Let $fcolon Mto M$ be a self-map of a compact manifold and $nin mathbb{N}$. The least number of $n$-periodic points in the smooth homotopy class of $f$ may be smaller than in the continuous homotopy class. We ask: for which self-maps $fcolon Mto M$ the two minima are the same, for each prescribed multiplicity? In the study of self-maps of tori and compact Lie groups a necessary condition appears. Here we give a criterion which helps to decide whether the necessary condition is also sufficient. We apply this result to show that for self-maps of the product of the lens space $M=L(3)times L(3)$ the necessary condition is also sufficient.
{"title":"Periodic points of self-maps of products of lens spaces $L(3)times L(3)$","authors":"J. Jezierski","doi":"10.12775/tmna.2022.053","DOIUrl":"https://doi.org/10.12775/tmna.2022.053","url":null,"abstract":"Let $fcolon Mto M$ be a self-map of a compact manifold and $nin mathbb{N}$.\u0000The least number of $n$-periodic points in the smooth homotopy class of $f$ may be smaller than in the continuous homotopy class. We ask: for which self-maps\u0000$fcolon Mto M$ the two minima are the same, for each prescribed multiplicity?\u0000 In the study of self-maps of tori and compact Lie groups a necessary condition appears.\u0000Here we give a criterion which helps to decide whether the necessary condition is also sufficient.\u0000We apply this result to show that for self-maps of the product of the lens space $M=L(3)times L(3)$ the necessary condition is also sufficient.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46267926","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the existence of multiple solutions for the $(p,q)$-quasilinear elliptic problem [ begin{cases} -Delta_p u -Delta_q u = g(x, u) + varepsilon h(x,u) & mbox{in } Omega, u=0 & mbox{on } partialOmega, end{cases} ] where $1< p< q< +infty$, $Omega$ is an open bounded domain of ${mathbb R}^N$, the nonlinearity $g(x,u)$ behaves at infinity as $|u|^{q-1}$, $varepsilonin{mathbb R}$ and $hin C(overlineOmegatimes{mathbb R},{mathbb R})$. In spite of the possible lack of a variational structure of this problem, from suitable assumptions on $g(x,u)$ and appropriate procedures and estimates, the existence of multiple solutions can be proved for small perturbations.
我们研究了$（p，q）$拟线性椭圆问题的多重解的存在性{cases}-Delta_p u-Delta_q u =g（x，u）+varepsilonh（x，u）&mbox｛in｝Omega，u=0&mbox｛on｝partial Omega，end｛cases｝]其中$1＜p＜q＜+infty$，$Omega$是$｛mathbb R｝^N$的开有界域，非线性$g（x，u）$在无穷大时表现为$|u|^｛q-1｝$，｛mathbb R｝$中的$varepsilonin和C中的$h（overlineOmegatimes｛math BB R｝，｛mathibb R｝）$。尽管这个问题可能缺乏变分结构，但通过对$g（x，u）$的适当假设以及适当的程序和估计，可以证明小扰动下存在多个解。
{"title":"Multiple solutions for perturbed quasilinear elliptic problems","authors":"R. Bartolo, A. M. Candela, A. Salvatore","doi":"10.12775/tmna.2022.069","DOIUrl":"https://doi.org/10.12775/tmna.2022.069","url":null,"abstract":"We investigate the existence of multiple solutions\u0000for the $(p,q)$-quasilinear elliptic problem\u0000[\u0000begin{cases}\u0000-Delta_p u -Delta_q u = g(x, u) + varepsilon h(x,u)\u0000& mbox{in } Omega,\u0000u=0 & mbox{on } partialOmega,\u0000 end{cases}\u0000]\u0000where $1< p< q< +infty$, $Omega$ is an open bounded domain of\u0000${mathbb R}^N$, the nonlinearity $g(x,u)$ behaves at infinity as $|u|^{q-1}$,\u0000$varepsilonin{mathbb R}$ and $hin C(overlineOmegatimes{mathbb R},{mathbb R})$.\u0000In spite of the possible lack of a variational structure of this problem,\u0000from suitable assumptions on $g(x,u)$ and\u0000appropriate procedures and estimates,\u0000the existence of multiple solutions can be proved for small perturbations.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42577350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $G$ be a torsion-free abelian group of rank two and let $phi$ be an endomorphism of $G$, called a rank-two emph{solenoidal endomorphism}. Then it is represented by a $2times 2$-matrix $M_phi$ with rational entries. The purpose of this article is to prove the following: The group, $mathrm{coker}(phi)$, of the cokernut of $phi$ is finite if and only if $M_phi$ is nonsingular, and if it is so, then we give an explicit formula for the order of $mathrm{coker}(phi)$, $[G:mathrm{im}(phi)]$, in terms of $p$-adic absolute values of the determinant of $M_phi$. Since $G$ is abelian, the Reidemeister number of $phi$ is equal to the order of the cokernut of $mathrm{id}-phi$ and, when it is finite, it is equal to the number of fixed points of the Pontryagin dual $widehatphi$ of $phi$. Thereby, we solve completely the problem raised in cite{Miles} of finding the possible sequences of periodic point counts for emph{all} endomorphisms of the rank-two solenoids.
{"title":"Rank-two solenoidal endomorphisms","authors":"K. Ha, Jong Bum Lee","doi":"10.12775/tmna.2022.063","DOIUrl":"https://doi.org/10.12775/tmna.2022.063","url":null,"abstract":"Let $G$ be a torsion-free abelian group of rank two and let\u0000$phi$ be an endomorphism of $G$, called a rank-two emph{solenoidal endomorphism}.\u0000Then it is represented by a $2times 2$-matrix $M_phi$ with rational entries.\u0000The purpose of this article is to prove the following:\u0000The group, $mathrm{coker}(phi)$, of the cokernut of $phi$ is finite\u0000if and only if $M_phi$ is nonsingular, and if it is so, then\u0000we give an explicit formula for the order of $mathrm{coker}(phi)$, $[G:mathrm{im}(phi)]$,\u0000in terms of $p$-adic absolute values of the determinant of $M_phi$.\u0000Since $G$ is abelian, the Reidemeister number of $phi$ is equal to the order of the cokernut of $mathrm{id}-phi$ and, when it is finite, \u0000it is equal to the number of fixed points of the Pontryagin dual $widehatphi$ of $phi$.\u0000Thereby, we solve completely the problem raised in cite{Miles} of finding the possible sequences of periodic point counts\u0000for emph{all} endomorphisms of the rank-two solenoids.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43163203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The inverse involution on a Lie group $G$ is the periodic $2$ transformation $gamma $ that sends each element $gin G$ to its inverse $g^{-1}$. The variety of the fixed point set $Fix(gamma )$ is of importance for the relevances with Morse function on the Lie group $G$, and the $G$-representations of the cyclic group $mathbb{Z}_{2}$. In this paper we develop an approach to calculate the diffeomorphism types of the fixed point sets $Fix(gamma)$ for the simple Lie groups.
{"title":"The fixed point set of the inverse involution on a Lie group","authors":"H. Duan, Shali Liu","doi":"10.12775/tmna.2022.012","DOIUrl":"https://doi.org/10.12775/tmna.2022.012","url":null,"abstract":"The inverse involution on a Lie group $G$ is the periodic $2$ transformation\u0000$gamma $ that sends each element $gin G$ to its inverse $g^{-1}$. The\u0000variety of the fixed point set $Fix(gamma )$ is of importance for the\u0000relevances with Morse function on the Lie group $G$, and the $G$-representations\u0000of the cyclic group $mathbb{Z}_{2}$. \u0000In this paper we develop an approach to calculate the diffeomorphism types of the fixed point sets $Fix(gamma)$ for the simple Lie groups.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48534835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The index $mathrm{ind}(mathbf{F})$ of a fixed point class $mathbf{F}$ is a classical invariant in the Nielsen fixed point theory. In the recent paper cite{ZZ}, the authors introduced a new invariant $mathrm{ichr}(mathbf{F})$ called the improved characteristic, and proved that $mathrm{ind}(mathbf{F})leq mathrm{ichr}(mathbf{F})$ for all fixed point classes of $pi_1$-injective selfmaps of connected finite graphs. In this note, we show that the two homotopy invariants mentioned above are exactly the same. Since the improved characteristic is totally determined by the endomorphism of the fundamental group, we give a group-theoretical approach to compute indices of fixed point classes of graph selfmaps. As a consequence, we give a new criterion of a fixed point, which extends the one due to Gaboriau, Jaeger, Levitt and Lustig.
{"title":"Fixed point indices and fixed words at infinity of selfmaps of graphs II","authors":"Qiang Zhang, Xuezhi Zhao","doi":"10.12775/tmna.2022.007","DOIUrl":"https://doi.org/10.12775/tmna.2022.007","url":null,"abstract":"The index $mathrm{ind}(mathbf{F})$ of a fixed point class $mathbf{F}$ is a classical invariant in the Nielsen fixed point theory. In the recent paper cite{ZZ}, the authors introduced a new invariant $mathrm{ichr}(mathbf{F})$ called the improved characteristic, and proved that $mathrm{ind}(mathbf{F})leq mathrm{ichr}(mathbf{F})$ for all fixed point classes of $pi_1$-injective selfmaps of connected finite graphs. In this note, we show that the two homotopy invariants mentioned above are exactly the same. Since the improved characteristic is totally determined by the endomorphism of the fundamental group, we give a group-theoretical approach to compute indices of fixed point classes of graph selfmaps. As a consequence, we give a new criterion of a fixed point, which extends the one due to Gaboriau, Jaeger, Levitt and Lustig.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136042197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that for a given finite graph $G$ without loop edges and isolated vertices, there exists an embedding of a closed orientable surface in $mathbb{R}^3$ such that the Reeb graph of the associated height function has the structure of $G$. In particular, this gives a positive answer to the corresponding question posed by Masumoto and Saeki in 2011. We also give a criterion for a given surface to admit such a realization of a given graph, and study the problem in the class of Morse functions and in the class of round Morse-Bott functions. In the case of realization up to homeomorphism, the height function can be chosen Morse-Bott; we estimate from below the number of its critical circles and the number of its isolated critical points in terms of the graph structure.
{"title":"Realization of a graph as the Reeb graph of a height function on an embedded surface","authors":"Irina Gelbukh","doi":"10.12775/tmna.2021.058","DOIUrl":"https://doi.org/10.12775/tmna.2021.058","url":null,"abstract":"We show that for a given finite graph $G$ without loop edges and isolated vertices, there exists an embedding of a closed orientable surface in $mathbb{R}^3$\u0000such that the Reeb graph of the associated height function has the structure of $G$.\u0000In particular, this gives a positive answer to the corresponding question posed by Masumoto and Saeki in 2011.\u0000We also give a criterion for a given surface to admit such a realization of a given graph, and study the problem in the class of Morse functions\u0000and in the class of round Morse-Bott functions.\u0000In the case of realization up to homeomorphism, the height function can be chosen Morse-Bott;\u0000we estimate from below the number of its critical circles and the number of its isolated critical points in terms of the graph structure.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":"55 6","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41267184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Francisco Odair de Paiva, O. Miyagaki, Adilson E. Presoto
We are concerned with of existence of solutions to the semilinear elliptic problem $$ begin{cases} - Delta u=lambda_{k}u+u^3 &text{in } Omega, u= 0 &text{on }partial Omega, end{cases} $$% in a bounded domain $Omega subset mathbb{R}^{4}$. Here $lambda_k$ is an eigenvalue of the $-Delta$ in $H_0^1(Omega)$. We prove that this problem has a nontrivial solution.
{"title":"Existence of solutions for the Brezis-Nirenberg problem","authors":"Francisco Odair de Paiva, O. Miyagaki, Adilson E. Presoto","doi":"10.12775/tmna.2022.029","DOIUrl":"https://doi.org/10.12775/tmna.2022.029","url":null,"abstract":"We are concerned with of existence of solutions to the semilinear elliptic problem\u0000$$\u0000 begin{cases}\u0000 - Delta u=lambda_{k}u+u^3 &text{in } Omega, \u0000 u= 0 &text{on }partial Omega,\u0000 end{cases}\u0000$$%\u0000in a bounded domain $Omega subset mathbb{R}^{4}$. Here $lambda_k$\u0000is an eigenvalue of the $-Delta$ in $H_0^1(Omega)$. We prove that this problem has a nontrivial solution.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43340171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The existence of solutions for a huge class of superlinear elliptic problems involving fourth-order elliptic problems defined on bounded domains under Navier boundary conditions is established. To this end we do not apply the well-known Ambrosetti-Rabinowitz condition. Instead, we assume that the nonlinear term is nonquadratic at infinity. Furthermore, the nonlinear term is a concave-superlinear function which can be indefinite in sign. In order to apply variational methods we employ some delicate arguments recovering some kind of compactness.
{"title":"Fourth-order elliptic problems involving concave-superlinear nonlinearities","authors":"T. Cavalcante, Edcarlos D. Silva","doi":"10.12775/tmna.2022.011","DOIUrl":"https://doi.org/10.12775/tmna.2022.011","url":null,"abstract":"The existence of solutions for a huge class of superlinear elliptic problems involving fourth-order elliptic problems defined on bounded domains\u0000under Navier boundary conditions is established. To this end we do not apply the well-known\u0000Ambrosetti-Rabinowitz condition. Instead, we assume that the nonlinear term\u0000is nonquadratic at infinity. Furthermore, the nonlinear term is a concave-superlinear function which can be indefinite in sign. \u0000In order to apply variational methods we employ some delicate arguments recovering some kind of compactness.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45984491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Anderson L. A. de Araujo, Luiz F. O. Faria, S. Alarcón, Leonelo Iturriaga
We study existence and multiplicity of radial solutions for some quasilinear elliptic problems involving the operator $L_N=Delta - xcdot nabla$ on $mathbb{R}^Nsetminus B_1$, where $Delta$ is the Laplacian, $xcdot nabla$ is an unbounded drift term, $Ngeq 3$ and $B_1$ is the unit ball centered at the origin. We consider: (i) Eigenvalue problems, and (ii) Problems involving a nonlinearity of concave and convex type. On the first class of problems we get a compact embedding result, whereas on the second, we address the well-known question of Ambrosetti, Brezis and Cerami from 1993 concerning the existence of two positive solutions for some problems involving the supercritical Sobolev exponent in symmetric domains for the Laplacian. Specifically, we providelinebreak a new approach of answering the ABC-question for elliptic problems with unbounded coefficients in exterior domains and we find asymptotic properties of the radial solutions. Furthermore, we study the limit case, namely when nonlinearity involves a sublinear term and a linear term. As far as we know, this is the first work that deals with such a case, even for the Laplacian. In our approach, we use both topological and variational arguments.
{"title":"On radial solutions for some elliptic equations involving operators with unbounded coefficients in exterior domains","authors":"Anderson L. A. de Araujo, Luiz F. O. Faria, S. Alarcón, Leonelo Iturriaga","doi":"10.12775/tmna.2022.026","DOIUrl":"https://doi.org/10.12775/tmna.2022.026","url":null,"abstract":"We study existence and multiplicity of radial solutions for some quasilinear elliptic\u0000 problems involving the operator $L_N=Delta - xcdot nabla$ on\u0000$mathbb{R}^Nsetminus B_1$, where $Delta$ is the Laplacian,\u0000 $xcdot nabla$ is an unbounded drift term, $Ngeq 3$ and $B_1$ is the unit ball centered at the origin.\u0000We consider: (i) Eigenvalue problems, and (ii) Problems involving a nonlinearity of concave and convex type. On the first class of problems we get a compact\u0000 embedding result, whereas on the second, we address the well-known question\u0000of Ambrosetti, Brezis and Cerami from 1993 concerning the existence of two positive\u0000 solutions for some problems involving the supercritical Sobolev exponent in symmetric domains for the Laplacian. Specifically, we providelinebreak a new approach of\u0000 answering the ABC-question for elliptic problems with unbounded coefficients in\u0000 exterior domains and we find asymptotic properties of the radial solutions. Furthermore, we study the limit case, namely when nonlinearity involves\u0000a sublinear term and a linear term. As far as we know, this is the first work that deals with such a case, even for the Laplacian. In our approach,\u0000 we use both topological and variational arguments.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47651785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yasir Arfat, P. Kumam, M. A. A. Khan, P. S. Ngiamsunthorn
In this paper, we consider an accelerated shrinking projection based parallel hybrid algorithm to study the split null point problem (SNPP) associated with the maximal monotone operators in Hilbert spaces. The analysis of the proposed algorithm provides strong convergence results under suitable set of control conditions as well as viability with the help of a numerical experiment. The results presented in this paper improve various existing results in the current literature.
{"title":"An accelerated variant of the projection based parallel hybrid algorithm for split null point problems","authors":"Yasir Arfat, P. Kumam, M. A. A. Khan, P. S. Ngiamsunthorn","doi":"10.12775/tmna.2022.015","DOIUrl":"https://doi.org/10.12775/tmna.2022.015","url":null,"abstract":"In this paper, we consider an accelerated shrinking projection based parallel hybrid algorithm to study the split null point problem (SNPP) associated with the maximal monotone operators in Hilbert spaces. The analysis of the proposed algorithm provides strong convergence results under suitable set of control conditions as well as viability with the help of a numerical experiment. The results presented in this paper improve various existing results in the current literature.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44746966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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