Ω-periodic Solutions Research Articles

Measure pseudo S -asymptotically ω-periodic solution in distribution for some stochastic differential equations with Stepanov pseudo S -asymptotically ω-periodic coefficients

In this research paper, a mathematical concept known as the ω-periodic process is discussed, and a new type of processes called the doubly measure pseudo -asymptotically ω-periodic in distribution process is set forward. Additionally, the properties of these processes are identified and invested to tackle the solution of a stochastic differential equation guided by Brownian motion. The basic objective of the current work resides in corroborating the existence and uniqueness of the solution which is doubly measure pseudo -asymptotically ω-periodic in distribution.

Non-constant periodic solutions of the Ricker model with periodic parameters

In this paper, we study the Ricker model x n + 1 = x n exp ⁡ [ r n ( 1 − x n ) ] , n = 0 , 1 , 2 … , ( ∗ ) where x 0 ≥ 0 , and { r n } n = 0 ∞ is a sequence of positive ω-periodic numbers. We obtain sufficient conditions for equation (*) to have at least two non-constant periodic solutions by transforming the existence problem of non-constant ω-periodic solutions of (*) into the existence problem of positive fixed points except 1 of some function. For the special case of period-two parameters, we show that ( ∗ ) has at most two non-constant 2-periodic solutions, and give necessary and sufficient conditions for (*) to have no non-constant 2-periodic solutions, to have a unique non-constant 2-periodic solution which or the equilibrium 1 is semi-stable, and to have exactly two non-constant 2-periodic solutions, respectively. Finally, some specific examples are also provided to illustrate our theoretical results.

S-asymptotically ω periodic solutions of heat equation

We mean by an S-asymptotically ω-periodic function any continuous and bounded function from the real axis to a Banach space that converges to a periodic function as t tends to infinity. We study in this paper the existence, uniqueness and differentiable dependence on the S-asymptotically ω-periodic mild solution of a heat equation on the space of continuous functions from a non-empty n-dimensional bounded domain with a Lipschitz boundary to the real axis. The dependence of the solution concerns the initial conditions, more precisely when the initial conditions is a S-asymptotically ω-periodic function we study the differentiable dependence of the S-asymptotically ω-periodic solution of heat equation. To show our main result in this work we introduce the properties of the superposition operator, or also called Nemytskii operator, in the space of S-asymptotically ω-periodic functions. The notion of derivation for the last operator will also be highlighted. We also use in this paper semi-groups which have become important tools for differential equations. In this study, our focus shifted from seeking an s-asymptotically w-periodic solution for our heat equation, which is a problem in dynamic systems, to one in functional analysis. More precisely, our strategy consists of applying the implicit function theorem on a certain operator that we constructed on our workspaces in order to achieve the objective described in our main theorem. In fact our theorem gives conditions to ensure that around a mild S-asymptotically ω-periodic solution of our heat equation with an initial value, there exists a regular (in the usual sens) mild S-asymptotically ω-periodic solution which depends on a neighboring initial condition.

S-asymptotically ω-Periodic Solutions of generalized Liénard

The S-asymptotically ω-periodic functions are a continuous and bounded functions from the real axis to a Banach space that converges to a periodic functions as tends to infinity. Starting from the zero solution, we prove in this work the existence and uniqueness of S-asymptotically ω-periodic solution of generalized Liénard's differential Equation. We stady after that the regular dependence of this solution with a certain parameter in Banach space, present in our equation, and with the forcing term hwo possesses a similar nature as the later. For this, our approach will be to use a perturbation method around an equilibrium. More precisely when the forcing is a Sasymptotically ω-periodic function we study the differentiable dependence of the S-asymptotically ω-periodic solution of Liénard equation. In this study, we changed our initial objective, which was to search for na S asymptotically ω-periodic solution for our Lienard equation, a problem relating to dynamical systems, towards an approach based on functional analysis. Concretely, we adopted a strategy consisting of using the implicit function theorem on a specific operator that we defined in our workspaces. This approach allowed us to achieve the objective stated in our main theorem. To realize our aim, we use the Nemytskii operators (also called superposition operators) and state some properties on these operators. We have also extended the well-established result on the almost periodic function of the derivative of an almost periodic function to the context of S-asymptotically ω-periodic cases. Finally, and to close our work, we give a corollary which presents a particular case of our main theorem.

On the S-asymptotically ω-periodic mild solutions for multi-term time fractional measure integro-differential equations

This paper deals with a class of nonlocal problem for multi-term time-fractional measure integro-differential evolution equations of mixed type in Hilbert spaces. Firstly, we introduce the concept of S-asymptotically ω-periodic mild solution, on the premise of by utilizing -resolvent family and measure functional (Henstock–Lebesgue–Stieltjes integral). Secondly, combining regulated functions and fixed point theorem, the existence S-asymptotically ω-periodic mild solution for a classes of measure-driven differential equation is discussed. Finally, as samples of applications, we consider the nonlocal problem to a class of time-fractional measure partial differential equation with nonlocal condition at the end of this paper.

Monotone iterative technique for delay elastic systems in ordered Banach space

We research extremal S-asymptotically ω-periodic mild solutions to elastic systems by using upper and lower solutions. In our first result, it is assumed that the semigroup is compact. In addition, in our second result, the semigroup does not need to be compact and not assuming lower and upper solutions. Our conclusion is drawn using measures of noncompactness. Finally, the existence of mild solutions for damped equations is discussed as an application.

Monotone iterative technique for evolution equations with delay and nonlocal conditions in ordered Banach space

In this paper, based on monotone iterative method in the presence of the lower and upper solutions, we investigate the existence and uniqueness of the S-asymptotically ω-periodic mild solutions to a class of nonlocal problems of evolution equations with delay in ordered Banach spaces. Firstly, we introduce the concept of lower S-asymptotically ω-periodic solution and upper S-asymptotically ω-periodic solution. Secondly, we construct monotone iterative method in the presence of the lower and upper solutions to evolution equations with delay, and obtain the existence of maximal and minimal S-asymptotically ω-periodic mild solutions for the mentioned system under wide monotone conditions and noncompactness measure condition of nonlinear term. Finally, as the application of abstract results, an example is given to illustrate our main results.

A study on mild solutions for multi-term time fractional measure differential equations

In this paper, we investigate the existence and uniqueness of the S-asymptotically ω-periodic mild solutions to a class of multi-term time-fractional measure differential equations with initial conditions in Banach spaces. Firstly, we look for a suitable concept of S-asymptotically ω-periodic mild solution to our concerned problem, by means of the Laplace transform and -resolvent family . Secondly, the existence of S-asymptotically ω-periodic mild solutions for the mentioned system is obtained by utilizing regulated functions and fixed point theorem. Finally, as the application of abstract results, an example is given to illustrate our main results.

A study on S-asymptotically ω-periodic positive mild solutions for damped elastic systems

The goal of this paper is to consider damped elastic systems with delay and nonlocal conditions in the framework of ordered Banach spaces. Firstly, we investigate the existence of minimal positive S-asymptotically ω-periodic mild solution for structural damped elastic systems with delay and nonlocal conditions on infinite interval. Secondly, based on monotone iterative technique coupled with fixed point theorem, the existence of minimal positive S-asymptotically ω-periodic mild solution is discussed without assuming the existence of upper and lower solutions. Finally, a concrete problem regarding the vibration equation of simply supported beam is given to illustrate the feasibility of our abstract results.

Dynamics of a multi-strain malaria model with diffusion in a periodic environment

This paper mainly explores the complex impacts of spatial heterogeneity, vector-bias effect, multiple strains, temperature-dependent extrinsic incubation period (EIP) and seasonality on malaria transmission. We propose a multi-strain malaria transmission model with diffusion and periodic delays and define the reproduction numbers and (i = 1, 2). Quantitative analysis indicates that the disease-free ω-periodic solution is globally attractive when , while if (), then strain i persists and strain j dies out. More interestingly, when and are greater than 1, the competitive exclusion of the two strains also occurs. Additionally, in a heterogeneous environment, the coexistence conditions of the two strains are and . Numerical simulations verify the analytical results and reveal that ignoring vector-bias effect or seasonality when studying malaria transmission will underestimate the risk of disease transmission.

On periodic boundary value problem for certain planar system of nonlinear ordinary differential equations

In this paper we establish sufficient conditions for existence of ω-periodic solutions of certain two-dimensional system of nonlinear differential equations.

The effects of vaccination, vector controls and media on dengue transmission dynamics with a seasonally varying mosquito population

Although numerous studies have attempted to investigate the effects of dengue intervention on its transmission dynamics, little work has considered the effects of implementing multiple dengue elimination strategies in a seasonally varying mosquitoes population, which is the focus of this work. A non-autonomous dengue mathematical model with a seasonally varying mosquito population, vector control, vaccination and media has been formulated and studied. We found that the disease-free equilibrium is globally asymptotically stable when the R0<1 and there exists at least a positive ω-periodic solution when R0>1. Furthermore, the implementation of continuous vaccination or both (vaccination and vector controls), and the efficacy of media campaign, which may raise the individual’s awareness, determines the reduction of dengue infections. Moreover, the rate of waning immunity is one of the influential factor contributing to an increase in dengue infections, which indicates the possibility of obtaining higher number of secondary infections.

Existence of ω-periodic solutions for second order delay differential equation in Banach spaces

The propose of the paper is devoted to study the existence of ?-periodic solutions for second order delay differential equation in abstract Banach space. Firstly, we build a new maximum principle for the ?-periodic solutions of the corresponding linear equation. Secondly, with the help of this maximum principle, we study the existence of the minimal and maximal periodic solutions for our concerns problem by means of perturbation method and monotone iterative technique of the lower and upper solutions. In addition, an example is presented to show the application of our main results.

Existence of positive S-asymptotically periodic solutions of the fractional evolution equations in ordered Banach spaces

In this paper, we discuss the asymptotically periodic problem for the abstract fractional evolution equation under order conditions and growth conditions. Without assuming the existence of upper and lower solutions, some new results on the existence of the positive S-asymptotically ω-periodic mild solutions are obtained by using monotone iterative method and fixed point theorem. It is worth noting that Lipschitz condition is no longer needed, which makes our results more widely applicable.

Monotone Iterative Technique for the Periodic Solutions of High-Order Delayed Differential Equations in Abstract Spaces

This paper deals with the existence of ω-periodic solutions for nth-order ordinary differential equation involving fixed delay in Banach space E. Lnu(t)=f(t,u(t),u(t−τ)),t∈R, where Lnu(t):=u(n)(t)+∑i=0n−1aiu(i)(t), ai∈R, i=0,1,⋯,n−1, are constants, f(t,x,y):R×E×E⟶E is continuous and ω-periodic with respect to t, τ&gt;0. By applying the approach of upper and lower solutions and the monotone iterative technique, some existence and uniqueness theorems are proved under essential conditions.

Existence of positive periodic solutions for a class of second-order neutral functional differential equations

Abstract In this paper, under some ordered conditions, we investigate the existence of positive ω-periodic solutions for a class of second-order neutral functional differential equations with delayed derivative in nonlinearity of the form (x(t) − cx(t − δ))″ + a(t)g(x(t))x(t) = λb(t)f(t, x(t), x(t − τ 1(t)), x′(t − τ 2(t))). By utilizing the perturbation method of a positive operator and the fixed point index theory in cones, some sufficient conditions are established for the existence as well as the non-existence of positive ω-periodic solutions.

S-Asymptotically ω-Periodic Solutions of Fractional-Order Complex-Valued Recurrent Neural Networks With Delays

In this paper, we consider the problem of the S-asymptotically ω-periodic synchronization of fractional-order complex-valued recurrent neural networks with time delays. Firstly, we can not explicitly decompose the fractional-order complex-valued systems into equivalent fractional-order real-valued systems, by means of the contraction mapping principle and some important features of Mittag-Leffler functions, we obtain some sufficient conditions for the existence and uniqueness of S-asymptotically ω-periodic solutions for this class of neural networks. Then, by constructing an appropriate Lyapunov functional, the theory of fractional differential equation, and some inequality techniques, sufficient conditions are obtained to guarantee the global Mittag-Leffler synchronization of the drive-response systems. Finally, two examples are given to illustrate the effectiveness and feasibility of our main results.

Positive periodic solutions for high-order differential equations with multiple delays in Banach spaces

This paper deals with the existence of positive ω-periodic solutions for nth-order ordinary differential equation with delays in Banach space E of the form Lnu(t)=f(t,u(t−τ1),…,u(t−τm)),t∈R,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$L_{n}u(t)=f\\bigl(t,u(t-\\tau_{1}),\\ldots,u(t- \\tau_{m})\\bigr),\\quad t\\in\\mathbb{R}, $$\\end{document} where L_{n}u(t)=u^{(n)}(t)+sum_{i=0}^{n-1}a_{i} u^{(i)}(t) is the nth-order linear differential operator, a_{i}inmathbb {R} (i=0,1,ldots,n-1) are constants, f: mathbb{R}times E^{m}rightarrow E is a continuous function which is ω-periodic with respect to t, and tau_{i}>0 (i=1,2,ldots,m) are constants which denote the time delays. We first prove the existence of ω-periodic solutions of the corresponding linear problem. Then the strong positivity estimation is established. Finally, two existence theorems of positive ω-periodic solutions are proved. Our discussion is based on the theory of fixed point index in cones.

Existence of [formula omitted]-periodic solutions for a delayed chemostat with periodic inputs

This paper proposes an ω-periodic version of the Ellermeyer model of delayed chemostat. We obtain a sufficient condition ensuring the existence of a positive ω-periodic solution. Our proof is based on the application of the generalized continuation theorem. In addition, as a consequence of the implicit function theorem, we obtain a uniqueness result for sufficiently small delays.

Pseudo S-asymptotically ω-periodic solution for a differential equation with piecewise constant argument in a Banach space

ABSTRACTIn this paper, we give sufficient conditions for the existence and uniqueness of Pseudo S-Asymptotically ω-periodic solutions for a differential equation with piecewise constant argument in a Banach space. This result is obtained using the Pseudo S-asymptotically ω-periodic sequences.