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Abstract: Abstract In order to unify the study of Besicovitch almost periodic solutions of continuous time and discrete-time stochastic differential equations, we first propose concepts of Besicovitch almost periodic stochastic processes in p-th mean and of Besicovitch almost periodic stochastic processes in distribution on time scales, and reveal the relationship between the two random processes. Then, taking a class of stochastic Clifford-valued neural networks with time-varying delays on time scales as an example of stochastic dynamic equations with delays, we establish the existence and stability of Besicovitch almost periodic solutions in distribution for this class of networks by using Banach’s fixed point theorem, time scale calculus theory and inequality techniques. PubDate: 2022-05-18

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Abstract: Abstract In this paper, we consider a fourth order parabolic equation with heat source function and homogeneous boundary conditions. We establish some sufficient conditions to guarantee that the solution \(u({\varvec{x}},t)\) blows up at finite time and obtain upper and lower bounds of the blow-up time by constructing the corresponding functionals. We generalize the heat source function and improve the proof of the critical inequality in the reference Philippin (Proc Am Math Soc 143(6):2507–2513, 2015). PubDate: 2022-05-14

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Abstract: Abstract In this paper, we study the bifurcation of limit cycles of near-Hamilton system with four zones separated by nonlinear switching curves. We derive the expression of the first order Melnikov function. As an application, we consider the cyclicity of the system \({\dot{x}}=y, {\dot{y}}=-x^{2m-1}\) , where (0, 0) is a global nilpotent center and \(2\le m\in {\mathbb {N}}^{+}\) , under the perturbations of piecewise smooth polynomials with four zones separated by \(y=\pm kx^{m}\) with \(k>0\) . By analyzing the first order Melnikov function, we obtain the exact bound of the number of limit cycles bifurcating from the period annulus if the first order Melnikov function is not identically zero. We also give some examples to illustrate our results. PubDate: 2022-05-14

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Abstract: Abstract In this paper, we study the the following problem 0.1 $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _N u-\mu \Delta _p u=\lambda (u^{r}+a \nabla u ^{s})+f(x,u)&{}\quad \mathrm{in} \; \Omega ,\\ u>0 &{}\quad \mathrm{in} \; \Omega ,\\ u=0 &{}\quad \mathrm{on} \; \partial \Omega . \end{array}\right. } \end{aligned}$$ The term f could be exponential growth at \(+\infty \) . The convection term involved with \(\nabla u\) makes the problem (0.1) nonvariational and the variational methods are not applicable. Under suitable conditions imposed on f, the approximation scheme is employed to obtain the existence of positive solutions for all \(\lambda \in (0,\lambda ^*]\) with \(\lambda ^*>0\) . PubDate: 2022-05-11

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Abstract: Abstract The oscillatory properties of solutions to the second order functional differential equation $$\begin{aligned} \mathcal {L}x(t)+f(t)x^\beta (\sigma (t))=0, t \ge t_0 >0 \end{aligned}$$ where \(\mathcal {L}x(t)=(\eta (t)x'(t))'\) is a noncanonical operator, are studied. The main idea is to transform the noncanonical equation into canonical form which simplifies the investigation of oscillation of the equation. The obtained criteria are new and complement to the existing results reported in the literature. Examples illustrating the main results are presented. PubDate: 2022-05-10

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Abstract: Abstract The Julia set of any quadratic rational map is either connected or a Cantor set. In this paper, we extend this dichotomy to any cubic rational map with all critical points escaping to an attracting fixed point. PubDate: 2022-05-10

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Abstract: Abstract We study the bifurcation phenomena for a class of nonlinear elliptic problems with Hardy potential and gradient term. Some results about global continua of positive solutions emanating from bifurcation point from zero or infinity are established. PubDate: 2022-05-09

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Abstract: Abstract This article is concerned with the existence of positive ground state solutions for an asymptotically periodic quasilinear Schrödinger equation. By using a Nehari-type constraint, we get the existence results which improve the ones in Shi and Chen (Comput Math Appl 71:849–858, 2016). Moreover, we give an application of our results, which extends the results in Li (Commun Pure Appl Anal 14:1803–1816, 2015). PubDate: 2022-05-07

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Abstract: Abstract In this paper, we adopt a new approach to study the controllability and observability of linear quaternion-valued systems (QVS) from the point of complex-valued systems, which is much different from the method used in the previous paper. We show the equivalence relation of complete controllability for linear QVS and its complex-valued system. Then we establish two effective criteria for controllability and observability of the linear QVS in the sense of complex representation. In addition, we give a direct method to solve the control function. Finally, we use numerical examples to illustrate our theoretical results. PubDate: 2022-05-06

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Abstract: Abstract In this paper we investigate the global dynamics for a degenerate linear differential system with symmetry and two paralleled switching lines. After analyzing the qualitative properties of all equilibria including infinity and the number of closed orbits, we obtain all global phase portraits on the Poincaré disc. From these main results, we find necessary and sufficient conditions for the existence of crossing limit cycles, crossing heteroclinic loops and sliding heteroclinic loops, respectively, and prove that the numbers of these three types of closed orbits are all at most 1. Moreover, switching lines maybe pseudo singular lines or boundary singular lines. PubDate: 2022-05-06

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Abstract: Abstract In this paper, we study a family of planar piecewise linear systems with saddle-saddle dynamics and sector-wise separation, which is formed by two rays starting from the same point. We mainly investigate the existence of four-crossing-points limit cycles, which intersect each of the two separation rays at two points. This is a new and complex type of limit cycle that cannot exist in planar piecewise linear systems with two zones separated by a straight line and their existence in planar sector-wise linear systems with saddle-saddle dynamics has not been studied until now. We first obtain some sufficient and necessary conditions for the existence of a special four-crossing-points limit cycle. Then based on this result, we give some sufficient conditions for the existence of general four-crossing-points limit cycles. Moreover, we show that the four-crossing-points limit cycle and two different types of two-crossing-points limit cycles can exist simultaneously by providing concrete examples. PubDate: 2022-05-03

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Abstract: Abstract In this paper, the global existence and uniqueness of an isolated periodic wave solution is investigated for the Burgers–Fisher equation in a very general form. The method is based on employing the monotonicity of the ratio of the Abelian integrals. However, the complexity of the nonlinear term increases the difficulty to prove its monotonicity. The obtained results generalize and improve the previous one (Zhang et al. Appl Math Lett 121:107353, 2021), and the new results in this paper have never appeared in the previous works including (Zhang et al. 2021). PubDate: 2022-05-03

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Abstract: Abstract In this paper, we propose a new type of boundary value problems for 2-interval Sturm-Liouville equations which differs from the classical periodic Sturm-Liouville problems in that, the boundary and transmission conditions depend on a positive parameter \(\alpha >0.\) We will call this problem \(\alpha \) -semi periodic Sturm-Liouville problem. It is important to note that our problem is not self-adjoint in the classical Hilbert space of square-integrable functions \(L_2[-\pi ,\pi ]\) when the parameter \(\alpha \ne 1\) . First by using an our own approach we investigated some properties of eigenvalues and their corresponding eigenfunctions. Then, for self-adjoint realization of the problem under consideration we define a different inner product in the classical Hilbert space in which we treated an operator-theoretic formulation. The results obtained generalize and extend similar results of the classical periodic Sturm-Liouville theory, since in the special case \(\alpha =1\) our problem is transformed into classical periodic Sturm-Liouville problems. PubDate: 2022-04-28

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Abstract: Abstract We show that a linear homeomorphism with the shadowing property of a Banach space is hyperbolic if and only if the set of points with bounded orbit is closed. The proof is based on an auxiliary type of shadowing called bounded shadowing property. We give examples where our result can be applied. PubDate: 2022-04-26

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Abstract: Abstract Burgers-type equations are seen in cosmology, hydrodynamics, plasma physics, hurricane dynamics, statistical dynamics, traffic modelling, etc. For an extended ( \(2+1\) )-dimensional coupled Burgers system in fluid mechanics, concerning the velocity components in the fluid-related problems, our symbolic computation brings forth an auto-Bäcklund transformation with some solitons, and two sets of the similarity reductions. Our results rely on the coefficients in the system. PubDate: 2022-04-25

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Abstract: Abstract We prove Liouville theorems for the double phase problem $$\begin{aligned} -\text {div} ( \nabla u ^{p-2} \nabla u + w(x) \nabla u ^{q-2} \nabla u) = f(x) u ^{r-1}u \quad \text { in } \mathbb {R}^N, \end{aligned}$$ where \(q\ge p\ge 2\) , \(r>q-1\) and \(w,f \in L^1_\mathrm{loc}(\mathbb {R}^N)\) are two nonnegative functions such that \(w(x) \le C_1 x ^a\) and \(f(x) \ge C_2 x ^b\) for all \( x > R_0\) , where \(R_0,C_1,C_2>0\) and \(a,b\in \mathbb {R}\) . Our Liouville results hold for stable solutions in dimension \(N<N^\sharp \) , where \(N^\sharp \) is explicitly computed. We also prove Liouville theorems for finite energy solutions as well as solutions stable outside a compact set when \(\frac{N+b}{r+1} > \max \left\{ \frac{N-p}{p}, \frac{N-q+a}{q}\right\} \) . Methods of integral estimates and a Pohožaev type identity for double phase problems are exploited in our proofs. PubDate: 2022-04-23

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Abstract: Abstract In this paper, we analyze the features of a stochastically perturbed two-species predator-prey patch-system with ratio-dependent functional response. We first prove that the system which we investigate has a unique global positive solution. Then, the sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system are presented by establishing a series of suitable Lyapunov functions. PubDate: 2022-04-22

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Abstract: Abstract In this paper, we prove some new dynamic inequalities of Opial type involving higher-order derivatives of two functions, with two different weights on time scales. From these inequalities, we will derive some special cases and give an improvement of some versions of recent results. PubDate: 2022-04-20

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Abstract: Abstract In this paper, we present a malaria transmission model with climatic factors to study the climatic transmission dynamics of malaria with the effect of the Serratia AS1 bacterium. It has been shown in controlled laboratory experiments that Serratia AS1 bacterium can rapidly disseminate throughout mosquito populations and efficiently inhibit development of malaria parasites in mosquitoes. We derive the basic reproduction ratio \(R_{0}\) . We introduce the basic reproduction ratio \(R_{0}\) which can be used as a threshold parameter in the global dynamical model. When \(R_{0}\le 1\) , the disease-free periodic solution of this model is globally asymptotically stable; and when \(R_{0}>1\) , the disease remains persistent. We analyze the sensitivity of \(R_{0}\) in terms of the vertical and horizontal transmission rates of Serratia AS1, and find that \(R_{0}\) can be small when the vertical transmission rate of Serratia AS1 tends to 1 and the influence of the horizontal transmission rate performs inversely proportional to \(R_{0}\) when the vertical transmission rate is less than 1. Based on the data of Luanda, we perform numerical simulations to illustrate our theoretical results, which indicate that treatment of Serratia AS1 provides us an effective measure in controlling malaria. PubDate: 2022-04-11

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Abstract: Abstract In this paper, for general parameters we investigate the global dynamics of a piecewise smooth system, which is a two-parametric unfolding of a normal form with a fold–cusp. The main difficulty comes from the global switching of vector fields on switching manifold and the non-locality of parameters because switching makes classic theory of qualitative analysis and bifurcations for smooth systems invalid. Analyzing the global structure of switching manifold including all singularities and determining the precise domain of Poincaré map on the whole switching manifold, we obtain the bifurcation diagram in the whole parameter space and all corresponding global phase portraits in Poincaré disc. In this bifurcation diagram, the fold–fold bifurcation curve intersects the sliding homoclinic bifurcation curve and the pseudo-saddle-node bifurcation curve at two certain nonlocal parameters, respectively. Such intersections correspond to a degenerate sliding homoclinic loop and a degenerate fold–fold. Moreover, a sliding limit cycle and a pseudo-equilibrium bifurcate from the former and two pseudo-equilibria bifurcate from the latter. This generalizes the bifurcation theory of sliding homoclinic loop and fold–fold to the degenerate case. PubDate: 2022-04-10