A method which is a combination of the Halpern method and proximal point method (PPA) is introduced in this paper. It is proved that the sequence of iterates generated by our method converges strongly to a point which is a common solution to some monotone inclusion problem and fixed point problem in CAT$(0)$ spaces under some appropriate conditions.
{"title":"Halpern-type proximal point algorithm in CAT(0) spaces","authors":"C. C. Okeke","doi":"10.12775/tmna.2022.038","DOIUrl":"https://doi.org/10.12775/tmna.2022.038","url":null,"abstract":"A method which is a combination of the Halpern method and proximal point method\u0000(PPA) is introduced in this paper. It is proved that the sequence of iterates generated by our method converges strongly \u0000to a point which is a common solution to some monotone inclusion problem and fixed point problem in CAT$(0)$ spaces under some appropriate conditions.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48696183","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 a 1961 paper by E. Rakotch it was shown that a uniformly locally contractive mapping has a fixed point. In the present paper we show that for such a mapping, the fixed point problem is well posed and that inexact iterates of such a mapping converge to its unique fixed point, uniformly on bounded sets. Using the porosity notion, we also show that most uniformly locally nonexpansive mappings are, in fact, uniformly locally contractive.
{"title":"Convergence and well-posedness properties of uniformly locally contractive mappings","authors":"S. Reich, A. Zaslavski","doi":"10.12775/tmna.2022.035","DOIUrl":"https://doi.org/10.12775/tmna.2022.035","url":null,"abstract":"In a 1961 paper by E. Rakotch it was shown that a uniformly locally contractive\u0000 mapping has a fixed point. In the present paper we show that for such a mapping,\u0000 the fixed point problem is well posed and that inexact iterates of such a mapping\u0000 converge to its unique fixed point, uniformly on bounded sets. Using the porosity\u0000 notion, we also show that most uniformly locally nonexpansive mappings are, in fact, uniformly locally contractive.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46706953","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 characterize the sequentially weakly lower semicontinuity of the parameter-depending energy functional associated with the critical Kirchhoff problem in context of (sub)Riemannian manifolds. We also present some spectral gap and convexity results.
{"title":"Lower semicontinuity of Kirchhoff-type energy functionals and spectral gaps on (sub)Riemannian manifolds","authors":"Csaba Farkas, S. Kajántó, C. Varga","doi":"10.12775/tmna.2022.034","DOIUrl":"https://doi.org/10.12775/tmna.2022.034","url":null,"abstract":"In this paper we characterize the sequentially weakly lower semicontinuity\u0000of the parameter-depending energy functional associated with the critical Kirchhoff\u0000problem in context of (sub)Riemannian manifolds.\u0000We also present some spectral gap and convexity results.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44069069","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}
Pub Date : 2023-07-13eCollection Date: 2023-01-01DOI: 10.3897/zookeys.1169.105901
So-Yeon Shin, Chang-Mok Lee, Jun-Haeng Heo, Young-Hyo Kim
A new species of the genus Pseudaeginella Mayer, 1890 belonging to the family Caprellidae Leach, 1814 was collected from the South Sea in Korea. Pseudaeginellacarinaspinosasp. nov. is morphologically similar to related congeners belonging to the genera Paradeutella Mayer, 1890 and Pseudaeginella, in having dorsal projections on pereonites, triarticulate mandibular palp, small or absent molar, and uniarticulate pereopods 3 and 4. However, this new species is distinguished from its congeners by the position and size of dorsal projection. This is the first record of Pseudaeginella from the Northwest Pacific region, including Korea, and a key to species of the genus Pseudaeginella is also provided.
{"title":"First record of the genus <i>Pseudaeginella</i> Mayer, 1890 (Crustacea, Amphipoda, Caprellidae) with a new species from Korean waters.","authors":"So-Yeon Shin, Chang-Mok Lee, Jun-Haeng Heo, Young-Hyo Kim","doi":"10.3897/zookeys.1169.105901","DOIUrl":"10.3897/zookeys.1169.105901","url":null,"abstract":"<p><p>A new species of the genus <i>Pseudaeginella</i> Mayer, 1890 belonging to the family Caprellidae Leach, 1814 was collected from the South Sea in Korea. <i>Pseudaeginellacarinaspinosa</i><b>sp. nov.</b> is morphologically similar to related congeners belonging to the genera <i>Paradeutella</i> Mayer, 1890 and <i>Pseudaeginella</i>, in having dorsal projections on pereonites, triarticulate mandibular palp, small or absent molar, and uniarticulate pereopods 3 and 4. However, this new species is distinguished from its congeners by the position and size of dorsal projection. This is the first record of <i>Pseudaeginella</i> from the Northwest Pacific region, including Korea, and a key to species of the genus <i>Pseudaeginella</i> is also provided.</p>","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":"9 1","pages":"163-174"},"PeriodicalIF":1.3,"publicationDate":"2023-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10838568/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88751188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we deal with the following Kirchhoff-type equation: begin{equation*} -bigg(1 +int_{mathbb{R}^{3}}|nabla u|^{2}dxbigg) Delta u +frac{A}{|x|^{alpha}}u =f(u),quad xinmathbb{R}^{3}, end{equation*} where $A> 0$ is a real parameter and $alphain(0,1)cup ({4}/{3},2)$. Remark that $f(u)=|u|^{2_{alpha}^{*}-2}u +lambda|u|^{q-2}u +|u|^{4}u$, where $lambda> 0$, $qin(2_{alpha}^{*},6)$, $2_{alpha}^{*}=2+{4alpha}/({4-alpha})$ is the embedding bottom index, and $6$ is the embedding top index and Sobolev critical exponent. We point out that the nonlinearity $f$ is the almost ``optimal'' choice. First, for $alphain({4}/{3},2)$, applying the generalized version of Lions-type theorem and the Nehari manifold, we show the existence of nonnegative Nehari-type ground sate solution for above equation. Second, for $alphain(0,1)$, using the generalized version of Lions-type theorem and the Pohov{z}aev manifold, we establish the existence of nonnegative Pohov{z}aev-type ground state solution for above equation. Based on our new generalized version of Lions-type theorem, our works extend the results in Li-Su [Z. Angew. Math. Phys. {bf 66} (2015)].
{"title":"Critical Kirchhoff-type equation with singular potential","authors":"Yujian Su, Senli Liu","doi":"10.12775/tmna.2022.051","DOIUrl":"https://doi.org/10.12775/tmna.2022.051","url":null,"abstract":"In this paper, we deal with the following Kirchhoff-type equation:\u0000begin{equation*}\u0000-bigg(1\u0000+int_{mathbb{R}^{3}}|nabla u|^{2}dxbigg)\u0000Delta u\u0000+frac{A}{|x|^{alpha}}u\u0000=f(u),quad xinmathbb{R}^{3},\u0000end{equation*}\u0000where $A> 0$ is a real parameter and $alphain(0,1)cup ({4}/{3},2)$.\u0000Remark that $f(u)=|u|^{2_{alpha}^{*}-2}u +lambda|u|^{q-2}u\u0000+|u|^{4}u$,\u0000where $lambda> 0$, $qin(2_{alpha}^{*},6)$,\u0000$2_{alpha}^{*}=2+{4alpha}/({4-alpha})$\u0000is the embedding bottom index, and $6$ is the embedding top index and Sobolev critical exponent.\u0000We point out that the nonlinearity $f$ is the almost ``optimal'' choice.\u0000First, for $alphain({4}/{3},2)$, applying the generalized version of Lions-type\u0000 theorem and the Nehari manifold, we show the existence of nonnegative\u0000Nehari-type ground sate solution for above equation. Second, for $alphain(0,1)$,\u0000 using the generalized version of Lions-type theorem and the Pohov{z}aev\u0000 manifold, we establish the existence of nonnegative Pohov{z}aev-type ground\u0000state solution for above equation. Based on our new generalized version\u0000of Lions-type theorem, our works extend the results in Li-Su [Z. Angew. Math. Phys. {bf 66} (2015)].","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42518484","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 use Morse theory to study existence of nontrivial solutions to the following Schrödinger system with linear and nonlinear couplings which arises from Bose-Einstein condensates: $$ begin{cases} -Delta u+lambda_{1} u+kappa v=mu_{1} u^{3}+beta uv^{2} & text{in } Omega, -Delta v+lambda_{2} v+kappa u=mu_{2} v^{3}+beta vu^{2} & text{in } Omega, u=v=0 & text{on } partialOmega, end{cases} $$ where $Omega$ is a bounded smooth domain in $mathbb{R}^{N}$($N=2,3$), $lambda_{1},lambda_{2},mu_{1},mu_{2} in mathbb{R} setminus { 0 }$, $beta, kappa in mathbb{R}$. In two cases of $kappa=0$ and $kappaneq 0$, by transferring an eigenvalue problem into an algebraic problem, we compute the Morse index and critical groups of the trivial solution. Furthermore, even when the trivial solution is degenerate, we show a local linking structure of energy functional at zero within a suitable parameter range and then get critical groups of the trivial solution. As an application, we use Morse theory to get an existence theorem on existence of nontrivial solutions under some conditions.
{"title":"Existence of nontrivial solutions to Schrödinger systems with linear and nonlinear couplings via Morse theory","authors":"Zhitao Zhang, Meng Yu, Xiaotian Zheng","doi":"10.12775/tmna.2022.032","DOIUrl":"https://doi.org/10.12775/tmna.2022.032","url":null,"abstract":"In this paper, we use Morse theory to study existence of nontrivial solutions to the following Schrödinger system with linear and nonlinear couplings which arises from Bose-Einstein condensates:\u0000$$\u0000begin{cases}\u0000-Delta u+lambda_{1} u+kappa v=mu_{1} u^{3}+beta uv^{2}\u0000& text{in } Omega,\u0000-Delta v+lambda_{2} v+kappa u=mu_{2} v^{3}+beta vu^{2}\u0000& text{in } Omega,\u0000u=v=0 & text{on } partialOmega,\u0000end{cases}\u0000$$\u0000where $Omega$ is a bounded smooth domain in $mathbb{R}^{N}$($N=2,3$),\u0000$lambda_{1},lambda_{2},mu_{1},mu_{2} in mathbb{R} setminus { 0 }$,\u0000$beta, kappa in mathbb{R}$.\u0000 In two cases of\u0000$kappa=0$ and $kappaneq 0$, by transferring an eigenvalue problem into an algebraic problem, we compute the Morse index and critical groups of the trivial\u0000 solution. Furthermore, even when the trivial solution is degenerate,\u0000we show a local linking structure of energy functional at zero within a suitable\u0000 parameter range and then get critical groups of the trivial solution.\u0000As an application, we use Morse theory to get an existence theorem on existence\u0000of nontrivial solutions under some conditions.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42699179","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 article, we establish the existence of positive and multiple sign-changing solutions to the fractional $p$-Laplacian equation with purely critical nonlinearity begin{equation} label{Ppomegas-a}tag{P$_{p,Omega}^{s}$} begin{cases} (-Delta)_{p}^s u =|u|^{p_s^*-2} u& text{in }Omega, u =0 & text{on }Omega^{c}, end{cases} end{equation} in a bounded domain $Omegasubset mathbb{R}^{N}$ for $sin (0,1)$, $pin (1,infty)$, and the fractional critical Sobolev exponent $p^{*}_{s}={Np}/({N-sp})$ under some symmetry assumptions. We study Struwe's type global compactness results for the Palais-Smale sequence in the presence of symmetries.
{"title":"Multiple solutions to Bahri-Coron problem involving fractional $p$-Laplacian in some domain with nontrivial topology","authors":"Uttam Kumar, Sweta Tiwari","doi":"10.12775/tmna.2022.033","DOIUrl":"https://doi.org/10.12775/tmna.2022.033","url":null,"abstract":"In this article, we establish the existence of positive and multiple\u0000 sign-changing solutions to the fractional $p$-Laplacian equation with purely critical nonlinearity\u0000 begin{equation}\u0000label{Ppomegas-a}tag{P$_{p,Omega}^{s}$}\u0000begin{cases}\u0000 (-Delta)_{p}^s u =|u|^{p_s^*-2} u& text{in }Omega, \u0000 u =0 & text{on }Omega^{c},\u0000 end{cases}\u0000end{equation}\u0000in a bounded domain $Omegasubset mathbb{R}^{N}$ for $sin (0,1)$,\u0000$pin (1,infty)$, and the fractional critical Sobolev exponent\u0000$p^{*}_{s}={Np}/({N-sp})$ under some symmetry assumptions.\u0000We study Struwe's type global compactness results for the Palais-Smale sequence\u0000in the presence of symmetries.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45702993","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 are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem [ begin{cases} -bigg({varepsilon}^2a+{varepsilon}bint_{mathbb{R}^3} |n u|^2dxbigg)Delta u+u=Q(x)|u|^{p-2}u, & xinmathbb{R}^3, uin H^1big(mathbb{R}^3big), quad u> 0, & xinmathbb{R}^3, end{cases} ] where $varepsilon> 0$ is a small parameter, $a,b> 0$ are constants, $4< p< 6$, $Q$ is a nonnegative continuous potential and does not satisfy any asymptotic condition. Combining Nehari manifold and concentration compactness principle, we study how the shape of the graph of $Q(x)$ affects the number of positive solutions.
{"title":"Multiplicity of positive solutions for a Kirchhoff type problem without asymptotic conditions","authors":"X. Qian","doi":"10.12775/tmna.2022.031","DOIUrl":"https://doi.org/10.12775/tmna.2022.031","url":null,"abstract":"In this paper, we are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem\u0000[\u0000begin{cases}\u0000-bigg({varepsilon}^2a+{varepsilon}bint_{mathbb{R}^3} |n u|^2dxbigg)Delta u+u=Q(x)|u|^{p-2}u, & xinmathbb{R}^3,\u0000uin H^1big(mathbb{R}^3big), quad u> 0, & xinmathbb{R}^3,\u0000end{cases}\u0000]\u0000where $varepsilon> 0$ is a small parameter, $a,b> 0$ are constants, $4< p< 6$, $Q$\u0000 is a nonnegative continuous potential and does not satisfy any asymptotic condition.\u0000 Combining Nehari manifold and concentration compactness principle, we study how the shape of the graph of $Q(x)$ affects the number of positive solutions.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48374354","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 %dedicated to the memory of Edward Fadell and Sufian Husseini we show how the notion of the Lusternik-Schnirelmann relative category can be used to study a multiplicity problem for brake orbits in a potential well which is homeomorphic to the $N$-dimensional unit disk. The estimate of the relative category of the set of chords with endpoints on the $(N-1)$-unit sphere was shown to the third author by Fadell and Husseini while he was visiting the University of Wisconsin at Madison.
{"title":"On the relative category in the brake orbits problem","authors":"Dario Corona, R. Giambò, F. Giannoni, P. Piccione","doi":"10.12775/tmna.2022.057","DOIUrl":"https://doi.org/10.12775/tmna.2022.057","url":null,"abstract":"In this paper %dedicated to the memory of Edward Fadell and Sufian Husseini\u0000we show how the notion of the Lusternik-Schnirelmann relative category can be used\u0000to study a multiplicity problem for brake orbits in a potential well\u0000which is homeomorphic to the $N$-dimensional unit disk.\u0000The estimate of the relative category of the set of chords with endpoints on the\u0000$(N-1)$-unit sphere was shown to the third author by\u0000Fadell and Husseini while he was visiting the University of Wisconsin at Madison.","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":"46213723","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 Reeb graph of a circle-valued function is a topological space obtained by contracting connected components of level sets (preimages of points) to points. For some smooth functions, the Reeb graph has the structure of a finite graph. This notion finds numerous applications in the theory of dynamical systems, as well as in the topological classification of circle-valued functions and the study of their homotopy properties. However, important theoretical facts on the topological properties of the Reeb graphs of circle-valued functions are scattered across numerous papers on different topics, according to the specific needs of the corresponding application. In this paper, we systematize the existing results on the Reeb graphs of circle-valued functions and generalize some of them to wider classes of functions or spaces. We also show how some results can be carried out from real-valued functions. Finally, we adapt some facts from the theory of foliations to the Reeb graphs of circle-valued functions. In particular, we analyze the cycle rank of the Reeb graph and address the problem of realization of a finite graph as a Reeb graph.
{"title":"Reeb graphs of circle-valued functions: A survey and basic facts","authors":"Irina Gelbukh","doi":"10.12775/tmna.2022.023","DOIUrl":"https://doi.org/10.12775/tmna.2022.023","url":null,"abstract":"The Reeb graph of a circle-valued function is a topological space obtained by contracting connected components of level sets (preimages of points) to points.\u0000For some smooth functions, the Reeb graph has the structure of a finite graph.\u0000This notion finds numerous applications in the theory of dynamical systems, as well as in the topological classification of circle-valued functions and the study of their homotopy properties.\u0000However, important theoretical facts on the topological properties of the Reeb graphs of circle-valued functions are scattered across numerous papers on different topics, according to the specific needs of the corresponding application.\u0000In this paper, we systematize the existing results on the Reeb graphs of circle-valued functions and generalize some of them to wider classes of functions or spaces.\u0000We also show how some results can be carried out from real-valued functions. Finally, we adapt some facts from the theory of foliations to the Reeb graphs of circle-valued functions.\u0000In particular, we analyze the cycle rank of the Reeb graph and address the problem of realization of a finite graph as a Reeb graph.","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":"44244132","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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