Semilinear Differential Equations Research Articles

Exponential dichotomy and invariant manifolds of semi-linear differential equations on the line

In this paper we investigate the homogeneous linear differential equation vi(t) = A(t)v(t) and the semi-linear differential equation vi(t) = A(t)v(t) + g(t, v(t)) in Banach space X, in which A : R → L(X) is a strongly continuous function, g : R × X → X is continuous and satisfies ϕ-Lipschitz condition. The first we characterize the exponential dichotomy of the associated evolution family with the homogeneous linear differential equation by space pair (E, E∞), this is a Perron type result. Applying the achieved results, we establish the robustness of exponential dichotomy. The next we show the existence of stable and unstable manifolds for the semi-linear differential equation and prove that each a fiber of these manifolds is differentiable submanifold of class C1. Mathematics Subject Classification (2010): 34C45, 34D09, 34D10. Received 14 June 2021; Accepted 09 September 2022

On a controllability problem for a feedback control system governed by a semilinear differential equation and a sweeping process

We consider a controllability problem for a feedback control system governed by a semilinear differential equation and a sweeping process in Hilbert spaces. We define the operator whose fixed points are generating solutions of the problem. By applying the methods of the fixed point theory for condensing maps we study the properties of this operator, in particular, we prove that under certain conditions it is condensing w.r.t. an appropriate measure of noncompactness. This allows to present the general controllability principle.

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.

Functional delay random semilinear differential equations

Functional delay random semilinear differential equations

The stability of interval two-dimensional semi-linear differential equations based on constrained interval arithmetic

In this study, the constrained interval arithmetic (CIA) is used as an effective mathematical tool for solving the stability analysis for interval two-dimensional semi-linear differential equations. Under certain assumptions, the origin is a focus of the interval semi-linear differential equations if it is a focus of the interval linear ones. Meanwhile, the origin can be a center, a center-focus or a focus of interval semi-linear differential equations if it is a center of the interval linear ones. On the other word, the types of equilibrium point are still determined by the linear part when a nonlinear disturbance is added to the interval linear differential equations. Based on CIA, the stability results of interval differential equations are the same as those of the real differential equations. At last, three illustrative examples validate the stability results of the origin for interval two-dimensional semi-linear differential equations.

Conditional Lipschitz Shadowing for Ordinary Differential Equations

AbstractWe introduce the notion of conditional Lipschitz shadowing, which does not aim to shadow every pseudo-orbit, but only those which belong to a certain prescribed set. We establish two types of sufficient conditions under which certain nonautonomous ordinary differential equations have such a property. The first criterion applies to a semilinear differential equation provided that its linear part is hyperbolic and the nonlinearity is small in a neighborhood of the prescribed set. The second criterion requires that the logarithmic norm of the derivative of the right-hand side with respect to the state variable is uniformly negative in a neighborhood of the prescribed set. The results are applicable to important classes of model equations including the logistic equation, whose conditional shadowing has recently been studied. Several examples are constructed showing that the obtained conditions are optimal.

Exact null controllability for semilinear differential equations with nonlocal conditions in Hilbert spaces

Exact null controllability for semilinear differential equations with nonlocal conditions in Hilbert spaces

Weyl almost anti-periodic solution to a neutral functional semilinear differential equation

<abstract><p>In this work, we first propose a concept of Weyl almost anti-periodic functions. Then, we make use of the contraction mapping principle and analysis techniques to research the existence of a unique Weyl almost anti-periodic solution to a neutral functional semilinear abstract differential equation. Finally, we give an example of a neutral functional partial differential equation to show the validity of the obtained results.</p></abstract>

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>

Ergodicity of a nonlinear stochastic reaction–diffusion equation with memory

We consider a class of semi-linear differential Volterra equations with memory terms, polynomial nonlinearities and random perturbation. For a broad class of nonlinearities, we show that the system in concern admits a unique weak solution. Also, any statistically steady state must possess regularity compatible with that of the solution. Moreover, if sufficiently many directions are stochastically forced, we employ the generalized coupling approach to prove that there exists a unique invariant probability measure and that the system is exponentially attractive. This extends ergodicity results previously established in Bonaccorsi et al., (2012).

Partial observability of stochastic semilinear systems

In this article, we consider an Itô stochastic semilinear differential equation with unknown initial state and a linear observation system. It is proved that under a certain condition on the observability Gramian, the initial state of the equation can be recovered. This result is demonstrated by an example.

Equilibrium point, exponential stability and synchronization of numerical fractional-order shunting inhibitory cellular neural networks with piecewise feature

In recent years, exponential Euler difference methods for first-order semi-linear differential equations have been developed rapidly, and various exponential Euler difference methods have become a very important research topic. For fractional-order shunting inhibitory cellular neural networks with time lags, this article first establishes a new difference method named Mittag-Leffler Euler difference, which includes exponential Euler difference. Second, the existence of unique global bounded solution and equilibrium point, exponential stability and synchronization of the derived difference model are investigated. Compared with the classical fractional-order Euler difference method, the Mittag-Leffler discrete shunting inhibitory cellular neural networks can better describe and maintain the dynamic properties of the corresponding continuous-time models. More importantly, it opens up a new way to study discrete-time fractional-order systems and establishes a set of theory and methods to study Mittag-Leffler discrete neural networks.

Analysis of a Fitted Finite Difference Scheme for a Semilinear System of Singularly Perturbed Reaction-Diffusion Equations Having Discontinuous Source Term

In this paper, a system of an arbitrary number of singularly perturbed semilinear differential equations of reaction-diffusion type having discontinuous source term is examined, for the case in which the diffusion parameters associated with each equation of the considered system are assumed to be different in magnitude. In addition to the occurrence of the boundary layers, the solution exhibits the overlapping and interacting internal layers due to the existence of discontinuity in the data. A central finite difference scheme is used in conjunction with a suitable piecewise uniform Shishkin mesh and a Bakhvalov mesh, to construct the numerical method. Using discrete Green’s function technique, the proposed numerical scheme has been proved to be an almost second-order of uniform convergent for the Shishkin mesh and second-order of uniform convergent for the Bakhvalov mesh, independent of the perturbation parameters. Numerical test examples are presented to demonstrate the performance of the numerical scheme.

Measure ω , c -Pseudo-Almost Periodic Functions and Lasota-Wazewska Model with Ergodic and Unbounded Oscillating Oxygen Demand

The primary aim of this work is to introduce a new class of functions called μ - ω , c -pseudo-almost periodic functions. Using the measure theory, we generalize in a natural way some recent works and study some properties of those μ - ω , c -pseudo-almost periodic functions including two new composition results which play a crucial role for the existence of some μ - ω , c -pseudo-almost periodic solutions of certain semilinear differential equations and partial differential equations. We also investigate the existence and uniqueness of the μ - ω , c -pseudo-almost periodic solutions for some models of Lasota-Wazewska equation with measure ω , c -pseudo-almost periodic coefficient and mixed delays.

Stability analysis and synchronized control of fuzzy Mittag-Leffler discrete-time genetic regulatory networks with time delays

Exponential Euler differences for semi-linear differential equations of first order have got rapid development in the past few years and a variety of exponential Euler difference methods have become very significant researching topics. In allusion to fuzzy genetic regulatory networks of fractional order, this paper firstly establishes a novel difference method called Mittag-Leffler Euler difference, which includes the exponential Euler difference. In the second place, the existence of a unique global bounded solution and equilibrium point, global exponential stability and synchronization of the derived difference models are investigated. Compared with the classical fractional Euler differences, fuzzy Mittag-Leffler discrete-time genetic regulatory networks can better depict and retain the dynamic characteristics of the corresponding continuous-time models. What’s more important is that it starts a new avenue for studying discrete-time fractional-order systems and a set of theories and methods is constructed in studying Mittag-Leffler discrete models.

Square-mean inertial manifolds for stochastic differential equations

Abstract In this paper, we prove the existence of a square-mean inertial manifold for a class of semilinear stochastic differential equations in a real separable Hilbert space. An example is given to illustrate our results.

Results on impulsive semilinear differential equations with control functions

The article deals with the sufficient conditions for the approximate controllability for the impulsive semilinear control systems. We have considered two different sets of sufficient conditions. In the first set, we derived the results by applying the theories related to the compactness of semigroup and Schauder's fixed point theorem. In the second set, we derived the results by applying Gronwall's inequality and avoids the use of fixed point theorem and compactness of semigroup. Under suitable assumptions, the discussion on the existence and uniqueness of the impulsive system is also proved. We have also included an example for the validation of theoretical results.

On the periodic boundary value problems for fractional nonautonomous differential equations with non-instantaneous impulses

In this paper, we investigate periodic boundary value problems for Caputo type fractional semilinear nonautonomous differential equations with non-instantaneous impulses. By using semigroup theory combined with the measure of noncompactness and some fixed point theorems, the existence of PC-mild solutions for the equations is established. At the end, an example is presented to illustrate the application of our main results.

A generalized change of variable formula for the Young integral

This article shows an Itô-Wentzell type formula adapted to Young integral. We apply this formula to study differential equations driven by α-Hölder paths with α∈121.We obtain a formula for the composition and the inverse of solution maps associated with a Young differential equation. Also, we prove a Leibniz rule for interchanging derivative and the Young integral, and a substitution formula for Young integrals.We also study the Cauchy problem for first-order semilinear differential equations driven by a Hölder path via the Method of Characteristics. We also prove a version of the structure theorem of characteristics in the Young context. We conclude by showing the explicit solution of a transport equation perturbed by the H-fractional Brownian motion with Hurst parameter 1/2 <H ≤ 1.

Exponential stability and synchronisation of fuzzy Mittag–Leffler discrete-time Cohen–Grossberg neural networks with time delays

Exponential Euler differences for semi-linear differential equations of first order have got rapid development in the past few years and a variety of exponential Euler difference methods have become very significant researching topics. Therefore, exponential Euler differences for fractional-order differential models are novel and significant. In allusion to fuzzy Cohen–Grossberg neural networks of fractional order, this paper establishes a Mittag–Leffler Euler difference model on the basis of the constant variation formula in fractional calculus. In the second place, the existence of a unique global bounded solution and equilibrium point is studied by using the contractive mapping principle. Thirdly, global exponential stability and synchronisation of the derived difference models are investigated by employing some inequality techniques and reductio. At last, some illustrative examples and numerical simulations are employed to demonstrate the main results of this paper.