Continuation Theorem Research Articles

Global exponential stability of positive almost periodic solutions for a class of two-layer Gilpin\u2013Ayala predator\u2013prey model with time delays

This paper mainly considers a class of two-layer Gilpin–Ayala predator–prey models with time delays. By means of Mawhin’s continuation theorem of coincidence degree theory, some new sufficient criteria for the existence of positive almost periodic solutions have been established. We also obtain the global exponential stability of the positive almost periodic solution for this system by constructing appropriate Lyapunov functionals and smart transformations. As an application, an example is given to illustrate the validity of our main results.

Mathematical model of fish, birds and tourists in wetlands: the impact of periodic fluctuations on the coexistence of species

The present paper deals with a harvested predator–prey model and the dynamics of tourists in wetlands. This model describes the coexistence and extinction of at least one of the species in the wetlands. By the Mahwin continuation theorem of coincidence degree theory, we prove the existence of a positive periodic solution (in place of an equilibrium) provided that the harvest rate of prey is in a suitable range. Further, by constructing a suitable Lyapunov functional, we obtain sufficient conditions for global asymptotic stability of the positive periodic solution. Finally numerical simulations are carried out to illustrate the feasibility of the mathematical results.

Existence of solutions for fractional differential equations with infinite point boundary conditions at resonance

In this paper, by using Mawhin’s continuation theorem, we establish some sufficient conditions for the existence of at least one solution for a class of fractional infinite point boundary value problem at resonance. Moreover, an example is given to illustrate our results.

Positive periodic solution for p-Laplacian neutral Rayleigh equation with singularity of attractive type

In this paper, we consider a kind of p-Laplacian neutral Rayleigh equation with singularity of attractive type, \t\t\t(ϕp(u(t)−cu(t−δ))′)′+f(t,u′(t))+g(t,u(t))=e(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}$$\\bigl(\\phi_{p} \\bigl(u(t)-cu(t-\\delta) \\bigr)' \\bigr)'+f \\bigl(t,u'(t) \\bigr)+g \\bigl(t, u(t) \\bigr)=e(t). $$\\end{document} By applications of an extension of Mawhin’s continuation theorem, sufficient conditions for the existence of periodic solution are established.

Continuity theorems for a class of convolution operators and applications to superoscillations

We prove a new theorem on the continuity of convolution operators with variable coefficients, and we use it to deduce that the limit of a superoscillating sequence maintains the superoscillatory behaviour for all values of time, when evolved according to Schrodinger equations with time-dependent potentials.

Study on the Influence of Acoustically Inhomogeneous Characteristics on Magneto-acoustic-electrical Tomography

Magneto-acoustic-electrical tomography (MAET) is a method of non-invasive conductivity imaging, with three couplings of magnetic field, acoustic field and electric field. It is suitable for the conductivity imaging of low-conductivity fluid or approximate fluid medium. MAET will have a broad application prospect in the field of bio-impedance imaging. Acoustic and electrical characteristics of biological tissue are important factors in magneto-acoustic-electrical tomography, it is essential to evaluate the effect of acoustic inhomogeneity on the sound pressure distribution and the electrical signal. In this paper, a numerical analysis of the positive problem of magneto-acoustic -electrical tomography of eight groups of human tissues with different density and sound velocity, such as blood and cerebrospinal fluid, is carried out. Firstly, establishing a two-dimensional plane model of pure fluid and applying ultrasonic excitation. Secondly, combining with the current continuity theorem and getting the target body potential distribution. Finally, the voltage signal can be calculated. The simulation results show that compared to the acoustic parameter value usually used in the approximate calculation, the voltage signal error of the muscle tissue is the largest of the seven groups of parameters under the same condition. The results of the study laid the foundation for the study of the influence of the acoustically inhomogeneous model of human tissue on the magneto–acoustic -electrical tomography.

Existence and global exponential stability of periodic solutions for quaternion-valued cellular neural networks with time-varying delays

Abstract In this paper, a class of quaternion-valued cellular neural networks (QVCNNs) with time-varying delays is considered. By using the continuation theorem of Mawhin’s coincidence degree theory, the existence of periodic solutions for QVCNNs is obtained. By constructing a suitable Lyapunov function, some sufficient conditions are derived to guarantee the global exponential stability of periodic solutions for QVCNNs. Finally, two examples are given to illustrate the effectiveness of the obtained results.

Global asymptotic stability of periodic solutions for delayed complex-valued Cohen–Grossberg neural networks by combining coincidence degree theory with LMI method

The paper is concerned with the existence and global asymptotic stability of periodic solutions for a class of delayed complex-valued Cohen–Grossberg neural networks. Without using the method of the a priori estimate of periodic solutions, by combining Mawhin’s continuation theorem of coincidence degree theory with LMI method and using inequality techniques, a novel LMI-based sufficient condition on the existence of periodic solutions is established for the complex-valued Cohen–Grossberg neural networks. Then by using inequality techniques, a novel sufficient condition on the global asymptotic stability of periodic solutions for the above complex-valued neural networks is established. Our results and method are new and complementary to the existing papers on the study of periodic solutions of neural networks.

Periodic solution for p-Laplacian Rayleigh equation with attractive singularity and time-dependent deviating argument

In this paper, we consider a p-Laplacian singular Rayleigh equation with time-dependent deviating argument \t\t\t(φp(x′(t)))′+f(t,x′(t))+g(t,x(t−σ(t)))=e(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}$$\\bigl(\\varphi_{p}\\bigl(x'(t)\\bigr)\\bigr)'+f \\bigl(t,x'(t)\\bigr)+g\\bigl(t,x\\bigl(t-\\sigma(t)\\bigr)\\bigr)=e(t), $$\\end{document} where g has an attractive singularity at x=0. Using the Manásevich–Mawhin continuation theorem, we prove that the equation has at least one T-periodic solution.

Existence of solutions for fractional multi-point boundary value problems at resonance with three-dimensional kernels

In this paper, by using Mawhin’s continuation theorem, we investigate the existence of solutions for a class of fractional differential equations with multi-point boundary value problems at resonance, and the dimension of the kernel for a fractional differential operator is three. An example is given to show our main result.

On a system of fuzzy fractional differential inclusions with projection operators

The aim of this paper is to introduce and study a class of fuzzy fractional differential inclusions with projection operators, which captures the desired features of both fuzzy differential inclusions and fractional-order projective dynamical systems within the same framework. The existence of solutions for the open situation is proved by using continuous selection theorem. Moreover, the existence of solutions for the closed situation is showed by employing Lipschitzean selection theorem and fixed point theorem, respectively. Furthermore, an application and a numerical example are also given.

Existence and space-time regularity for stochastic heat equations on p.c.f. fractals

We define linear stochastic heat equations (SHE) on p.c.f.s.s. sets equipped with regular harmonic structures. We show that if the spectral dimension of the set is less than two, then function-valued "random-field" solutions to these SPDEs exist and are jointly H\"older continuous in space and time. We calculate the respective H\"older exponents, which extend the well-known results on the H\"older exponents of the solution to SHE on the unit interval. This shows that the "curse of dimensionality" of the SHE on $\mathbb{R}^n$ depends not on the geometric dimension of the ambient space but on the analytic properties of the operator through the spectral dimension. To prove these results we establish generic continuity theorems for stochastic processes indexed by these p.c.f.s.s. sets that are analogous to Kolmogorov's continuity theorem. We also investigate the long-time behaviour of the solutions to the fractal SHEs.

Short proofs in extrema of spectrally one sided Lévy processes

We provide short and simple proofs of the continuous time ballot theorem for processes with cyclically interchangeable increments and Kendall’s identity for spectrally positive Levy processes. We obtain the later result as a direct consequence of the former. The ballot theorem is extended to processes having possible negative jumps. Then we prove through straightforward arguments based on the law of bridges and Kendall’s identity, Theorem 2.4 in [20] which gives an expression for the law of the supremum of spectrally positive Levy processes. An analogous formula is obtained for the supremum of spectrally negative Levy processes.

Periodic solutions for indefinite singular equations with singularities in the spatial variable and non-monotone nonlinearity

TWe prove the existence of $T$-periodic solutions for the second order non-linear equation $ {\left( {\frac{{u'}}{{\sqrt {1 - {{u'}^2}} }}} \right)^\prime } = h(t)g(u), $ where the non-linear term $g$ has two singularities and the weight function $h$ changes sign. We find a relation between the degeneracy of the zeroes of the weight function and the order of one of the singularities of the non-linear term. The proof is based on the classical Leray-Schauder continuation theorem. Some applications to important mathematical models are presented.

Currents carried by the subgradient graphs of semi-convex functions and applications to Hessian measures

In this paper we study integer multiplicity rectifiable currents carried by the subgradient (subdifferential) graphs of semi-convex functions on an n-dimensional convex domain, and show a weak continuity theorem with respect to pointwise convergence for such currents. As an application, the structure theorem of the Lagrangian currents for semi-convex functions is given and the k-Hessian measures are calculated by a different method in terms of currents.

A new result on the existence of periodic solutions for Rayleigh equation with a singularity

In this paper, we study the existence of periodic solutions for Rayleigh equation with a singularity of repulsive type \t\t\tx″(t)+f(x′(t))+φ(t)x(t)−1xα(t)=p(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}$$x''(t)+f\\bigl(x'(t)\\bigr)+\\varphi (t)x(t)-\\frac{1}{x^{\\alpha }(t)}=p(t), $$\\end{document} where alpha geqslant 1 is a constant, and φ and p are T-periodic functions. The proof of the main result relies on a known continuation theorem of coincidence degree theory. The interesting point is that the sign of the function varphi (t) is allowed to change for tin [0,T].

Global Asymptotic Periodic Synchronization for Delayed Complex-Valued BAM Neural Networks via Vector-Valued Inequality Techniques

In this paper, we are concerned with a class of delayed complex-valued BAM neural networks. In stead of using the priori estimate method of periodic solutions, by means of combining Mawhin’s continuation theorem of coincidence degree theory with novel LMI method and some analysis techniques, a novel LMI-based sufficient condition is obtained for the existence of periodic solutions of the delayed complex-valued BAM neural networks. Then by using novel LMI method, a novel sufficient condition on global asymptotic periodic synchronization of above complex-valued BAM neural networks is established.

Lindeberg theorem for Gibbs–Markov dynamics

A dynamical array consists of a family of functions and a family of initial times . For a dynamical system we identify distributional limits for sums of the formfor suitable (non-random) constants and . We derive a Lindeberg-type central limit theorem for dynamical arrays. Applications include new central limit theorems for functions which are not locally Lipschitz continuous and central limit theorems for statistical functions of time series obtained from Gibbs–Markov systems. Our results, which hold for more general dynamics, are stated in the context of Gibbs–Markov dynamical systems for convenience.

Global existence of solutions for a fractional Caputo nonlocal thermistor problem

We begin by proving a local existence result for a fractional Caputo nonlocal thermistor problem. Then additional existence and continuation theorems are obtained, ensuring global existence of solutions.

Solitary Wave and Periodic Wave Solutions of Generalized Neutral-Type Neural Networks with Delays

In this paper, we are concerned with the generalized neutral-type neural networks with delays. By employing the continuation theorem and some analysis techniques, several sufficient conditions of the existence of solitary wave and periodic wave solutions for the considered neural networks are proposed. Moreover, we also consider the asymptotic behavior results of periodic wave solutions via the Lyapunov function. Finally, a numerical example is given to illustrate the effectiveness of our obtained results.