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Our main goal is to explore the existence of positive solutions for a class of nonlinear fractional Schrödinger equation involving supercritical growth given by $$ (- \Delta)^{\alpha} u + V(x)u=p(u),\quad x\in \mathbb{R^N},\ N \geq 1. $$ We analyze two types of problems, with $V$ being periodic and asymptotically periodic; for this we use a variational method, a truncation argument and a concentration compactness principle.

In this article, we address the delay-dependent robust stability of uncertain fractional order neutral-type (FONT) systems with distributed delays, nonlinear perturbations, and input saturation. With the aid of the Lyapunov–Krasovskii functional, criteria on asymptotic robust stability of FONT, expressed in terms of linear matrix inequalities, are constructed to compute the state-feedback controller gains. The controller gains are determined subject to maximizing the domain of attraction via the cone complementarity linearization algorithm. The theoretical results are validated using numerical simulations.

In this paper, we study the existence of solutions for integrodifferential Schrodinger equations of the form −LKu+V(x)u=f(x,u)in R, where −LK is a nonlocal operator with a measurable kernel which s...

We study the behavior of an exponential dichotomy for a nonautonomous linear delay equation under sufficiently small linear and nonlinear perturbations. In particular, we establish a partial version of the Grobman–Hartman theorem and we use the characterization of an exponential dichotomy in terms of an admissibility property to establish the robustness property.

This paper addresses the problem of distributed simultaneous fault detection and control (SFDC) of multi-weighted and multi-delayed (MWMD) large-scale interconnected systems which are subjected to event-triggered communication, nonlinear perturbations, measured output quantization, redundant channels, and stochastic deception attacks. The large-scale systems under consideration have multiple coupling links between neighboring subsystems, and all the links are considered to have different coupling weights and delays. A distributed fault detector and controller (FDC) module is designed to guarantee the exponential mean square stability of the overall closed-loop system along with a prescribed extended dissipative control performance and H ∞ fault detection performance. The gain parameters of the distributed fault detector and controller module are determined by using the cone complementarity linearization (CCL) algorithm. In the end, a numerical example involving a continuous stirred tank reactor (CSTR) system is described to prove the validity of the presented results.

In this paper, we aim at addressing the problem of robust quantized control for singular system with time delay, actuator saturation and nonlinear perturbations. Based on Lyapunov Krasovskii approach, a novel delay-dependent sufficient condition is given to guarantee the corresponding system is admissible in terms of Linear Matrix Inequalities (LMIs). The submitted results are derived by exploiting three integral inequalities. A robust quantized controller are further developed as well as the domain of attraction. Finally, examples are utilized to demonstrate the merit of the proposed results.

We study finite-time stability (FTS) for discrete-time systems with time-varying delay in this paper. Effect of nonlinear perturbations is also studied. Novel weighted summation inequalities, which generalize the discrete Jensen-based inequality, are established. These inequalities are used to analyze a Lyapunov-Krasovskii functional with a certain weight. New FTS criteria are derived to guarantee that the state of a system stays within a given threshold on a given interval. Numerical examples show that the new criteria can yield better results than others given in previous works.

This paper is concerned with the robust stability and stabilization problems of discrete-time singularly perturbed systems (DTSPSs) with nonlinear perturbations. A proper sufficient condition via the fixed-point principle is proposed to guarantee that the given system is in a standard form. Then, based on the singular perturbation approach, a linear matrix inequality (LMI) based sufficient condition is presented such that the original system is standard and input-to-state stable (ISS) simultaneously. Thus, it can be easily verified for it only depends on the solution of an LMI. After that, for the case where the nominal system is unstable, the problem of designing a control law to make the resulting closed-loop system ISS is addressed. To achieve this, a sufficient condition is proposed via LMI techniques for the purpose of implementation. The criteria presented in this paper are independent of the small parameter and the stability bound can be derived effectively by solving an optimal problem. Finally, the effectiveness of the obtained theoretical results is illustrated by two numerical examples.

Background and objective: The stability problem in double delay differential diabetes system is affected by delay term and meal perturbation. This paper is addressed to assess the stability of such double diabetes system with unknown nonlinear perturbation and bounded time varying delay. Method: The stability of a double diabetes system following a meal function as an unknown nonlinear perturbation function has been analyzed by using a Lyapunov–Krasovskii function. The stability matrices according to linear matrix inequality (LMI) were constructed and solved using YALMIP. Results: The solution of the stability theorem gives semi definite matrices which fulfilled the stability criteria. Simulations observed such oscillation in the early meal intake. Conclusion: Thus, it shows that for given values (30–51 min delay and unknown time varying perturbation) the double diabetes system is stable.

This paper focuses on the problem of h-stability of certain classes of dynamic perturbed systems on time scales using time scale versions of some Gronwall type inequalities. We prove under certain conditions on the nonlinear perturbations that the resulting perturbed nonlinear initial value problem still acquire h-stable, if the associated abstract dynamic equation has already owned this property. The paper ends up with two illustrative examples to highlight the utility of our results.

This paper investigates the problem of finite-time extended dissipative control for T–S fuzzy time-varying delay systems with nonlinear perturbations via sampled-data and quantized controller. The definition of finite-time bounded mixed extended dissipative of fuzzy systems is first proposed. Based on the constructed Lyapunov–Krasovskii functional(LKF) and Peng–Parks integral inequality, some sufficient conditions are obtained in the form of linear matrix inequalities(LMIs), which are less conservative than other papers. By combining the input delay approach and dynamic quantizer, the sampled-data and quantized controller is designed to guarantee that the T–S fuzzy system is finite-time bounded mixed extended dissipative. Finally, some numerical examples and practical examples are presented to verify the feasibility and effectiveness of the proposed methods.

The long time effect of nonlinear perturbation to oscillatory linear systems can be characterized by the averaging method, and we consider first-order averaging for its simplest applicability to high-dimensional problems. Instead of the classical approach, in which one uses the pullback of linear flow to isolate slow variables and then approximate the effective dynamics by averaging, we propose an alternative coordinate transform that better approximates the mean of oscillations. This leads to a simple improvement of the averaged system, which will be shown both theoretically and numerically to provide a more accurate approximation. Three examples are then provided: in the first, a new device for wireless energy transfer modeled by two coupled oscillators was analyzed, and the results provide design guidance and performance quantification for the device; the second is a classical coupled oscillator problem (Fermi-Pasta-Ulam), for which we numerically observed improved accuracy beyond the theoretically justified timescale; the third is a nonlinearly perturbed first-order wave equation, which demonstrates the efficacy of improved averaging in an infinite dimensional setting., Comment: Comments are welcomed

New algorithms for construction of asymptotic expansions for exponential and power-exponential moments of hitting times for nonlinearly perturbed semi-Markov processes are presented. The algorithms ...

This paper formulates two stochastic nonautonomous SIRI epidemic systems with nonlinear perturbations. The main aim of this study is to investigate stochastic dynamics of the two SIRI epidemic systems and obtain their thresholds. For the nonautonomous stochastic SIRI epidemic system with white noise, the authors provide analytic results regarding the stochastic boundedness, stochastic permanence and persistence in mean. Moreover, the authors prove that the system has at least one nontrivial positive T-periodic solution by using Lyapunov function and Hasminskii’s theory. For the system with Markov conversion, the authors establish sufficient conditions for positive recurrence and existence of ergodic stationary distribution. In addition, sufficient conditions for the extinction of disease are obtained. Finally, numerical simulations are introduced to illustrate the main results.

In this paper, we study the following nonlinear Schrodinger equation \begin{document}$ \begin{eqnarray} \sqrt{-1}u_{t}-u_{xx}+V*u+\epsilon f(x)|u|^4u = 0, \ x\in\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}, ~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)\end{eqnarray} $\end{document} where \begin{document}$ V* $\end{document} is the Fourier multiplier defined by \begin{document}$ \widehat{(V* u})_n = V_{n}\widehat{u}_n, V_n\in[-1, 1] $\end{document} and \begin{document}$ f(x) $\end{document} is Gevrey smooth. It is shown that for \begin{document}$ 0\leq|\epsilon|\ll1 $\end{document} , there is some \begin{document}$ (V_n)_{n\in\mathbb{Z}} $\end{document} such that, the equation (1) admits a time almost periodic solution (i.e., full dimensional KAM torus) in the Gevrey space. This extends results of Bourgain [ 7 ] and Cong-Liu-Shi-Yuan [ 8 ] to the case that the nonlinear perturbation depends explicitly on the space variable \begin{document}$ x $\end{document} . The main difficulty here is the absence of zero momentum of the equation.

We prove that small nonlinear perturbations of random linear dynamics admitting a tempered exponential dichotomy have a random version of the shadowing property. As a consequence, if the exponential dichotomy is uniform, we get that the random linear dynamics is Hyers-Ulam stable. Moreover, we apply our results to study the conservation of Lyapunov exponents of the random linear dynamics subjected to nonlinear perturbations., Comment: Revised version. To appear in Fundamenta Mathematicae

We formulate sufficient conditions under which a large class of semilinear nonautonomous difference equations with infinite delay is Hyers–Ulam stable. These conditions require that the nonautonomous linear part admits an exponential dichotomy and the nonlinear perturbations are uniformly Lipschitz continuous with a sufficiently small Lipschitz constant. In the more general case when the linear part admits a shifted exponential dichotomy, we are able to provide sufficient conditions for the existence of a certain weighted form of the shadowing property.

We study the inverse problem for the fractional Laplace equation with multiple nonlinear lower order terms. We show that the direct problem is well-posed and the inverse problem is uniquely solvable. More specifically, the unknown nonlinearities can be uniquely determined from exterior measurements under suitable settings., 17 pages

We establish the future stability of nonlinear perturbations of a class of homogeneous solutions to the relativistic Euler equations with a linear equation of state $p=K\rho$ on exponentially expanding FLRW spacetimes for the equation of state parameter values $1/3 < K < 1/2$., Comment: Comments typos corrected; agrees with the published version

A partially hyperbolic dynamical system is said to have the quasi-shadowing property if every pseudotrajectory can be shadowed by a sequence of points $(x_n)_{n\in \Z}$ such that $x_{n+1}$ is obtained from the image of $x_n$ by moving it by a small factor in the central direction. In the present paper, we prove that a small nonlinear perturbation of a partially dichotomic sequence of (not necessarily invertible) linear operators acting on an arbitrary Banach space has the quasi-shadowing property. We also get obtain a continuous time version of this result. As an application of our main result, we prove that a certain class of partially dichotomic sequences of linear operators is stable up to the movement in the central direction., Minor changes. Accepted for publication in Journal of Mathematical Analysis and Applications. arXiv admin note: text overlap with arXiv:1905.08251

The aim of this paper is to study a boundary value problem of the hybrid differential equation with linear and nonlinear perturbations. It generalizes the existing problem of second type. The existence result is constructed using the Leray–Schauder alternative, and the uniqueness is guaranteed by Banach’s fixed-point theorem. Towards the end of this paper, an example is provided to illustrate the obtained results.

We deal with the problem of identifying a source term in the Rayleigh-Stokes type equation with a nonlinear perturbation, where the nonlinearity may have a superlinear growth and the additional measurement is given at final time and depends on the state. Our aim is to prove the unique solvability and stability of solution. Furthermore, we show that the obtained solution is differentiable and it is the strong one.

In this paper, we propose a stochastic predator–prey model with hunting cooperation and nonlinear perturbation of white noise. The sufficient criteria for the existence of a unique ergodic stationary distribution are established by constructing suitable Lyapunov functions. It reveals that the white noise has a significant impact on the dynamical behavior of the model.

We study a class of nonlocal partial differential equations with nonlinear perturbations, which is a general model for some equations arose from fluid dynamics. Our aim is to analyze some sufficient conditions ensuring the global solvability, regularity and stability of solutions. Our analysis is based on the theory of completely positive kernel functions, local estimates and a new Gronwall type inequality.

We study the generalized Rayleigh-Stokes problem involving a fractional derivative and nonlinear perturbation. Our aim is to analyze some sufficient conditions ensuring the global solvability, regularity and asymptotic stability of solutions. In particular, if the nonlinearity is Lipschitzian then the mild solution of the mentioned problem becomes a classical one and its convergence to equilibrium point is proved.

We prove the existence of ground state solutions by variational methods to the nonlinear Choquard equations with a nonlinear perturbation \[ -{\Delta}u+ u=\big(I_\alpha*|u|^{\frac{\alpha}{N}+1}\big)|u|^{\frac{\alpha}{N}-1}u+f(x,u)\qquad \text{ in } \mathbb{R}^N \] where $N\geq 1$, $I_\alpha$ is the Riesz potential of order $\alpha \in (0, N)$, the exponent $\frac{\alpha}{N}+1$ is critical with respect to the Hardy--Littlewood--Sobolev inequality and the nonlinear perturbation $f$ satisfies suitable growth and structural assumptions., Comment: 18 pages

This paper investigates a new impulsive stochastic chemostat model with nonlinear perturbation in a polluted environment. We present the analysis and the criteria of the extinction of the microorganisms, and establish sufficient conditions for the existence of a unique ergodic stationary distribution of the model via Lyapunov functions method. The results show that both stochastic noise and impulsive toxicant input have great effects on the survival and extinction of the microorganisms. Moreover, we provide a series of numerical simulations to illustrate the analytical results.

In this paper, we formulate and analyze a stochastic predator–prey model with additional food and nonlinear perturbation. Firstly, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of an ergodic stationary distribution of the positive solution to the model. Then we obtain sufficient conditions for extinction of the predator species in two cases, one is the prey species surviving and the predator species extinction; the other is both the prey and predator species are extinct. Finally, some examples together with numerical simulations are provided to illustrate the analytical results.

Analytical properties of the well-known FitzHugh–Nagumo model are studied. It is shown that the standard FitzHugh–Nagumo model does not pass the Painleve test in the general case and does not have any meromorphic solutions because there are not any expansions of the general solution in the Laurent series. We demonstrate that the introduction of a nonlinear perturbation into the standard system of equations does not lead to the Painleve property as well. However, in this case there are expansions of the general solution of the system of equations in the Laurent series for some values of parameters. This allows us to look for some exact solutions of the system of the perturbed FitzHugh–Nagumo model. We find some exact solutions of the perturbed FitzHugh–Nagumo system of equations in the form of kinks. These exact solutions can be used for testing numerical simulations of the system of equations corresponding to the FitzHugh–Nagumo model.

This paper focuses on the problem of delay-dependent state feedback control for a class of networked control systems (NCSs) with nonlinear perturbations and two delay components. Based on the dynamic delay interval (DDI) method and the Wirtinger integral inequality, some improved delay-dependent stability analysis are obtained. Furthermore, the results are extended to the conditions of NCSs with one time delay, and the corresponding stability analysis results and state feedback controller are obtained. Finally, some numerical examples and simulations are given to show the effectiveness of the proposed methods.

In this paper, we analyze a stochastic SIR model with nonlinear perturbation. By the Lyapunov function method, we establish sufficient conditions for the existence of a unique ergodic stationary distribution of the model. Moreover, sufficient conditions for extinction of the disease are also obtained.

We prove the existence of weak solutions of variational inequalities for general quasilinear parabolic operators of order m = 2 with strongly nonlinear perturbation term. The result is based on a priori bound for the time derivatives of the solutions.

In this paper, we investigate the problem of exponential stability and passivity analysis of a class of switched systems with interval time-varying delays and nonlinear perturbations. By constructing an improved Lyapunov---Krasovskii functional combining with novel refined Jensen-based inequalities, some improved sufficient conditions for exponential stability are proposed for a class of switching signals with average dwell time. Moreover, a new sufficient condition for passivity analysis of switched continuous-time systems with an interval time-varying delay is also derived. These conditions are delay dependent and are given in the form of linear matrix inequalities, which therefore can be efficiently solved by existing convex algorithms. Lastly, four examples are provided to demonstrate the effectiveness of our results.

Summary This paper deals with the problem of delay-dependent H∞ exponential stability for neutral Markovian jump systems with mixed delays and nonlinear perturbations. Based on Lyapunov stability theory and linear matrix inequality method, some new H∞ exponential stability criteria are presented. The difference between this paper and other existing results is that the lower bounds of the neutral delay, the upper bounds of the neutral delay and discrete delay are considered, which will obtain some less conservative stability analysis results. Numerical examples are given to show that the proposed criteria improve the existing results.

This paper deals with the following Choquard equation with a local nonlinear perturbation: $$\begin{aligned} \left\{ \begin{array}{ll} - \Delta u+V(x)u=(I_{\alpha }*F(u))f(u)+g(u), &{}\quad x\in \mathbb {R}^N; \\ u\in H^1(\mathbb {R}^N), \end{array} \right. \end{aligned}$$where $$I_{\alpha }: \mathbb {R}^N\rightarrow \mathbb {R}$$ is the Riesz potential, $$N\ge 3$$, $$\alpha \in (0, N)$$, $$F(t)=\int _{0}^{t}f(s)\mathrm {d}s\ge 0\ (\not \equiv 0)$$, $$V\in {\mathcal {C}}^1(\mathbb {R}^N, [0, \infty ))$$ and $$f, g\in {\mathcal {C}}(\mathbb {R}, \mathbb {R})$$ satisfying the subcritical growth. Under some suitable conditions on V, we prove that the above problem admits ground state solutions without super-linear conditions near infinity or monotonicity properties on f and g. In particular, some new tricks are used to overcome the combined effects and the interaction of the nonlocal nonlinear term and the local nonlinear term. Our results improve and extends the previous related ones in the literature.

This paper is concerned with the problem of mean-square exponential stability for uncertain stochastic neutral systems with nonlinear perturbations and distributed time delays. By introducing an auxiliary vector, it is proved that Lyapunov-Krasovskii functional can be extended to stochastic distributed time-delay systems. Based on the linear matrix inequality (LMI) method, a new mean-square exponential stability criterion for time-delay correlation of stochastic neutral systems is obtained. Finally, the effectiveness of this method is proved by numerical examples.

In this paper, we consider the following magnetic nonlinear Choquard equation \[-(\nabla+iA(x))^2u+ V(x)u = \left(\frac{1}{|x|^{\alpha}}*|u|^{2_{\alpha}^*}\right) |u|^{2_{\alpha}^*-2} u + \lambda f(u)\ \textrm{ in }\ \R^N,\] where $2_{\alpha}^{*}=\frac{2N-\alpha}{N-2}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, $\lambda>0$, $N\geq 3$, $0, Comment: 21 pages

Despite the recent introduction of novel solution methods for Dynamic Stochastic General Equilibrium (DSGE), perturbation methods are still among the most popular and widely used solution techniques for DSGE models. Unfortunately, nonlinear perturbation solutions produce paths with stochastic properties that invalidate the econometric analysis. This paper proposes a correction that renders the econometric analysis valid and sound. The proposed correction is simple to implement in existing software packages such as Dynare, it does not add any significant computational effort and, as a result, does not impact computational times. The corrected solution retains the same approximation properties as standard higher-order perturbation methods and, in contrast to those methods, generates stable sample paths that are stationary, geometrically ergodic and absolutely regular. Additionally, moments are shown to be bounded. Transformed perturbation solutions are an alternative to the pruning method as proposed in Kim et al. (2008). The advantages of our approach are that, unlike pruning, we do not need to sacrifice accuracy around the steady-state by ignoring higher-order effects, and furthermore, we also deliver a policy function. Moreover, the newly proposed solution is always more accurate globally than standard perturbation methods. We demonstrate the superior accuracy of our method in a range of simple examples.

The problem of delay-range-dependent stability analysis for linear systems with distributed time-varying delays and nonlinear perturbations is studied without using the model transformation and delay-decomposition approach. The less conservative stability criteria are obtained for the systems by constructing a new augmented Lyapunov–Krasovskii functional and various inequalities, which are presented in terms of linear matrix inequalities (LMIs). Four numerical examples are demonstrated for the results given to illustrate the effectiveness and improvement over other methods.

The exponential stability problem of uncertain neutral complex system with Markovian switching is investigated in the presence of nonlinear perturbations and partial information on transition rates. The study begins to consider the related nominal systems and construct a novel augmented stochastic Lyapunov functional which contains some triple-integral terms to reduce the conservatism. Then the exponential stability criteria are developed by utilizing Lyapunov stability theory, reciprocally convex lemma and free-weighting matrices. The results are further extended to the corresponding uncertain case. Finally, numerical examples are given to illustrate the effectiveness of the proposed methods.

In this paper, we proposes and analyzes the mixed 4th-order Runge-Kutta scheme of conditional nonlinear perturbation (CNOP) approach for the EI Niño-Southern Oscillation (ENSO) model. This method consists of solving the ENSO model by using a mixed 4th-order Runge-Kutta method. Convergence, the local and global truncation error of this mixed 4th-order Runge-Kutta method are proved. Furthermore, optimal control problem is developed and the gradient of the cost function is determined.

We consider a nonlinear system of equations governing the motion of a viscosity-layered fluid with a free surface in long-wave approximation. Using the semi-Lagrangian coordinates, we rewrite the governing equations in the integro-differential form and obtain necessary and sufficient hyperbolicity conditions. We approximate the integro-differential model by a finite-dimensional system of differential conservation laws and propose a model of propagation of nonlinear perturbations in a viscosity-stratified fluid.

In this paper, we study the solitary wave solution and numerical simulation for the generalized Rosenau–Kawahara-RLW equation with generalized Novikov type nonlinear perturbation, which is an extension of our recent work He and Pan (Appl Math Comput 271:323–336, 2015), He (Nonlinear Dyn 82:1177–1190, 2015). We first derive the exact solitary wave solution for the newly proposed perturbed Rosenau–Kawahara-RLW equation with power law nonlinearity and then develop a three-level linearly implicit difference scheme for solving the equation. We prove that the proposed scheme is energy-conserved, unconditionally stable and second-order convergent both in time and space variables. Finally, numerical experiments are carried out to confirm the energy conservation, the convergence rates of the scheme and effectiveness for long-time simulation.

In this paper, the problem of finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations is investigated. By constructing a novel Lyapunov-Krasovskii functional and employing a new summation inequality named discrete Wirtinger-based inequality, reciprocally convex approach and zero equality, the improved finite-time stability criteria are derived to guarantee that the state of the system with time-varying delay does not exceed a given threshold when fixed time interval. Furthermore, the obtained conditions are formulated in forms of linear matrix inequalities which can be solved by using some standard numerical packages. Finally, three numerical examples are given to show the effectiveness and less conservatism of the proposed method.

This paper deals with the following Choquard equation with a local nonlinear perturbation: − Δ u + u = I α ∗ | u | α N + 1 | u | α N − 1 u + f ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , where I α : R N → R is the Riesz potential, N ≥ 3 , α ∈ ( 0 , N ) , the exponent α N + 1 is critical with respect to the Hardy–Littlewood–Sobolev inequality, and the nonlinear perturbation f is only required to satisfy some weak assumptions near 0 and ∞ . Our results improve the previous related ones in the literature.

The novel delay-dependent asymptotic stability of a differential and Riemann-Liouville fractional differential neutral system with constant delays and nonlinear perturbation is studied. We describe the new asymptotic stability criterion in the form of linear matrix inequalities (LMIs), using the application of zero equations, model transformation and other inequalities. Then we show the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with constant delays. Furthermore, we not only present the improved delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with single constant delay but also the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral equation with constant delays. Numerical examples are exploited to represent the improvement and capability of results over another research as compared with the least upper bounds of delay and nonlinear perturbation.