Showing papers on "C0-semigroup published in 2009"
TL;DR: In this article, sufficient conditions for the existence of mild solutions for some densely defined semilinear functional differential equations and inclusions involving the Riemann-Liouville fractional derivative are established.
Abstract: We establish sufficient conditions for the existence of mild solutions for some densely defined semilinear functional differential equations and inclusions involving the Riemann-Liouville fractional derivative. Our approach is based on the -semigroups theory combined with some suitable fixed point theorems.
240 citations
01 Jan 2009
TL;DR: This paper systematically investigates the exponential stability of the solution for \(I\hat{t}o\) equations, presenting the comparison criterions of stochastic exponential stability, exponential p-stability and almost surely exponential stability.
Abstract: This paper systematically investigates the exponential stability of the solution for \(I\hat{t}o\) equations, presenting the comparison criterions of stochastic exponential stability, exponential p-stability and almost surely exponential stability. These comparison criterions generalize the corresponding research results by Nevel’son and Has’minskii.
219 citations
TL;DR: In this paper, the existence results in a Banach space for a nonlocal boundary value problem involving a nonlinear differential equation of fractional order given by,,,,,, and, were studied.
Abstract: We study some existence results in a Banach space for a nonlocal boundary value problem involving a nonlinear differential equation of fractional order given by , , , , , , , . Our results are based on the contraction mapping principle and Krasnoselskii's fixed point theorem.
142 citations
TL;DR: In this paper, the authors discuss the solvability of integral equations associated with initial value problems for a nonlinear differential equation of fractional order, and obtain the existence of at least one solution for integral equations using the Leray-Schauder Nonlinear Alternative for several types of initial-value problems.
Abstract: We discuss the solvability of integral equations associated with initial value problems for a nonlinear differential equation of fractional order. The differential operator is the Caputo fractional derivative and the inhomogeneous term depends on the fractional derivative of lower orders. We obtain the existence of at least one solution for integral equations using the Leray–Schauder Nonlinear Alternative for several types of initial value problems. In addition, using the Banach contraction principle, we establish sufficient conditions for unique solutions. Our approach in obtaining integral equations is the “reduction” of the fractional order of the integro-differential equations based on certain semigroup properties of the Caputo operator.
136 citations
TL;DR: In this article, the existence of mild solutions to the Cauchy Problem for the fractional differential equation with nonlocal conditions was studied, where 0 < q<1, where q is the infinitesimal generator of a C.............. 0-semigroup of bounded linear operators on a Banach space X.
Abstract: We are concerned in this paper with the existence of mild solutions to the Cauchy Problem for the fractional differential equation with nonlocal conditions: D
q
x(t)=Ax(t)+t
n
f(t,x(t),Bx(t)), t∈[0,T], n∈ℤ+, x(0)+g(x)=x
0, where 0
135 citations
TL;DR: In this paper, the existence of solutions for a class of initial value problems for impulsive fractional differential equations involving the Caputo fractional derivative in a Banach space is studied.
Abstract: This paper is devoted to study the existence of solutions for a class of initial value problems for impulsive fractional differential equations involving the Caputo fractional derivative in a Banach space. The arguments are based upon
123 citations
Book•
05 May 2009
TL;DR: In this article, the authors present results from functional analysis of well-posed and ill-posed expressions and their regularization regularized approximations using regularized approximation methods.
Abstract: Basic Results from Functional Analysis Well-Posed Equations and Their Approximations Ill-Posed Equations and Their Regularization Regularized Approximation Methods.
76 citations
TL;DR: In this paper, the existence of mild solutions for a class of impulsive second-order partial neutral functional differential equations with infinite delay in a Banach space is established, and mild solutions can be found for all these problems.
Abstract: We establish the existence of mild solutions for a class of impulsive second-order partial neutral functional differential equations with infinite delay in a Banach space.
75 citations
TL;DR: A sufficient condition for the controllability of a first-order semilinear differential system with nonlocal initial conditions in Banach spaces is established using the Sadovskii fixed-point theorem combined with operator semigroups.
Abstract: In this note, we establish a sufficient condition for the controllability of a first-order semilinear differential system with nonlocal initial conditions in Banach spaces. The approach used is the Sadovskii fixed-point theorem combined with operator semigroups. Particularly, the compactness of the operator semigroups is not needed in this article.
66 citations
TL;DR: The existence of integral solutions for nonlinear differential equations with nonlocal initial conditions under the assumptions of the Hausdorff measure of noncompactness in separable and uniformly smooth Banach spaces was shown in this article.
Abstract: In this paper we give the existence of integral solutions for nonlinear differential equations with nonlocal initial conditions under the assumptions of the Hausdorff measure of noncompactness in separable and uniformly smooth Banach spaces.
60 citations
Journal Article•
TL;DR: In this article, the authors give conditions for the parabolic evolution operator to be an-athetic with respect to a coecient operator, and show that the solution of a homogeneous parabolized evolution equation is analytic.
Abstract: We give conditions for the parabolic evolution operator to be an- alytic with respect to a coecient operator. We also show that the solution of a homogeneous parabolic evolution equation is analytic with respect to the coecient operator and to the initial data. We apply our results to example that can not be studied by the standard methods.
SIDI1
TL;DR: In this article, the existence of solutions for functional partial perturbed hyperbolic differential equations with fractional order was investigated based upon a fixed point theorem for the sum of contraction and compact operators.
TL;DR: In this paper, a class of boundary value problems for fractional differential equations involving nonlinear integral conditions involving non-compactness is investigated, and the main tool used in their considerations is the technique associated with measures of noncompactity.
Abstract: The aim of this paper is to investigate a class of boundary value problem for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of noncompactness.
TL;DR: In this article, the existence and uniqueness of pseudo-almost-automorphic solutions for some neutral partial functional differential equations in Banach spaces is investigated, assuming that the undelayed part is not necessarily densely defined and satisfies the well-known Hille-Yosida condition.
Abstract: This work aims to investigate the existence and uniqueness of pseudo-almost-automorphic solutions for some neutral partial functional differential equations in Banach spaces. Recall that the new concept of pseudo-almost-automorphy generalizes the one of the pseudo-almost-periodicity and it has been recently introduced in the literature. Here we assume that the undelayed part is not necessarily densely defined and satisfies the well-known Hille–Yosida condition, the delayed parts are assumed to be pseudo-almost-automorphic with respect to the first argument and Lipschitz continuous with respect to the second argument.
TL;DR: In this paper, the existence, uniqueness and continuous dependence of a mild solution of an impulsive neutral functional differential evolution nonlocal Cauchy problem in general Banach spaces are studied, by using the fixed point technique and semigroup of operators.
Abstract: The existence, uniqueness and continuous dependence of a mild solution of an impulsive neutral functional differential evolution nonlocal Cauchy problem in general Banach spaces are studied, by using the fixed point technique and semigroup of operators.
TL;DR: The results in Chalishajar and Chang are only valid for ordinary differential control systems and the examples provided cannot be recovered as applications of the abstract results.
Abstract: We show the results in Chalishajar [Controllability of mixed Volterra–Fredholm-type integro-differential systems in Banach space, J. Franklin Inst. 344(1) (2007) 12–21] and Chang and Chalishajar [Controllability of mixed Volterra–Fredholm type integro-differential systems in Banach space, J. Franklin Inst., doi: 10.1016/j.jfranklin.2008.02.002 ] are only valid for ordinary differential control systems. As a result the examples provided cannot be recovered as applications of the abstract results.
TL;DR: In this article, the authors investigate properties of particular Banach spaces of Lipschitz functions on a metric space and semigroups defined on their (pre)duals.
Abstract: Interpretation, derivation and application of a variation of constants formula for measure-valued functions motivate our investigation of properties of particular Banach spaces of Lipschitz functions on a metric space and semigroups defined on their (pre)duals. Spaces of measures densely embed into these preduals. The metric space embeds continuously in these preduals, even isometrically in a specific case. Under mild conditions, a semigroup of Lipschitz transformations on the metric space then embeds into a strongly continuous semigroups of positive linear operators on these Banach spaces generated by measures.
TL;DR: In this article, the authors modify recently developed sequential subspace optimization methods in such a way that the employed Bregman projections onto hyperplanes are replaced by Bregmans projections onto stripes whose width is in the order of the noise level.
Abstract: We are concerned with fast computations of regularized solutions of linear operator equations in Banach spaces in case only noisy data are available. To this end we modify recently developed sequential subspace optimization methods in such a way that the therein employed Bregman projections onto hyperplanes are replaced by Bregman projections onto stripes whose width is in the order of the noise level.
TL;DR: In this article, the existence of mild solutions without the assumption of compactness or equicontinuity on the associated linear semigroup is obtained using tools involving the measure of noncompactness and fixed point theory.
Abstract: This paper is concerned with semilinear differential equations with nonlocal conditions in Banach spaces. Using the tools involving the measure of noncompactness and fixed point theory, existence of mild solutions is obtained without the assumption of compactness or equicontinuity on the associated linear semigroup.
TL;DR: In this article, the authors consider partial differential equations with discrete (concentrated) state-dependent delays in the space of continuous functions and prove the existence of a compact global attractor.
Abstract: Partial differential equations with discrete (concentrated) state-dependent delays in the space of continuous functions are investigated. In general, the corresponding initial value problem is not well-posed. So we find an additional assumption on the state-dependent delay function to guarantee the well-posedness. For the constructed dynamical system we study the long-time asymptotic behavior and prove the existence of a compact global attractor.
TL;DR: Using the resolvent operator technique associated with H(.,.)-accretive operator, the existence of the solution for the system of inclusions is proved and a step-controlled iterative algorithm is developed to approach the unique solution.
TL;DR: In this paper, the existence of a global attractor in the phase space H 0 1 ( Ω ) is established and the Brinkman-forchheimer equation is studied.
Abstract: This study focuses on the Brinkman–Forchheimer equation u t = γ △ u − a u − b | u | u − c | u | β u − ∇ p + f in an open, bounded domain Ω and the existence of a global attractor in the phase space H 0 1 ( Ω ) is established.
TL;DR: In this article, the authors consider pseudo-monotone parametric variational inequalities on a class of Banach spaces and show that such problems admit continuous (with respect to the parameter) solutions, as long as generic existence and uniqueness conditions for these solutions are satisfied.
TL;DR: In this paper, the Castaing representation theorem for the set-valued stochastic integral becomes a submartingale, which is different from the single-valued case.
TL;DR: The concept of lushness was introduced recently as a Banach space property, which ensures that the space has numerical index as discussed by the authors, and it has been shown that for Asplund spaces lushness is actually equivalent to numerical index.
Abstract: The concept of lushness was introduced recently as a Banach space property, which ensures that the space has numerical index 1 We prove that for Asplund spaces lushness is actually equivalent to numerical index 1 We prove that every separable Banach space containing an isomorphic copy of c0 can be renormed equivalently to be lush, and thus to have numerical index 1 The rest of the paper is devoted to the study of lushness just as a property of Banach spaces We prove that lushness is separably determined, is stable under ultraproducts, and we characterize those spaces of the form X = (R, ‖ · ‖) with absolute norm such that X-sum preserves lushness of summands, showing that this property is equivalent to lushness of X
TL;DR: In this article, a stochastic integral of Banach space valued deterministic functions with respect to Levy processes is defined, and the integral is used to prove a Levy-Ito decomposition for Banach spaces valued Levy processes.
TL;DR: In this article, a hybrid projection algorithm was proposed for finding a common element of the set of solutions of the equilibrium problem and the variational inequality for an inverse-strongly monotone operator in a Banach space.
Abstract: The purpose of this paper is to introduce a new hybrid projection algorithm for finding a common element of the set of solutions of the equilibrium problem and the set of the variational inequality for an inverse-strongly monotone operator and the set of fixed points of relatively quasi-nonexpansive mappings in a Banach space. Then we show a strong convergence theorem. Using this result, we obtain some applications in a Banach space.
TL;DR: In this article, the existence and regularity of solutions for partial functional integrodifferential equations in Banach spaces was studied and sufficient conditions for the existence of strict solutions were given.
Abstract: In this work, we study the existence and regularity of solutions for some partial functional integrodifferential equations in Banach spaces. We suppose that the undelayed part admits a resolvent operator in the sense given by Grimmer in [R. Grimmer, Resolvent operators for integral equations in a Banach space, Transaction of American Mathematical Society 273 (1982) 333–349]. The delayed part is assumed to be locally Lipschitz. Firstly, we show the existence of the mild solutions. Secondly, we give sufficient conditions ensuring the existence of the strict solutions.
TL;DR: In this paper, the existence of mild solutions for a class of impulsive abstract partial neutral functional differential equations with state-dependent delay was studied by using Leray-Schauder Alternative fixed point theorem.
Abstract: In this article, we study the existence of mild solutions for a class of impulsive abstract partial neutral functional differential equations with state-dependent delay. The results are obtained by using Leray-Schauder Alternative fixed point theorem. Example is provided to illustrate the main result.
TL;DR: In this article, the authors investigate the asymptotic behavior of solutions to the initial boundary value problem for a one-dimensional mixture of thermoviscoelastic solids.