Showing papers on "C0-semigroup published in 1986"
Book•
01 Mar 1986
TL;DR: In this article, Spectral theory of positive semigroups on Co(X) and asymptotics of positive operators on Co (X) were presented. But the results on the spectral properties of the positive operators were not considered.
Abstract: Basic results on semigroups on banach spaces.- Characterization of semigroups on banach spaces.- Spectral theory.- Asymptotics of semigroups on banach spaces.- Basic results on spaces Co(X).- Characterization of positive semigroups on Co(X).- Spectral theory of positive semigroups on Co(X).- Asymptotics of positive semigroups on Co(X).- Basic results on banach lattices and positive operators.- Characterization of positive semigroups on banach lattices.- Spectral theory of positive semigroups on banach lattices.- Asymptotics of positive semigroups on banach lattices.- Basic results on semigroups and operator algebras.- Characterization of positive semigroups on w*-algebras.- Spectral theory of positive semigroups on w*-algebras and their preduals.- Asymptotics of positive semigroups on c*-and w*-algebras.
1,001 citations
TL;DR: In this paper, the authors obtained sufficient conditions for an autonomous functional differential equation to generate a strongly monotone semi-low on a suitable state space, and established a very striking relationship between such functional differential equations and corresponding ordinary differential equations.
182 citations
TL;DR: A proof of the mesh-independence principle for a general class of operator equations and discretizations covers the earlier results and extends them well beyond the cases that have been considered before.
Abstract: The mesh-independence principle asserts that, when Newton’s method is applied to a nonlinear equation between some Banach spaces as well as to some finite-dimensional discretization of that equation, then the behavior of the discretized process is asymptotically the same as that for the original iteration and, as a consequence, the number of steps required by the two processes to converge to within a given tolerance is essentially the same. So far this result has been proved only for certain classes of boundary value problems. In this paper a proof is presented for a general class of operator equations and discretizations. It covers the earlier results and extends them well beyond the cases that have been considered before.
136 citations
TL;DR: In this article, a unified theory of periodicity of dissipative ordinary and functional differential equations in terms of uniform boundedness is discussed, and sufficient conditions for the uniform boundedess are given by means of Liapunov functionals having a weighted norm as an upper bound.
Abstract: We discuss a unified theory of periodicity of dissipative ordinary and functional differential equations in terms of uniform boundedness. Sufficient conditions for the uniform boundedness are given by means of Liapunov functionals having a weighted norm as an upper bound. The theory is developed for ordinary differential equations, equations with bounded delay, and equations with infinite delay.
59 citations
TL;DR: In this article, the Radon-Nikodym property of Banach spaces has been shown to be equivalent to Bourgain's theorem of Bourgain in terms of optimization results, and some old ideas of Smulian can be used to give another proof of a theorem.
48 citations
TL;DR: In this article, the stability analysis of composite hybrid dynamical feedback systems of the type depicted in Fig. 1 has been studied, consisting of a block (usually the plant) which is described by an operator L and of a finite-dimensional block described by a system of ODEs.
Abstract: We address the stability analysis of composite hybrid dynamical feedback systems of the type depicted in Fig. 1, consisting of a block (usually the plant) which is described by an operator L and of a finite-dimensional block described by a system of ordinary differential equations (usually the controller). We establish results for the well-posedness, attractivity, asymptotic stability, uniform boundedness, asymptotic stability in the large, and exponential stability in the large for such systems. The hypotheses of these results are phrased in terms of the I/O properties of L and in terms of the Lyapunov stability properties of the subsystem described by the indicated ordinary differential equations. The applicability of our results is demonstrated by means of general specific examples (involving C 0 -semigroups, partial differential equations, or integral equations which determine L ).
33 citations
TL;DR: In this paper, linear integrodifferential equations in general Banach space are studied and applications are given to linear integro-differential partial differential equations in Banach spaces.
31 citations
TL;DR: In this paper, the Cayley-Hilbert metric is defined for a real Banach space containing a closed cone, and the Banach contraction mapping theorem is used to prove the existence of a unique fixed point of the operator with explicit upper and lower bounds.
29 citations
29 citations
TL;DR: In this paper, a class of weakly integrable semigroups on locally convex spaces is introduced and studied, which include fractional powers of a closed operator and spectral local semigroup.
01 Mar 1986
TL;DR: In this article, an infinite-dimensional superreflexive real Banach space which does not admit complex structure and consequently is not isomorphic to the Cartesian square of any Banach spaces is constructed.
Abstract: We construct an infinite-dimensional superreflexive real Banach space which does not admit complex structure and consequently is not isomorphic to the Cartesian square of any Banach space. We also construct a variant of Bourgain's example of a complex Banach space with nonunique complex structure and state a number of open problems about structure of Banach spaces and their linear groups.
TL;DR: In this article, the operator semigroups of the title are studied on Hardy spaces and conditions for strong continuity are found for the infinitesimal generator and its point spectrum are identified.
TL;DR: The decomposition method of Adomian has recently been generalized in a number of directions and is now applicable to wide classes of linear and nonlinear, deterministic and stochastic differential, partial differential, and differential delay equations as well as algebraic equations of all types including polynomial equations, matrix equations, equations with negative or nonintegral powers, and random algebraic expressions.
Abstract: The decomposition method of Adomian, which was developed to solve nonlinear stochastic differential equations, has recently been generalized in a number of directions and is now applicable to wide classes of linear and nonlinear, deterministic and stochastic differential, partial differential, and differential delay equations as well as algebraic equations of all types including polynomial equations, matrix equations, equations with negative or nonintegral powers, and random algebraic equations. This paper will demonstrate applicability to transcendental equations as well. The decomposition method basically considers operator equations of the form Fu = g where g may be a number, a function, or even a stochastic process. F is an operator which in general is nonlinear. (If it involves stochastic processes as well, we use a script letter F). The operator F may be a differential or algebraic operator. In this paper we will concentrate on the latter. The authors have thus developed a useful system for realistic solutions of real‐world problems.
01 Oct 1986
TL;DR: In this paper, the authors give an elementary introduction to certain types of stochastic differential equations in infinite dimensional spaces, such as Ornstein-Uhlenbeck type processes with a nuclear valued martingale as a driving term.
Abstract: : These lectures aim at giving an elementary introduction to certain types of stochastic differential equations in infinite dimensional spaces. One lecture introduces countably Hilbertian Nuclear (CHN) spaces and give some examples to illustrate why these infinite dimensional spaces are convenient for the study of some practical problems, e.g. those occuring in stochastic evolutions. This lecture assumes a complete probability spade with a right continuous filtration. It also assumes a given Countably Hilbertian nuclear space. Ornstein-Uhlenbeck stochastic differential equations on duals of nuclear spaces introduces a special class of linear stochastic differential equations with values in duals of nuclear spaces, namely Ornstein-Uhlenbeck type processes with a nuclear valued martingale as a driving term. Weak Convergence of Solutions: now consider the weak convergence of the solutions of to the corresponding stochastic differential equations driven by a Gaussian noise. This last lecture gives an outline of recent works on stochastic evolution equations and nonlinear stochastic differential equations on the dual of a Countably Hilbert nuclear space.
01 Jan 1986
01 Feb 1986
TL;DR: In this paper, it was shown that P > 6/L in a Hilbert space with constant L in a Lipschitz space and that P ≥ 27r/L with constant l in a Banach space.
Abstract: Let f be Lipschitz with constant L in a Banach space and let x(t) be a P-periodic solution of x'(t) = f(x(t)). We show that P > 6/L. An example is given with P = 27r/L, 8o the bound is nearly strict. We also give a short proof that P > 27r/L in a Hilbert space.
TL;DR: In this article, a generalized Fourier transform operator for the complex functions on finite non-Abelian groups is defined and a linear harmonic differential equation with constant coefficients is given.
Abstract: In this note we define a differential operator for the complex functions on finite non-Abelian groups.For the characterization of this differential operator we use the coefficients of the generalized Fourier transforms on groups.Using this operator we define the linear harmonic differential equations with constant coefficients and give the general solution of these equations
TL;DR: In this article, the authors give an approach to the study of both differential inclusions and ordinary differential equations in a Banach space X. The central point concerns the question of the existence and properties of the solution set of a differential inclusion whose right-hand side has the weak Scorza Dragoni property.
Abstract: This article gives an approach to the study of both differential inclusions and ordinary differential equations in a Banach space X. The central point concerns the question of the existence and properties of the solution set of a differential inclusion whose right-hand side has the weak Scorza Dragoni property.Bibliography: 37 titles.
Journal Article•
TL;DR: In this paper, a class of operators from a Banach lattice to the Banach space was studied, which map positive sequences in weak-l^p$-spaces with values in X$ into sequences in l^q$ -spaces having values in B$ and obtain some different characterizations of them.
Abstract: In this paper we study a class of operators from a Banach lattice $X$ into a Banach space $B$. These operators map positive sequences in $weak-l^p$-spaces with values in $X$ into sequences in $l^q$-spaces with values in $B$. We obtain some different characterizations of them an we consider, in particular, the case $X=1^r$.
TL;DR: In this article, the Hopf bifurcation for fully nonlinear evolution equations in Banach spaces is studied and a Hopf partitioning method is proposed to solve the problem.
Abstract: We study Hopf bifurcation for some fully nonlinear evolution equations in Banach spaces.
01 Jan 1986
TL;DR: In this article, an extension of Ando-Krieger's theorem for positive, irreducible, order continuous Harris operators on Dedekind complete Banach lattices is presented.
Abstract: We prove an extension of Ando-Krieger's theorem for positive, irreducible, order continuous Harris operators on Dedekind complete Banach lattices.
TL;DR: In this paper, it was proved that the resolution problem of an operator boundary value problem for a second-order differential operator equation with constant coefficients is solved in terms of solutions of certain algebraic operator equations.
TL;DR: In this article, the decomposition method is applied to the solution of nonlinear differential equations of the form Ly + Ny = x ( t ), where L is a linear differential operator and Ny is a nonlinear term of the forms y γ with γ a decimal number.
TL;DR: In this article, it was shown that the Banach space of X -valued analytic functions on a polydisc P ⊂ C n with absolutely convergent Taylor series has a closed range.