Showing papers on "C0-semigroup published in 1992"
TL;DR: In this article, the authors studied the initial value problem for the abstract evolution equation A(du/dt)+B(u) ∋f, where A and B are maximal monotone operators from a Banach space W to its dual space W *, with A bounded and B unbounded.
Abstract: The initial value problem is studied for the abstract evolution equationA(du/dt)+B(u) ∋f, whereA andB are maximal monotone operators from a Banach spaceW to its dual spaceW
*, withA bounded andB unbounded. Assuming suitable coerciveness conditions, the existence of a solution is established when at least one of the operators is the subdifferential of a proper convex lower semicontinuous function. The existence theorems are shown by introducing a suitable time discretization of the problem and then passing to the limit by monotonicity and compactness. Uniqueness is proved whenA orB is linear and symmetric and one of them is strictly monotone. Applications are indicated for classes of nonlinear partial differential equations and systems.
120 citations
TL;DR: The problem is to evaluate the complexity of solving the equation to a given accuracy, i.e., given a class 2l of instances, to point out the best possible upper bound of the number of calls of the oracle sufficient to find an e-solution to each of the instances.
104 citations
TL;DR: In this paper, the duality equations of stochastic partial differential equations are solved in the Sobolev space H m = W 2 m (R d ) and the H m -norm estimates of the solutions are obtained.
88 citations
TL;DR: In this paper, the existence and uniqueness of anti-periodic solutions to a class of abstract nonlinear second-order differential equations are studied. But their results rely on the theory of m-accretive operators in Banach or Hilbert spaces.
85 citations
35 citations
01 Jan 1992
TL;DR: In this paper, a sequential quadratic programming method for optimization problems in Banach spaces is proposed and the results are applied to various optimization problems and sufficient conditions for local convergence of the method are given.
Abstract: We analyze a sequential quadratic programming method for optimization problems in Banach spaces. We give sufficient conditions for local quadratic convergence of the method. The results are applied to various optimization problems.
29 citations
TL;DR: In this article, the fixed point theory is used to prove two existence proofs of positive solutions for the impulsive Fredholm integral Equations in Banach spaces, and then they offer some applications to the two-point boundary value problems for the second order impulsive differential equations.
Abstract: In this paper, we first use the fixed point theory to prove two existence
theorems of positive solutions for the impulsive Fredholm integral Equations
in Banach spaces. And then, we offer some applications to the two-point
boundary value problems for the second order impulsive differential equations
in Banach spaces.
20 citations
20 citations
TL;DR: In this paper, the authors studied the mean-field limit of a sequence of dynamical semigroups on the n-fold tensor products of a C*-algebra with itself.
Abstract: We study a notion of the mean-field limit of a sequence of dynamical semigroups on the n-fold tensor products of a C*-algebra with itself. In analogy with the theory of semigroups on Banach spaces we give abstract conditions for the existence of these limits. These conditions are verified in the case of semigroups whose generators are determined by the successive resymmetrizations of a fixed operator, as well as generators which can be approximated by generators of this type. This includes the time evolutions of the mean-field versions of quantum lattice systems. In these cases the limiting dynamical semigroup is given by a continuous flow on the state space of . For a class of such flows we show stability by constructing a Liapunov function. We also give examples where the limiting evolution is given by a diffusion, rather than a flow on the state space of .
19 citations
TL;DR: The exact order of the e-complexity in a power scale for some class of equations z = Hz + f in Hilbert space is found for classes of Fredholm equations, Volterra equations, and weakly singular integral equations.
17 citations
TL;DR: Boundary value problems for second order operator differential equations with two boundary value conditions are studied in this paper, and explicit expressions of the solutions in terms of data problems are given by means of the application of algebraic techniques.
TL;DR: In this article, the authors characterised equations of the type u ∆( t ) + Bu ∆ + Au ∆ = 0 in a Banach space where A, B are densely defined closed linear operators and whose solutions are all almost periodic.
TL;DR: In this article, a notion of T -regularity is introduced to generalize a well known fixed point theorem of Browder, and some related results in this direction are given in this paper.
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TL;DR: In this article, the current state of the homogeneous Banach space problem is reviewed, and several questions arise naturally from this problem, some of which seem to be fundamental but new.
Abstract: We review the current state of the homogeneous Banach space problem. We then formulate several questions which arise naturally from this problem, some of which seem to be fundamental but new. We give many examples defining the bounds on the problem. We end with a simple construction showing that every infinite dimensional Banach space contains a subspace on which weak properties have become stable (under passing to further subspaces). Implications of this construction are considered.
TL;DR: In this paper, the existence of bounded solutions to nonlinear nonautonomous ordinary differential equations is studied by associating the given equation to non-linear autonomous ones by means of a family of skew-product flows related by homotopy.
Abstract: The existence of bounded solutions to nonlinear nonautonomous ordinary differential equations is studied. This is done by associating the given equation to nonlinear autonomous ones by means of a family of skew-product flows related by homotopy. The existence of a bounded solution to the original differential equation is then related to the nontriviality of a certain Conley index associated with the autonomous differential equations. The existence of nontrivial bounded solutions is also considered. The differential equations studied are perturbations of homogeneous ones
01 Jan 1992
TL;DR: In this article, the authors proved existence results on abstract differential equations of the type d{Bu)/dt + A(u) = f and gave some applications of them to partial differential equations.
Abstract: — We prove two existence results on abstract differential equations of the type d{Bu)/dt + A(u) = f and we give some applications of them to partial differential equations.
01 Mar 1992
TL;DR: In this paper, the authors construct an operator with the properties mentioned in the title and show that without the regularity assumption such an example was first discovered by U. Krengel.
Abstract: We construct an operator with the properties mentioned in the title. Without the regularity assumption such an example was first discovered by U. Krengel.
TL;DR: In this article, it was shown that a topological product of infinitely many infinite-dimensional Frechet spaces, each not isomorphic to, does not belong to the class of locally complete locally convex spaces such that an existence theorem holds for the linear Cauchy problem with respect to functions.
Abstract: Let be the class of sequentially complete locally convex spaces such that an existence theorem holds for the linear Cauchy problem , , with respect to functions . It is proved that if , then for an arbitrary set . It is also proved that a topological product of infinitely many infinite-dimensional Frechet spaces, each not isomorphic to , does not belong to .
TL;DR: In this paper, the vector field formulation of and the Sato-Segal-Wilson approach to soliton equations are related to each other in the sense that they derive homogeneous Banach manifolds of solutions on which these equations are realized by vector fields.
Abstract: The vector field formulation of and the Sato-Segal-Wilson approach to soliton equations are related to each other in this paper. From Banach Lie groups associated with the MKdV hierarchy of differential equations, we derive homogeneous Banach manifolds of solutions on which these equations are realized by vector fields. In the same way, we obtain homogeneous Banach manifolds of solutions to the sine-Gordon equation. The scattering and inverse scattering maps in this set-up are also discussed.
TL;DR: In this article, the existence of a common continuous selection of finitely many multivalued mappings with values in a space of Bochner-integrable functions is proved.
Abstract: A continuous version of a theorem of Lyapunov on convexity for measures with values in a Banach space is proved, and then used to obtain two results on the existence of a common continuous selection of finitely many multivalued mappings with values in a space of Bochner-integrable functions. These results are applied to the investigation of properties of solutions of differential inclusions with -accretive operators.
TL;DR: In this paper, the existence theorems for nonlinear random equations and inequalities involving operators of monotone type in Banach spaces are studied and a random Hammerstein integral equation is also studied.
Abstract: In this paper we give some new existence theorems for nonlinear random equations and inequalities involving operators of monotone type in Banach spaces. A random Hammerstein integral equation is also studied. In order to obtain random solutions we use some results from the existing deterministic theory as well as from the theory of measurable multifunctions and, in particular, the measurable selection theorems of Kuratowski/Ryll-Nardzewski and of Saint-Beuve.
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TL;DR: The class of countably intersected families of sets is defined in this article, and a subspace of a W.C.G. space not containing $\ell^{1}(\NN )$ is defined.
Abstract: The class of countably intersected families of sets is defined. For any such family we define a Banach space not containing $\ell^{1}(\NN )$. Thus we obtain counterexamples to certain questions related to the heredity problem for W.C.G. Banach spaces. Among them we give a subspace of a W.C.G. Banach space not containing $\ell^{1}(\NN )$ and not being itself a W.C.G. space.
01 Jan 1992
TL;DR: In this article, the concept of exponential dichotomy for nonlinear evolution operators is introduced and necessary and sufficient conditions for exponential dichotomies are given. But these results are generalizations of well-known results of R. Datko (1973) and A. Ichikawa (1984) about exponential stability.
Abstract: This paper introduces a concept of exponential dichotomy for a general class of nonlinear evolution operators. Necessary and sufficient conditions for exponential dichotomy are given. Obtained results are generalizations of well-known results of R. Datko (1973) and A. Ichikawa (1984) about exponential stability.