Showing papers on "C0-semigroup published in 2004"
Book•
31 Mar 2004
TL;DR: De Gruyter et al. as discussed by the authors proposed a unified procedure for the analysis of boundary-value problems for functional differential equations in abstract spaces, and showed the existence of solutions of linear and nonlinear differential and difference systems bounded on the entire axis.
Abstract: 01/07 This title is now available from Walter de Gruyter. Please see www.degruyter.com for more information. The problems of development of constructive methods for the analysis of linear and weakly nonlinear boundary-value problems for a broad class of functional differential equations traditionally occupy one of the central places in the qualitative theory of differential equations. The authors of this monograph suggest some methods for the construction of the generalized inverse (or pseudo-inverse) operators for the original linear Fredholm operators in Banach (or Hilbert) spaces for boundary-value problems regarded as operator systems in abstract spaces. They also study basic properties of the generalized Green's operator. In the first three chapters some results from the theory of generalized inversion of bounded linear operators in abstract spaces are given, which are then used for the investigation of boundary-value problems for systems of functional differential equations. Subsequent chapters deal with a unified procedure for the investigation of Fredholm boundary-value problems for operator equations; analysis of boundary-value problems for standard operator systems; and existence of solutions of linear and nonlinear differential and difference systems bounded on the entire axis.
222 citations
TL;DR: In this article, a modified proximal point algorithm for maximal monotone operators in a Banach space was proposed and a strong convergence theorem for resolvents was obtained.
Abstract: We first introduce a modified proximal point algorithm for
maximal monotone operators in a Banach space. Next, we obtain a
strong convergence theorem for resolvents of maximal monotone
operators in a Banach space which generalizes the previous result
by Kamimura and Takahashi in a Hilbert space. Using this result,
we deal with the convex minimization problem and the variational
inequality problem in a Banach space.
138 citations
TL;DR: In this article, it was shown that the abstract Cauchy problem can be solved by a closed linear operator on a Banach space, and it is shown that this is the case for any linear operator.
Abstract: Let X be a Banach space and let A be a closed linear operator on X. It is shown that the abstract Cauchy problem
136 citations
TL;DR: In this paper, the fundamental solutions for linear fractional evolution equations are obtained and the continuous dependence of solutions on the initial conditions is studied, and a mixed problem of general parabolic partial differential equations with fractional order is given as an application.
Abstract: The fundamental solutions for linear fractional evolution equations are obtained. The coefficients of these equations are a family of linear closed operators in the Banach space. Also, the continuous dependence of solutions on the initial conditions is studied. A mixed problem of general parabolic partial differential equations with fractional order is given as an application.
114 citations
TL;DR: In this article, approximate and exact controllability for semilinear stochastic functional differential equations in Hilbert spaces is studied and sufficient conditions are established for each of these types of controllabilities.
95 citations
TL;DR: It is proved that a time discretization based on the trapezium rule can preserve the asymptotic stability for the considered set of test problems, and the stability region of the fully continuous problem is analyzed first.
Abstract: This paper is concerned with the study of the stability of ordinary and partial differential equations with both fixed and distributed delays, and with the study of the stability of discretizations of such differential equations. We start with a delay-dependent asymptotic stability analysis of scalar ordinary differential equations with real coefficients. We study the exact stability region of the continuous problem as a function of the parameters of the model. Next, it is proved that a time discretization based on the trapezium rule can preserve the asymptotic stability for the considered set of test problems. In the second part of the paper, we study delay partial differential equations. The stability region of the fully continuous problem is analyzed first. Then a semidiscretization in space is applied. It is shown that the spatial discretization leads to a reduction of the stability region when the standard second-order central difference operator is employed to approximate the diffusion operator. Finally we consider the delay-dependent stability of the fully discrete problem, where the partial differential equation is discretized both in space and in time. Some numerical examples and further discussions are given.
83 citations
TL;DR: In this article, operator-valued Fourier multiplier theorems are used to establish maximal regularity results for an integro-differential equation with infinite delay in Banach spaces.
Abstract: Operator-valued Fourier multiplier theorems are used to establish maximal regularity results for an integro-differential equation with infinite delay in Banach spaces Results are obtained under general conditions for periodic solutions in the vector-valued Lebesgue and Besov spaces The latter scale includes in particular the Holder spaces $C^{\alpha},\,0\,{
63 citations
TL;DR: In this article, the existence, smoothness and attractivity of invariant manifolds for evolutionary processes on general Banach spaces were investigated for systems for which the nonlinear perturbation has a small global Lipschitz constant and locally Ck-smooth near the trivial solution.
60 citations
TL;DR: More than 13 years have passed since the fundamental survey [16] was prepared, which, as the author intended, should be the first part of a large work devoted to abstract differential equations and methods for solving them as discussed by the authors.
Abstract: More than 13 years have passed since the fundamental survey [16] was prepared, which, as the author intended, should be the first part of a large work devoted to abstract differential equations and methods for solving them. However, the troubles being in the Russian science during the whole this period have influenced also on the authors, and instead of two years supposed, the preparation of the second part has occupied considerably more time. During the last 10 years, the work in the field of differential equations in abstract spaces was very active (in foreign countries), and every year several books and a heavy number of papers devoted to this direction appear in the world (of course, the most of them are not available for the Russian reader). At the same time, only two books [33, 75] of such a type appeared being translated by the authors of the present survey and [20], which were edited by Yu. A. Daletskii. Therefore, the work whose second part is proposed to the reader will be undoubtedly useful for the Russian reader. Its style coincides with that of [16], i.e., the material is often presented without proofs, and the main attention is paid to the structure of presentation, although we present certain proofs from foreign sources that are almost inaccessible for Russian readers. From our viewpoint, this allows us to demonstrate clearly the philosophy, to describe the results obtained, and to indicate the main directions of the development of the theory in the framework of a limited volume of the survey.
59 citations
TL;DR: In this article, the existence of the solution of the variational inequality is studied by applying the generalized projection operator π K :B ∗ →B, where B is a Banach space with dual space B ∗ and using the well-known FanKKM Theorem.
58 citations
TL;DR: In this paper, the authors discuss "plank problems" for complex Banach spaces and in particular for the classical spaces and give an estimate in the case of a real Hilbert space.
Abstract: In this work we discuss "plank problems" for complex Banach spaces and in particular for the classical spaces. In the case we obtain optimal results and for finite dimensional complex Banach spaces, in a special case, we have improved an early result by K. Ball [3]. By using these results, in some cases we are able to find best possible lower bounds for the norms of homogeneous polynomials which are products of linear forms. In particular, we give an estimate in the case of a real Hilbert space which seems to be a difficult problem. We have also obtained some results on the so-called n-th (linear) polarization constant of a Banach space which is an isometric property of the space. Finally, known polynomial inequalities have been derived as simple consequences of various results related to plank problems.
Journal Article•
TL;DR: In this article, the uniform boundedness principles that arise when one considers exact sequences of Banach spaces and several elements of homological algebra applied to the construction of a nontrivial twisted sum of a Banach space were studied.
Abstract: We construct a Banach space X admitting an uncomplemented copy of l1 so that X/l1 = c0. To do that we study the uniform boundedness principles that arise when one considers exact sequences of Banach spaces; as well as several elements of homological algebra applied to the construction of nontrivial twisted sum of Banach spaces. The combination of both elements allows one to determine the existence of nontrivial twisted sums for almost all combinations of classical Banach spaces.
TL;DR: In this paper, it was shown that the shift semigroup does not fit into the framework of C0-semigroups, nevertheless S clearly describes an evolution system on R. This leads us to weaken the notion of C 0-Semigroup, thus allowing to consider such "pathological" cases, these therefore turn out to be less unpleasant than thought previously.
Abstract: initial value problems (Cauchy problems) are usually studied via operator semigroups. In many cases, the well-developed theory of C0-semigroups, i.e., one-parameter operator semigroups which are strongly continuous for the norm on a Banach space X , suffices and provides a powerful machinery to study such problems. The applications range from partial differential equations, Volterra integro-differential equations and dynamic boundary problems to delay equations. It seems that a linear (autonomous) evolution equation like { u′(t) = Au(t) u(0) = x, (EE) that is an equation that describes a system evolving from an initial state ”in time”, can be handled via the theory of C0-semigroups. However, as the most trivial example shows this is not the case. Consider the left shift semigroup S on the space of bounded, continuous functions Cb(R) S(t)f(s) := f(t+ s). It is clear that the orbit t 7→ S(t)f is continuous for the supremum norm, if and only if f is uniformly continuous. This shows that the shift semigroup does not fit into the framework of C0-semigroups, nevertheless S clearly describes an evolution system on R. The situation is not so bad either. If we replace the norm topology by the topology τc of uniform convergence on compact sets, then the orbits become τc-continuous. This leads us to weakening the notion of C0-semigroups, thus allowing to consider such ”pathological” cases, these therefore turn out to be less unpleasant than thought previously. There are numerous generalisations of the theory of C0-semigroups. Among these we find the approach of introducing new continuity notions, this method is we want to follow. Investigations on semigroups on locally convex spaces were started fairly long ago, see e.g., [14], [51], [52] and [64]. A nice exposition on the historical aspects can be found in [56]. To exploit the Banach space structure and the same time to introduce coarser topologies, the notion of bi-continuous semigroups was introduced recently by Kuhnemund in [56]. The theory developed therein covers a large part of previously known results, and puts these concrete examples in an abstract framework. The general theory is then applicable in concrete cases.
01 Jan 2004
TL;DR: In this article, the Hyers-Ulam-Rassias stability of the linear functional equation in Banach modules over unital Banach algebras was proved, and it was shown that the functional equation can be expressed as
Abstract: We prove the Hyers-Ulam-Rassias stability of the linear functional equation in Banach modules over a unital Banach algebra.
TL;DR: In this article, the authors show that the canonical inclusion of Figa-Talamanca-Herz algebra Ap(G) is completely bounded (with cb-norm at most K G 2, where K G is Grothendieck's constant).
TL;DR: In this article, the authors considered the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space and showed that if A is the generator of an analytic semigroup, then there exists a family of regularizing operators for such a problem by using the quasi-reversibility method, fractional powers and semigroups of linear operators.
01 Jan 2004
01 Jan 2004
TL;DR: In this article, the DSM (Dynamical Systems Method) for solving equation (*) consists of finding and solving a Cauchy problem: u = Φ(t, u), u(0) = u0, t ≥ 0, which has the following properties: 1) it has a global solution u(t), 2) this solution tends to a limit as time tends to infinity, 3) this limit solves the equation B(u) = 0, i.e., u(∞) exists, and 4)
Abstract: Consider an operator equation (*) B(u) + u = 0 in a real Hilbert space, where > 0 is a small constant. The DSM (Dynamical Systems Method) for solving equation (*) consists of finding and solving a Cauchy problem: u = Φ(t, u), u(0) = u0, t ≥ 0, which has the following properties: 1) it has a global solution u(t), 2) this solution tends to a limit as time tends to infinity, i.e., u(∞) exists, 3) this limit solves the equation B(u) = 0, i.e., B(u(∞)) = 0. Existence of the unique solution is proved by the DSM for equation (*) with operators B defined on all of H and satisfying a spectral assumption: ||[B′(u) + I]−1|| ≤ c/ for any u ∈ H, where c > 0 is a constant independent of u and ∈ (0, 0). If = 0 and equation (**) B(u) = 0 is solvable, the DSM yields a solution to (**). The case when B is a monotone, hemicontinuous, defined on all of H operator is also studied, and DSM is justified for this case, that is, above properties 1),2), and 3) are proved. A sufficient condition for surjectivity of a nonlinear map is given. Meyer’s generalization of the Hadamard theorem about global homeomorphisms is proved by the DSM. The DSM method is justified for non-differentiable, hemicontinuous, monotone, defined on all of H operators. 2000 AMS Subject Classification:34R30, 35R25, 35R30, 37C35, 37L05, 37N30, 47A52, 47J06, 65M30, 65N21.
TL;DR: In this article, the mixed initial boundary value problem for complete second order (in time) linear differential equations in Banach spaces, in which time-derivatives occur in the boundary conditions, is studied.
TL;DR: A new algorithm for computing exponential solutions of differential operators with rational function coefficients is presented, which uses a combination of local and modular computations, which allows to reduce the number of possibilities in the combinatorial part of the algorithm.
TL;DR: In this paper, the existence and uniqueness of coupled lower and upper solutions for the general n th problem in time scales with linear dependence on the i th Δ-derivatives for i = 1,2,…,n, together with antiperiodic boundary value conditions was proved.
Abstract: We prove existence and uniqueness results in the presence of coupled lower and upper solutions for the general n th problem in time scales with linear dependence on the i th Δ-derivatives for i = 1,2,…,n, together with antiperiodic boundary value conditions. Here the nonlinear right-hand side of the equation is defined by a function f(t,x) which is rd-continuous in t and continuous in x uniformly in t. To do that, we obtain the expression of the Green's function of a related linear operator in the space of the antiperiodic functions.
TL;DR: It is proved that the sequence of all generalized eigenvectors of the system principal operator forms a Riesz basis for the state Hilbert space.
Abstract: The Riesz basis property of the generalized eigenvector system of a Timoshenko beam with boundary feedback controls applied to two ends is studied in the present paper. The spectral property of the operator A determined by the closed loop system is investigated. It is shown that operator A has compact resolvent and generates a C0 semigroup, and its spectrum consists of two branches and has two asymptotes under some conditions. Furthermore it is proved that the sequence of all generalized eigenvectors of the system principal operator forms a Riesz basis for the state Hilbert space.
TL;DR: In this article, the generalized Hyers-Ulam-Rassias stability of generalized A-quadratic mappings of type (P) in Banach modules over a Banach ∗-algebra was proved.
TL;DR: This paper gives some sufficient conditions and some necessary conditions for the system to have exponential stability and is based on the operator semigroup technique, the multiplier technique, and the contradiction argument of the frequency domain method.
Abstract: In this paper, we are concerned with a boundary feedback system of a class of nonuniform undamped Euler--Bernoulli beam with both ends free. We give some sufficient conditions and some necessary conditions for the system to have exponential stability. Our method is based on the operator semigroup technique, the multiplier technique, and the contradiction argument of the frequency domain method.
TL;DR: The class of the so-called asymptotically C (n) -almost periodic functions is introduced and some of their properties are given and some applications to ordinary as well as partial differential equations are presented.
Abstract: We deal with C (n) -almost periodic functions taking values in a Banach space. We give several properties of such functions, in particular, we investigate their behavior in view of differentiation as well as integration. The superposition operator acting in the space of such functions is also under consideration. Some applications to ordinary as well as partial differential equations are presented. Moreover, we introduce the class of the so-called asymptotically C (n) -almost periodic functions and give some of their properties.
TL;DR: In this article, a unified treatment of the mixed initial boundary value problem for second order (in time) parabolic linear differential equations in Banach spaces, whose boundary conditions are of a dynamical nature, is presented.
Abstract: In this paper, we exhibit a unified treatment of the mixed initial boundary value problem for second order (in time) parabolic linear differential equations in Banach spaces, whose boundary conditions are of a dynamical nature. Results regarding existence, uniqueness, continuous dependence (on initial data) and regularity of classical and strict solutions are established. Moreover, several examples are given as samples for possible applications.
TL;DR: In this article, the authors obtained the existence of multiple positive solutions for a boundary value problem of a class of nth-order nonlinear impulsive integro-differential equations on an infinite interval in a Banach space by means of the fixed point index theory of completely continuous operators.
Abstract: In this paper, the author obtains the existence of multiple positive solutions for a boundary value problem of a class of nth-order nonlinear impulsive integro-differential equations on an infinite interval in a Banach space by means of the fixed point index theory of completely continuous operators.
TL;DR: In this article, analytic solutions are obtained for iterative functional differential equations that are natural extensions of x' = 1/x × x. The authors make use of neutral functional differential equation with proportional delays as well as neutral differential-difference equations for achieving their purposes.
TL;DR: In this paper, it was shown that a Banach space X has numerical index 0 if and only if it is the direct sum of a real space X 0 and nonzero complex spaces X 1,...,X n.
Abstract: We prove that a finite-dimensional Banach space X has numerical index 0 if and only if it is the direct sum of a real space X 0 and nonzero complex spaces X 1 ,...,X n in such a way that the equality ∥x 0 + e iq1ρ x 1 +...+e iqnρ x n ∥ = ∥x 0 +...+ x n ∥ holds for suitable positive integers q 1 ,...,q n , and every ρ ∈ R and every x j ∈ X j (j = 0, 1,...,n). If the dimension of X is two, then the above result gives X = C, whereas dim(X) = 3 implies that X is an absolute sum of R and C. We also give an example showing that, in general, the number of complex spaces cannot be reduced to one.