Time-optimal solution of a linear evolution equation in Banach spaces

The problem is that of finding trajectories of a linear evolution equation connecting two prescribed sets of states, initial and terminal, in the shortest possible time. Necessary conditions for the existence of solutions, i.e., time-optimum trajectories, are given in the form of a maximum principle.     

On some nonlocal evolution equations in Banach spaces

This note is concerned with the initial value problem for the abstract nonlocal equation where A is a maximal monotone operator from a reflexive Banach space E to its dual E^*, while B is a nonlocal maximal monotone operator from . Under proper boundedness and coercivity assumptions on the operators, a solution is achieved by means of a discretization argument. Uniqueness and continuous dependence are also discussed and we prove some estimates for the discretization error. Finally, we deal with the approximation of linear Volterra integrodifferential operators.     

On nonuniform exponential dichotomy of evolution operators in Banach spaces

The problem of nonuniform exponential dichotomy of evolution operators in Banach spaces is considered. Connections between this concept and admissibility of the pair (C ₀,C ₀) are established. Generalizations to the nonuniform case of some results of Van Min, Räbiger and Schnaubelt ([MRS]) are obtained. It is shown that an implication from the uniform case is not true for nonuniform exponential dichotomy. The results are applicable for general time-varying linear equations with unbounded coefficients in Banach spaces.     

Semilinear evolution equations in Banach spaces

We study the evolution equation u′(t) = Au(t) + J(u(t)), t ? 0, where e^tA is a C₀ semi-group on a Banach space E, and J is a “singular” non-linear mapping defined on a subset of E. In Sections 1 and 2 of the paper we suppress the map J and instead consider maps K_t: E → E, t > 0, which heuristically are just e^tAJ. Under certain integrability conditions on the K_t we prove existence and uniqueness of local solutions to the integral equation u(t) = e^tAφ + ∝₀^tK_{t ? s}(u(s)) ds for all φ in E, and investigate the regularity of the solutions. Conditions which insure existence of global solutions are given. In Section 3 we recover the map J from the maps K_t, and show that the generator of the semi-flow on E induced by the integral equation has dense domain. Finally, we apply these results to a large class of examples which includes polynomial perturbations to elliptic operators on a domain in Rⁿ.     

Nonlinear evolution equations in Banach spaces

The evolution problem 0∈du/dt+A(t)u(t),u(s)=x, where theA(t) are nonlinear operators acting in a Banach space, is studied. Evolution operators are constructed from theA(t) under various assumptions. Basic properties of these evolution operators are established and their relationship to the evolution equation is determined. The results obtained extend several known existence theorems and provide generalized solutions of the evolution equation in more general cases. Sponsored by the United States Army under Contract No. DA-31-124-ARO-D-462. and in part by the Office of Naval Research under Contract N000-14-69-A-0200-4022. Reproduction in whole or in part is permitted for any purpose of the United States Government. This research was partially supported by the NSF Grant #GP-18127.     

On weak exponential stability of evolution operators in Banach spaces

The aim of this paper is to give some characterizations for weak exponential stability properties of evolution operators in Banach spaces. Variants for weak exponential stability of some well-known results in uniform stability theory (Bu?e and Niculescu (2009) [1], Daleckij and Krein (1974) [2], Datko (1973) [3], Rolewicz (1986) [7], Stoica and Megan (2009) [8]) are obtained.     

On a singular two dimensional nonlinear evolution equation with nonlocal conditions

This paper deals with a nonclassical initial boundary value problem for a two dimensional parabolic equation with Bessel operator. We prove the existence and uniqueness of the weak solution of the given nonlinear problem. We start by solving the associated linear problem. After writing this latter in its operator form, we establish an a priori bound from which we deduce the uniqueness of the strong solution. For the solvability of the associated linear problem, we prove that the range of the operator generated by the considered problem is dense. On the basis of the obtained results of the linear problem, we apply an iterative process to establish the existence and uniqueness of the nonlinear problem.     

On the aronszajn property for a differential equation of fractional order in Banach spaces

In this paper we investigate some topological properties of solution sets of a differential equation of fractional order in Banach spaces. Our assumptions and proofs are expressed in terms of measures of noncompactness.     

Cauchy problem for a degenerate heat equation in Banach spaces

We study the solvability of a degenerate heat equation with closed linear operators B multiplying the time derivative and A multiplying the Laplace operator in the class of generalized functions in Banach spaces. Under various assumptions on the operator pencil λB-A (it can be Fredholm of index zero, Fredholm, spectrally bounded, sectorial, or radial), we construct the fundamental operator function for the differential operator Bδ′(t) × δ ( $ \bar x $ ) ? Aδ(t) × Δδ ( $ \bar x $ ) and use it for the closed-form construction of the desired generalized solution of the Cauchy problem for the equation in question. We single out uniqueness classes for these solutions and analyze the relationship between continuous and generalized solutions.     

Approximate controllability of the non-autonomous impulsive evolution equation with state-dependent delay in Banach spaces

In this paper, we consider the non-autonomous semilinear impulsive differential equations with state-dependent delay. The approximate controllability results of the first-order systems are obtained in a separable reflexive Banach space, which has a uniformly convex dual. In order to establish sufficient conditions of the approximate controllability of such a system, we have used the theory of linear evolution systems, properties of the resolvent operator and Schauder’s fixed point theorem. Finally, we provide two concrete examples to validate our results.     

Mild solution of semilinear impulsive integro‐differential evolution equation in Banach spaces

Through solving the problem step by step and by applying the method of a C₀ semigroup of operators combined with the Banach contraction theorem, we investigate the existence and uniqueness of a mild solution of semilinear impulsive integro‐differential evolution equation in Banach spaces. In addition, an explicit iterative approximation sequence of the mild solution is derived. The assumed conditions in the present theorems are weaker and more general, and the results obtained are the generalizations and improvements of some known results. Examples are also given to illustrate our main results. Copyright © 2017 John Wiley & Sons, Ltd.     

Singularly perturbed evolution equations in Banach spaces

The singular perturbation method is applied to systems of linear evolution equations in Banach spaces with one of the time derivatives multiplied by a small parameter. An asymptotic solution of any given order is shown to consist of inner and outer terms obtained by a proper choice of initial conditions, and to converge uniformly to the exact solution.     

Stochastic evolution equations in UMD Banach spaces

We discuss existence, uniqueness, and space-time Hölder regularity for solutions of the parabolic stochastic evolution equation