Pseudo-almost periodic solutions of neutral integral and differential equations with applications

The existence and uniqueness of pseudo-almost periodic solutions to general neutral integral equations with deviations are obtained. For this, pseudo-almost periodic functions in two variables are considered. The results extend the corresponding ones to the convolution type integral equations. They are used to study pseudo-almost periodic solutions of general neutral differential equations and to the so-called scalar neutral logistic equation version.     

Existence of solutions for a class of abstract differential equations with nonlocal conditions

We discuss the existence of mild, classical and strict solutions for a class of abstract differential equations with nonlocal conditions. Our technical approach allows the study of partial differential equations with nonlocal conditions involving partial derivatives or nonlinear expressions of the solution. Some concrete applications to partial differential equations are considered.     

Existence of solutions of non-autonomous second order functional differential equations with infinite delay

In this paper the existence of solutions of a non-autonomous abstract retarded functional differential equation of second order with infinite delay is considered. Assuming the existence of an evolution operator corresponding to the associate abstract Cauchy problem of second order, we establish the existence of mild solutions of the functional equation. Furthermore, we study the existence of classical solutions of the abstract Cauchy problem of second order and we apply these results to establish the existence of classical solutions of the functional equation. Finally, we apply our results to study the existence of solutions of the non-autonomous wave equation with delay.     

Hyperbolicity of linear partial differential equations with delay

Robust hyperbolicity and stability results for linear partial differential equations with delay will be given and, as an application, the effect of small delays to the asymptotic properties of feedback systems will be analyzed.The author thanks W. Desch (Graz), I. Gyri (Veszprém) and R. Schnaubelt (Halle) for helpful discussions.     

Existence of nondensely defined evolution equations with nonlocal conditions

This paper is concerned with the existence for nondensely defined evolution equations with nonlocal conditions. Using the techniques of fixed point theory and approximate solutions, existence results are obtained, for integral solutions, when the nonlocal item is Lipschitz continuous or continuous, respectively. Examples are also given to illustrate our results.     

Nonlocal differential equations with multivalued perturbations in Banach spaces

We discuss the existence of mild solutions for nonlocal differential inclusions with multivalued perturbations in Banach spaces and establish new existence theorems for related Cauchy problems, which extend some existing results in this area. Using the established results, we investigate a special nonlocal problem. Finally, we also consider a partial functional differential equation.     

Impulsive problems for semilinear differential equations with nonlocal conditions

This paper is concerned with the existence of impulsive semilinear differential equations with nonlocal conditions. Using the technique of fixed point theory, existence results are obtained, for mild solutions, when the nonlocal item is Lipschitz, is not Lipschitz and not compact, respectively.     

Existence results of semilinear differential equations with nonlocal initial conditions in Banach spaces

In this paper, we establish the existence results for semilinear differential systems with nonlocal initial conditions in Banach spaces. The approaches used are fixed point theorems combined with convex-power condensing operators. The first result obtained will be applied to a class of semilinear parabolic equations.     

On small solutions of delay equations in infinite dimensions

LetX be a Banach space and 1p<. LetL be a bounded linear operator fromL ^p ([–1,0],X) intoX. Consider the delay differential equationu(t)=Lu _t ,u(0)=x,u ₀=f on the state spaceL ^p ([–1,0],X). We prove that a mild solutionu(t)=u(t;x,f) is a small solution if and only if the Laplace transform ofu(t;x,f) extends to an entire function. The same result holds for the state spaceC([–1,0],X).This paper was written while the authors were affiliated with the University of Tübingen. It is part of a research project supported by the Deutsche Forschungsgemeinschaft DFG. The authors warmly thank Professor Rainer Nagel and the AG Funktionalanalysis for the stimulating and enjoyable working environment.Support by DAAD is gratefully acknowledged.Support by an Individual Fellowship from the Human Capital and Mobility programme of the European Community is gratefully acknowledged.     

On a class of abstract neutral functional differential equations

By using the theory of semigroups of growth α, we discuss the existence of mild solutions for a class of abstract neutral functional differential equations. A concrete application to partial neutral functional differential equations is considered.     

Abstract quasilinear equations of second order with Wentzell boundary conditions

We introduce a general framework to treat abstract quasilinear equations of second order with Wentzell boundary conditions. As an example we study a wave equation for a second order quasilinear differential operator on with Wentzell boundary conditions.     

Circular spectrum and bounded solutions of periodic evolution equations

We consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form u^′=A(t)u+?H(t,u)+f(t), where A(t) is, in general, an unbounded operator depending 1-periodically on t, H is 1-periodic in t, ? is small, and f is a bounded and continuous function that is not necessarily uniformly continuous. We propose a new approach to the spectral theory of functions via the concept of “circular spectrum” and then apply it to study the linear equations u^′=A(t)u+f(t) with general conditions on f. For small ? we show that the perturbed equation inherits some properties of the linear unperturbed one. The main results extend recent results in the direction, saying that if the unitary spectrum of the monodromy operator does not intersect the circular spectrum of f, then the evolution equation has a unique mild solution with its circular spectrum contained in the circular spectrum of f.     

Classical solutions for partial functional differential equations with nonautonomous past

We study partial functional differential equations with infinite delay where the history function is modified by a backward evolution family. Under appropriate assumptions and using semigroup techniques we prove the existence of a unique classical solution. Die endgültige Fassung ging am 20. 6. 2001 einein     

Dissipativity of the delay semigroup

Under mild conditions a delay semigroup can be transformed into a (generalized) contraction semigroup by modifying the inner product on the (Hilbert) state space into an equivalent inner product. Applications to stability of differential equations with delay and stochastic differential equations with delay are given as examples.     

Wellposedness of abstract differential equations with state‐dependent delay

We study the existence and uniqueness of solutions, and the wellposedness of a general class of second order abstract differential equations with state‐dependent delay. Some examples related to partial differential equations with state‐dependent delay are presented.     

Periodic solutions of abstract neutral functional differential equations with infinite delay

We establish the existence of mild solutions and periodic mild solutions for a class of abstract first-order non-autonomous neutral functional differential equations with infinite delay in a Banach space. This research was supported by FONDECYT-CONICYT, Grant 1050314.     

Moment boundedness of linear stochastic delay differential equations with distributed delay

This paper studies the moment boundedness of solutions of linear stochastic delay differential equations with distributed delay. For a linear stochastic delay differential equation, the first moment stability is known to be identical to that of the corresponding deterministic delay differential equation. However, boundedness of the second moment is complicated and depends on the stochastic terms. In this paper, the characteristic function of the equation is obtained through techniques of the Laplace transform. From the characteristic equation, sufficient conditions for the second moment to be bounded or unbounded are proposed.     

A population dynamics model with nonautonomous past

Abstract

In this paper we study a two-phase population model, which distinguishes the population by two different stages

By the standard technique of characteristics, this population equation is transformed as the ordinary differential equation with nonautonomous past

where 1 ≤ p < ∞ and I = [?r, 0] (finite delay) or I = (?∞, 0] (infinite delay), E a Banach space, Φ : W^1,p(I, E) → E a linear delay operator and B a nonlinear operator on E. The main result of this paper is the well-posedness of this delay equation by using the (right) multiplicative perturbation result of Desch and Schappacher in [8].     

The Laplacian on $$ C(\overline \Omega) $$ with generalized Wentzell boundary conditions

In this note we prove that the Laplacian with generalized Wentzell boundary conditions on an open bounded regular domain in defined by generates an analytic semigroup of angle on for every > 0 and (for the definition of cf. (1.3)).Received: 13 July 2002