Spatial Dynamics of a Nonlocal Dispersal Population Model in a Shifting Environment

This paper is concerned with the spatial dynamics of a nonlocal dispersal population model in a shifting environment where the favorable region is shrinking. It is shown that the species becomes extinct in the habitat if the speed of the shifting habitat edge \(c>c^*(\infty )\), while the species persists and spreads along the shifting habitat at an asymptotic speed \(c^*(\infty )\) if \(c<c^*(\infty )\), where \(c^*(\infty )\) is determined by the nonlocal dispersal kernel, diffusion rate and the maximum linearized growth rate. Moreover, we demonstrate that for any given speed of the shifting habitat edge, the model system admits a nondecreasing traveling wave with the wave speed at which the habitat is shifting, which indicates that the extinction wave phenomenon does happen in such a shifting environment.     

Asymptotic behavior of the Cahn-Hilliard-Oono equation

Our aim in this article is to study the asymptotic behavior, in terms of finite-dimensional&nbsp;attractors, of the Cahn-Hilliard-Oono equation. This equation differs from the usual Cahn-Hilliard&nbsp;equation by the presence of a term of the form \( \epsilon u,\ \epsilon &gt;0\), which takes into account long-ranged interactions. In particular, we prove the existence of a robust family of exponential attractors as \(\epsilon\) goes to \(0\).     

Bifurcation of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems of degree 5 with a cusp

This paper deals with small perturbations of a class of hyper-elliptic Hamiltonian system, which is a Li &eacute; nard system of the form \(\dot{x}=y,\)&nbsp; \(\;\dot{y}=Q_1(x)+\varepsilon yQ_2(x)\) with \(Q_1\) and \(Q_2\) polynomials of degree 4 and 3, respectively. It is shown that this system can undergo degenerated Hopf bifurcation and Poincar &eacute; bifurcation, which emerge at most three limit cycles for \(\varepsilon\) sufficiently small.     

Existence and Stability of Traveling Waves for Degenerate Reaction–Diffusion Equation with Time Delay

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction–diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of \(c\ge c^*\) for the degenerate reaction–diffusion equation without delay, where \(c^*>0\) is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay \(\tau >0\). Furthermore, we prove the global existence and uniqueness of \(C^{\alpha ,\beta }\)-solution to the time-delayed degenerate reaction–diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted \(L^1\)-space. The exponential convergence rate is also derived.     

Multilicity results for some nonlinear ellitic roblems with asymtotically $${{\varvec{}}}$$-linear terms

Taking any \(p > 1\), we consider the asymptotically p-linear problem

$$\begin{aligned} \left\{ \begin{array}{ll} - {{\mathrm{div}}}(a(x,u,\nabla u)) + A_t(x,u,\nabla u)\ = \ \lambda ^\infty |u|^{p-2}u + g^\infty (x,u) &{}\quad \hbox {in}\;\Omega ,\\ u\ = \ 0 &{}\quad \hbox {on}\;\partial \Omega , \end{array} \right. \end{aligned}$$

where \(\Omega \) is a bounded domain in \(\mathbb R^N\), \(N\ge 2\), \(A(x,t,\xi )\) is a real function on \(\Omega \times \mathbb R\times \mathbb R^N\) which grows with power p with respect to \(\xi \) and has partial derivatives \(A_t(x,t,\xi ) = \frac{\partial A}{\partial t}(x,t,\xi )\), \(a(x,t,\xi ) = \nabla _\xi A(x,t,\xi )\). If \(A(x,t,\xi ) \rightarrow A^\infty (x,t)\) and \(\frac{g^\infty (x,t)}{|t|^{p-1}} \rightarrow 0\) as \(|t| \rightarrow +\infty \), suitable assumptions, variational methods and either the cohomological index theory or its related pseudo-index one, allow us to prove the existence of multiple nontrivial bounded solutions in the non-resonant case, i.e. if \(\lambda ^\infty \) is not an eigenvalue of the operator associated to \(\nabla _\xi A^\infty (x,\xi )\). In particular, while in [14] the model problem \(A(x,t,\xi ) = \mathcal{A}(x,t) |\xi |^p\) with \(p > N\) is studied, here our goal is twofold: extending such results not only to a more general family of functions \(A(x,t,\xi )\), but also to the more difficult case \(1 < p \le N\).

     

On the number of \(n\)-dimensional invariant spheres in polynomial vector fields of \(\mathbb{C}^{n+1}\)

We study the polynomial vector&nbsp;fields \(\mathcal{X}= \displaystyle \sum_{i=1}^{n+1} P_i(x_1,\ldots,x_{n+1}) \frac{\partial}{\partial x_i}\) in \(\mathbb{C}^{n+1}\) with \(n\geq 1\) . Let \(m_i\) be the degree of the&nbsp;polynomial \(P_i\). We call \((m_1,\ldots,m_{n+1})\) the degree of \(\mathcal{X}\). For these polynomial vector fields \(\mathcal{X}\) and&nbsp;in function of their degree we provide upper bounds, first for the&nbsp;maximal number of invariant \(n\)-dimensional spheres, and&nbsp;second for&nbsp;the maximal number of \(n\)-dimensional concentric invariant spheres.     

Global Stability for a Dynamic model of Hepatitis B with Antivirus Treatment

An epidemic model on the basis of therapy of chronic Hepatitis B with antivirus treatment was introduced in this paper. By applying a comparison theorem and analyzing the corresponding characteristic equations, we obtain sufficient conditions on the parameters for the global stability of the disease-free state. It's proved that if the basic reproduction number \(R_0 &lt; 1\) , the disease-free equilibrium is globally asymptotically stable. If \(R_0 &gt; 1\), the disease-free equilibrium is unstable and the disease is uniformly permanent. Moreover, if \(R_0 &gt; 1\), sufficient conditions are obtained for the global stability of the endemic equilibrium.     

Compactness and sequential completeness in some spaces of operators

Let \(X\) be a completely regular Hausdorff space and \(C_b(X)\) be the Banach lattice of all real-valued bounded continuous functions on \(X\) , endowed with the strict topologies \(\beta _\sigma ,\) \(\beta _\tau \) and \(\beta _t\) . Let \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) \((z=\sigma ,\tau ,t)\) stand for the space of all \((\beta _z,\xi )\) -continuous linear operators from \(C_b(X)\) to a locally convex Hausdorff space \((E,\xi ),\) provided with the topology \(\mathcal{T}_s\) of simple convergence. We characterize relative \(\mathcal{T}_s\) -compactness in \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) in terms of the representing Baire vector measures. It is shown that if \((E,\xi )\) is sequentially complete, then the spaces \((\mathcal{L}_{\beta _z,\xi }(C_b(X),E),\mathcal{T}_s)\) are sequentially complete whenever \(z=\sigma \) ; \(z=\tau \) and \(X\) is paracompact; \(z=t\) and \(X\) is paracompact and ?ech complete. Moreover, a Dieudonné–Grothendieck type theorem for operators on \(C_b(X)\) is given.     

Propagation Speed of the Bistable Traveling Wave to the Lotka–Volterra Competition System in a Periodic Habitat

We study propagation direction of the traveling wave for the diffusive Lotka–Volterra competition system with bistable nonlinearity in a periodic habitat. By directly proving the strong stability of two semitrivial equilibria, we establish a new and sharper result on the existence of traveling wave. Using the method of upper and lower solutions, we provide two comparison theorems concerning the direction of traveling wave propagation. Several explicit sufficient conditions on the determination of the speed sign are established. In addition, an interval estimation of the bistable-wave speed reveals the relations among the bistable speed and the spreading speeds of two monostable subsystems. Biologically, our idea and insight provide an effective approach to find or control the direction of wave propagation for a system in heterogeneous environments.

     

Groups with a ternary equivalence relation

We consider in a group \((G,\cdot )\) the ternary relation

$$\begin{aligned} \kappa := \{(\alpha , \beta , \gamma ) \in G^3 \ | \ \alpha \cdot \beta ^{-1} \cdot \gamma = \gamma \cdot \beta ^{-1} \cdot \alpha \} \end{aligned}$$

and show that \(\kappa \) is a ternary equivalence relation if and only if the set \( \mathfrak Z \) of centralizers of the group G forms a fibration of G (cf. Theorems 2, 3). Therefore G can be provided with an incidence structure

$$\begin{aligned} \mathfrak G:= \{\gamma \cdot Z \ | \ \gamma \in G , Z \in \mathfrak Z(G) \}. \end{aligned}$$

We study the automorphism group of \((G,\kappa )\), i.e. all permutations \(\varphi \) of the set G such that \( (\alpha , \beta , \gamma ) \in \kappa \) implies \((\varphi (\alpha ),\varphi (\beta ),\varphi (\gamma ))\in \kappa \). We show \(\mathrm{Aut}(G,\kappa )=\mathrm{Aut}(G,\mathfrak G)\), \(\mathrm{Aut} (G,\cdot ) \subseteq \mathrm{Aut}(G,\kappa )\) and if \( \varphi \in \mathrm{Aut}(G,\kappa )\) with \(\varphi (1)=1\) and \(\varphi (\xi ^{-1})= (\varphi (\xi ))^{-1}\) for all \(\xi \in G\) then \(\varphi \) is an automorphism of \((G,\cdot )\). This allows us to prove a representation theorem of \(\mathrm{Aut}(G,\kappa )\) (cf. Theorem 6) and that for \(\alpha \in G \) the maps

$$\begin{aligned} \tilde{\alpha }\ : \ G \rightarrow G;~ \xi \mapsto \alpha \cdot \xi ^{-1} \cdot \alpha \end{aligned}$$

of the corresponding reflection structure \((G, \widetilde{G})\) (with \( \tilde{G} := \{\tilde{\gamma }\ | \ \gamma \in G \}\)) are point reflections. If \((G ,\cdot )\) is uniquely 2-divisible and if for \(\alpha \in G\), \(\alpha ^{1\over 2}\) denotes the unique solution of \(\xi ^2=\alpha \) then with \(\alpha \odot \beta := \alpha ^{1\over 2} \cdot \beta \cdot \alpha ^{1\over 2}\), the pair \((G,\odot )\) is a K-loop (cf. Theorem 5).

     

Space–Time Fractional Equations and the Related Stable Processes at Random Time

In this paper, we consider the general space–time fractional equation of the form \(\sum _{j=1}^m \lambda _j \frac{\partial ^{\nu _j}}{\partial t^{\nu _j}} w(x_1, \ldots , x_n ; t) = -c^2 \left( -\varDelta \right) ^\beta w(x_1, \ldots , x_n ; t)\), for \(\nu _j \in \left( 0,1 \right] \) and \(\beta \in \left( 0,1 \right] \) with initial condition \(w(x_1, \ldots , x_n ; 0)= \prod _{j=1}^n \delta (x_j)\). We show that the solution of the Cauchy problem above coincides with the probability density of the n-dimensional vector process \(\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) \), \(t>0\), where \(\varvec{S}_n^{2\beta }\) is an isotropic stable process independent from \(\mathcal {L}^{\nu _1, \ldots , \nu _m}(t)\), which is the inverse of \(\mathcal {H}^{\nu _1, \ldots , \nu _m} (t) = \sum _{j=1}^m \lambda _j^{1/\nu _j} H^{\nu _j} (t)\), \(t>0\), with \(H^{\nu _j}(t)\) independent, positively skewed stable random variables of order \(\nu _j\). The problem considered includes the fractional telegraph equation as a special case as well as the governing equation of stable processes. The composition \(\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) \), \(t>0\), supplies a probabilistic representation for the solutions of the fractional equations above and coincides for \(\beta = 1\) with the n-dimensional Brownian motion at the random time \(\mathcal {L}^{\nu _1, \ldots , \nu _m} (t)\), \(t>0\). The iterated process \(\mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t)\), \(t>0\), inverse to \(\mathfrak {H}^{\nu _1, \ldots , \nu _m}_r (t) =\sum _{j=1}^m \lambda _j^{1/\nu _j} \, _1H^{\nu _j} \left( \, _2H^{\nu _j} \left( \, _3H^{\nu _j} \left( \ldots \, _{r}H^{\nu _j} (t) \ldots \right) \right) \right) \), \(t>0\), permits us to construct the process \(\varvec{S}_n^{2\beta } \left( c^2 \mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t) \right) \), \(t>0\), the density of which solves a space-fractional equation of the form of the generalized fractional telegraph equation. For \(r \rightarrow \infty \) and \(\beta = 1\), we obtain a probability density, independent from t, which represents the multidimensional generalization of the Gauss–Laplace law and solves the equation \(\sum _{j=1}^m \lambda _j w(x_1, \ldots , x_n) = c^2 \sum _{j=1}^n \frac{\partial ^2}{\partial x_j^2} w(x_1, \ldots , x_n)\). Our analysis represents a general framework of the interplay between fractional differential equations and composition of processes of which the iterated Brownian motion is a very particular case.     

Traveling wavefronts of a delayed lattice reaction-diffusion model

We investigate a system of delayed lattice differential system which is a model of pioneer-climax species distributed on one dimensional discrete space. We show that there exists a constant $c^*&gt;0$, such that the model has traveling wave solutions connecting a boundary equilibrium to a co-existence equilibrium for $c\geq c^*$. We also argue that $c^*$ is the minimal wave speed and the delay is harmless. The Schauder's fixed point theorem combining with upper-lower solution technique is used for showing the existence of wave solution.     

Analytic Conjugation, Global Attractor, and the Jacobian

It is proved that the dilation \(\lambda f\) of an analytic map \(f\) on \({\bf C}^n$\) with \(f(0)=0,f'(0)=I, |\lambda|&gt;1\) has an analytic conjugation to its linear part \(\lambda x\) if and only if \(f\) is an analytic automorphism on \({\bf C}^n\) and \(x=0\) is a global attractor for the inverse \((\lambda f)^{-1}\). This result is used to show that the dilation of the Jacobian polynomial of [12] is analyticly conjugate to its linear part.     

Explicit soliton solutions of the Kaup-Kupershmidt equation through the dynamical system approach

In this paper, we study the traveling wave solutions of&nbsp;the Kaup-Kupershmidt (KK)&nbsp;equation through using the dynamical system approach, which is an integrable fifth-order&nbsp;wave equation. Based on Cosgrove's work [3] and the phase analysis method of dynamical&nbsp;systems, infinitely many soliton solutions are presented in an explicit form. To guarantee&nbsp;the existence of soliton solutions, we discuss the parameters range as well as geometrical&nbsp;explanation of soliton solutions.     

Duality theory of regularized resolvent operator family

Let \( k \in C(R^+)\), A be a closed linear densely defined operator in the Banach&nbsp;space \(X\) and \( \{R(t)\}_{t\geq 0} \) be an exponentially bounded \(k\)-regularized resolvent operator families&nbsp;generated by A. In this paper, we mainly study pseudo k-resolvent and duality theory&nbsp;of k-regularized resolvent operator families. The conditions that pseudo k-resolvent&nbsp;become k-resolvent of the closed linear densely defined operator A are given. The&nbsp;some relations between the duality of the regularized resolvent operator families and&nbsp;the generator A are gotten. In addition, the corresponding results of duality of \(k\)-regularized resolvent operator families in Favard space are educed.     

New complexity analysis of a full-Newton step feasible interior-point algorithm for $$P_*(\kappa )$$-LCP

In this paper, we consider a full-Newton step feasible interior-point algorithm for \(P_*(\kappa )\)-linear complementarity problem. The perturbed complementarity equation \(xs=\mu e\) is transformed by using a strictly increasing function, i.e., replacing \(xs=\mu e\) by \(\psi (xs)=\psi (\mu e)\) with \(\psi (t)=\sqrt{t}\), and the proposed interior-point algorithm is based on that algebraic equivalent transformation. Furthermore, we establish the currently best known iteration bound for \(P_*(\kappa )\)-linear complementarity problem, namely, \(O((1+4\kappa )\sqrt{n}\log \frac{n}{\varepsilon })\), which almost coincides with the bound derived for linear optimization, except that the iteration bound in the \(P_{*}(\kappa )\)-linear complementarity problem case is multiplied with the factor \((1+4\kappa )\).     

Strong extensions for q-summing o perators acting in p-convex Banach function s paces for $$1 \le p \le q$$1≤ p≤ q

Let \(1\le p\le q<\infty \) and let X be a p-convex Banach function space over a \(\sigma \)-finite measure \(\mu \). We combine the structure of the spaces \(L^p(\mu )\) and \(L^q(\xi )\) for constructing the new space \(S_{X_p}^{\,q}(\xi )\), where \(\xi \) is a probability Radon measure on a certain compact set associated to X. We show some of its properties, and the relevant fact that every q-summing operator T defined on X can be continuously (strongly) extended to \(S_{X_p}^{\,q}(\xi )\). Our arguments lead to a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provided the known (strong) factorizations for q-summing operators through \(L^q\)-spaces when \(1 \le q \le p\). Thus, our result completes the picture, showing what happens in the complementary case \(1\le p\le q\).     

Minimum wave speed for a diffusive competition model with time delay

In this paper we study mono-stable traveling&nbsp;wave solutions for a Lotka-Volterra reaction-diffusion competition model with&nbsp;time delay. By constructing upper and lower solutions, we obtain the precise minimum&nbsp;wave speed of traveling waves under certain conditions. Our results also extend the known results on&nbsp;the minimum wave speed for Lotka-Volterra competition model without delay.     

On the existence of bubble-type solutions of nonlinear singular problems

Considered in this paper is a class of singular boundary value&nbsp;problem, arising&nbsp;in hydrodynamics and nonlinear field theory, when centrally bubble-type solutions are&nbsp;sought: \((p(t)u0)0 = c(t)p(t)f(u); u0(0) = 0; u(+1) = L &gt; 0\) in the half-line \([0;+1)\), where \(p(0) = 0\). We are interested in strictly increasing solutions&nbsp;of this problem in \([0;1)\) having just one zero in \((0;+1) \)and finite limit at zero, which has&nbsp;great importance in applications or pure and applied mathematics. Su&plusmn;cient conditions&nbsp;of the existence of such solutions are obtained by applying the critical point theory and&nbsp;by using shooting argument [9,10] to better analysis the properties of certain solutions&nbsp;associated with the singular di&reg;erential equation. To the authors' knowledge, for the first&nbsp;time, the above problem is dealt with when f satis&macr;es non-Lipschitz condition. Recent&nbsp;results in the literature are generalized and signi&macr;cantly improved.     

Associate space with respect to a locally $$\sigma $$-finite measure on a $$\delta $$δ-ring and applications to spaces of integrable functions defined by a vector measure

We show that for a locally \(\sigma \)-finite measure \(\mu \) defined on a \(\delta \)-ring, the associate space theory can be developed as in the \(\sigma \)-finite case, and corresponding properties are obtained. Given a saturated \(\sigma \)-order continuous \(\mu \)-Banach function space E, we prove that its dual space can be identified with the associate space \(E ^\times \) if, and only if, \(E^\times \) has the Fatou property. Applying the theory to the spaces \(L^p (\nu )\) and \(L_w^p (\nu )\), where \(\nu \) is a vector measure defined on a \(\delta \)-ring \(\mathcal {R}\) and \(1 \le p < \infty \), we establish results corresponding to those of the case when the vector measure is defined on a \(\sigma \)-algebra.