Periodic Mild Solutions Research Articles

Square-mean S-Asymptotically $$\omega $$-Periodic Solutions for Some Stochastic Delayed Integrodifferential Inclusions

AbstractIn this work, by the resolvent operator theory, stochastic analysis theory and the fixed-point technique in Fréchet spaces, we present sufficient conditions for the existence of square-mean S-asymptotically $$\omega $$ ω -periodic mild solutions of abstract stochastic integrodifferential inclusions with infinite delay in separable Hilbert spaces and provide an example to illustrate the abstract results.

(N,λ)-periodic solutions to abstract difference equations of convolution type

This work primarily focuses on (N,λ)-periodic sequences and their applications. To begin, we provide a brief overview of (N,λ)-periodic sequences and introduce several results. Secondly, in terms of applications and main objective, we establish sufficient criteria for both the existence and uniqueness of (N,λ)-periodic mild solutions for abstract difference equations of convolution type. Furthermore, we present illustrative examples to highlight our key findings.

Measure pseudoasymptotically Bloch periodic functions in the sense of Stepanov and applications

UDC 517.9 We focus on the measures of Stepanov-like pseudoasymptotically Bloch τ -periodicity and its applications. First, we define a new notion of measures of Stepanov-like pseudoasymptotically Bloch periodic functions and diccuss some of its fundamental properties. Then the obtained results are applied to investigate the existence and uniqueness of the measure Stepanov-like pseudoasymptotically Bloch periodic mild solutions to semilinear delay differential equation in Banach spaces. Finally, an application is presented to illustrate the efficiency of the results.

Periodic solutions of the parabolic–elliptic Keller–Segel system on whole spaces

AbstractIn this paper, we investigate to the existence and uniqueness of periodic solutions for the parabolic–elliptic Keller–Segel system on whole spaces detailized by Euclidean space and real hyperbolic space . We work in framework of critical spaces such as on weak‐Lorentz space to obtain the results for the Keller–Segel system on and on for to obtain those on . Our method is based on the dispersive and smoothing estimates of the heat semigroup and fixed point arguments. This work provides also a fully comparison between the asymptotic behaviors of periodic mild solutions of the Keller–Segel system obtained in and the one in .

Periodic solutions for Boussinesq systems in weak-Morrey spaces

We prove the existence and polynomial stability of periodic mild solutions for Boussinesq systems in critical weak-Morrey spaces for dimension n⩾3. Those systems are derived via the Boussinesq approximation and describe the movement of an incompressible viscous fluid under natural convection filling the whole space Rn. Using certain dispersive and smoothing properties of heat semigroups on Morrey-Lorentz spaces as well as Yamazaki-type estimate on block spaces, we prove the existence of bounded mild solutions for the linear systems corresponding to the Boussinesq systems. Then, we establish a Massera-type theorem to obtain the existence and uniqueness of periodic solutions to corresponding linear systems on the half line time-axis by using a mean-ergodic method. Next, using fixed point arguments, we can pass from linear systems to prove the existence uniqueness and polynomial stability of such solutions for Boussinesq systems. Finally, we apply the results to Navier-Stokes equations.

Existence and asymptotic behavior of square-mean $ S $-asymptotically periodic solutions of fractional stochastic evolution equations

This paper investigates the abstract fractional stochastic evolution equations. A new existence result of the square-mean $ S $-asymptotically periodic mild solutions are obtained under the assumption that the nonlinear terms only satisfy some growth conditions. Moreover, the uniqueness and asymptotic stability results of the square-mean $ S $-asymptotically periodic solution are presented when the nonlinear functions satisfy the general Lipschitz condition. Finally, two examples are given to illustrate our main results.

On the $S$-asymptotically $\omega$-periodic mild solutions for multi-term time fractional measure differential equations

In this paper, based on regulated functions and fixed point theorem, a class of nonlocal problem of multi-term time-fractional measure differential equations involving nonlocal conditions in Banach spaces. Firstly, we introduce the concept of $S$-asymptotically $\omega$-periodic mild solution, on the premise of by utilizing $(\beta,\gamma_k)$-resolvent family and measure functional (Henstock-Lebesgue-Stieltjes integral), the existence of $S$ -asymptotically $\omega$ periodic mild solutions for the mentioned system are obtained. Finally, as the application of abstract results, the existence $S$-asymptotically $\omega$-periodic mild solution for a classes of measure driven differential equation are discussed.

Asymptotically almost periodic mild solutions for some partial integrodifferential inclusions using scale of Banach spaces

Abstract We are interested in the existence of mild solutions for a class of partial integrodifferential inclusions in infinite dimensional Banach spaces. First, we show the existence of mild solutions with the help of a scale of Banach spaces, the theory of resolvent operators, and the fixed point theory for the measure of non-compactness. Moreover, we examine the existence of asymptotically almost periodic solutions for our problem. Finally, an example of the abstract results is provided.

Weighted pseudo S-asymptotically Bloch type periodic solutions for a class of mean field stochastic fractional evolution equations

This paper concerns a class of mean-field stochastic fractional evolution equations. Initially, we establish some auxiliary results for weighted pseudo $S$-asymptotically Bloch type periodic stochastic processes. Without a compactness assumption on the resolvent operator and some additional conditions on forced terms, the existence and uniqueness of weighted pseudo $S$-asymptotically Bloch type periodic mild solutions on the real line of the referred equation are obtained. In addition, we show the existence of weighted pseudo $S$-asymptotically Bloch type periodic mild solutions with sublinear growth assumptions on the drift term and compactness conditions. Finally, an example is provided to verify the main outcomes.

Sv-asymptotically T-periodic solutions for some class of neutral differential equations — Model of the heat equation

In this paper, we aim to present the concept of [Formula: see text]-asymptotically T-periodic functions taking values in a Banach space and investigate some of their properties. Also, we establish conditions under which semi linear evolution equations in a Banach space have a unique global mild [Formula: see text]-asymptotically T-periodic solution. Then, we apply the results obtained to prove the existence and uniqueness of [Formula: see text]-asymptotically [Formula: see text]-periodic mild solutions to a nonautonomous semi linear differential equations.

On the Study of Pseudo S-Asymptotically Periodic Mild Solutions for a Class of Neutral Fractional Delayed Evolution Equations

The goal of this paper is to investigate the existence and uniqueness of pseudo S-asymptotically periodic mild solutions for a class of neutral fractional evolution equations with finite delay. We essentially use the fractional powers of closed linear operators, the semigroup theory and some classical fixed point theorems. Furthermore, we provide an example to illustrate the applications of our abstract results.

On S-asymptotically $$\omega$$-periodic mild solutions of some integrodifferential inclusions of Volterra-type

On S-asymptotically $$\omega$$-periodic mild solutions of some integrodifferential inclusions of Volterra-type

Existence and global asymptotic behavior of S-asymptotically periodic solutions for fractional evolution equation with delay

This paper discusses the S-asymptotically periodic problem of fractional evolution equation with delay. By introducing a new noncompact measure theory involving infinite interval, we study the existence of S-asymptotically periodic mild solutions under the situation that the relevant semigroup is noncompact and the nonlinear term satisfies more general growth conditions instead of Lipschitz-type conditions. Moreover, by establishing a new Gronwall-type integral inequality corresponding to fractional differential equation with delay, we consider the global asymptotic behavior of S-asymptotically periodic mild solutions, which will make up for the blank of this field.

Existence and exponential stability of the piecewise pseudo almost periodic mild solution for some partial impulsive stochastic neutral evolution equations

In the present paper, we first introduce the concept of double measure ‐mean piecewise pseudo almost periodic for stochastic processes for . Next, we make extensive use of the exponential dichotomy techniques and a fixed point strategy with stochastic analysis theory to obtain the existence of doubly measure ‐mean piecewise pseudo almost periodic mild solutions for a class of impulsive non‐autonomous partial stochastic evolution equations in Hilbert spaces. In addition, we study the exponential stability of ‐mean piecewise pseudo almost periodic mild solutions. Finally, we give an example to confirm the reliability and feasibility of our findings.

Pseudo almost periodicity for stochastic differential equations in infinite dimensions

In this article, we introduce the concept of p-mean θ-pseudo almost periodic stochastic processes, which is slightly weaker than p-mean pseudo almost periodic stochastic processes. Using the operator semigroup theory and stochastic analysis theory, we obtain the existence and uniqueness of square-mean θ-pseudo almost periodic mild solutions for a semilinear stochastic differential equation in infinite dimensions. Moreover, we prove that the obtained solution is also pseudo almost periodic in path distribution. It is noteworthy that the ergodic part of the obtained solution is not only ergodic in square-mean but also ergodic in path distribution. Our main results are even new for the corresponding stochastic differential equations (SDEs) in finite dimensions.

(omega ,c)-periodic solutions for a class of fractional integrodifferential equations

In this paper we investigate the following fractional order in time integrodifferential problem Dtαu(t)+Au(t)=f(t,u(t))+∫−∞tk(t−s)g(s,u(s))ds,t∈R.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathbb{D}_{t}^{\\alpha}u(t)+Au(t)=f \\bigl(t,u(t) \\bigr)+ \\int _{-\\infty}^{t} k(t-s)g \\bigl(s,u(s) \\bigr)\\,ds, \\quad t \\in \\mathbb{R}. $$\\end{document} Here, mathbb{D}_{t}^{alpha} is the Caputo derivative. We obtain results on the existence and uniqueness of (omega ,c)-periodic mild solutions assuming that −A generates an analytic semigroup on a Banach space X and f, g, and k satisfy suitable conditions. Finally, an interesting example that fits our framework is given.

(omega ,Q)-periodic mild solutions for a class of semilinear abstract differential equations and applications to Hopfield-type neural network model

In this paper, we investigate the existence and uniqueness of (omega ,Q)-periodic mild solutions for the following problem x′(t)=Ax(t)+f(t,x(t)),t∈R,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} x'(t)=Ax(t)+f(t,x(t)),\\quad t\\in \\mathbb {R}, \\end{aligned}$$\\end{document}on a Banach space X. Here, A is a closed linear operator which generates an exponentially stable C_0-semigroup and the nonlinearity f satisfies suitable properties. The approaches are based on the well-known Banach contraction principle. In addition, a sufficient criterion is established for the existence and uniqueness of (omega ,Q)-periodic mild solutions to the Hopfield-type neural network model.

Existence and asymptotic behavior of square-mean S-asymptotically periodic solutions for fractional stochastic evolution equation with delay

This paper studies a class of the fractional stochastic evolution equation with delay. With the aid of the compact semigroup theory and Schauder fixed point theorem, the existence of square-mean S-asymptotically periodic mild solutions is obtained under the situation that the nonlinear functions satisfy certain growth conditions. Moreover, by establishing a new Grönwall integral inequality corresponding to fractional differential equation with delay, the global asymptotic stability of the square-mean S-asymptotically periodic mild solutions are discussed. Finally, an example is given to illustrate our abstract results.

Optimal (ω,c)-asymptotically periodic mild solutions to some fractional evolution equations

Optimal (ω,c)-asymptotically periodic mild solutions to some fractional evolution equations

Existence of $ S $-asymptotically $ \omega $-periodic solutions for non-instantaneous impulsive semilinear differential equations and inclusions of fractional order $ 1 < \alpha < 2 $

<abstract><p>It is known that there is no non-constant periodic solutions on a closed bounded interval for differential equations with fractional order. Therefore, many researchers investigate the existence of asymptotically periodic solution for differential equations with fractional order. In this paper, we demonstrate the existence and uniqueness of the $ S $-asymptotically $ \omega $-periodic mild solution to non-instantaneous impulsive semilinear differential equations of order $ 1 < \alpha < 2 $, and its linear part is an infinitesimal generator of a strongly continuous cosine family of bounded linear operators. In addition, we consider the case of differential inclusion. Examples are given to illustrate the applicability of our results.</p></abstract>