Class Of Non-autonomous Evolution Equations Research Articles

Solutions to nonlocal evolution equations governed by non-autonomous forms and demicontinuous nonlinearities

We deal with the existence of solutions having L^2-regularity for a class of non-autonomous evolution equations. Associated with the equation, a general non-local condition is studied. The technique we used combines a finite dimensional reduction together with the Leray–Schauder continuation principle. This approach permits to consider a wide class of nonlinear terms by allowing demicontinuity assumptions on the nonlinearity.

Non-autonomous Evolution Equations of Parabolic Type with Non-instantaneous Impulses

In this paper, we study the Cauchy problem to a class of non-autonomous evolution equations of parabolic type with non-instantaneous impulses in Banach spaces, where the operators in linear part (possibly unbounded) depend on time t and generate an evolution family. New existence result of piecewise continuous mild solutions is established under more weaker conditions. At last, as a sample of application, the abstract result is applied to a class of non-autonomous partial differential equation of parabolic type with non-instantaneous impulses. The result obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area.

Existence of mild solutions for a class of non-autonomous evolution equations with nonlocal initial conditions

Existence of mild solutions for a class of non-autonomous evolution equations with nonlocal initial conditions

New composition theorems of Stepanov-like weighted pseudo almost automorphic functions and applications to nonautonomous evolution equations

In this paper, we establish some new composition theorems for Stepanov-like weighted pseudo almost automorphic functions. We then apply the obtained results to investigate the existence and uniqueness of weighted pseudo almost automorphic mild solutions to a class of nonautonomous evolution equations with Sp-weighted pseudo almost automorphic coefficients.

Sell’s conjecture for non-autonomous dynamical systems

This paper is dedicated to the study of the G. Sell’s conjecture for general non-autonomous dynamical systems. We give a positive answer for this conjecture and we apply this result to different classes of non-autonomous evolution equations: Ordinary Differential Equations, Functional Differential Equations and Semi-linear Parabolic Equations.

Pseudo-almost periodic solutions for some classes of nonautonomous partial evolution equations

In this paper upon making some suitable assumptions such as the Acquistapace–Terreni conditions and exponential dichotomy we obtain the existence of pseudo-almost periodic solutions to some classes of nonautonomous evolution equations of Sobolev-type. An example is given at the end of the paper to illustrate our abstract result.

A Trotter–Kato product formula for a class of non-autonomous evolution equations

In this article dedicated to Professor V. Lakshmikantham on the occasion of the celebration of his 84th birthday, we announce new results concerning the existence and various properties of an evolution system U A + B ( t , s ) 0 ≤ s ≤ t ≤ T generated by the sum − ( A ( t ) + B ( t ) ) of two linear, time-dependent and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B . In particular, writing L ( B ) for the algebra of all linear bounded operators on B , we can express U A + B ( t , s ) 0 ≤ s ≤ t ≤ T as the strong limit in L ( B ) of a product of the holomorphic contraction semigroups generated by − A ( t ) and − B ( t ) , thereby getting a product formula of the Trotter–Kato type under very general conditions which allow the domain D ( A ( t ) + B ( t ) ) to evolve with time provided there exists a fixed set D ⊂ ∩ t ∈ [ 0 , T ] D ( A ( t ) + B ( t ) ) everywhere dense in B . We then mention several possible applications of our product formula to various classes of non-autonomous parabolic initial-boundary value problems, as well as to evolution problems of Schrödinger type related to the theory of time-dependent singular perturbations of self-adjoint operators in quantum mechanics. We defer all the proofs and all the details of the applications to a separate publication.

Inertial manifolds for nonautomous infinite dimensional dynamical systems

In this paper, the long time behavior of nonautonomous infinite dimensional dynamical systems is discussed. Under the spectral gap condition, It is proved that there exist inertial manifolds for a class of nonautonomous evolution equations.