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C0-semigroup

About: C0-semigroup is a research topic. Over the lifetime, 6301 publications have been published within this topic receiving 133595 citations.


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Journal ArticleDOI
TL;DR: In this article, the BakerHausdorff formula for non-commuting variables is proved in Section I1 of the present paper together with some applications which concern the addition theorem of the exponential function for noncommuting variable.
Abstract: The present investigation was stimulated by a recent paper of K. 0. Friedrichs 113, who arrived at some purely algebraic problems in connection with the theory of linear operators in quantum mechanics. In particuIar, Friedrichs used a theorem by which the Lie elements in a free associative ring can be characterized. This theorem is proved in Section I1 of the present paper together with some applications which concern the addition theorem of the exponential function for non-commuting variables, the so-called BakerHausdorff formula. Section I contains some algebraic preliminaries. It is of a purely expository character and so is part of Section 111. Otherwise, Section 111 deals with the following problem, also considered by Friedrichs: Let A( t ) be a linear operator depending on a real variable 1. Let Y(t) be a second operator satisfying the differential equation

2,203 citations

Journal ArticleDOI
TL;DR: In this paper, the existence, uniqueness and stability results for ordinary differential equations with coefficients in Sobolev spaces were derived from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions.
Abstract: We obtain some new existence, uniqueness and stability results for ordinary differential equations with coefficients in Sobolev spaces. These results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions.

2,075 citations

Book
06 Apr 1976
TL;DR: In this article, the authors consider boundary value problems for second order nonlinear differential equations in Banach spaces, and show that the boundary value problem can be expressed as a boundary value maximization problem.
Abstract: I Preliminaries.- 1. Metric properties of normed spaces.- 1.1 Duality mappings.- 1.2 Strictly convex normed spaces.- 1.3 Uniformly convex Banach spaces.- 2. Vectorial functions defined on real intervals.- 2.1 Absolutely continuous vectorial functions.- 2.2 Vectorial distributions and Wk,p spaces.- 2.3 Sobolev spaces.- 3. Semigroups of continuous linear operators.- 3.1 Semigroups of class (C0). Hille-Yosida theorem.- 3.2 Analytic semigroups.- 3.3 Nonhomogeneous linear differential equations.- II Nonlinear Operators in Banach Spaces.- 1. Maximal monotone operators.- 1.1 Definitions and fundamental concepts.- 1.2 A general perturbation theorem.- 1.3 A nonlinear elliptic boundary problem.- 2. Subdifferential mappings.- 2.1 Lower semicontinuous convex functions.- 2.2 Subdifferentials of convex functions.- 2.3 Some examples of cyclically monotone operators.- 3. Dissipative sets in Banach spaces.- 3.1 Basic properties of dissipative sets.- 3.2 Perturbations of dissipative sets.- 3.3 Riccati equations in Hilbert spaces.- Bibliographical notes.- III Differential Equations in Banach Spaces.- 1. Semigroups of nonlinear contractions in Banach spaces.- 1.1 General properties of nonlinear semigroups.- 1.2 The exponential formula.- 1.3 Convergence theorems.- 1.4 Generation of nonlinear semigroups.- 2. Quasi-autonomous differential equations.- 2.1 Existence theorems.- 2.2 Periodic solutions.- 2.3 Examples.- 3. Differential equations associated with continuous dissipative operators.- 3.1 A general existence result.- 3.2 Continuous perturbations of m-dissipative operators.- 3.3 Semi-linear second-order elliptic equations in L1.- 4. Time-dependent nonlinear differential equations.- 4.1 Evolution equations associated with dissipative sets.- 4.2 Evolution equations associated with nonlinear monotone hemicon-tinuous operators.- Bibliographical notes.- IV Nonlinear Differential Equations in Hilbert Spaces.- 1. Nonlinear semigroups in Hilbert spaces.- 1.1 Nonlinear version of the Hille-Yosida theorem.- 1.2 Exponential formulae.- 1.3 Invariant sets with respect to nonlinear semigroups.- 2. Smoothing effect on initial data.- 2.1 The case in which A = ? ?.- 2.2 The case in which int D(A) ? ?.- 2.3 Applications.- 3. Variational evolution inequations.- 3.1 Unilateral conditions on u(t).- 3.2 Unilateral conditions on $$ \frac{{du}}{{dt}}(t) $$.- 3.3 A class of nonlinear variational inequations.- 3.4 Applications.- 4. Nonlinear Volterra equations with positive kernels in Hilbert spaces.- 4.1 Positive kernels.- 4.2 Equation (4.1) with A = ? ?.- 4.3 Equation (4.1) with A demicontinuous.- 4.4 A class of integro-differential equations.- 4.5 Further investigation of the preceding case.- Bibliographical notes.- V Second Order Nonlinear Differential Equations.- 1. Nonlinear differential equations of hyperbolic type.- 1.1 The equation $$ \frac{{{d^2}u}}{{d{t^2}}} + Au + M\left( {\frac{{du}}{{dt}}} \right) \mathrel\backepsilon f $$.- 1.2 Further investigation of the preceding case.- 1.3 Examples.- 1.4 Singular perturbations and hyperbolic variational inequations.- 1.5 Nonlinear wave equation.- 2. Boundary value problems for second order nonlinear differential equations.- 2.1 A class of two-point boundary value problems.- 2.2 Examples.- 2.3 A boundary value problem on half-axis.- 2.4 The square root of a nonlinear maximal monotone operator.- Bibliographical notes.

1,763 citations

Journal ArticleDOI
TL;DR: In this paper, the structure of the solution set for a large class of nonlinear eigenvalue problems in a Banach space is investigated, and the existence of continua, i.e., closed connected sets, of solutions of these equations is demonstrated.

1,749 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202323
202248
20212
20207
201910
201820