Topological Methods in Nonlinear Analysis Articles (Project Euclid)
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The latest articles from Topological Methods in Nonlinear Analysis on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2016 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Mon, 21 Mar 2016 15:31 EDTMon, 21 Mar 2016 15:31 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Coincidence of maps on torus fiber bundles over the circle
http://projecteuclid.org/euclid.tmna/1458588650
<strong>João Peres Vieira</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Volume 46, Number 2, 507--548.</p><p><strong>Abstract:</strong><br/> The main purpose of this work is to study coincidences of
fibre-preserving self-maps over the circle $S^1$ for spaces which
are fibre bundles over $S^1$ and the fibre is the torus $T$. We
classify all pairs of self-maps over $S^1$ which can be deformed
fibrewise to a pair of coincidence free maps. </p>projecteuclid.org/euclid.tmna/1458588650_20160321153103Mon, 21 Mar 2016 15:31 EDTNew results of mixed monotone operator equationshttps://projecteuclid.org/euclid.tmna/1552356036<strong>Tian Wang</strong>, <strong>Zhaocai Hao</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 19 pp..</p><p><strong>Abstract:</strong><br/>
In this article, we study the existence and uniqueness of fixed points for some mixed monotone operators and monotone operators with perturbation. These mixed monotone operators and monotone operators are $e$-concave-convex operators and $e$-concave operators respectively. Without using compactness or continuity, we obtain the existence and uniqueness of fixed points by monotone iterative techniques and properties of cones. Our main results extended and improved some existing results. Also, we applied the results to some differential equations.
</p>projecteuclid.org/euclid.tmna/1552356036_20190311220052Mon, 11 Mar 2019 22:00 EDTOn positive viscosity solutions of fractional Lane-Emden systemshttps://projecteuclid.org/euclid.tmna/1554170692<strong>Edir Junior Ferreira Leite</strong>, <strong>Marcos Montenegro</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 19 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we discuss the existence, nonexistence and uniqueness of positive viscosity solution for the following coupled system involving fractional Laplace operator on a smooth bounded domain $\Omega$ in $\mathbb R^n$: \[ \begin{cases} (-\Delta)^{s}u = v^p & \text{\rm in } \Omega,\\ (-\Delta)^{s}v = u^q & \text{\rm in } \Omega,\\ u= v=0 & \text{\rm in } \mathbb R^n\setminus\Omega. \end{cases} \] By means of an appropriate variational framework and a H\"{o}lder regularity result for suitable weak solutions of the above system, we prove that such a system admits at least one positive viscosity solution for any $0 < s < 1$, provided that $p, q > 0$, $pq \neq 1$ and the couple $(p,q)$ is below the critical hyperbole \[ \frac{1}{p + 1} + \frac{1}{q + 1} = \frac{n - 2s}{n} \] whenever $n > 2s$. Moreover, by using the maximum principles for the fractional Laplace operator, we show that uniqueness occurs whenever $pq < 1$. Lastly, assuming $\Omega$ is star-shaped, by using a Rellich type variational identity, we prove that no such a solution exists if $(p,q)$ is on or above the critical hyperbole. A crucial point in our proofs is proving, given a critical point $u \in W_{0}^{ s, ({p+1})/{p}}({\Omega}) \cap W^{ 2s, ({p+1})/{p}}(\Omega)$ of a related functional, that there is a function $v$ in an appropriate Sobolev space (Proposition 2.1) so that $(u,v)$ is a weak solution of the above system and a bootstrap argument can be applied successfully in order to establish its H\"{o}lder regularity (Proposition 3.1). The difficulty is caused mainly by the absence of a $L^p$ Calderón-Zygmund theory with $p > 1$ associated to the operator $(-\Delta)^{s}$ for $0 < s < 1$.
</p>projecteuclid.org/euclid.tmna/1554170692_20190401220458Mon, 01 Apr 2019 22:04 EDTLower and upper bounds for the waists of different spaceshttps://projecteuclid.org/euclid.tmna/1554170693<strong>Arseniy Akopyan</strong>, <strong>Alfredo Hubard</strong>, <strong>Roman Karasev</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 34 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we prove several new results around Gromov's waist theorem. We give a simple proof of Vaaler's theorem on sections of the unit cube using the Borsuk-Ulam-Crofton technique, consider waists of real and complex projective spaces, flat tori, convex bodies in Euclidean space; and establish waist-type results in terms of the Hausdorff measure.
</p>projecteuclid.org/euclid.tmna/1554170693_20190401220458Mon, 01 Apr 2019 22:04 EDTAbout positive $W_{\rm loc}^{1,\Phi}(\Omega)$-solutions to quasilinear elliptic problems with singular semilinear termhttps://projecteuclid.org/euclid.tmna/1554170694<strong>Carlos Alberto Santos</strong>, <strong>José Valdo Gonçalves</strong>, <strong>Marcos Leandro Carvalho</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 27 pp..</p><p><strong>Abstract:</strong><br/>
This paper deals with the existence, uniqueness and regularity of positive $W_{\rm loc}^{1,\Phi}(\Omega)$-solutions of singular elliptic problems on a smooth bounded domain with Dirichlet boundary conditions involving the $\Phi$-Laplacian operator. The proof of the existence is based on a variant of the generalized Galerkin method that we developed inspired by ideas of Browder [4] and a comparison principle. By the use of a kind of Moser's iteration scheme we show the $L^{\infty}(\Omega)$-regularity for positive solutions.
</p>projecteuclid.org/euclid.tmna/1554170694_20190401220458Mon, 01 Apr 2019 22:04 EDTOn directional derivatives for cone-convex functionshttps://projecteuclid.org/euclid.tmna/1554170695<strong>Krzysztof Leśniewski</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 13 pp..</p><p><strong>Abstract:</strong><br/>
We investigate the relationship between the existence of directional derivatives for cone-convex functions with values in a Banach space $Y$ and isomorphisms between $Y$ and $c_0$.
</p>projecteuclid.org/euclid.tmna/1554170695_20190401220458Mon, 01 Apr 2019 22:04 EDTFormal barycenter spaces with weights: the Euler characteristichttps://projecteuclid.org/euclid.tmna/1554170696<strong>Sadok Kallel</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 23 pp..</p><p><strong>Abstract:</strong><br/>
We compute the Euler characteristic with compact supports $\chi_c$ of the formal barycenter spaces with weights of some locally compact spaces, connected or not. This reduces to the topological Euler characteristic $\chi$ when the weights of the singular points are less than one. As foresighted by Andrea Malchiodi, our formula is related to the Leray-Schauder degree for mean field equations on a compact Riemann surface obtained by C.C. Chen and C.S. Lin.
</p>projecteuclid.org/euclid.tmna/1554170696_20190401220458Mon, 01 Apr 2019 22:04 EDTExtreme partitions of a Lebesgue space and their application in topological dynamicshttps://projecteuclid.org/euclid.tmna/1557367217<strong>Wojciech Bułatek</strong>, <strong>Brunon Kamiński</strong>, <strong>Jerzy Szymański</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 0 pp..</p><p><strong>Abstract:</strong><br/>
It is shown that any topological action $\Phi$ of a countable orderable and amenable group $G$ on a compact metric space $X$ and every $\Phi$-invariant probability Borel measure $\mu$ admit an extreme partition $\zeta$ of $X$ such that the equivalence relation $R_{\zeta}$ associated with $\zeta$ contains the asymptotic relation $A(\Phi)$ of $\Phi$. As an application of this result and the generalized Glasner theorem it is proved that $A(\Phi)$ is dense for the set $E_{\mu}(\Phi)$ of entropy pairs.
</p>projecteuclid.org/euclid.tmna/1557367217_20190508220028Wed, 08 May 2019 22:00 EDTNonautonomous Conley index theory the connecting homomorphismhttps://projecteuclid.org/euclid.tmna/1557453830<strong>Axel Jänig</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 20 pp..</p><p><strong>Abstract:</strong><br/>
Attractor-repeller decompositions of isolated invariant sets give rise to so-called connecting homomorphisms. These homomorphisms reveal information on the existence and strucuture of connecting trajectories of the underlying dynamical system.
To give a meaningful generalization of this general principle to nonautonomous problems, the nonautonomous homology Conley index is expressed as a direct limit. Moreover, it is shown that a nontrivial connecting homomorphism implies, on the dynamical systems level, a sort of uniform connectedness of the attractor-repeller decomposition.
</p>projecteuclid.org/euclid.tmna/1557453830_20190509220401Thu, 09 May 2019 22:04 EDTOn ground state solutions for the nonlinear Kirchhoff type problems with a general critical nonlinearityhttps://projecteuclid.org/euclid.tmna/1557453831<strong>Weihong Xie</strong>, <strong>Haibo Chen</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 27 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we are concerned with the following Kirchhoff type problem with critical growth: \begin{equation*} -\bigg(a+b\int_{\mathbb R^3}|\nabla u|^2dx\bigg)\Delta u+V(x)u=f(u)+|u|^4u, \quad u\in H^1(\mathbb R^3), \end{equation*} where $a,b > 0$ are constants. Under certain assumptions on $V$ and $f$, we prove that the above problem has a ground state solution of Nehari-Pohozaev type and a least energy solution via variational methods. Furthermore, we also show that the mountain pass value gives the least energy level for the above problem. Our results improve and extend some recent ones in the literature.
</p>projecteuclid.org/euclid.tmna/1557453831_20190509220401Thu, 09 May 2019 22:04 EDTAmenability and Hahn-Banach extension property for set-valued mappingshttps://projecteuclid.org/euclid.tmna/1557453832<strong>Anthony To-Ming Lau</strong>, <strong>Liangjin Yao</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 27 pp..</p><p><strong>Abstract:</strong><br/>
Amenability is an important notion in harmonic analysis on groups and semigroups, and their associated Banach algebras. In this paper, we present some characterizations of a semitopological semigroup $S$ on the existence of a left invariant mean on ${\rm LUC}(S)$, ${\rm AP}(S)$ and ${\rm WAP}(S)$ in terms of Hahn-Banach extension theorem, which extend the first author's early results in 1970s. Moreover, we refine and extend the well known Day's result and Mitchell's results on fixed point properties for set-valued mappings. As an application, we give an application of our result to a class of the Banach algebras related to amenability of groups and semigroups.
</p>projecteuclid.org/euclid.tmna/1557453832_20190509220401Thu, 09 May 2019 22:04 EDTMultiplicity results for fractional $p$-Laplacian problems with Hardy term and Hardy-Sobolev critical exponent in $\mathbb{R}^N$https://projecteuclid.org/euclid.tmna/1557453833<strong>Hadi Mirzaee</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 19 pp..</p><p><strong>Abstract:</strong><br/>
This paper is devoted to the study of a class of singular fractional $p$-Laplacian problems of the form $$ (-\Delta)_p^su-\mu\frac{|u|^{p-2}u}{|x|^{ps}} =\alpha\frac{|u|^{ p_{s}^{*}(b)-2 }u}{|x|^b} +\beta f(x)|u|^{q-2}u\quad \text{in }\mathbb{R}^N $$% where $0 < s< 1$, $0\leq b< ps< N$, $1< q< p_{s}^{*}(b)$, $\alpha, \beta> 0$, $\mu\in \mathbb{R}$, and $f(x)$ is a given function which satisfies some appropriate condition. By using variational methods, we prove the existence of infinitely many solutions under different conditions.
</p>projecteuclid.org/euclid.tmna/1557453833_20190509220401Thu, 09 May 2019 22:04 EDTPositive least energy solutions for coupled nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponenthttps://projecteuclid.org/euclid.tmna/1557540142<strong>Song You</strong>, <strong>Qingxun Wang</strong>, <strong>Peihao Zhao</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 35 pp..</p><p><strong>Abstract:</strong><br/>
\begin{cases} \displaystyle -\Delta u+\nu_{1}u=\mu_{1}\left(\frac{1}{|x|^{4}}\ast u^{2}\right)u +\beta \left(\frac{1}{|x|^{4}}\ast v^{2}\right) u, & x \in \Omega,\\ -\Delta v+\nu_{2}v=\mu_{2}\left(\frac{1}{|x|^{4}}\ast v^{2}\right)v +\beta\left(\frac{1}{|x|^{4}}\ast u^{2}\right)v, & x \in \Omega,\\ u,v \geq 0 \quad\text{in }\Omega, \qquad u=v=0 \quad \text{on } \partial\Omega. \end{cases} \end{equation*} Here $\Omega\subset\mathbb{R}^{N}$ is a smooth bounded domain, $-\lambda_{1}(\Omega)< \nu_{1},\nu_{2}< 0, \lambda_{1}(\Omega)$ is the first eigenvalue of $ (-\Delta, H_{0}^{1}(\Omega))$, $\mu_{1},\mu_{2}> 0$ and $\beta\neq 0$ is a coupling constant. We show that the critical nonlocal elliptic system has a positive least energy solution under appropriate conditions on parameters via variational methods. For the case in which $\nu_{1}=\nu_{2}$, we obtain the classification of the positive least energy solutions. Moreover, the asymptotic behaviors of the positive least energy solutions as $\beta\rightarrow 0$ are studied.
</p>projecteuclid.org/euclid.tmna/1557540142_20190510220242Fri, 10 May 2019 22:02 EDTStrong convergence of bi-spatial random attractors for parabolic equations on thin domains with rough noisehttps://projecteuclid.org/euclid.tmna/1557540143<strong>Fuzhi Li</strong>, <strong>Yangrong Li</strong>, <strong>Renhai Wang</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 24 pp..</p><p><strong>Abstract:</strong><br/>
This article concerns bi-spatial random dynamics for the stochastic reaction-diffusion equation on a thin domain, where the noise is described by a general stochastic process instead of the usual Wiener process. A bi-spatial attractor is obtained when the non-initial state space is the $p$-times Lebesgue space, meanwhile, measurability of the attractor in the Banach space is proved by using measurability of both cocycle and absorbing set. Finally, the $p$-norm convergence of attractors is obtained when the thin domain collapses onto a lower dimensional domain. The method of symbolical truncation is applied to provide some uniformly asymptotic estimates.
</p>projecteuclid.org/euclid.tmna/1557540143_20190510220242Fri, 10 May 2019 22:02 EDTOn finding the ground state solution to the linearly coupled Brezis-Nirenberg system in high dimensions: the cooperative casehttps://projecteuclid.org/euclid.tmna/1557540144<strong>Yuanze Wu</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 33 pp..</p><p><strong>Abstract:</strong><br/>
Consider the following elliptic system \begin{cases} -\Delta u_i+\mu_i u_i=|u_i|^{2^*-2}u_i+\lambda \sum\limits_{j=1,j\not=i}^ku_j &\text{in }\Omega,\\ u_i=0,\quad i=1,\dots,k,&\text{on }\partial\Omega, \end{cases} \end{equation*} where $k\geq2$, $\Omega\subset\mathbb R^N$ ($N\geq4$) is a bounded domain with smooth boundary $\partial\Omega$, $2^*={2N}/({N-2})$ is the Sobolev critical exponent, $\mu_i\in\mathbb R$ for all $i=1,\dots,k$ are constants and $\lambda\in\mathbb R$ is a parameter. By the variational method, we mainly prove that the above system has a ground state for all $\lambda> 0$. Our results reveal some new properties of the above system that imply that the parameter $\lambda$ plays the same role as in the following well-known Brezis-Nirenberg equation \begin{equation*} \begin{cases} -\Delta u =\lambda u+ |u|^{2^*-2}u &\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega, \end{cases} \end{equation*} and this system has a very similar structure of solutions as the above Brezis-Nirenberg equation for $\lambda$.
</p>projecteuclid.org/euclid.tmna/1557540144_20190510220242Fri, 10 May 2019 22:02 EDTExistence of positive solutions for Hardy nonlocal fractional elliptic equations involving critical nonlinearitieshttps://projecteuclid.org/euclid.tmna/1557540145<strong>Somayeh Rastegarzadeh</strong>, <strong>Nemat Nyamoradi</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 16 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we have used variational methods to study existence of solutions for the following critical nonlocal fractional Hardy elliptic equation \begin{equation*} (- \Delta)^s u - \gamma \frac{u}{|x|^{2 s}} = \frac{|u|^{2_s^*(b) - 2} u}{|x|^{b}} + \lambda f (x, u ), \quad \text{in } \mathbb{R}^N, \end{equation*} where $N > 2 s $, $ 0< s< 1 $, $ \gamma, \lambda $ are real parameters, $(- \Delta)^s$ is the fractional Laplace operator, $2_s^*(b) = {2 (N - b)}/(N - 2s)$ is a critical Hardy-Sobolev exponent with $b \in [0, 2s)$ and $ f \in C(\mathbb R^{N} \times \mathbb{R}, \mathbb{R})$.
</p>projecteuclid.org/euclid.tmna/1557540145_20190510220242Fri, 10 May 2019 22:02 EDTGlobal secondary bifurcation, symmetry breaking and period-doublinghttps://projecteuclid.org/euclid.tmna/1557540146<strong>Rainer Mandel</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 22 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we provide a criterion for global secondary bifurcation via symmetry breaking. As an application, the occurrence of period-doubling bifurcations for the Lugiato-Lefever equation is proved.
</p>projecteuclid.org/euclid.tmna/1557540146_20190510220242Fri, 10 May 2019 22:02 EDTOn the topological degree of planar maps avoiding normal coneshttps://projecteuclid.org/euclid.tmna/1557540147<strong>Alessandro Fonda</strong>, <strong>Giuliano Klun</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 21 pp..</p><p><strong>Abstract:</strong><br/>
The classical Poincaré-Bohl theorem provides the existence of a zero for a function avoiding external rays. When the domain is convex, the same holds true when avoiding normal cones. We consider here the possibility of dealing with nonconvex sets having inward corners or cusps, in which cases the normal cone vanishes. This allows us to deal with situations where the topological degree may be strictly greater than $1$.
</p>projecteuclid.org/euclid.tmna/1557540147_20190510220242Fri, 10 May 2019 22:02 EDTReidemeister spectra for solvmanifolds in low dimensionshttps://projecteuclid.org/euclid.tmna/1558663227<strong>Karel Dekimpe</strong>, <strong>Sam Tertooy</strong>, <strong>Iris Van den Bussche</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 27 pp..</p><p><strong>Abstract:</strong><br/>
The Reidemeister number of an endomorphism of a group is the number of twisted conjugacy classes determined by that endomorphism. The collection of all Reidemeister numbers of all automorphisms of a group $G$ is called the Reidemeister spectrum of $G$. In this paper, we determine the Reidemeister spectra of all fundamental groups of solvmanifolds up to Hirsch length 4.
</p>projecteuclid.org/euclid.tmna/1558663227_20190523220047Thu, 23 May 2019 22:00 EDTBlow-up solutions for a $p$-Laplacian elliptic equation of logistic type with singular nonlinearityhttps://projecteuclid.org/euclid.tmna/1558663228<strong>Claudianor O. Alves</strong>, <strong>Carlos Alberto Santos</strong>, <strong>Jiazheng Zhou</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 31 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we deal with existence, uniqueness and exact rate of boundary behavior of blow-up solutions is for a class of logistic type quasilinear problems in a smooth bounded domain involving the $p$-Laplacian operator, where the nonlinearity can have a singular behavior. In the proof of the existence of solution, we have used the sub and super solution method in conjunction with variational techniques and comparison principles. Related to the rate on boundary and uniqueness, we combine comparison principle with our result of existence of solution.
</p>projecteuclid.org/euclid.tmna/1558663228_20190523220047Thu, 23 May 2019 22:00 EDT$L^{p}$-pullback attractors for non-autonomous reaction-diffusion equations with delayshttps://projecteuclid.org/euclid.tmna/1562551221<strong>Kaixuan Zhu</strong>, <strong>Yongqin Xie</strong>, <strong>Feng Zhou</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 19 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the non-autonomous reaction-diffusion equations with hereditary effects and the nonlinear term $f$ satisfying the polynomial growth of arbitrary order $p-1$ $(p\geq2)$. The delay term may be driven by a function with very weak assumptions, namely, just measurability. We extend the asymptotic a priori estimate method (see [29]) to our problem and establish a new existence theorem for the pullback attractors in $C_{L^{p}(\Omega)}$ $(p> 2)$ (see Theorem 2.12), which generalizes the results obtained in [12].
</p>projecteuclid.org/euclid.tmna/1562551221_20190707220037Sun, 07 Jul 2019 22:00 EDTConley index continuation for a singularly perturbed periodic boundary value problemhttps://projecteuclid.org/euclid.tmna/1562551222<strong>Maria C. Carbinatto</strong>, <strong>Krzysztof P. Rybakowski</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 30 pp..</p><p><strong>Abstract:</strong><br/>
We establish spectral convergence and Conley index continuation results for a class of singularly perturbed periodic boundary value problems.
</p>projecteuclid.org/euclid.tmna/1562551222_20190707220037Sun, 07 Jul 2019 22:00 EDTAsymptotically almost automorphic solutions of dynamic equations on time scaleshttps://projecteuclid.org/euclid.tmna/1562551223<strong>Carlos Lizama</strong>, <strong>Jaqueline G. Mesquita</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 22 pp..</p><p><strong>Abstract:</strong><br/>
In the present work, we introduce the concept of asymptotically almost automorphic functions on time scales and study their main properties. We study nonautonomous dynamic equations on time scales given by $x^{\Delta} (t) = A(t) x(t) + f(t)$ and $x^{\Delta} (t) = A(t) x(t) + f(t, x(t))$, $t \in \mathbb T$, where $\mathbb T$ is an invariant under translations time scale and $A \in \mathcal{R}(\mathbb T, \mathbb R^{n \times n})$. We give new criteria ensuring the existence of an asymptotically almost automorphic solution for both equations.
</p>projecteuclid.org/euclid.tmna/1562551223_20190707220037Sun, 07 Jul 2019 22:00 EDTClassification of radial solutions to Hénon type equation on the hyperbolic spacehttps://projecteuclid.org/euclid.tmna/1562551224<strong>Shoichi Hasegawa</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 28 pp..</p><p><strong>Abstract:</strong><br/>
We devote this paper to classifying radial solutions of a weighted semilinear elliptic equation on the hyperbolic space. More precisely, for a weighted Lane-Emden equation on the hyperbolic space, we shall study the sign and asymptotic behavior of the radial solutions. We shall also show the existence of fast-decay sign-changing solutions to the Lane-Emden equation on the hyperbolic space.
</p>projecteuclid.org/euclid.tmna/1562551224_20190707220037Sun, 07 Jul 2019 22:00 EDTTopological characteristics of solution sets for fractional evolution equations and applications to control systemshttps://projecteuclid.org/euclid.tmna/1562551225<strong>Shouguo Zhu</strong>, <strong>Zhenbin Fan</strong>, <strong>Gang Li</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 26 pp..</p><p><strong>Abstract:</strong><br/>
This paper explores an abstract Riemann-Liouville fractional evolution model with a weighted delay initial condition. We develop the resolvent technique, a generalization of semigroup method, to formulate an appropriate notion of mild solutions to this abstract system and present the topological characteristics of the corresponding solution set in a weighted space. Furthermore, in view of the topological characteristics, we analyze the approximate controllability of the abstract system without Lipschitz assumption. We end up addressing an infinite dimensional fractional delay diffusion control system and a finite dimensional fractional ordinary differential control system by utilizing our theoretical findings.
</p>projecteuclid.org/euclid.tmna/1562551225_20190707220037Sun, 07 Jul 2019 22:00 EDTA weighted Trudinger-Moser type inequality and its applications to quasilinear elliptic problems with critical growth in the whole Euclidean spacehttps://projecteuclid.org/euclid.tmna/1563242553<strong>Francisco S. B. Albuquerque</strong>, <strong>Sami Aouaoui</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 22 pp..</p><p><strong>Abstract:</strong><br/>
We establish a version of the Trudinger-Moser inequality involving unbounded or decaying radial weights in weighted Sobolev spaces. In the light of this inequality and using a minimax procedure we also study existence of solutions for a class of quasilinear elliptic problems involving exponential critical growth.
</p>projecteuclid.org/euclid.tmna/1563242553_20190715220257Mon, 15 Jul 2019 22:02 EDTKrasnosel'skiĭ-Schaefer type method in the existence problemshttps://projecteuclid.org/euclid.tmna/1563242554<strong>Calogero Vetro</strong>, <strong>Dariusz Wardowski</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 9 pp..</p><p><strong>Abstract:</strong><br/>
We consider a general integral equation satisfying algebraic conditions in a Banach space. Using Krasnosel'skiĭ-Schaefer type method and technical assumptions, we prove an existence theorem producing a periodic solution of some nonlinear integral equation.
</p>projecteuclid.org/euclid.tmna/1563242554_20190715220257Mon, 15 Jul 2019 22:02 EDTThe limit cycles of a class of quintic polynomial vector fieldshttps://projecteuclid.org/euclid.tmna/1563242555<strong>Jaume Llibre</strong>, <strong>Tayeb Salhi</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 11 pp..</p><p><strong>Abstract:</strong><br/>
Using the inverse integrating factor we study the limit cycles of a class of polynomial vector fields of degree $5$.
</p>projecteuclid.org/euclid.tmna/1563242555_20190715220257Mon, 15 Jul 2019 22:02 EDTExistence, localization and stability of limit-periodic solutions to differential equations involving cubic nonlinearitieshttps://projecteuclid.org/euclid.tmna/1563242556<strong>Jan Andres</strong>, <strong>Denis Pennequin</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 20 pp..</p><p><strong>Abstract:</strong><br/>
We will prove, besides other things like localization and (in)stability, that the differential equations $x'+x^3-\lambda x=\varepsilon r(t)$, $\lambda> 0$, and $x''+x^3-x=\varepsilon r(t)$, where $r\colon\mathbb{R}\to\mathbb{R}$ are uniformly limit-periodic functions, possess for sufficiently small values of $\varepsilon > 0$ uniformly limit-periodic solutions, provided $r$ in the first-order equation is strictly positive. As far as we know, these are the first nontrivial effective criteria, obtained for limit-periodic solutions of nonlinear differential equations, in the lack of global lipschitzianity restrictions. A simple illustrative example is also indicated for difference equations.
</p>projecteuclid.org/euclid.tmna/1563242556_20190715220257Mon, 15 Jul 2019 22:02 EDTSolutions for quasilinear elliptic systems with vanishing potentialshttps://projecteuclid.org/euclid.tmna/1563242557<strong>Billel Gheraibia</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 23 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we study the following strongly coupled quasilinear elliptic system: $$ \begin{cases} -\Delta_{p} u+\lambda a(x)|u|^{p-2}u=\dfrac{\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta}, & x\in {\mathbb R}^{N}, \\ -\Delta_{p} v+\lambda b(x)|v|^{p-2}v=\dfrac{\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v, & x\in {\mathbb R}^{N}, \\ u,v\in D^{1,p}(\mathbb R^{N}), \end{cases} $$ where $N\geq 3$, $\lambda> 0$ is a parameter, $p< \alpha+\beta< p^{*}:={Np}/({N-p})$. Under some suitable conditions which are given in section 1, we use variational methods to obtain both the existence and multiplicity of solutions for the system on an appropriated space when the parameter $\lambda$ is sufficiently large. Moreover, we study the asymptotic behavior of these solutions when $\lambda\rightarrow\infty$.
</p>projecteuclid.org/euclid.tmna/1563242557_20190715220257Mon, 15 Jul 2019 22:02 EDTGeneralized fractional differential equations and inclusions equipped with nonlocal generalized fractional integral boundary conditionshttps://projecteuclid.org/euclid.tmna/1563242558<strong>Sotiris K. Ntouyas</strong>, <strong>Bashir Ahmad</strong>, <strong>Madeaha Alghanmi</strong>, <strong>Ahmed Alsaedi</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 23 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we establish sufficient criteria for the existence of solutions for generalized fractional differential equations and inclusions supplemented with generalized fractional integral boundary conditions. We make use of the standard fixed point theorems for single-valued and multivalued maps to obtain the desired results, which are well illustrated with the aid of examples.
</p>projecteuclid.org/euclid.tmna/1563242558_20190715220257Mon, 15 Jul 2019 22:02 EDTPeriodic solutions for a singular Liénard equation with indefinite weighthttps://projecteuclid.org/euclid.tmna/1563242559<strong>Shiping Lu</strong>, <strong>Runyu Xue</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 16 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, the existence of positive periodic solutions is studied for a singular Liénard equation where the weight function has an indefinite sign. Due to the lack of a priori estimates over the set of all possible positive periodic solutions in this equation, a new method is proposed for estimating a priori bounds of positive periodic solutions. By the use of a continuation theorem of the Mawhin coincidence degree, new conditions for existence of positive periodic solutions to the equation are obtained. The main results show that the singularity of coefficient function associated to the friction term at $x=0$ may help the existence of periodic solutions.
</p>projecteuclid.org/euclid.tmna/1563242559_20190715220257Mon, 15 Jul 2019 22:02 EDTInfinitely many solutions for a class of critical Choquard equation with zero masshttps://projecteuclid.org/euclid.tmna/1563242560<strong>Fashun Gao</strong>, <strong>Minbo Yang</strong>, <strong>Carlos Alberto Santos</strong>, <strong>Jiazheng Zhou</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 14 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we investigate the following nonlinear Choquard equation $$ -\Delta u =\bigg(\int_{\mathbb{R}^N}\frac{G(y,u)}{|x-y|^{\mu}}\,dy\bigg)g(x,u)\quad \textrm{in}\ \mathbb{R}^N, $$ where $0< \mu< N$, $N\geq3$, $g(x,u)$ is of critical growth in the sense of the Hardy-Littlewood-Sobolev inequality and $G(x,u)=\int^u_0g(x,s)\,ds$. By applying minimax procedure and perturbation technique, we obtain the existence of infinitely many solutions.
</p>projecteuclid.org/euclid.tmna/1563242560_20190715220257Mon, 15 Jul 2019 22:02 EDTOn exact multiplicity for a second order equation with radiation boundary conditionshttps://projecteuclid.org/euclid.tmna/1563242561<strong>Pablo Amster</strong>, <strong>Mariel P. Kuna</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 14 pp..</p><p><strong>Abstract:</strong><br/>
A second order ordinary differential equation with a superlinear term $g(x,u)$ under radiation boundary conditions is studied. Using a shooting argument, all the results obtained in the previous work [2] for a Painlevé II equation are extended. It is proved that the uniqueness or multiplicity of solutions depend on the interaction between the mapping $\frac {\partial g}{\partial u}(\cdot,0)$ and the first eigenvalue of the associated linear operator. Furthermore, two open problems posed in [2] regarding, on the one hand, the existence of sign-changing solutions and, on the other hand, exact multiplicity are solved.
</p>projecteuclid.org/euclid.tmna/1563242561_20190715220257Mon, 15 Jul 2019 22:02 EDTThe continuity of additive and convex functions which are upper bounded on non-flat continua in $\mathbb R^n$https://projecteuclid.org/euclid.tmna/1563242562<strong>Taras Banakh</strong>, <strong>Eliza Jabłońska</strong>, <strong>Wojciech Jabłoński</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 10 pp..</p><p><strong>Abstract:</strong><br/>
We prove that for a continuum $K\subset \mathbb R^n$ the sum $K^{+n}$ of $n$ copies of $K$ has non-empty interior in $\mathbb R^n$ if and only if $K$ is not flat in the sense that the affine hull of $K$ coincides with $\mathbb R^n$. Moreover, if $K$ is locally connected and each non-empty open subset of $K$ is not flat, then for any (analytic) non-meager subset $A\subset K$ the sum $A^{+n}$ of $n$ copies of $A$ is not meager in $\mathbb R^n$ (and then the sum $A^{+2n}$ of $2n$ copies of the analytic set $A$ has non-empty interior in $\mathbb R^n$ and the set $(A-A)^{+n}$ is a neighbourhood of zero in $\mathbb R^n$). This implies that a mid-convex function $f\colon D\to\mathbb R$ defined on an open convex subset $D\subset\mathbb R^n$ is continuous if it is upper bounded on some non-flat continuum in $D$ or on a non-meager analytic subset of a locally connected nowhere flat subset of $D$.
</p>projecteuclid.org/euclid.tmna/1563242562_20190715220257Mon, 15 Jul 2019 22:02 EDTNonlinear vector Duffing inclusions with no growth restriction on the orientor fieldhttps://projecteuclid.org/euclid.tmna/1563242563<strong>Nikolaos S. Papageorgiou</strong>, <strong>Calogero Vetro</strong>, <strong>Francesca Vetro</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 19 pp..</p><p><strong>Abstract:</strong><br/>
We consider nonlinear multivalued Dirichlet Duffing systems. We do not impose any growth condition on the multivalued perturbation. Using tools from the theory of nonlinear operators of monotone type, we prove existence theorems for the convex and the nonconvex problems. Also we show the existence of extremal trajectories and show that such solutions are $C_0^1(T,\mathbb{R}^N)$-dense in the solution set of the convex problem (strong relaxation theorem).
</p>projecteuclid.org/euclid.tmna/1563242563_20190715220257Mon, 15 Jul 2019 22:02 EDTRemoving isolated zeroes by homotopyhttps://projecteuclid.org/euclid.tmna/1563242564<strong>Adam Coffman</strong>, <strong>Jiří Lebl</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 22 pp..</p><p><strong>Abstract:</strong><br/>
Suppose that the inverse image of the zero vector by a continuous map $f\colon {\mathbb R}^n\to{\mathbb R}^q$ has an isolated point $P$. The existence of a continuous map $g$ which approximates $f$ but is nonvanishing near $P$ is equivalent to a topological property we call ``local inessentiality of zeros'', generalizing the notion of index zero for vector fields, the $q=n$ case. We consider the problem of constructing such an approximation $g$ and a continuous homotopy $F(x,t)$ from $f$ to $g$ through locally nonvanishing maps. If $f$ is a semialgebraic map, then there exists $F$ also semialgebraic. If $q=2$ and $f$ is real analytic with a locally inessential zero, then there exists a Hölder continuous homotopy $F(x,t)$ which, for $(x,t)\ne(P,0)$, is real analytic and nonvanishing. The existence of a smooth homotopy, given a smooth map $f$, is stated as an open question.
</p>projecteuclid.org/euclid.tmna/1563242564_20190715220257Mon, 15 Jul 2019 22:02 EDTMultiple normalized solutions for Choquard equations involving Kirchhoff type perturbationhttps://projecteuclid.org/euclid.tmna/1563760817<strong>Zeng Liu</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 23 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we study the existence of critical points of the $C^1$ functional \begin{equation*} E(u)=\frac{a}{2}\int_{\mathbb R^N}|\nabla u|^2dx+\frac{b}{4}\bigg(\int_{\mathbb R^N}|\nabla u|^2dx\bigg)^2-\frac{1}{2p}\int_{\mathbb R^N}(I_\alpha*|u|^p)|u|^pdx \end{equation*} under the constraint \begin{equation*} S_c=\bigg\{u\in H^1(\mathbb R^N)\ \bigg\vert\ \int_{\mathbb R^N}|u|^2dx=c^2\bigg\}, \end{equation*} where $a> 0$, $b> 0$, $N\geq3$, $\alpha\in(0,N)$, $ (N+\alpha)/{N}< p< (N+\alpha)/(N-2)$ and $I_{\alpha}$ is the Riesz Potential. When $p$ belongs to different ranges, we obtain the threshold values separating the existence and nonexistence of critical points of $E$ on $S_c$. We also study the behaviors of the Lagrange multipliers and the energies corresponding to the constrained critical points when $c\to 0$ and $c\to +\infty$, respectively.
</p>projecteuclid.org/euclid.tmna/1563760817_20190721220033Sun, 21 Jul 2019 22:00 EDTDecay rates for a viscoelastic wave equation with Balakrishnan-Taylor and frictional dampingshttps://projecteuclid.org/euclid.tmna/1563760818<strong>Baowei Feng</strong>, <strong>Yong Han Kang</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 23 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we are concerned with a viscoelastic wave equation with Balakrishnan-Taylor damping and frictional damping. By using the multiplier method and some properties of convex functions, we establish general energy decay rates of the equation without imposing any growth assumption near the origin on the frictional term and strongly weakening the usual assumptions on the relaxation term. Our stability result generalizes the earlier related results.
</p>projecteuclid.org/euclid.tmna/1563760818_20190721220033Sun, 21 Jul 2019 22:00 EDTA global multiplicity result for a very singular critical nonlocal equationhttps://projecteuclid.org/euclid.tmna/1563760819<strong>Jacques Giacomoni</strong>, <strong>Tuhina Mukherjee</strong>, <strong>Konijeti Sreenadh</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 26 pp..</p><p><strong>Abstract:</strong><br/>
In this article we show the global multiplicity result for the following nonlocal singular problem \begin{equation*} (-\Delta)^s u = u^{-q} + \lambda u^{{2^*_s}-1}, \quad u> 0 \quad \text{in } \Omega,\quad u = 0 \quad \mbox{in } \mathbb R^n \setminus\Omega, \tag{${\rm P}_\lambda$} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial \Omega$, $n > 2s$, $ s \in (0,1)$, $ \lambda > 0$, $q> 0$ satisfies $q(2s-1)< (2s+1)$ and $2^*_s=2n/(n-2s)$. Employing the variational method, we show the existence of at least two distinct weak positive solutions for $({\rm P}_\lambda)$ in $X_0$ when $\lambda \in (0,\Lambda)$ and no solution when $\lambda> \Lambda$, where $\Lambda> 0$ is appropriately chosen. We also prove a result of independent interest that any weak solution to $({\rm P}_\lambda)$ is in $C^\alpha(\mathbb R^n)$ with $\alpha=\alpha(s,q)\in (0,1)$. The asymptotic behaviour of weak solutions reveals that this result is sharp.
</p>projecteuclid.org/euclid.tmna/1563760819_20190721220033Sun, 21 Jul 2019 22:00 EDTTopologically Anosov plane homeomorphismshttps://projecteuclid.org/euclid.tmna/1563760820<strong>Gonzalo Cousillas</strong>, <strong>Jorge Groisman</strong>, <strong>Juliana Xavier</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 12 pp..</p><p><strong>Abstract:</strong><br/>
This paper deals with classifying the dynamics of topologically Anosov plane homeomorphisms. We prove that a topologically Anosov homeomorphism $f\colon\mathbb{R}^2 \to \mathbb{R}^2$ is conjugate to a homothety if it is the time one map of a flow. We also obtain results for the cases when the nonwandering set of $f$ reduces to a fixed point, or if there exists an open, connected, simply connected proper subset $U$ such that $\overline {f(U)} \subset {\rm Int} (U)$, and such that $$ \bigcup_{n\leq 0} f^n (U)= \mathbb{R}^2.$$ In the general case, we prove a structure theorem for the $\alpha$-limits of orbits with empty $\omega$-limit (or the $\omega$-limits of orbits with empty $\alpha$-limit).
</p>projecteuclid.org/euclid.tmna/1563760820_20190721220033Sun, 21 Jul 2019 22:00 EDTNonlocal Schrödinger equations for integro-differential operators with measurable kernelshttps://projecteuclid.org/euclid.tmna/1563760821<strong>Ronaldo C. Duarte</strong>, <strong>Marco A. S. Souto</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 24 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we investigate the existence of positive solutions for the problem $$ -\mathcal{L}_{K}u+V(x)u=f(u) $$% in $\mathbb R^N$, where $-\mathcal{L}_{K}$ is an integro-differential operator with measurable kernel $K$. Under apropriate hypotheses, we prove by variational methods that this equation has a nonnegative solution.
</p>projecteuclid.org/euclid.tmna/1563760821_20190721220033Sun, 21 Jul 2019 22:00 EDTA periodic bifurcation problem depending on a random variablehttps://projecteuclid.org/euclid.tmna/1564365634<strong>Mikhail Kamenskiĭ</strong>, <strong>Paolo Nistri</strong>, <strong>Paul Raynaud de Fitte</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 21 pp..</p><p><strong>Abstract:</strong><br/>
We consider an abstract bifurcation equation $P(x)+\varepsilon Q(x,\varepsilon, \omega)=0$, where $P$ and $Q$ are operators, $\varepsilon$ is the bifurcation parameter, $\omega \in \Omega$, is the random variable and $(\Omega, \mathcal{F})$ is a measurable space. The aim of the paper is to provide conditions on $P$ and $Q$ to ensure the existence, for any $\omega \in \Omega$, of a branch of solutions originating from the zeros of the operator $P$. We show that the considered abstract bifurcation is the model of a random autonomous periodically perturbed differential equation having the property that the unperturbed equation corresponding to $\varepsilon = 0$ has a limit cycle. As a consequence we obtain the existence, for any $\omega \in \Omega$, of a branch of periodic solutions of the perturbed equation emanating from the limit cycle.
</p>projecteuclid.org/euclid.tmna/1564365634_20190728220103Sun, 28 Jul 2019 22:01 EDTA three solution theorem for a singular differential equation with nonlinear boundary conditionshttps://projecteuclid.org/euclid.tmna/1564365635<strong>Rajendran Dhanya</strong>, <strong>Ratnasingham Shivaji</strong>, <strong>Byungjae Son</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 13 pp..</p><p><strong>Abstract:</strong><br/>
We study positive solutions to singular boundary value problems of the form: \begin{equation*} \begin{cases} -u'' = h(t) \dfrac{f(u)}{u^\alpha} &\text{for } t \in (0,1), \\ u(0) = 0, \\ u'(1) + c(u(1)) u(1) = 0, \end{cases} \end{equation*} where $0< \alpha< 1$, $h\colon(0,1]\rightarrow(0,\infty)$ is continuous such that $h(t)\leq {d}/{t^\beta}$ for some $d> 0$ and $\beta\in[0,1-\alpha)$ and $c\colon [0,\infty)\rightarrow [0,\infty)$ is continuous such that $c(s)s$ is nondecreasing. We assume that $f\colon[0,\infty)\rightarrow(0,\infty)$ is continuously differentiable such that $[(f(s)-f(0))/s^\alpha]+\tau s$ is strictly increasing for some $\tau\geq 0$ for $s\in(0,\infty)$. When there exists a pair of sub-supersolutions $(\psi,\phi)$ such that $0\leq \psi\leq\phi$, we first establish a minimal solution $\underline u$ and a maximal solution $\overline u$ in $[\psi,\phi]$. When there exist two pairs of sub-supersolutions $(\psi_1,\phi_1)$ and $(\psi_2,\phi_2)$ where $0\leq \psi_1 \leq \psi_2 \leq \phi_1$, $\psi_1 \leq \phi_2 \leq \phi_1$ with $\psi_2\not \leq \phi_2$, and $\psi_2$, $\phi_2$ are not solutions, we next establish the existence of at least three solutions $u_1$, $u_2$ and $u_3$ satisfying $u_1\in [\psi_1,\phi_2], u_2\in [\psi_2,\phi_1]$ and $u_3\in [\psi_1,\phi_1]\setminus ([\psi_1,\phi_2]\cup [\psi_2,\phi_1])$.
</p>projecteuclid.org/euclid.tmna/1564365635_20190728220103Sun, 28 Jul 2019 22:01 EDTSymmetric topological complexity for finite spaces and classifying spaceshttps://projecteuclid.org/euclid.tmna/1564365636<strong>Kohei Tanaka</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 17 pp..</p><p><strong>Abstract:</strong><br/>
We present a combinatorial approach to the symmetric motion planning in polyhedra using finite spaces. For a finite space $P$ and a positive integer $k$, we introduce two types of combinatorial invariants, $\mathrm{CC}^{S}_k(P)$ and $\mathrm{CC}^{\Sigma}_k(P)$, that are closely related to the design of symmetric robotic motions in the $k$-iterated barycentric subdivision of the associated simplicial complex $\mathcal{K}(P)$. For the geometric realization $\mathcal{B}(P)=|\mathcal{K}(P)|$, we show that the first $\mathrm{CC}^{S}_k(P)$ converges to Farber-Grant's symmetric topological complexity $\mathrm{TC}^{S}(\mathcal{B}(P))$ and the second $\mathrm{CC}^{\Sigma}_k(P)$ converges to Basabe-González-Rudyak-Tamaki's symmetrized topological complexity $\mathrm{TC}^{\Sigma}(\mathcal{B}(P))$ as $k$ becomes larger.
</p>projecteuclid.org/euclid.tmna/1564365636_20190728220103Sun, 28 Jul 2019 22:01 EDTAsymptotically almost periodic solutions of limit periodic difference systems with coefficients from commutative groupshttps://projecteuclid.org/euclid.tmna/1564365637<strong>Petr Hasil</strong>, <strong>Michal Veselý</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 21 pp..</p><p><strong>Abstract:</strong><br/>
We study the behaviour of solutions of limit periodic difference systems over (infinite) fields with absolute values. The considered systems are described by the coefficient matrices that belong to commutative groups whose boundedness is not required. In particular, we are interested in special systems with solutions which vanish at infinity or which are not asymptotically almost periodic. We obtain a transparent condition on the matrix groups which ensures that the special systems form a dense subset in the space of all considered systems, i.e. that, in any neighbourhood of any considered limit periodic system, there exists asystem which have non-asymptotically almost periodic or vanishing solutions. The presented results improve and extend known ones.
</p>projecteuclid.org/euclid.tmna/1564365637_20190728220103Sun, 28 Jul 2019 22:01 EDTThe long-time behavior of weighted $p$-Laplacian equationshttps://projecteuclid.org/euclid.tmna/1564365638<strong>Shan Ma</strong>, <strong>Hongtao Li</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 16 pp..</p><p><strong>Abstract:</strong><br/>
In this work we study weighted $p$-Laplacian equations in a bounded domain with a variable and generally non-smooth diffusion coefficient having at most a finite number of zeroes. The main attention is focused on the case that the diffusion coefficient $a(x)$ in such equations satisfies the inequality $\liminf_{x\to z}|x-z|^{-p}a(x)> 0$ for every $ z\in \overline\Omega$. We show the existence of weak solutions and global attractors in $L^2(\Omega)$, $L^q(\Omega)$ $(q\geq 2)$ and $D_0^{1,p}(\Omega)$, respectively.
</p>projecteuclid.org/euclid.tmna/1564365638_20190728220103Sun, 28 Jul 2019 22:01 EDTTwo homoclinic orbits for some second-order Hamiltonian systemshttps://projecteuclid.org/euclid.tmna/1569808829<strong>Patricio Cerda</strong>, <strong>Luiz F.O. Faria</strong>, <strong>Eduard Toon</strong>, <strong>Pedro Ubilla</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 18 pp..</p><p><strong>Abstract:</strong><br/>
This paper is concerned with the existence of homoclinic orbits for a class of second order Hamiltonian systems considering a non-periodic potential and a weaker Ambrosetti-Rabinowitz condition. By considering an auxiliary problem, we show the existence of two different approximative sequences of periodic solutions, the first one of mountain pass type and the second one of local minima. We obtain two different homoclinic orbits by passing to the limit in such sequences. As a relevant application, we obtain another homoclinic solution for the Hamiltonian system studied in [5].
</p>projecteuclid.org/euclid.tmna/1569808829_20190929220039Sun, 29 Sep 2019 22:00 EDTReminiscences about Professor Andrzej Granashttps://projecteuclid.org/euclid.tmna/1569808854<strong>Lech Górniewicz</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Volume 54, Number 1, 1--8.</p><p><strong>Abstract:</strong><br/>
n/a
</p>projecteuclid.org/euclid.tmna/1569808854_20190929220120Sun, 29 Sep 2019 22:01 EDTFixed point results in set $P_{h,e}$ with applications to fractional differential equationshttps://projecteuclid.org/euclid.tmna/1570413617<strong>Lingling Zhang</strong>, <strong>Hui Wang</strong>, <strong>Xiaoqiang Wang</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 30 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, without assuming operators to be continuous or compact, by employing monotone iterative technique on ordered Banach space, we at first establish new fixed point theorems for some kinds of nonlinear mixed monotone operators in set $P_{h,e}$. Then, we study a new form of fractional two point boundary value problem depending on a certain constant and give the existence and uniqueness of solutions. We also show that the unique solution exists in set $P_{h,e}$ or $P_{h}$ and can be uniformly approximated by constructing two iterative sequences for any initial values. At the end, a concrete example is given to illustrate our abstract results. The conclusions of this article enrich the fixed point theorems and provide new methods to deal with nonlinear differential equations.
</p>projecteuclid.org/euclid.tmna/1570413617_20191006220038Sun, 06 Oct 2019 22:00 EDTOn the existence of skyrmions in planar liquid crystalshttps://projecteuclid.org/euclid.tmna/1570413618<strong>Carlo Greco</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 20 pp..</p><p><strong>Abstract:</strong><br/>
The study of topologically nontrivial field configurations is an important topic in many branches of physics and applied sciences. In this paper we are interested to the existence of such structures, the so-called skyrmions, in the context of liquid crystals. More precisely, we consider a two-dimensional nematic or cholesteric liquid crystal. In the nematic case we use a Bogomol'nyi type decomposition in order to get a topological lower bound on the configurations with a given degree for the full Oseen-Frank energy functional, and so we can find a global minimum of degree $\pm 1$ for the energy. Then we consider the cholesteric case in presence of an electric field under the one constant approximation assumption, and, by using the concentration-compactness method, we prove the existence of a minimum again on the configurations of degree $\pm 1$, for sufficiently large electric fields.
</p>projecteuclid.org/euclid.tmna/1570413618_20191006220038Sun, 06 Oct 2019 22:00 EDTQuasilinear elliptic problems on non-reflexive Orlicz-Sobolev spaceshttps://projecteuclid.org/euclid.tmna/1570413619<strong>Edcarlos D. Silva</strong>, <strong>Marcos L. M. Carvalho</strong>, <strong>Kaye Silva</strong>, <strong>José V. Gonçalves</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 26 pp..</p><p><strong>Abstract:</strong><br/>
In the paper the existence, uniqueness and the multiplicity of solutions for a quasilinear elliptic problems driven by the $\Phi$-Laplacian operator is established. Here we consider the non-reflexive case taking into account the Orlicz and Orlicz-Sobolev framework. The non-reflexive case occurs when the $N$-function $\widetilde{\Phi}$ does not verify the $\Delta_{2}$-condition. In order to prove our main results we employ variational methods, regularity results and truncation arguments.
</p>projecteuclid.org/euclid.tmna/1570413619_20191006220038Sun, 06 Oct 2019 22:00 EDTSubspaces of interval maps related to the topological entropyhttps://projecteuclid.org/euclid.tmna/1570413620<strong>Xiaoxin Fan</strong>, <strong>Jian Li</strong>, <strong>Yini Yang</strong>, <strong>Zhongqiang Yang</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 14 pp..</p><p><strong>Abstract:</strong><br/>
For $a\in [0,+\infty)$, the function space $E_{\geq a}$ ($E_{> a}$; $E_{\leq a}$; $E_{< a}$) of all continuous maps from $[0,1]$ to itself whose topological entropies are larger than or equal to $a$ (larger than $a$; smaller than or equal to $a$; smaller than $a$) with the supremum metric is investigated. It is shown that the spaces $E_{\geq a}$ and $E_{> a}$ are homeomorphic to the Hilbert space $l_2$ and the spaces $E_{\leq a}$ and $E_{< a}$ are contractible. Moreover, the subspaces of $E_{\leq a}$ and $E_{< a}$ consisting of all piecewise monotone maps are homotopy dense in them, respectively.
</p>projecteuclid.org/euclid.tmna/1570413620_20191006220038Sun, 06 Oct 2019 22:00 EDT