TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS

 Vol. 46, No. 1           September 2015

Title and Author(s) Page
Positive solutions for nonlinear nonhomogeneous parametric equations
Nikolaos S. Papageorgiou and George Smyrlis
 ABSTRACT. We consider a nonlinear parametric Dirichlet problem driven by a nonhomogeneous differential operator which includes as special cases the $p$-Laplacian, the $(p,q)$-Laplacian and the generalized $p$-mean curvature operator. Using variational methods, we prove a bifurcation-type theorem describing the dependence of positive solutions on the parameter.
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An averaging formula for Reidemeister traces
Jiyou Liu and Xuezhi Zhao
 ABSTRACT. In this note, we shall give an averaging formula for Reidemeister traces, which is a simple relation among Reidemeister traces of a self-map and those of its liftings with respect to a finite-fold regular covering.
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Analytic invariant manifolds for nonautonomous equations
Luis Barreira and Claudià Valls
 ABSTRACT. We construct real analytic stable invariant manifolds for sufficiently small perturbations of a linear equation $v'=A(t)v$ admitting a~nonuniform exponential dichotomy. As a byproduct of our approach we obtain an exponential control not only of the trajectories on the invariant manifolds, but also of all their derivatives.
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On solutions of semilinear elliptic equation with linear growth nonlinearity in $\mathbb{R}^{N}$
Rong Cheng and Jianhua Hu
 ABSTRACT. We study nontrivial solutions for a class of semilinear elliptic equation which could be resonant at infinity. We establish the existence of solutions for the equation by considering the modified non-resonant problem associated with the original equation through Morse theory. Moreover, only linear growth assumption is imposed on the nonlinearity and condition on the potential is weaker than the coercive assumption.
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A partial positive solution to a conjecture of Ricceri
Francisco Javier Garcia-Pacheco and Justin R. Hill
 ABSTRACT. In this manuscript we introduce a new class of convex sets called quasi-absolutely convex and show that a Hausdorff locally convex topological vector space satisfies the weak anti-proximinal property if and only if every totally anti-proximinal quasi-absolutely convex subset is not rare. This improves results from \cite{GPtop} and provides a partial positive solution to a Ricceri's Conjectured posed in \cite{R} with many applications to the theory of partial differential equations. We also study the intrinsic structure of totally anti-proximinal convex subsets proving, among other things, that the absolutely convex hull of a linearly bounded totally anti-proximinal convex set must be finitely open. Finally, a new characterization of barrelledness in terms of comparison of norms is provided.
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The blow-up and global existence of solutions of Cauchy problems for a time fractional diffusion equation
Quan-Guo Zhang and Hong-Rui Sun
 ABSTRACT. In this paper, we investigate the blow-up and global existence of solutions to the following time fractional nonlinear diffusion equations {_0^C D_t^\alpha u}-\triangle u=|u|^{p-1}u, & x\in \Bbb{R}^N,\ t>0, u(0,x)=u_0(x), & x\in \Bbb{R}^N, where $0<\alpha<1$, $p>1$, $u_0\in C_0(\Bbb{R}^N)$ and ${_0^CD_t^\alpha u}=({\partial}/{\partial t}){_0^{}I_t^{1-\alpha}(u(t,x)-u_0(x))}$, ${_0^{}I_t^{1-\alpha}}$ denotes left Riemann--Liouville fractional integrals of order $1-\alpha$. We prove that if $1 69 A further generalization of midpoint convexity of multimaps towards common fixed point theorems and applications Irene Benedetti and Anna Martellotti  ABSTRACT. We furtherly generalize midpoint convexity for multivalued maps and derive Fixed Point Theorems and Common Fixed Point Theorems without requiring strong compactness. As an application we obtain some Best Approximation results, and minimax and variational inequalities. 93 On properties of solutions for a functional equation Zeqing Liu and Shin Min Kang  ABSTRACT. his paper studies properties of solutions for a functional equation arising in dynamic programming of multistage decision processes. Using the Banach fixed point theorem and the Mann iterative methods, we prove the existence and uniqueness of solutions and convergence of sequences generated by the Mann iterative methods for the functional equation in the Banach spaces$BC(S)$and$B(S)$and the complete metric space$BB(S)$, and discuss behaviors of solutions for the functional equation in the complete metric space$BB(S)$. Four examples illustrating the results presented in this paper are also provided. 113 Eigenvalue, bifurcation and convex solutions for Monge-Ampère equations Guowei Dai, Ruyun Ma and Xiaoling Han  ABSTRACT. In this paper we study the following eigenvalue boundary value problem for Monge-Amp\`{e}re equations \det(D^2u)=\lambda^N f(-u)& \text{in } \Omega, u=0 &\text{on } \partial \Omega. We establish global bifurcation results for the problem with$f(u)=u^N+g(u)$and$\Omega$being the unit ball of$\mathbb{R}^N$. More precisely, under some natural hypotheses on the perturbation function$g\colon[0,+\infty)\rightarrow[0,+\infty)$, we show that$(\lambda_1,0)$is a bifurcation point of the problem and there exists an unbounded continuum of convex solutions, where$\lambda_1$is the first eigenvalue of the problem with$f(u)=u^N$. As the applications of the above results, we consider with determining interval of$\lambda$, in which there exist convex solutions for this problem in unit ball. Moreover, we also get some results about the existence and nonexistence of convex solutions for this problem on general domain by domain comparison method. 135 Existence of Anosov diffeomorphisms on infra-nilmanifolds modeled on free nilpotent Lie groups Karel Dekimpe and Jonas Deré  ABSTRACT. An infra-nilmanifold is a manifold which is constructed as a~quotient space$\Gamma\setminus G$of a simply connected nilpotent Lie group$G$, where$\Gamma$is a discrete group acting properly discontinuously and cocompactly on~$G$via so called affine maps. The manifold$\Gamma\setminus G$is said to be modeled on the Lie group~$G$. This class of manifolds is conjectured to be the only class of closed manifolds allowing an Anosov diffeomorphism. However, it is far from obvious which of these infra-nilmanifolds actually do admit an Anosov diffeomorphism. In this paper we completely solve this question for infra-nilmanifolds modeled on a free$c$-step nilpotent Lie group. 165 Partially symmetric solutions of the generalized Hénon equation in symmetric domains Ryuji Kajikiya  ABSTRACT. We study the generalized H\'{e}non equation in a symmetric domain$\Omega$. Let$H$and$G$be closed subgroups of the orthogonal group such that$H \varsubsetneq G$and$\Omega$is$G$invariant. Then we show the existence of a positive solution which is$H$invariant but$G$non-invariant under suitable assumptions of$H$,$G$and the coefficient function of the equation. 191 Index 1 fixed points of orientation reversing planar homeomorphisms Francisco R. Ruiz del Portal and José M. Salazar  ABSTRACT. Let $$U \subset {\mathbb R}^2$$ be an open subset, $$f\colon U \rightarrow f(U) \subset {\mathbb R}^2$$ be an orientation reversing homeomorphism and let $$0 \in U$$ be an isolated, as a~periodic orbit, fixed point. The main theorem of this paper says that if the fixed point indices $$i_{{\mathbb R}^2}(f,0)=i_{{\mathbb R}^2}(f^2,0)=1$$ then there exists an orientation preserving dissipative homeomorphism$\varphi\colon {\mathbb R}^2 \rightarrow {\mathbb R}^2$such that $$f^2=\varphi$$ in a~small neighbourhood of $$0$$ and $$\{0\}$$ is a~global attractor for $$\varphi$$. As a corollary we have that for orientation reversing planar homeomorphisms a~fixed point, which is an isolated fixed point for $$f^2$$, is asymptotically stable if and only if it is stable. We also present an application to periodic differential equations with symmetries where orientation reversing homeomorphisms appear naturally. 223 A predator-prey model of Holling-type II with state dependent impulsive effects Changming Ding and Zhongxin Zhang  ABSTRACT. We investigate a predator-prey model with state dependent impulsive effects, which is based on a modified version of the Leslie-Gower scheme and on the Holling-type II scheme. Using topological methods, we present some sufficient conditions to guarantee the existence and asymptotical stability of semi-trivial periodic solutions and positive periodic solutions, respectively. 247 Harmonic perturbations with delay of periodic separated variables differential equations Luca Bisconti and Marco Spadini  ABSTRACT. We study the structure of the set of harmonic solutions to perturbed, nonautonomous,$T$-periodic, separated variables ODEs on manifolds. The perturbing term, supposed to be$T$-periodic in time, is allowed to contain a finite delay. Our main result extends those of \cite{FS09} and \cite{spaSepVar} but it cannot be simply deduced from them: It emerges from of a combination of the techniques exposed in those two papers. 261 P-regular nonlinear dynamics Beata Medak and Alexey A. Tret'yakov  ABSTRACT. In this paper we generalize the notion of$p$-factor operator which is the basic notion of the so-called$p$-regularity theory for nonlinear and degenerated operators. We prove a theorem related to a new construction of$p$-factor operator. The obtained results are illustrated by an example concerning nonlinear dynamical system. 283 On the stability of new impulsive differential equations Jinrong Wang, Zeng Lin and Yong Zhou  ABSTRACT. In this paper, we study new impulsive ordinary differential equations and apply fixed point approach to establish existence and uniqueness theorem and derive an interesting stability result in the sense of generalized$\beta$-Ulam--Hyers--Rassias. At last, two examples are given to demonstrate the applicability of our result. 303 Weak and strong convergence theorems for$m$-generalized hybrid mappings in Hilbert spaces Sattar Alizadeh and Fridoun Moradlou  ABSTRACT. In this paper, we prove a weak convergence theorem of Ishikawa's type for$m$-generalized hybrid mappings in a Hilbert space. Further, by using a new modification of Ishikawa iteration, we prove a strong convergence theorem for$m$-generalized hybrid mappings in a Hilbert space. 315 Nontrivial solutions for a mixed boundary problem for Schrödinger equations with an external magnetic field Claudianor O. Alves, Rodrigo C. M. Nemer and Sergio H. Monari Soares  ABSTRACT. We study the existence of solutions for a class of nonlinear Schr\"{o}dinger equations involving a magnetic field with mixed Dirichlet--Neumann boundary conditions. We use Lusternik--Shnirelman category and the Morse theory to estimate the number of nontrivial solutions in terms of the topology of the part of the boundary where the Neumann condition is prescribed. 329 Geometric proof of strong stable/unstable manifolds with application to the restricted three body problem Maciej J. Capiński and Anna Wasieczko-Zając  ABSTRACT. We present a method for establishing strong stable/unstable manifolds of fixed points for maps and ODEs. The method is based on cone conditions, suitably formulated to allow for application in computer assisted proofs. In the case of ODEs, assumptions follow from estimates on the vector field, and it is not necessary to integrate the system. We apply our method to the restricted three body problem and show that for a given choice of the mass parameter, there exists a homoclinic orbit along matching strong stable/unstable manifolds of one of the libration points. 363 On the degree for oriented quasi-Fredholm maps: its uniqueness and its effective extension of the Leray-Schauder degree Alessandro Calamai, Pierlugi Benevieri and Massimo Furi  ABSTRACT. In a previous paper, the first and third author developed a~degree theory for oriented locally compact perturbations of$C\sp{1}$Fredholm maps of index zero between real Banach spaces. In the spirit of a celebrated Amann--Weiss paper, we prove that this degree is unique if it is assumed to satisfy three axioms: Normalization, Additivity and Homotopy invariance. Taking into account that any compact vector field has a canonical orientation, from our uniqueness result we shall deduce that the above degree provides an effective extension of the Leray--Schauder degree. 401 Positive solutions of one-dimensional$p$-Laplacian equations and applications to population models of one species Kunquan Lan, Xiaojing Yang and Guangchong Yang  ABSTRACT. We prove new results on the existence of positive solutions of one-dimensional$p$-Laplacian equations under sublinear conditions involving the first eigenvalues of the corresponding homogeneous Dirichlet boundary value problems. To the best of our knowledge, this is the first paper to use fixed point index theory of compact maps to give criteria involving the first eigenvalue for one-dimensional$p$-Laplacian equations with$p\ne 2$. Our results generalize some previous results where either$p$is required to be greater than$2$or the nonlinearities satisfy stronger conditions. We shall apply our results to tackle a logistic population model arising in mathematical biology. 431 Set-valued perturbation for time dependent subdifferential operator Soumia Saïdi and Mustapha Fateh Yarou  ABSTRACT. In a separable Hilbert space, we consider an evolution inclusion involving time-dependent subdifferential of a proper convex lower semicontinuous function with a set-valued perturbation depending on both time and state variable. We prove, under a compactness condition on the perturbation, that there exists at least one absolutely continuous solution. 447 Hopf-bifurcation theorem and stability for the magneto-hydrodynamics equations Weiping Yan  ABSTRACT. This paper is devoted to the study of the dynamical behavior for the 3D viscous Magneto-hydrodynamics equations. We first prove that this system under smooth external forces possesses time dependent periodic solutions, bifurcating from a steady solution. If the time periodic solution is smooth, then the linear stability of the time periodic solution implies nonlinear stability is obtained in$\textbf{L}^p$for all$p\in(3,\infty)$. 471 Equivalence and nonexistence of standing waves for coupled Schrodinger equations with Chern-Simons gauge fields Hyungjin Huh and Jinmyoung Seok  ABSTRACT. This paper is devoted to the study of standing waves for so-called$N=2$supersymmetric Chern--Simons--Schrodinger equations, a coupled system of Schr\"odinger equations in which Chern--Simons gauge fields are incorporated. We show that there is no nontrivial standing wave to the$N=2$supersymmetric Chern--Simons--Schrodinger equations when the coupling constants are less than critical numbers. We also prove that static$N=2\$ supersymmetric Chern--Simons--Schrodinger equations are equivalent to their first order self-dual system when the coupling constants are critical.
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