Continuation Theorem Research Articles

Global Asymptotic Stability of Periodic Solutions for Neutral-Type BAM Neural Networks with Delays

In this paper, we study the neutral-type BAM neural networks with time-varying delays. By applying the continuation theorem and some analysis techniques, some sufficient conditions to guarantee the neutral-type BAM neural networks have at least one periodic solution are proposed. Moreover, we also consider the asymptotic behaviours of periodic solutions by Lyapunov function and inequality $$2ab\le a^{2}+b^{2}$$. At last, an example is given to illustrate the effectiveness and feasibility of the obtain results.

Existence and global exponential stability of anti-periodic solutions for generalised inertial competitive neural networks with time-varying delays

ABSTRACTIn this paper, a class of generalised inertial competitive neural networks with time-varying delays is proposed. The existence and global exponential stability of anti-periodic solutions for this class of neural networks are investigated. By using a continuation theorem of coincidence degree theory, the Wirtinger inequality and constructing an appropriate Lyapunov function, some sufficient conditions are derived to guarantee the existence, uniqueness, and global exponential stability of anti-periodic solutions for the considered networks. Our results are completely new.

Continuation theorems for Mönch countable compactness-type set-valued maps

A number of Leray–Schauder-type continuation theorems for general classes of maps are presented in this paper. Our maps will satisfy a countable compactness-type condition.

Periodic solutions for a singular Liénard equation with indefinite weight

In this paper, the existence of positive periodic solutions is studied for a singular Liénard equation where the weight function has an indefinite sign. Due to the lack of a priori estimates over the set of all possible positive periodic solutions in this equation, a new method is proposed for estimating a priori bounds of positive periodic solutions. By the use of a continuation theorem of the Mawhin coincidence degree, new conditions for existence of positive periodic solutions to the equation are obtained. The main results show that the singularity of coefficient function associated to the friction term at $x=0$ may help the existence of periodic solutions.

On some of convergence domains of multidimensional S-fractions with independent variables

The convergence of multidimensional S-fractions with independent variables is investigated using the multidimensional generalization of the classical Worpitzky's criterion of convergence, the criterion of convergence of the branched continued fractions with independent variables, whose partial quotients are of the form $\frac{q_{i(k)}^{i_k}q_{i(k-1)}^{i_k-1}(1-q_{i(k-1)})z_{i(k)}}{1}$, and the convergence continuation theorem to extend the convergence, already known for a small domain (open connected set), to a larger domain. It is shown that the union of the intersections of the parabolic and circular domains is the domain of convergence of the multidimensional S-fraction with independent variables, and that the union of parabolic domains is the domain of convergence of the branched continued fraction with independent variables, reciprocal to it.

Weighted self-avoiding walks

We study the connective constants of weighted self-avoiding walks (SAWs) on infinite graphs and groups. The main focus is upon weighted SAWs on finitely generated, virtually indicable groups. Such groups possess so-called height functions, and this permits the study of SAWs with the special property of being bridges. The group structure is relevant in the interaction between the height function and the weight function. The main difficulties arise when the support of the weight function is unbounded, since the corresponding graph is no longer locally finite. There are two principal results, of which the first is a condition under which the weighted connective constant and the weighted bridge constant are equal. When the weight function has unbounded support, we work with a generalized notion of the ‘length’ of a walk, which is subject to a certain condition. In the second main result, the above equality is used to prove a continuity theorem for connective constants on the space of weight functions endowed with a suitable distance function.

Forward and Backward Mean-Field Stochastic Partial Differential Equation and Optimal Control

This paper is mainly concerned with the solutions to both forward and backward mean-field stochastic partial differential equation and the corresponding optimal control problem for mean-field stochastic partial differential equation. The authors first prove the continuous dependence theorems of forward and backward mean-field stochastic partial differential equations and show the existence and uniqueness of solutions to them. Then they establish necessary and sufficient optimality conditions of the control problem in the form of Pontryagin’s maximum principles. To illustrate the theoretical results, the authors apply stochastic maximum principles to study the infinite-dimensional linear-quadratic control problem of mean-field type. Further, an application to a Cauchy problem for a controlled stochastic linear PDE of mean-field type is studied.

On Caputo–Hadamard fractional differential equations

ABSTRACTIn this paper, the existence and uniqueness of solution to Caputo–Hadamard fractional differential equation (FDE) are studied. The continuation theorem is established too. Then, Euler and predictor–corrector methods are built up to solve Caputo–Hadamard FDE. The stability and error analysis of the derived numerical schemes are investigated as well. At last, a numerical example is carried out to verify the numerical algorithm.

Dynamics of two-species delayed competitive stage-structured model described by differential-difference equations

Abstract Overf the last few years, by utilizing Mawhin’s continuation theorem of coincidence degree theory and Lyapunov functional, many scholars have been concerned with the global asymptotical stability of positive periodic solutions for the non-linear ecosystems. In the real world, almost periodicity is usually more realistic and more general than periodicity, but there are scarcely any papers concerning the issue of the global asymptotical stability of positive almost periodic solutions of non-linear ecosystems. In this paper we consider a kind of delayed two-species competitive model with stage structure. By means of Mawhin’s continuation theorem of coincidence degree theory, some sufficient conditions are obtained for the existence of at least one positive almost periodic solutions for the above model with nonnegative coefficients. Furthermore, the global asymptotical stability of positive almost periodic solution of the model is also studied. The work of this paper extends and improves some results in recent years. An example and simulations are employed to illustrate the main results of this paper.

New result of existence of periodic solutions for a generalized p-Laplacian Liénard type differential equation with a variable delay

In this paper, we propose a generalized [Formula: see text]-Laplacian Liénard type differential equation with a variable delay. By applying the Mawhin continuation theorem, we established a set of sufficient conditions on the existence of at least one periodic solution with period [Formula: see text]. It is significant that the growth degree with respect to the variables [Formula: see text] imposed on [Formula: see text] is allowed to be greater than [Formula: see text] and the growth degree with respect to the variables [Formula: see text] imposed on [Formula: see text] is allowed to be greater than [Formula: see text] so the result not only improves but also generalizes. Some examples are provided to illustrate the results.

A topological degree approach to a nonlocal Neumann problem for a system at resonance

In this paper, we investigate the existence of solutions for a system of second-order differential equations with two nonlocal Neumann boundary conditions at resonance. The existence results are obtained by applying the Mawhin continuation theorem, where the required a priori estimates are derived using some conditions of the Nirenberg type.

Autonomous evolution of electron speeds in a thermostatted system: exact results

We investigate a dynamical system consisting of N particles moving on a d-dimensional torus under the action of an electric field E with a Gaussian thermostat to keep the total energy constant. The particles are also subject to stochastic collisions which randomize direction but do not change the speed. We prove that in the van Hove scaling limit, and , the trajectory of the speeds is described by a stochastic differential equation corresponding to diffusion on a constant energy sphere. This verifies previously conjectured behavior.Our results are based on splitting the system’s evolution into a ‘slow’ process and an independent ‘noise’. We show that the noise, suitably rescaled, converges to a Brownian motion, enhanced in the sense of rough paths. Then we employ the Itô–Lyons continuity theorem to identify the limit of the slow process.

Positive periodic solution for indefinite singular Li\xe9nard equation with p-Laplacian

The efficient conditions guaranteeing the existence of positive T-periodic solution to the p-Laplacian–Liénard equation \t\t\t(ϕp(x′(t)))′+f(x(t))x′(t)+α1(t)g(x(t))=α2(t)xμ(t),\\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}$$\\bigl(\\phi _{p}\\bigl(x'(t)\\bigr) \\bigr)'+f \\bigl(x(t)\\bigr)x'(t)+\\alpha _{1}(t)g\\bigl(x(t)\\bigr)= \\frac{ \\alpha _{2}(t)}{x^{\\mu }(t)}, $$\\end{document} are established in this paper. Here phi _{p}(s)=|s|^{p-2}s, p>1, alpha _{1},alpha _{2}in L([0,T],{R}) , fin C({R}_{+},{R}) ({R} _{+} stands for positive real numbers) with a singularity at x=0, g(x) is continuous on (0;+infty ), μ is a constant with mu >0, the signs of alpha _{1} and alpha _{2} are allowed to change. The approach is based on the continuation theorem for p-Laplacian-like nonlinear systems obtained by Manásevich and Mawhin in (J. Differ. Equ. 145:367–393, 1998).

The existence of solutions for mixed fractional resonant boundary value problem with p(t)-Laplacian operator

In this paper, we consider the following mixed fractional resonant boundary value problem with p(t)-Laplacian operator $$\begin{aligned} \left\{ \begin{array}{lll} ^{C} D^{\beta }_{0^{+}}\varphi _{p(t)}(D^{\alpha }_{0^{+}}u(t))=f(t, u(t), D^{\alpha }_{0^{+}}u(t)),~~t\in [0, T],\\ t^{1-\alpha }u(t)\mid _{t=0}=0,~~D^{\alpha }_{0^{+}}u(0)=D^{\alpha }_{0^{+}}u(T), \end{array}\right. \end{aligned}$$ where \(^{C} D^{\beta }_{0^{+}}\) is Caputo fractional derivative, \(D^{\alpha }_{0^{+}}\) is Riemann–Liouville fractional derivative, \(\varphi _{p(t)}\) is p(t)-Laplacian operator, \(p(t)>1\), \(p(t)\in C^{1}[0, T]\) with \(p(0)=p(T)\). Under the appropriate conditions of the nonlinear term, the existence of solutions for the above mixed fractional resonant boundary value problem is obtained by using the continuation theorem of coincidence degree theory, which enrich the existing literatures. In addition, an example is included to demonstrate the main result.

Formalization of functional variation in HOL Light

Functional variation is an important fundamental mathematical theory widely used in engineering analysis, especially in dynamics based design. However, its formalization still remains unexplored and this makes the verification of dynamic properties of engineering structures quite cumbersome. The present paper aims at initiating the formalization of this theory. For this purpose, the function-space theory is first formalized and then it is formally proved that this continuous function space is a Banach space. Then, linear functionals based on this function space are formalized. Further, the functional variation in the Fréchet derivative form and its main properties are formalized. Then the necessary condition of functional extrema is formally verified. Lastly, we formally verify the continuous functional and mean value theorem of functional variation for demonstrating the application of the formalized functional variation. The present formalization work can serve as a foundation for the formal verification of dynamic behavior of mechanical systems.

A mechanism for holography for non-interacting fields on anti-de Sitter spacetimes

In the setting of non-interacting Klein–Gordon fields on asymptotically anti-de Sitter spacetimes, we show that algebras of observables localized in a neighborhood of the boundary are subalgebras of a boundary algebra. The underlying mechanism is directly related to holography for classical fields. In particular, the proof relies on unique continuation theorems at the conformal boundary.

Periodic solution for ϕ-Laplacian neutral differential equation

Abstract This paper is devoted to the existence of a periodic solution for ϕ-Laplacian neutral differential equation as follows $$\begin{array}{} (\phi(x(t)-cx(t-\tau))')'=f(t,x(t),x'(t)). \end{array}$$ By applications of an extension of Mawhin’s continuous theorem due to Ge and Ren, we obtain that given equation has at least one periodic solution. Meanwhile, the approaches to estimate a priori bounds of periodic solutions are different from the corresponding ones of the known literature.

Continuity, positivity and simplicity of the Lyapunov exponents for quasi-periodic cocycles

An analytic quasi-periodic cocycle is a linear cocycle over a fixed ergodic torus translation of one or several variables, where the fiber action depends analytically on the base point. Consider the space of all such cocycles of any given dimension and endow it with the uniform norm. Assume that the translation vector satisfies a generic Diophantine condition. We establish large deviation type estimates for the iterates of such cocycles, which, moreover, are stable under small perturbations of the cocycle. As a consequence of these uniform estimates, we obtain continuity properties of the Lyapunov exponents regarded as functions on this space of cocycles. This result builds upon our previous work on this topic and its proof uses an abstract continuity theorem for the Lyapunov exponents which we derived in a recent monograph. The new feature of this paper is extending the availability of such results to cocycles that are identically singular (i.e. non-invertible everywhere), in the several variables torus translation setting. This feature is exactly what allows us, through a simple limiting argument, to obtain criteria for the positivity and simplicity of the Lyapunov exponents of such cocycles. Specializing to the family of cocycles corresponding to a block Jacobi operator, we derive consequences on the continuity, positivity and simplicity of its Lyapunov exponents, and on the continuity of its integrated density of states.

Existence of positive periodic solutions for a neutral Liénard equation with a singularity of repulsive type

The periodic problem is studied in this paper for the neutral Lienard equation with a singularity of repulsive type $$\begin{aligned} (x(t)-cx(t-\sigma ))''+f(x(t))x'(t)+\varphi (t)x(t-\tau )-\frac{r(t)}{x^{\mu }(t)}=h(t), \end{aligned}$$ where $$f:[0,+\infty )\rightarrow R$$ is continuous, $$r: R\rightarrow (0,+\infty )$$ and $$\varphi :R \rightarrow R$$ are continuous with T-periodicity in the t variable, $$c,\mu ,\sigma ,\tau $$ are constants with $$|c|>1,\mu >1,0<\sigma ,\tau <T$$ . Many authors obtained the existence of periodic solutions under the condition $$|c|<1$$ , and we extend their results to the case of $$|c|>1$$ . The proof of the main result relies on a continuation theorem of coincidence degree theory established by Mawhin.

Unique continuation theorems for biharmonic maps.

We prove several unique continuation results for biharmonic maps between Riemannian manifolds.