Showing papers on "C0-semigroup published in 1983"
01 Jan 1983
85 citations
TL;DR: In this paper, the authors consider a class of feedback operators for a linear differential difference equation of the neutral type and show that if a system has unstable neutral root chains, any stabilizing feedback operator in this class must contain a derivative of the delayed state component.
Abstract: In this paper we consider a class of feedback operators for a linear differential difference equation of the neutral type. We show that if a system has unstable neutral root chains, any stabilizing feedback operator in this class must contain a derivative of the delayed state component. Under the assumption of exact controllability, we construct a stabilizing feedback operator.
73 citations
TL;DR: In this article, a differential inclusion is considered, where the mapping takes values in the family of all nonempty compact convex subsets of a Banach space, is upper semicontinuous with respect to for almost every, and has a strongly measurable selection for every.
Abstract: In this article a differential inclusion is considered, where the mapping takes values in the family of all nonempty compact convex subsets of a Banach space, is upper semicontinuous with respect to for almost every , and has a strongly measurable selection for every . Under certain compactness conditions on proofs are given for a theorem on the existence of solutions, a theorem on the upper semicontinuous dependence of solutions on the initial conditions, and an analogue of the Kneser-Hukuhara theorem on connectedness of the solution set.Bibliography: 20 titles.
66 citations
TL;DR: In this article, the authors define two geometric concepts of a Banach space, property α and β, which generalize in a certain way the geometric situation of l andc o. They show that many Banach spaces may (3+e)-equivalently be renormed to have property α.
Abstract: We define two geometric concepts of a Banach space, property α and β, which generalize in a certain way the geometric situation ofl andc
o. These properties have been used by J. Lindenstrauss and J. Partington in the study of norm attaining operators. J. Partington has shown that every Banach space may (3+e)-equivalently be renormed to have property β. We show that many Banach spaces (e.g., every WCG space) may (3+e)-equivalently be renormed to have property α. However, an example due to S. Shelah shows that not every Banach space is isomorphic to a Banach space with property α.
65 citations
63 citations
TL;DR: In this paper, the central limit theorem in B and the weak law of large numbers (for the sum of the squares of the random vectors) in another Banach lattice B(2) were shown to be equivalent.
Abstract: For B a type 2 Banach lattice, we obtain a relationship between the central limit theorem in B and the weak law of large numbers (for the sum of the squares of the random vectors) in another Banach lattice B(2). We then obtain some two-sided estimates for E∥Sn∥pwhich in lpspaces, 1≦p<∞, give n.a.s.c. for the weak law of large numbers. As a consequence of these estimates we also solve the domain of attraction problem in lp, p<2. Several examples and counterexamples are provided.
52 citations
TL;DR: In this paper, the weakly compactly generated (WCG) Banach space admits a Ck-smooth function with bounded support, and partitions of unity of unity.
51 citations
44 citations
TL;DR: In this article, a method for the approximate numerical solution of operator field equations on a lattice is proposed, where the solution of the operator field equation on the lattice can be expressed as
Abstract: A method is proposed for the approximate numerical solution of operator field equations on a lattice.
TL;DR: In this paper, a locally convergent expansion of a nonlinear integral equation in terms of u was shown to converge in a normed space of bounded continuous n-vector valued functions defined on [0, \infty] and involves terms that are sums of Volterra-like iterated integrals.
Abstract: Expansion theorems, and related results, concerning nonlinear integral equations are proved, and are applied to systems of differential equations of the form \dot{x} = f(x, u, t) , almost all t \geq 0, x continuous on [0, \infty), x(0) = x_{0} in which the solution x is n -vector valued. In particular, we show the existence of, and show how to obtain, a locally convergent expansion for x in terms of u , when certain reasonable conditions are met, including the condition that an associated system of linear differential equations is bounded-input bounded-output stable. The expansion converges in a normed space of bounded continuous n -vector valued functions defined on [0, \infty) , and involves terms that are sums of Volterra-like iterated integrals.
01 Jan 1983
TL;DR: In this paper, the author introduced a definition about the non-degeneracy of critical points of a differ-entiable functional defined on a Banach space and established the Morse theory to these functionals on a differentiable norm.
Abstract: The author introduces a definition about the nondegeneracy of critical points of a differ- entiable functional defined on a Banach space.Thus the Morse theory is estabished to these functionals on a Banach space with an equivalent differentiable norm.And by use of the Morse inequalities an exension of three critical point theorem due to Krasnoselski,Castro, Lazer and the author is provided.As an application,the multiple solutions of a quasilinear elliptic boundary value problems studied.
TL;DR: In this paper, the authors prove the lower semi-continuity of the functions r(·) and b(·), where r(T) is the supremum of all e≥O such that the operator T - λI is bounded below for |λ|
Abstract: Let T be a bounded linear operator on a Banach space. Denote by r(T) the supremum of all e≥O such that the operator T - λI is surjective for |λ|
TL;DR: In this article, the authors derived the equations which describe the surface wave phenomenon and proved the existence of a nontrivial branch of solutions for three periodic structures: rolls, squares and hexagons.
Abstract: Consider a slab of ferrofluid bounded below by a fixed boundary and above by a vacuum. If the fluid is subjected to a vertically directed magnetic field of sufficient strength, surface waves appear.The equations which describe this phenomenon are derived. In the physical space no natural Banach space structure is available due to the free surface. In order to use the available bifurcation theory, a transformation of coordinates is made, mapping the surface flat. In the new coordinate system the equations define an operator between Banach spaces. The minimum eigenvalue of the linearized operator is the critical magnetic field strength where the planar surface loses stability.Using a generalized inverse of the Frechet derivative of the operator and the implicit function theorem, the existence of a nontrivial branch of solutions is proved. A local stability criterion is also obtained and applied to three periodic structures: rolls, squares and hexagons.
01 Jan 1983
TL;DR: In this article, the Moore-Penrose inverse of A is defined, which is a closed, but unbounded linear operator defined on D(A*) = R(A) + R (A)⊥.
Abstract: Throughout this note, let X and Y be real Hilbert spaces, A: X → Y a bounded linear operator with non-closed range R(A). By A† we denote the Moore-Penrose inverse of A, which is a closed, but unbounded linear operator defined on D(A†) = R(A) + R(A)⊥. For properties of A† we use see [11] or [5].
TL;DR: In this article, an existence-uniqueness theorem and a regularity theorem are given for the Cauchy problem for quasi-linear equations of evolution in nonreflexive Banach spaces.
Abstract: An existence-uniqueness theorem and a regularity theorem are given for the Cauchy problem for quasi-linear equations of evolution in nonreflexive Banach spaces. As an application, C1-solutions are constructed for hyperbolic systems of partial differential equations in the “Schauder canonical form” (which include generic equations in two independent variables.).
TL;DR: In this article, a boundary value problem can be put into the form where A is a nonlinear operator, V is a reflexive Banach space, and B(t) is a continuous linear operator from V to V' which may vanish.
Abstract: Many initial boundary value problems can be put into the form where A is a nonlinear operator, V is a reflexive Banach space, and B(t) is a continuous linear operator from V to V' which may vanish. Existence and uniqueness theorems for such an equation are given.
TL;DR: In this article, a semigroup approach to differential-delay equations is developed which reduces such equations to ordinary differential equations on a Banach space of histories and seems more suitable for certain partial integro-differential equations than the standard theory.
TL;DR: In this article, it was proved that a Wiener-Hopf operator Tp (A) on a Banach space PX is generalized invertible iff A has a cross factorization with respect to X and P. If X is a separable Hilbert space, then a criterion for the weak factorization of A can be concluded.
Abstract: It is proved that a Wiener-Hopf operator Tp (A) on a Banach space PX is generalized invertible iff A has a cross factorization with respect toX and P. IfX is a separable Hilbert space, then a criterion for the weak factorization of A can be concluded.
TL;DR: In this paper, the order structure of the space of continuous linear operators on an ordered Banach space is studied and the main topic is the Robinson property, that is, the norm of a positive linear operator is attained on the positive unit cone.
Abstract: The order structure of the space of all continuous linear operators on an ordered Banach space is studied. The main topic is the Robinson property, that is, the norm of a positive linear operator is attained on the positive unit cone.
01 Jan 1983
TL;DR: In this paper, a monotone iterative technique was developed for delay differential equations in a Banach space by utilizing the method of upper and lower solutions, which is used to solve the problem of delay delay.