Periodic Mild Solutions Research Articles

Pseudo almost periodic solutions in the alpha-norm and Stepanov's sense for some evolution equations

UDC 517.9 Our aim is to present the concept of double-measure ergodic and double-measure pseudo almost periodic functions in Stepanov's sense. In addition, we present numerous interesting results, such as the composition theorems and completeness properties for these two spaces of the considered functions. We also establish the existence and uniqueness for the double-measure pseudo almost periodic mild solutions in Stepanov's sense for some evolution equations.

Sensitivity analysis for a fractional stochastic differential equation with $$S^{p}$$-weighted pseudo almost periodic coefficients and infinite delay

In this paper, we study the parameter sensitivity analysis a class of fractional impulsive stochastic differential equation with \(S^{p}\)-weighted pseudo almost periodic coefficients and infinite delay in Hilbert spaces. Firstly, a more appropriate concept of piecewise \(S^{p}\)-weighted pseudo almost periodic in distribution for stochastic processes of class \(\infty \) is introduced. The existence of piecewise weighted pseudo almost periodic mild solutions in distribution is presented by using \(\alpha \)-order fractional resolvent operator theory, the stochastic analysis techniques and a fixed-point theorem. Secondly, the sensitivity properties of these infinite delay systems under non-Lipschitz conditions is also obtained. Finally, two examples are given for demonstration.

Integration in some new concept of ergodic functions and application to some epidemiological models

In a former work, we introduced a new class of functions, which we call measure pseudo ‐asymptotically omega periodic functions and give some properties of those functions as the topological properties of the space of such functions, the invariance by translation of this space, and the convolution product in this space. In this paper, we consider the same class of functions and give contributions as the stability by the integral operator; we present some examples and composition results. Then we apply the obtained results to investigate the existence of measure pseudo ‐asymptotically omega periodic mild solutions for some nonlinear infinite delay integral equations arising in epidemiology; exactly, we consider the nonlinear infinite delay integral equation: Then we apply these results to a finite delay integral equation when the delay is time dependent

Asymptotically almost periodic solutions to parabolic equations on the real hyperbolic manifold

In this work, we study the existence and stability of asymptotically almost periodic mild solutions of vectorial parabolic equations on the real hyperbolic manifold Hd(R) (d⩾2). We will consider the equations associated with Ebin-Marsden's Laplace operator. Our method is based on certain dispersive and smoothing estimates of the semigroup associated with the linearized vectorial heat equation and the fixed point argument. First, we prove the existence and uniqueness of the asymptotically almost periodic mild solution for the linear equation. Next, using the fixed point argument we can pass from the linear equation to the semi-linear equation to prove the existence and uniqueness of such solution. We prove also the exponential stability of solution by using the cone inequality. Finally, abstract results will be applied to Navier-Stokes equations and the semi-linear vectorial heat equation.

On Periodic Solutions of the Incompressible Navier–Stokes Equations on Non-compact Riemannian Manifolds

In this paper, we study the existence, uniqueness and stability of the time periodic mild solutions to the incompressible Navier–Stokes equations on the non-compact manifolds with negative Ricci curvature tensor. In our strategy, we combine the dispersive and smoothing estimates for Stokes semigroups and Massera-type theorem to establish the existence and uniqueness of the time periodic mild solution to Stokes equation on Riemannian manifolds. Then using fixed point arguments, we can pass to semilinear equations to obtain the existence and uniqueness of the periodic solution to the imcompressible Navier–Stokes equations under the action of a periodic external force. The stability of the solution is also proved by using the cone inequality.

Stochastic Nicholson’s blowflies model with delays

In this research work, we introduce a new concept of double measure ergodic processes to identify the spaceof double measure pseudo almost periodic (or pap) processes in the pth mean sense. We display some findings regarding the completeness as well as the composition theorems and the invariance of the space consisting in double measure pap processes. Motivated by the above-mentioned results, the Banach fixed point theorem and the stochastic analysis techniques, we prove the existence, uniqueness and the global exponential stability of doubly measure pseudo almost periodic mild solution for a class of stochastic Nicholson’s blowflies model with mixed delays in a separable real Hilbert space. The concluding part of the work is crowned with an example to confirm the reliability and feasibility of our findings.

Generalized Bloch type periodicity and applications to semi-linear differential equations in banach spaces

AbstractIn this paper, we mainly introduce some new notions of generalized Bloch type periodic functions namely pseudo Bloch type periodic functions and weighted pseudo Bloch type periodic functions. A Bloch type periodic function may not be Bloch type periodic under certain small perturbations while it can be quasi Bloch type periodic in sense of generalized Bloch type periodic functions. We firstly show the completeness of spaces of generalized Bloch type periodic functions and establish some further properties such as composition and convolution theorems of such functions. We then apply these results to investigate existence results for generalized Bloch type periodic mild solutions to some semi-linear differential equations in Banach spaces. The obtained results show that for each generalized Bloch type periodic input forcing disturbance, the output mild solutions to reference evolution equations remain generalized Bloch type periodic.

Periodic solutions and their stability to the Navier-Stokes equations on a half space

Consider the Navier-Stokes equations in a half space. It is shown that this system admits a unique, $ T $-periodic mild solution in some suitable Lorentz space provided the outer force is $ T $-periodic and small enough. This result is then extended to the case of almost periodic functions. Moreover, it is shown that these solutions are polynomially stable.

Periodic problem for non-instantaneous impulsive partial differential equations

<abstract><p>We obtain a new maximum principle of the periodic solutions when the corresponding impulsive equation is linear. If the nonlinear is quasi-monotonicity, we study the existence of the minimal and maximal periodic mild solutions for impulsive partial differential equations by using the perturbation method, the monotone iterative technique and the method of upper and lower solution. We give an example in last part to illustrate the main theorem.</p></abstract>

Weighted pseudo asymptotically Bloch periodic solutions to nonlocal Cauchy problems of integrodifferential equations in Banach spaces

AbstractThis paper is mainly concerned with some new asymptotic properties on mild solutions to a nonlocal Cauchy problem of integrodifferential equation in Banach spaces. Under some well-imposed conditions on the nonlocal Cauchy, the neutral and forced terms, respectively, we establish some existence results for weighted pseudoS-asymptotically (ω,k)-Bloch periodic mild solutions to the referenced equation onR+${\mathbb{R}}_{+}$by suitable superposition theorems. The results show that the strict contraction of the nonlocal Cauchy and the neutral terms with the state variable has an appreciable effect on the existence and uniqueness of such a solution compared with the forced term. As an auxiliary result, the existence of weighted pseudoS-asymptotically (ω,k)-Bloch periodic mild solutions is deduced under the sublinear growth condition on the force term with its state variable. The existence of weighted pseudoS-asymptoticallyω-antiperiodic mild solution is also obtained as a special example.

(ω, c)-periodic and asymptotically (ω, c)-periodic mild solutions to fractional Cauchy problems

In this paper, we establish some new properties of -periodic and asymptotically -periodic functions, then we apply them to study the existence and uniqueness of mild solutions of these types to the following semilinear fractional differential equations: (1) and (2) where stands for the Caputo derivative and A is a linear densely defined operator of sectorial type on a complex Banach space and the function is -periodic or asymptotically -periodic with respect to the first variable. Our results are obtained using the Leray–Schauder alternative theorem, the Banach fixed point principle and the Schauder theorem. Then we illustrate our main results with an application to fractional diffusion-wave equations.

Some new asymptotic properties on solutions to fractional evolution equations in Banach spaces

In this paper, we mainly investigate some new asymptotic properties on mild solutions to a fractional evolution equation in Banach spaces. Under local, global and mixed Lipschitz type conditions on the second variable for neutral and forced functions respectively, we establish some existence results for pseudo -Bloch periodic and pseudo S-asymptotically -Bloch periodic mild solutions to the referenced equation on by suitable superposition theorems. The results show that the strict contraction of the neutral function for its second variable takes a dominated part in the existence and uniqueness of such solutions compared with the forced function. As subordinate results, we derive existence results of pseudo (S-asymptotically) -Bloch periodic mild solutions for the sublinear growth of forced function with its second variable. As special cases, we also deduce some existence results for pseudo ω-antiperiodic and pseudo S-asymptotically ω-antiperiodic mild solutions to the considered equation on .

Existence of p-mean Almost Periodic Mild Solution for Fractional Stochastic Neutral Functional Differential Equation

Existence of p-mean Almost Periodic Mild Solution for Fractional Stochastic Neutral Functional Differential Equation

Periodic Mild Solutions of Infinite Delay not Instantaneous Impulsive Evolution Inclusions

This paper deals with the existence of periodic mild solutions for a class of functional evolution inclusions. We use a multivalued fixed point theorem in Banach spaces combined with the technique of measure of noncompactness. We show that the Poincare operator is a condensing operator with respect to Kuratowski’s measure of noncompactness in a determined phase space, and then derive periodic solutions from bounded solutions by using Sadovskii’s fixed point theorem.

Time optimal control to a partial stochastic differential system with pseudo almost periodic coefficients

In this paper, we study the time optimal control of a class of partial stochastic impulsive functional differential systems with pseudo almost periodic coefficients in the α-norm functional spaces. Firstly, the existence of p-mean piecewise pseudo almost periodic mild solutions for the impulsive stochastic control systems is investigated by utilising stochastic analysis, analytic semigroup, the fixed-point techniques with fractional power arguments. Secondly, the existence of time optimal pairs of a system governed by partial stochastic impulsive functional differential equations is also obtained. Finally, two examples are provided to illustrate the time optimal control problems of parabolic control system with pseudo almost periodic coefficients and impulses.

(ω,c)-Periodic Mild Solutions to Non-Autonomous Abstract Differential Equations

We investigate the semi-linear, non-autonomous, first-order abstract differential equation x′(t)=A(t)x(t)+f(t,x(t),φ[α(t,x(t))]),t∈R. We obtain results on existence and uniqueness of (ω,c)-periodic (second-kind periodic) mild solutions, assuming that A(t) satisfies the so-called Acquistapace–Terreni conditions and the homogeneous associated problem has an integrable dichotomy. A new composition theorem and further regularity theorems are given.

S-Asymptotically Bloch Type Periodic Solutions to Some Semi-Linear Evolution Equations in Banach Spaces

This paper is mainly concerned with the S-asymptotically Bloch type periodicity. Firstly, we introduce a new notion of S-asymptotically Bloch type periodic functions, which can be seen as a generalization of concepts of S-asymptotically ω-periodic functions and S-asymptotically ω-anti-periodic functions. Secondly, we establish some fundamental properties on S-asymptotically Bloch type periodic functions. Finally, we apply the results obtained to investigate the existence and uniqueness of S-asymptotically Bloch type periodic mild solutions to some semi-linear differential equations in Banach spaces.

Pseudo $S$-Asymptotically Bloch Type Periodicity with Applications to Some Evolution Equations

This paper is mainly focused upon the pseudo S -asymptotically Bloch type periodicity and its applications. Firstly, a new notion of pseudo S -asymptotically Bloch type periodic functions is introduced, and some fundamental properties on pseudo S -asymptotically Bloch type periodic functions are established. Then, the notion and properties of weighted pseudo S -asymptotically Bloch type periodic functions are similarly presented. Finally, the obtained results are applied to investigate the existence and uniqueness of pseudo S -asymptotically Bloch type periodic mild solutions for some semi-linear evolution equations in Banach spaces.

Square-mean asymptotically almost periodic solutions of second order nonautonomous stochastic evolution equations

In this paper, we study the existence of square-mean asymptotically almost periodic mild solutions for a class of second order nonautonomous stochastic evolution equations in Hilbert spaces. By using the principle of Banach contractive mapping principle, the existence and uniqueness of square-mean asymptotically almost periodic mild solutions of the equation are obtained. To illustrate the abstract result, a concrete example is given.

Measure pseudo almost periodic solution for a class of nonlinear delayed stochastic evolution equations driven by Brownian motion

In this work, we present a new concept of measure-ergodic process to define the space of measure pseudo almost periodic process in the p-th mean sense. We show some results regarding the completeness, the composition theorems and the invariance of the space consisting in measure pseudo almost periodic process. Motivated by above mentioned results, the Banach fixed point theorem and the stochastic analysis techniques, we prove the existence, uniqueness and the global exponential stability of doubly measure pseudo almost periodic mild solution for a class of nonlinear delayed stochastic evolution equations driven by Brownian motion in a separable real Hilbert space. We provide an example to illustrate the effectiveness of our results.