Pseudo-almost Periodic Solutions Research Articles

Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equations

We study the existence of almost periodic (resp., pseudo-almost periodic) mild solutions for fractional differential and integro-differential equations in the case when the forcing term belongs to the class of Stepanov almost (resp., Stepanov-like pseudo-almost) periodic functions.

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.

Composition Theorems of Stepanov Almost Periodic Functions and Stepanov-Like Pseudo-Almost Periodic Functions

We establish a composition theorem of Stepanov almost periodic functions, and, with its help, a composition theorem of Stepanov-like pseudo almost periodic functions is obtained. In addition, we apply our composition theorem to study the existence and uniqueness of pseudo-almost periodic solutions to a class of abstract semilinear evolution equation in a Banach space. Our results complement a recent work due to Diagana (2008).

Pseudo-almost periodic solutions of a class of semilinear fractional differential equations

We study existence and uniqueness of a pseudo-almost periodic (of class infinity) mild solution to the semilinear fractional equation \(\partial_{t}^{\alpha}u=Au+\partial_{t}^{\alpha-1}f(\cdot,u),1<\alpha<2\), where A is a linear operator of sectorial negative type.

WEIGHTED PSEUDO-ALMOST PERIODIC SOLUTIONS FOR SOME ABSTRACT DIFFERENTIAL EQUATIONS WITH UNIFORM CONTINUITY

AbstractIn this work, we give some theorems on (mild) weighted pseudo-almost periodic solutions for some abstract semilinear differential equations with uniform continuity. To facilitate this we give a new composition theorem of weighted pseudo-almost periodic functions. Our composition theorem improves the known one by making use of a uniform continuity condition instead of the Lipschitz condition.

Existence of weighted pseudo-almost periodic solutions to some classes of nonautonomous partial evolution equations

In this paper, under Acquistapace–Terreni conditions, we make extensive use of interpolation spaces and exponential dichotomy techniques to obtain the existence of weighted pseudo-almost periodic solutions to some classes of nonautonomous partial evolution equations. Applications include the existence of weighted pseudo-almost periodic solutions to a nonautonomous heat equation with gradient coefficients.

Optimal mild solutions and weighted pseudo-almost periodic classical solutions of fractional integro-differential equations

In this paper, we investigate a class of fractional integro-differential equations given by d α x ( t ) d t α = A x ( t ) + f ( t , x ( t ) , G x ( t ) ) . Our main results concern the existence, uniqueness of optimal mild solutions and weighted pseudo-almost periodic classical solutions. An example is given to illustrate our results.

ASYMPTOTIC EQUIVALENCE OF ALMOST PERIODIC SOLUTIONS FOR A CLASS OF PERTURBED ALMOST PERIODIC SYSTEMS

AbstractThe solutions of a perturbed linear ordinary differential equation (ODE) system are studied. Provided that some integrability and oddness conditions are satisfied, we show that they are asymptotically equivalent at t = ±∞ to the solutions of the unperturbed one. This fact is used to determine the existence of almost periodic or pseudo-almost periodic solutions of the perturbed system.

Stepanov-Like Pseudo-Almost Periodicity and Semilinear Differential Equations with Uniform Continuity

By using the method of the invariant subspaces for unbounded linear operators and Schauder’s fixed point theorem, we give an existence theorem of mild pseudo-almost periodic solutions for some semilinear differential equations with a Stepanov-like pseudo-almost periodic term under some suitable assumptions. For this purpose, we show a new composition theorem of Stepanov-like pseudo-almost periodic functions. As applications, we examine the existence of mild pseudo-almost periodic solutions to some second-order hyperbolic equations. Our work is done under a “uniform continuity” condition instead of the “Lipschitz” condition assumed in the literature.

Weighted pseudo almost periodic solutions of neutral functional differential equations

In this paper, we study weighted pseudo almost periodic solutions of neutral functional differential equations. By applying the properties of weighted pseudo almost periodic functions and the exponential dichotomy of linear systems as well as Krasnoselskii’s fixed point theorem, we establish the conditions for the existence of weighted pseudo-almost periodic solution of the equations.

Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations

We study the existence and uniqueness of a weighted pseudo-almost periodic (mild) solution to the semilinear fractional equation ∂ t α u = A u + ∂ t α − 1 f ( ⋅ , u ) , 1 < α < 2 , where A is a linear operator of sectorial negative type. This article also deals with the existence of these types of solutions to abstract partial evolution equations.

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.

Asymptotically almost periodic, almost periodic and pseudo-almost periodic mild solutions for neutral differential equations

In this paper we establish the existence and uniqueness of almost periodic, asymptotically almost periodic and pseudo-almost periodic mild solutions for neutral differential equations in Banach spaces.

Existence of weighted pseudo-almost periodic solutions to some classes of differential equations with [formula omitted]-weighted pseudo-almost periodic coefficients

In this paper we introduce the concept of weighted pseudo-almost periodicity in the sense of Stepanov, also called S p -weighted pseudo-almost periodicity and study its properties. Next, we present a result on the existence of weighted pseudo-almost periodic solutions to the N -dimensional heat equation with S p -weighted pseudo-almost periodic coefficients.

Stepanov-like pseudo-almost periodic mild solutions to perturbed nonautonomous evolution equations with infinite delay

In this paper, we prove the invariance of Stepanov-like pseudo-almost periodic functions under bounded linear operators. Furthermore, we obtain existence and uniqueness theorems of pseudo-almost periodic mild solutions to evolution equations u ′ ( t ) = A ( t ) u ( t ) + h ( t ) and u ′ ( t ) = A ( t ) u ( t ) + f ( t , B u ( t ) ) + ∫ − ∞ t C ( t , s ) u ( s ) d s + F ( t ) on R , assuming that A ( t ) satisfy “Acquistapace–Terreni” conditions, that the evolution family generated by A ( t ) has exponential dichotomy, that R ( λ 0 , A ( ⋅ ) ) is almost periodic, that B , C ( t , s ) t ≥ s are bounded linear operators, that f is Lipschitz with respect to the second argument uniformly in the first argument and that h , f , F are Stepanov-like pseudo-almost periodic for p > 1 and continuous. To illustrate our abstract result, a concrete example is given.

Weighted pseudo-almost periodic solutions of a class of abstract differential equations

For abstract linear functional differential equations with a weighted pseudo-almost periodic forcing term, we prove that the existence of a bounded solution on R + implies the existence of a weighted pseudo-almost periodic solution. Our results extend the classical theorem due to Massera on the existence of periodic solutions for linear periodic ordinary differential equations. To illustrate the results, we consider the Lotka–Volterra model with diffusion.

Pseudo-almost periodic solutions for abstract partial functional differential equations

In this work we study the existence and uniqueness of pseudo-almost periodic solutions for a first-order abstract functional differential equation with a linear part dominated by a Hille–Yosida type operator with a non-dense domain.

Weighted pseudo-almost periodic solutions to a neutral delay integral equation of advanced type

This paper studies suitable sufficient conditions to ensure the existence and uniqueness of weighted pseudo-almost periodic solutions to a neutral delay integral equation of advanced type introduced by T.A. Burton in the literature. The abstract results are then utilized to characterize weighted pseudo-almost periodic solutions to the well-known logistic equation.

Stepanov-like pseudo-almost periodicity and its applications to some nonautonomous differential equations

The paper revisits the concept of S p -pseudo-almost periodicity recently introduced by the author. In particular, we study the existence of pseudo-almost periodic solutions to some nonautonomous differential equations in the case when the semilinear forcing term is both continuous and S p -pseudo-almost periodic for p > 1 . Applications include the existence of pseudo-almost periodic solutions to the heat equation with a forcing term, which is S p -pseudo-almost periodic and jointly continuous.

The existence of pseudo-almost periodic solutions for some nonlinear differential equations in Banach spaces

Let ( E , ‖ . ‖ ) be a real Banach, where [ x , h ] − denotes the limit of the quotient ( ‖ x ‖ − ‖ x − t h ‖ ) / t as 0 < t → 0 . Consider the differential equation d x d t = A ( t , x ) + f ( t ) where we assume, amongst other properties, that A is strongly dissipative in the following sense: [ x − y , A ( t , x ) − A ( t , y ) ] − ≤ p ( t ) ‖ x − y ‖ θ ( ‖ x − y ‖ ) where θ is a function of the type u α ( α ≥ 0 ) and p is a function majorized by an almost periodic function with negative mean value. First, we show that this equation has a unique pseudo-almost periodic solution. Then we apply this result to the class of the following equations: d x d t + q ( t ) ‖ x ‖ α x = f ( t ) ( α ≥ 0 ) , where q is a pseudo-almost periodic function with positive mean value.