Solutions For Nonautonomous Evolution Equations Research Articles

Positive almost periodic solutions of nonautonomous evolution equations and application to Lotka–Volterra systems

The aim of this paper is to establish some sufficient conditions ensuring the existence and uniqueness of positive (Bohr) almost periodic solutions to a class of semilinear evolution equations of the form: . We assume that the family of closed linear operators on a Banach lattice satisfies the “Acquistapace–Terreni” conditions, so that the associated evolution family is positive and has an exponential dichotomy on . The nonlinear term , acting on certain real interpolation spaces, is assumed to be almost periodic only in a weaker sense (i.e., in Stepanov's sense) with respect to , and Lipschitzian in bounded sets with respect to the second variable. Moreover, we prove a new composition result for Stepanov almost periodic functions by assuming only continuity of with respect to the second variable (see the condition Lemma 1‐(ii)). Finally, we provide an application to a system of Lotka–Volterra predator–prey type model with diffusion and time–dependent parameters in a generalized almost periodic environment.

Existence and uniqueness of global solutions for non-autonomous evolution equations with state-dependent nonlocal conditions

In this paper, we consider the existence and uniqueness of global solutions for non-autonomous evolution equations with state-dependent nonlocal conditions, in which the undelayed part admits an evolution operator. We discuss the problems by utilizing theory of evolution operators, Schauder fixed point theorem and Banach fixed point theorem. Some new results on existence and uniqueness of solutions of the considered equation are obtained on the infinite internal [0,+?). In the end, the obtained results are applied to a class of non-autonomous heat equations with state-dependent nonlocal conditions.

Composition of Stepanov-like pseudo almost automorphic functions and applications to nonautonomous evolution equations

In this paper, we establish a new composition theorem about Stepanov-like pseudo almost automorphic functions under the local Lipschitz condition. Using this composition theorem, we also study the existence and uniqueness of pseudo almost automorphic solutions for nonautonomous evolution equations. Our results extend many recent known ones on these topics.

Positive periodic solutions of parabolic evolution problems: a translation along trajectories approach

A translation along trajectories approach together with averaging procedure and topological degree are used to derive effective criteria for existence of periodic solutions for nonautonomous evolution equations with periodic perturbations. It is shown that a topologically nontrivial zero of the averaged right hand side is a source of periodic solutions for the equations with increased frequencies. Our setting involves equations on closed convex cones, therefore it enables us to study positive solutions of nonlinear parabolic partial differential equations.

Almost automorphic solutions of nonautonomous evolution equations

In this paper, we establish some new theorems about the existence of almost automorphic solutions to nonautonomous evolution equations u ′ ( t ) = A ( t ) u ( t ) + f ( t ) and u ′ ( t ) = A ( t ) u ( t ) + f ( t , u ( t ) ) in Banach spaces. As we will see, our results allow for a more general A ( t ) to some extent. An example is also given to illustrate our results. In addition, by means of an example, we show that one cannot ensure the existence of almost automorphic solutions to u ′ ( t ) = A ( t ) u ( t ) + f ( t ) even if the evolution family U ( t , s ) generated by A ( t ) is exponentially stable and f ∈ A A ( X ) .