associated scalar differential equation is oscillatory.
168 citations
86 citations
TL;DR: In this paper, it was shown that the set of continuous f is also meager (first category) in X in the scnsc of the Ba&e Category Theorem, where a solution z(*) of (I) will be a solution through us = (to, x0) if (2) is satisfied.
78 citations
01 Jan 1973
TL;DR: In this paper, saddle point analysis for an ODE in a Banach space is described and an application to dynamic buckling of a beam is discussed. But it is assumed in the chapter that both the equation may be written in variation of constants form and that an exponential decomposition holds for the linearized equation.
Abstract: Publisher Summary This chapter describes saddle point analysis for an ordinary differential equation in a Banach space and an application to dynamic buckling of a beam. Saddle point analysis originated in the context of an ordinary differential equation (ODE) in Rn, but has been extended to neutral functional differential equations. The chapter also presents an extension to a class of ODE in a Banach space As hypotheses, it is assumed in the chapter that both the equation may be written in variation of constants form and that an exponential decomposition holds for the linearized equation. With these hypotheses, certain proofs for ODE in Rn carry over to ODE in a Banach space almost word for word. Whether a solution to an ODE in a Banach space satisfies the corresponding variation of constants formula, and vice-versa, seem to be delicate questions. The combined use of an invariance principle and saddle point analysis may find other applications to problems in continuum mechanics, whose characteristic feature is a multiplicity of equilibrium states, steady states, or periodic solutions.
47 citations
TL;DR: For clarity of exposition, the underlying motivation is indicated at the end of this article under concluding Remarks 3, rather than at the beginning as mentioned in this paper, which is the starting point for this article.
Abstract: For clarity of exposition, the underlying motivation is indicated at the end of this article under concluding Remarks 3, rather than at the beginning.
39 citations
TL;DR: In this article, a result previously known only for certain ordered Banach spaces is generalized to arbitrary real Banach algebras, where the adjoint operation maps ℒ (U) onto ℳ (U), where U is a weak projection of a real real space.
Abstract: A result previously known only for certain ordered Banach spaces is generalized to arbitrary real Banach spaces Let ℒ be the Banach algebra of operators generated by theL-projections of a real Banach spaceU, and let ℳ (U
* be the bounded operators on the dual spaceU
* with adjoint in ℒ(U
** Then the adjoint operation maps ℒ (U) onto ℳ (U
*) In particular, anyM-projection ofU
* is weak* continuous
TL;DR: In this paper, a general class of approximate solutions to linear operator equations is studied, in which the domain and range of the operator are both Hilbert spaces possessing continuous reproducing kernels.
TL;DR: In this article, the boundary value problems for these classes of equations were presented and proved to be well-posed, based on a solvability theorem for the operator equation, where is a closed operator.
Abstract: On the half-line we investigate the following equation in a Banach space: (1)where are closed operators which commute with . We consider the following classes of equations: parabolic, inverse parabolic, hyperbolic, quasi-elliptic, and quasi-hyperbolic. We present boundary value problems for these classes and prove that they are well-posed. The proofs are based on a solvability theorem for the operator equation , where is a closed operator.Bibliography: 20 items.
TL;DR: In this paper, a special bounded topology is generated on a collection of absolutely continuous functions with essentially bounded derivatives, and an application to a class of nonlinear neutral functional differential equations due to Driver (1965) is presented.
TL;DR: For a contingent differential equation that takes values in the closed, convex, nonempty subsets of a Banach space, this paper proved an existence theorem and investigated the extendability of solutions and the closedness and continuity properties of solution funnels.
Abstract: For a contingent differential equation that takes values in the closed, convex, nonempty subsets of a Banach space E, we prove an existence theorem and we investigate the extendability of solutions and the closedness and continuity properties of solution funnels. We consider first a space E that is separable and reflexive and then a space E with a separable second dual space. We also consider the special case of a pointvalued or ordinary differential equation. 0. Introduction. Consider the contingent differential equation
TL;DR: In this paper, a system of differential equations on the semiaxis is considered with operator coefficients in a Hilbert space, and the coefficients of the system depend on and for are stabilized in a certain sense.
Abstract: A system of differential equations on the semiaxis is considered with operator coefficients in a Hilbert space. The coefficients of the system depend on and for are stabilized in a certain sense. The spectrum of the limit operator consists of normal eigenvalues and is contained inside a certain double angle with opening less than which contains the imaginary axis. Asymptotic formulas are derived for the solution, and the contribution which a multiple eigenvalue of the limiting operator pencil makes to the asymptotic expressions is investigated.Bibliography: 17 titles.
01 Jan 1973
TL;DR: In this paper, it was shown that for Bochner almost-periodic functions, any stepanov-bounded solution of the differential equation (ddt) u(t) − Bu(t = g(t)) is also almostperiodic.
TL;DR: The main purpose of this paper is to point out that many of the techniques used in the theory of monotone operators can be applied in a more general situation.
Abstract: where A is a continuous function from E into E. In particular, sufficient conditions are established to ensure that (1) has a unique critical point which is globally asymptotically stable and, with additional conditions on A, an iterative method is developed which converges to this critical point. The main purpose of this paper is to point out that many of the techniques used in the theory of monotone operators can be applied in a more general situation (see, for example, Hartman [8]). Instead of using the norm on E to study the solutions to (1), we assume the existence of a function V from E× E into [0, or) which has the following basic properties:
TL;DR: In this article, the authors investigated the question of whether the projection constant of every n-dimensional Banach space is strictly less than Θ(n 2 ) and showed that this is so when n = 2.
Abstract: We discuss various asymmetry constants of finite-dimensional Banach spaces in a more generalized frame than that of [2], and solve a problem raised in [7] by finding an increasing sequence of Banach spaces whose diagonal asymmetry constants tend to infinity. We investigate the question of whether the projection constant of everyn-dimensional Banach space is strictly less than\(\sqrt n \), and show that this is so whenn=2.
TL;DR: In this paper, it is shown how the operator can be factorized into two Volterra operators using a matrix Riccati equation, and the proofs of smoothing and optimal control under known disturbances are in this way especially clear and simple.
01 Feb 1973
TL;DR: In this article, a general theorem on the completeness of the eigenvectors of linear operators in a Banach space was proved and asymptotic estimates for the Green's functions of two-point boundary value problems were derived for a wide class of such problems in the spaces LP(O, 1), 1 p < oo.
Abstract: We prove a general theorem on the completeness of the eigenvectors of linear operators in a Banach space. We then derive asymptotic estimates for the Green's functions of two-point boundary value problems which allow us to apply the above theorem to a wide class of such problems in the spaces LP(O, 1), 1 p< oo.