In this paper, we study the relation between the Lusternik-Schnirelmann category and the topological complexity of two closed oriented manifolds connected by a degree one map.
{"title":"Maps of degree one, LS category and higher topological complexities","authors":"Yuli B. Rudyak, Soumen Sarkar","doi":"10.12775/tmna.2021.051","DOIUrl":"https://doi.org/10.12775/tmna.2021.051","url":null,"abstract":"In this paper, we study the relation between the Lusternik-Schnirelmann category\u0000and the topological complexity of two closed oriented manifolds connected by a degree one map.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49353195","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 $Sigma$ be a $C^3$ compact symmetric convex hypersurface in $mathbb{R}^{8}$. We prove that when $Sigma$ carries exactly four geometrically distinct closed characteristics, then there are at least two irrationally elliptic closed characteristics on $Sigma$.
{"title":"Irrationally elliptic closed characteristics on symmetric compact convex hypersurfaces in R^8","authors":"Wen Wang","doi":"10.12775/tmna.2021.057","DOIUrl":"https://doi.org/10.12775/tmna.2021.057","url":null,"abstract":"Let $Sigma$ be a $C^3$ compact symmetric convex hypersurface in\u0000$mathbb{R}^{8}$. We prove that when $Sigma$ carries exactly four\u0000geometrically distinct closed characteristics, then there are at least two\u0000irrationally elliptic closed characteristics on $Sigma$.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47487568","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}
Nikolaos S. Papageorgiou, Vicentiu D. Rădulescu, Dušan D. Repovš
We consider a Neumann boundary value problem driven by the anisotropic $(p,q)$-Laplacian plus a parametric potential term. The reaction is ``superlinear". We prove a global (with respect to the parameter) multiplicity result for positive solutions. Also, we show the existence of a minimal positive solution and finally, we produce a nodal solution.
{"title":"Global multiplicity for parametric anisotropic Neumann (p,q)-equations","authors":"Nikolaos S. Papageorgiou, Vicentiu D. Rădulescu, Dušan D. Repovš","doi":"10.12775/TMNA.2022.010","DOIUrl":"https://doi.org/10.12775/TMNA.2022.010","url":null,"abstract":"We consider a Neumann boundary value problem driven by the anisotropic\u0000 $(p,q)$-Laplacian plus a parametric potential term. \u0000The reaction is ``superlinear\". We prove a global (with respect to the parameter) multiplicity result for positive solutions. \u0000Also, we show the existence of a minimal positive solution and finally, we produce\u0000 a nodal solution.","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":"44218099","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 consider a geodesic problem in a manifold endowed with a Randers-Kropina metric. This is a type of a singular Finsler metric arising both in the description of the lightlike vectors of a spacetime endowed with a causal Killing vector field and in the Zermelo's navigation problem with a wind represented by a vector field having norm not greater than one. By using Lusternik-Schnirelman theory, we prove existence of infinitely many geodesics between two given points when the manifold is not contractible. Due to the type of non-holonomic constraints that the velocity vectors must satisfy, this is achieved thanks to some recent results about the homotopy type of the set of solutions of an affine control system associated with a totally non-integrable distribution.
{"title":"Multiple connecting geodesics of a Randers-Kropina metric via homotopy theory for solutions of an affine control system","authors":"E. Caponio, M. Javaloyes, A. Masiello","doi":"10.12775/tmna.2022.066","DOIUrl":"https://doi.org/10.12775/tmna.2022.066","url":null,"abstract":"We consider a geodesic problem in a manifold endowed with\u0000a Randers-Kropina metric. This is a type of a singular Finsler metric arising both\u0000in the description of the lightlike vectors of a spacetime endowed with a causal Killing vector field and in the Zermelo's navigation problem with a wind represented\u0000by a vector field having norm not greater than one.\u0000By using Lusternik-Schnirelman theory, we prove existence of infinitely many\u0000geodesics between two given points when the manifold is not contractible.\u0000Due to the type of non-holonomic constraints that the velocity vectors must satisfy,\u0000this is achieved thanks to some recent results about the homotopy type of the set of solutions of an affine control system associated with \u0000a totally non-integrable distribution.","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":"43067152","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}
In this paper, we construct multiple normalized solutions of the following from quasi-linear Schrödinger equation: -Delta u-Delta(|u|^{2})u-mu u=|u|^{p-2}u, quadtext{in } mathbb{R}^N, subject to a mass-subcritical constraint. In order to overcome non-smoothness of the associated variational formulation we make use of the dual approach. The constructed solutions possess energies being clustered at $0$ level which makes it difficult to use existing methods for non-smooth variational problems such as the variational perturbation approach.
{"title":"Multiple normalized solutions for a quasi-linear Schrödinger equation via dual approach","authors":"Lin Zhang, Yongqing Li, Zhi-Qiang Wang","doi":"10.12775/tmna.2022.052","DOIUrl":"https://doi.org/10.12775/tmna.2022.052","url":null,"abstract":"In this paper, we construct multiple normalized solutions of the following from quasi-linear Schrödinger equation:\u0000\u0000-Delta u-Delta(|u|^{2})u-mu u=|u|^{p-2}u, quadtext{in } mathbb{R}^N,\u0000\u0000subject to a mass-subcritical constraint. In order to overcome non-smoothness of the associated variational formulation we make use of the dual approach.\u0000The constructed solutions possess energies being clustered at $0$ level which makes it difficult to use existing methods \u0000for non-smooth variational problems such as the variational perturbation approach.","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":"41422826","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 prove a Brooks type coincidence minimization result for boundary preserving maps on compact surfaces with boundary. As an application we obtain non-boundary Wecken results for pairs of maps $f,gcolon (X,partial X) to (X,partial X)$ for most surfaces $X$.
{"title":"The Wecken problem for coincidences of boundary preserving surface maps","authors":"M. R. Kelly","doi":"10.12775/tmna.2022.061","DOIUrl":"https://doi.org/10.12775/tmna.2022.061","url":null,"abstract":"We prove a Brooks type coincidence minimization result for boundary preserving maps on compact surfaces with boundary. \u0000 As an application we obtain non-boundary Wecken results for pairs of maps \u0000 $f,gcolon (X,partial X) to (X,partial X)$ for most surfaces $X$.","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":"42674431","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}
Assuming that there is a known (trivial) branch of solutions of a parameterized family of equations, topological bifurcation studies the topological invariants of the linearized equations along the trivial branch whose nonvanishing entails the appearance of bifurcation from the trivial branch. We introduce here some refined topological invariants for semilinear elliptic boundary value problems equivariant with respect to the action of the circle $U(1)$ allowing to improve, in this case, some previously obtained bifurcation criteria for general nonlinear elliptic boundary value problems.
{"title":"Bifurcation of solutions of $U(1)$-equivariant semilinear boundary value problems","authors":"J. Pejsachowicz","doi":"10.12775/tmna.2022.056","DOIUrl":"https://doi.org/10.12775/tmna.2022.056","url":null,"abstract":"Assuming that there is a known (trivial) branch of solutions of a parameterized family\u0000of equations, topological bifurcation studies the topological invariants of the linearized equations along the trivial branch whose nonvanishing entails the\u0000appearance of bifurcation from the trivial branch. We introduce here some refined topological invariants for semilinear elliptic boundary value problems equivariant\u0000with respect to the action of the circle $U(1)$ allowing to improve, in this case, \u0000some previously obtained bifurcation criteria for general nonlinear elliptic boundary value problems.","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":"42997016","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}
In this paper we describe the equivariant cobordism classification of smooth actions $(M^m,phi)$ of the group $G=mathbb{Z}_2^k$ on closed smooth $m$-dimensional manifolds $M^m$, for which the fixed point set of the action is a connected manifold of dimension n and $2^k n - 2^{k-1} leq m < 2^k n$. Here, $mathbb{Z}_2^k$ is considered as the group generated by $k$ commuting smooth involutions defined on $M^m$. This generalizes a previous result of 2008 of the second author, who obtained this type of classification for $k=2$ and $m=4n-1$ or $m=4n-2$.
{"title":"$Z_2^k$-actions with connected fixed point set","authors":"J. C. Costa, P. Pergher, Renato M. Moraes","doi":"10.12775/tmna.2022.048","DOIUrl":"https://doi.org/10.12775/tmna.2022.048","url":null,"abstract":"In this paper we describe the equivariant cobordism classification of smooth actions $(M^m,phi)$ of the group $G=mathbb{Z}_2^k$ on closed smooth\u0000$m$-dimensional manifolds $M^m$, for which the fixed point set of the action is a connected manifold of dimension n and $2^k n - 2^{k-1} leq m < 2^k n$.\u0000Here, $mathbb{Z}_2^k$ is considered as the group generated by $k$ commuting smooth involutions defined on $M^m$. \u0000This generalizes a previous result of 2008 of the second author, who obtained this type of classification for $k=2$ and $m=4n-1$ or $m=4n-2$.","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":"42770974","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}
A root of an $n$-valued map $varphi colon X to D_n(Y)$ at $a in Y$ is a point $x in X$ such that $a in varphi(x)$. We lift the map $varphi$ to a split $n$-valued map of finite covering spaces and its single-valued factors are defined to be the lift factors of $varphi$. We describe the relationship between the root classes at $a$ of the lift factors and those of $varphi$. We define the Reidemeister root number $RR (varphi)$ in terms of the Reidemeister root numbers of the lift factors. We prove that the Reidemeister root number is a homotopy invariant upper bound for the Nielsen root number $NR(varphi)$, the number of essential root classes, and we characterize essentiality by means of an equivalence relation called the $Phi$-relation. A theorem of Brooks states that a single-valued map to a closed connected manifold is root-uniform, that is, its root classes are either all essential or all inessential. It follows that if $Y$ is a closed connected manifold, then the lift factors are root-uniform and we relate this property to the root-uniformity of $varphi$. If $X$ and $Y$ are closed connected oriented manifolds of the same dimension then, by means of the lift factors, we define an integer-valued index of a root class of $varphi$ that is invariant under $Phi$-relation and this implies that if its index is non-zero, then the root class is essential.
{"title":"Lift factors for the Nielsen root theory on $n$-valued maps","authors":"RobertF Brown, D. Gonçalves","doi":"10.12775/tmna.2022.017","DOIUrl":"https://doi.org/10.12775/tmna.2022.017","url":null,"abstract":"A root of an $n$-valued map $varphi colon X to D_n(Y)$ at $a in Y$\u0000 is a point $x in X$ such that $a in varphi(x)$. We lift the map \u0000$varphi$ to a split $n$-valued map of finite covering spaces and\u0000 its single-valued factors are defined to be the lift factors of \u0000$varphi$. We describe the relationship between the root classes at $a$ \u0000of the lift factors and those of $varphi$. We define the \u0000Reidemeister root number $RR (varphi)$ in terms \u0000of the Reidemeister root numbers of the lift factors. We prove that the\u0000 Reidemeister root number is a homotopy invariant upper bound for \u0000the Nielsen root number $NR(varphi)$, the number of essential root classes,\u0000and we characterize essentiality by means of an \u0000equivalence relation called the $Phi$-relation. A theorem of Brooks states that \u0000a single-valued map to a closed connected manifold is root-uniform, that is,\u0000 its root classes are either all essential or all inessential. It \u0000follows that if $Y$ is a closed connected manifold, then the lift factors are \u0000root-uniform and we relate this property to the root-uniformity of $varphi$. \u0000 If $X$ and $Y$ are closed connected oriented manifolds of the same dimension then, by means of the lift factors, we define an integer-valued \u0000index of a root class of $varphi$ that is invariant under $Phi$-relation and this \u0000implies that if its index is non-zero, then the root class is essential.","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":"49352063","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}
In this paper, we consider a class of $alpha$-$(h,e)$-convex operators defined in set $P_{h,e}$ and applications with $alpha> 1$. Without assuming the operator to be completely continuous or compact, by employing cone theory and monotone iterative technique, we not only obtain the existence and uniqueness of fixed point of $alpha$-$(h,e)$-convex operators, but also construct two monotone iterative sequences to approximate the unique fixed point. At last, we investigate the existence-uniqueness of a nontrivial solution for Riemann-Liouville fractional differential equations integral boundary value problems by employing $alpha$-$(h,e)$-convex operators fixed point theorem.
{"title":"$alpha$-$(h,e)$-convex operators and applications for Riemann-Liouville fractional differential equations","authors":"Bibo Zhou, Lingling Zhang","doi":"10.12775/tmna.2022.014","DOIUrl":"https://doi.org/10.12775/tmna.2022.014","url":null,"abstract":"In this paper, we consider a class of $alpha$-$(h,e)$-convex operators defined\u0000 in set $P_{h,e}$ and applications with $alpha> 1$. Without assuming the operator\u0000to be completely continuous or compact, by employing cone theory and monotone\u0000 iterative technique, we not only obtain the existence and uniqueness of fixed point\u0000of $alpha$-$(h,e)$-convex operators, but also construct two monotone iterative\u0000 sequences to approximate the unique fixed point. At last, we investigate the\u0000 existence-uniqueness of a nontrivial solution for Riemann-Liouville fractional differential equations integral boundary value problems by employing\u0000$alpha$-$(h,e)$-convex operators fixed point theorem.","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":"45209777","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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