# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

All Issues

## Discrete & Continuous Dynamical Systems - B

February 2020 , Volume 25 , Issue 2

Special issue on the occasion of the 60th birthday of Professor Juan J. Nieto

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2020, 25(2): ⅰ-ⅳ doi: 10.3934/dcdsb.2019269 +[Abstract](309) +[HTML](120) +[PDF](88.82KB)
Abstract:
2020, 25(2): 489-505 doi: 10.3934/dcdsb.2019250 +[Abstract](353) +[HTML](70) +[PDF](368.6KB)
Abstract:

The main purpose of this paper is to establish the existence and multiplicity of positive solutions for a fourth-order boundary value problem with integral condition. By using a new technique of construct a positive cone, we apply the Krasnoselskii compression/expansion and Leggett-Williams fixed point theorems in cones to show our multiplicity results. Finally, a particular case is studied, and the existence of multiple solutions is proved for two different particular functions.

2020, 25(2): 507-528 doi: 10.3934/dcdsb.2019251 +[Abstract](397) +[HTML](94) +[PDF](438.55KB)
Abstract:

The existence and uniqueness of mild solution of an impulsive stochastic system driven by a Rosenblatt process is analyzed in this work by using the Banach fixed point theorem and the theory of resolvent operator developed by R. Grimmer in [12]. Furthermore, the exponential stability in mean square for the mild solution to neutral stochastic integro–differential equations with Rosenblatt process is obtained by establishing an integral inequality. Finally, an example is exhibited to illustrate the abstract theory.

2020, 25(2): 529-544 doi: 10.3934/dcdsb.2019252 +[Abstract](284) +[HTML](60) +[PDF](374.39KB)
Abstract:

A Favard type theorem for Hurwitz polynomials is proposed. This result is a sufficient condition for a sequence of polynomials of increasing degree to be a sequence of Hurwitz polynomials. As in the Favard celebrated theorem, the three-term recurrence relation is used. Some examples of Hurwitz sequences are also presented. Additionally, a characterization of constructing a family of orthogonal polynomials on \begin{document}$[0, \infty)$\end{document} by two couples of numerical sequences \begin{document}$({A_{1, j}, B_{1, j}})$\end{document} and \begin{document}$({A_{2, j}, B_{2, j}})$\end{document} is stated.

2020, 25(2): 545-554 doi: 10.3934/dcdsb.2019253 +[Abstract](131) +[HTML](55) +[PDF](447.65KB)
Abstract:

We deal with the existence of nonnegative and nontrivial \begin{document}$T$\end{document}–periodic solutions for the equation \begin{document}$x'' = r(t)x^{\alpha}-s(t)x^{\beta}$\end{document} where \begin{document}$r$\end{document} and \begin{document}$s$\end{document} are continuous \begin{document}$T$\end{document}–periodic functions and \begin{document}$0<\alpha<\beta<1$\end{document}. This equation has been studied in connection with the valveless pumping phenomenon and we will take advantage of its variational structure in order to guarantee its solvability by means of the mountain pass theorem of Ambrosetti and Rabinowitz.

2020, 25(2): 555-567 doi: 10.3934/dcdsb.2019254 +[Abstract](156) +[HTML](56) +[PDF](321.62KB)
Abstract:

The aim of this paper is the study of existence of homoclinic solutions for a nonlinear difference equation involving \begin{document}$p$\end{document}-Laplacian. Under suitable growth conditions we prove that the considered problem has at least one homoclinic solution. The proof is based on the mountain-pass theorem with Cerami's condition, Brezis-Lieb lemma and variational method.

2020, 25(2): 569-597 doi: 10.3934/dcdsb.2019255 +[Abstract](219) +[HTML](186) +[PDF](518.41KB)
Abstract:

This work is concerned with well-posedness results for nonlocal semi-linear wave equations involving the fractional Laplacian and fractional derivative operator taken in the sense of Caputo. Representations for solutions, existence of classical solutions, and some \begin{document}$L^{p}$\end{document}-estimates are derived, by considering a quasi-stationary elliptic problem that comes from the realisation of the fractional Laplacian as the Dirichlet-to-Neumann map for a non-uniformly elliptic problem posed on a semi-infinite cylinder. We derive some properties such as existence of global weak solutions of the extended semi-linear integro-differential equations.

2020, 25(2): 599-615 doi: 10.3934/dcdsb.2019256 +[Abstract](153) +[HTML](68) +[PDF](1127.65KB)
Abstract:

A discussion on local bifurcations of codimension one and two is presented for generic unfoldings of Hopf-Bogdanov-Takens singularities of codimension three. Among all identified bifurcations, we focus on Hopf-Zero and Hopf-Hopf bifurcations, since, in certain cases, they can explain the emergence of chaotic dynamics. Moreover, numerical simulations are provided to illustrate that strange attractors appear at least when the second order normal form of the unfolding is considered.

2020, 25(2): 617-633 doi: 10.3934/dcdsb.2019257 +[Abstract](149) +[HTML](51) +[PDF](387.16KB)
Abstract:

We present existence and multiplicity principles for second–order discontinuous problems with nonlinear functional conditions. They are based on the method of lower and upper solutions and a recent extension of the Leray–Schauder topological degree to a class of discontinuous operators.

2020, 25(2): 635-650 doi: 10.3934/dcdsb.2019258 +[Abstract](195) +[HTML](76) +[PDF](369.4KB)
Abstract:

We present a new method to study the stability of one-dimensional discrete-time models, which is based on studying the graph of a certain family of functions. The method is closely related to exponent analysis, which the authors introduced to study the global stability of certain intricate convex combinations of maps. We show that the new strategy presented here complements and extends some existing conditions for the global stability. In particular, we provide a global stability condition improving the condition of negative Schwarzian derivative. Besides, we study the relation between this new method and the enveloping technique.

2020, 25(2): 651-670 doi: 10.3934/dcdsb.2019259 +[Abstract](190) +[HTML](54) +[PDF](569.87KB)
Abstract:

We present a systematic methodology to determine and locate analytically isolated periodic points of discrete and continuous dynamical systems with algebraic nature. We apply this method to a wide range of examples, including a one-parameter family of counterexamples to the discrete Markus-Yamabe conjecture (La Salle conjecture); the study of the low periods of a Lotka-Volterra-type map; the existence of three limit cycles for a piecewise linear planar vector field; a new counterexample of Kouchnirenko conjecture; and an alternative proof of the existence of a class of symmetric central configuration of the \begin{document}$(1+4)$\end{document}-body problem.

2020, 25(2): 671-690 doi: 10.3934/dcdsb.2019260 +[Abstract](188) +[HTML](60) +[PDF](422.47KB)
Abstract:

We establish the boundedness of some Schrödinger type operators on local generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.

2020, 25(2): 691-699 doi: 10.3934/dcdsb.2019261 +[Abstract](152) +[HTML](44) +[PDF](276.83KB)
Abstract:

We discuss the existence and non-existence of non-negative, non-decreasing solutions of certain perturbed Hammerstein integral equations with derivative dependence. We present some applications to nonlinear, second order boundary value problems subject to fairly general functional boundary conditions. The approach relies on classical fixed point index theory.

2020, 25(2): 701-713 doi: 10.3934/dcdsb.2019262 +[Abstract](149) +[HTML](60) +[PDF](1623.1KB)
Abstract:

In an attempt to explain experimental evidence of chaotic oscillations in blood cell population, A. Lasota suggested in 1977 a discrete-time one-dimensional model for the production of blood cells, and he showed that this equation allows to model the behavior of blood cell population in many clinical cases. Our main aim in this note is to carry out a detailed study of Lasota's equation, in particular revisiting the results in the original paper and showing new interesting phenomena. The considered equation is also suitable to model the dynamics of populations with discrete reproductive seasons, adult survivorship, overcompensating density dependence, and Allee effects. In this context, our results show the rich dynamics of this type of models and point out the subtle interplay between adult survivorship rates and strength of density dependence (including Allee effects).

2020, 25(2): 715-732 doi: 10.3934/dcdsb.2019263 +[Abstract](141) +[HTML](51) +[PDF](381.5KB)
Abstract:

The aim of this paper is to propose a new methodology in the construction of spaces of test functions used in a weak formulation of the Boundary Value Problem. The proposed construction is based on the so called "two dimensional" approach where at first, we select a partition of a domain and second, a dimension of an approximating functional subspace on each partition element. The main advantage consists in the independent selection of the key parameters, aiming at achieving a requested quality of approximation with a reasonable complexity. We give theoretical justification and illustration on examples that confirm our methodology.

2020, 25(2): 733-747 doi: 10.3934/dcdsb.2019264 +[Abstract](157) +[HTML](65) +[PDF](306.11KB)
Abstract:

Starting from some classical results of R. Conti, A. Haimovici and K. Iseki, and from a more recent result of S. Reich and A.J. Zaslavski, we present several theorems of approximation of the fixed points for non-self mappings on metric spaces. Both metric and topological conditions are involved. Some of the results are generalized to the multi-valued case. An application is given to a class of implicit first-order differential systems leading to a fixed point problem for the sum of a completely continuous operator and a nonexpansive mapping.

2020, 25(2): 749-760 doi: 10.3934/dcdsb.2019265 +[Abstract](156) +[HTML](59) +[PDF](625.25KB)
Abstract:

We study the passive particle transport generated by a circular vortex path in a 2D ideal flow confined in a circular domain. Taking the strength and angular velocity of the vortex path as main parameters, the bifurcation scheme of relative equilibria is identified. For a perturbed path, an infinite number of orbits around the centers are persistent, giving rise to periodic solutions with zero winding number.

2020, 25(2): 761-779 doi: 10.3934/dcdsb.2019266 +[Abstract](236) +[HTML](66) +[PDF](409.37KB)
Abstract:

We consider an optimal control problem of an advection-diffusion equation with Caputo time-fractional derivative. By convex duality method we obtain as optimality condition a forward-backward coupled system. We then prove the existence of a solution to this coupled system using Schauder fixed point theorem. The uniqueness of the solution is also established under certain monotonicity condition on the cost functional.

2020, 25(2): 781-798 doi: 10.3934/dcdsb.2019267 +[Abstract](209) +[HTML](70) +[PDF](367.02KB)
Abstract:

In this paper, we introduce the concepts of Bochner and Bohr almost automorphic functions on the semigroup induced by complete-closed time scales and their equivalence is proved. Particularly, when \begin{document}$\Pi = \mathbb{R}^{+}$\end{document} (or \begin{document}$\Pi = \mathbb{R}^{-}$\end{document}), we can obtain the Bochner and Bohr almost automorphic functions on continuous semigroup, which is the new almost automorphic case on time scales compared with the literature [20] (W.A. Veech, Almost automorphic functions on groups, Am. J. Math., Vol. 87, No. 3 (1965), pp 719-751) since there may not exist inverse element in a semigroup. Moreover, when \begin{document}$\Pi = h\mathbb{Z}^{+},\,h>0$\end{document} (or \begin{document}$\Pi = h\mathbb{Z}^{-},\,h>0$\end{document}), the corresponding automorphic functions on discrete semigroup can be obtained. Finally, we establish a theorem to guarantee the existence of Bochner (or Bohr) almost automorphic mild solutions of dynamic equations on semigroups induced by time scales.

2020, 25(2): 799-813 doi: 10.3934/dcdsb.2019268 +[Abstract](150) +[HTML](60) +[PDF](366.11KB)
Abstract:

Two Hopfield-type neural lattice models are considered, one with local \begin{document}$n$\end{document}-neighborhood nonlinear interconnections among neurons and the other with global nonlinear interconnections among neurons. It is shown that both systems possess global attractors on a weighted space of bi-infinite sequences. Moreover, the attractors are shown to depend upper semi-continuously on the interconnection parameters as \begin{document}$n \to \infty$\end{document}.

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