Pub Date : 2024-01-19DOI: 10.1007/s11784-023-01092-6
Abstract
We show that for a generic Riemannian or reversible Finsler metric on a compact differentiable manifold M of dimension at least three all closed geodesics are simple and do not intersect each other. Using results by Contreras (Ann Math 2(172):761–808, 2010; in: Proceedings of International Congress Mathematicians (ICM 2010) Hyderabad, India, pp 1729–1739, 2011) this shows that for a generic Riemannian metric on a compact and simply-connected manifold all closed geodesics are simple and the number N(t) of geometrically distinct closed geodesics of length (le t) grows exponentially.
摘要 我们证明，对于至少三维的紧凑可变流形 M 上的一般黎曼或可逆芬斯勒度量，所有闭合大地线都是简单且互不相交的。利用孔特雷拉斯的结果（Ann Math 2(172):761-808, 2010; in：Proceedings of International Congress Mathematicians (ICM 2010) Hyderabad, India, pp 1729-1739, 2011）表明，对于紧凑且简单连接流形上的一般黎曼度量，所有闭大地线都是简单的，且长度为 (le t) 的几何上不同的闭大地线的数量 N(t) 呈指数增长。
{"title":"Simple closed geodesics in dimensions $$ge 3$$","authors":"","doi":"10.1007/s11784-023-01092-6","DOIUrl":"https://doi.org/10.1007/s11784-023-01092-6","url":null,"abstract":"<h3>Abstract</h3> <p>We show that for a generic Riemannian or reversible Finsler metric on a compact differentiable manifold <em>M</em> of dimension at least three all closed geodesics are simple and do not intersect each other. Using results by Contreras (Ann Math 2(172):761–808, 2010; in: Proceedings of International Congress Mathematicians (ICM 2010) Hyderabad, India, pp 1729–1739, 2011) this shows that for a generic Riemannian metric on a compact and simply-connected manifold all closed geodesics are simple and the number <em>N</em>(<em>t</em>) of geometrically distinct closed geodesics of length <span> <span>(le t)</span> </span> grows exponentially.</p>","PeriodicalId":54835,"journal":{"name":"Journal of Fixed Point Theory and Applications","volume":"6 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139508315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-10DOI: 10.1007/s11784-023-01093-5
Oliver Fabert
We show how methods from Hamiltonian Floer theory can be used to establish lower bounds for the number of different time-periodic measures of time-periodic Hamiltonian systems with diffusion. After proving the existence of closed random periodic solutions and of the corresponding Floer curves for Hamiltonian systems with random walks with step width 1/n for every (nin mathbb {N}), we show that, after passing to a subsequence, they converge in probability distribution as (nrightarrow infty ). Besides using standard results from Hamiltonian Floer theory and about convergence of tame probability measures, we crucially use that sample paths of Brownian motion are almost surely Hölder continuous with Hölder exponent (0<alpha <frac{1}{2}).
{"title":"Cuplength estimates for time-periodic measures of Hamiltonian systems with diffusion","authors":"Oliver Fabert","doi":"10.1007/s11784-023-01093-5","DOIUrl":"https://doi.org/10.1007/s11784-023-01093-5","url":null,"abstract":"<p>We show how methods from Hamiltonian Floer theory can be used to establish lower bounds for the number of different time-periodic measures of time-periodic Hamiltonian systems with diffusion. After proving the existence of closed random periodic solutions and of the corresponding Floer curves for Hamiltonian systems with random walks with step width 1/<i>n</i> for every <span>(nin mathbb {N})</span>, we show that, after passing to a subsequence, they converge in probability distribution as <span>(nrightarrow infty )</span>. Besides using standard results from Hamiltonian Floer theory and about convergence of tame probability measures, we crucially use that sample paths of Brownian motion are almost surely Hölder continuous with Hölder exponent <span>(0<alpha <frac{1}{2})</span>.</p>","PeriodicalId":54835,"journal":{"name":"Journal of Fixed Point Theory and Applications","volume":"9 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139413375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-28DOI: 10.1007/s11784-023-01094-4
Yosuke Morita
{"title":"Correction to: Conley index theory without index pairs. I: The point-set level theory","authors":"Yosuke Morita","doi":"10.1007/s11784-023-01094-4","DOIUrl":"https://doi.org/10.1007/s11784-023-01094-4","url":null,"abstract":"","PeriodicalId":54835,"journal":{"name":"Journal of Fixed Point Theory and Applications","volume":"214 5","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139152960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-10DOI: 10.1007/s11784-023-01091-7
R. Pardo
{"title":"Correction to: L∞(Ω)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L^infty (Omega )$$end{document} a priori estima","authors":"R. Pardo","doi":"10.1007/s11784-023-01091-7","DOIUrl":"https://doi.org/10.1007/s11784-023-01091-7","url":null,"abstract":"","PeriodicalId":54835,"journal":{"name":"Journal of Fixed Point Theory and Applications","volume":"2 2","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138585212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-20DOI: 10.1007/s11784-023-01086-4
Jingzhou Liu, Carlos García-Azpeitia, Wieslaw Krawcewicz
In this paper, we prove the existence of non-radial solutions to the problem (-triangle u= f(x,u)), (u|_{partial Omega }=0) on the unit ball (Omega :={xin {mathbb {R}}^3: Vert xVert <1}) with (u(x)in {mathbb {R}}^s), where f is a sub-linear continuous function, differentiable with respect to u at zero and satisfying (f(gx,u) = f(x,u)) for all (gin O(3)), ( f(x,-u)=- f(x,u)). We investigate symmetric properties of the corresponding non-radial solutions. The abstract result is supported by a numerical example.
{"title":"Existence of non-radial solutions to semilinear elliptic systems on a unit ball in $${mathbb {R}}^3$$","authors":"Jingzhou Liu, Carlos García-Azpeitia, Wieslaw Krawcewicz","doi":"10.1007/s11784-023-01086-4","DOIUrl":"https://doi.org/10.1007/s11784-023-01086-4","url":null,"abstract":"<p>In this paper, we prove the existence of non-radial solutions to the problem <span>(-triangle u= f(x,u))</span>, <span>(u|_{partial Omega }=0)</span> on the unit ball <span>(Omega :={xin {mathbb {R}}^3: Vert xVert <1})</span> with <span>(u(x)in {mathbb {R}}^s)</span>, where <i>f</i> is a sub-linear continuous function, differentiable with respect to <i>u</i> at zero and satisfying <span>(f(gx,u) = f(x,u))</span> for all <span>(gin O(3))</span>, <span>( f(x,-u)=- f(x,u))</span>. We investigate symmetric properties of the corresponding non-radial solutions. The abstract result is supported by a numerical example.</p>","PeriodicalId":54835,"journal":{"name":"Journal of Fixed Point Theory and Applications","volume":"46 30 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-13DOI: 10.1007/s11784-023-01088-2
Yehuda John Levy
Abstract This paper uses tools on the structure of the Nash equilibrium correspondence of strategic-form games to characterize a class of fixed-point correspondences, that is, correspondences assigning, for a given parametrized function, the fixed-points associated with each value of the parameter. After generalizing recent results from the game-theoretic literature, we deduce that every fixed-point correspondence associated with a semi-algebraic function is the projection of a Nash equilibrium correspondence, and hence its graph is a slice of a projection, as well as a projection of a slice, of a manifold that is homeomorphic, even isotopic, to a Euclidean space. As a result, we derive an illustrative proof of Browder’s theorem for fixed-point correspondences.
{"title":"Slicing the Nash equilibrium manifold","authors":"Yehuda John Levy","doi":"10.1007/s11784-023-01088-2","DOIUrl":"https://doi.org/10.1007/s11784-023-01088-2","url":null,"abstract":"Abstract This paper uses tools on the structure of the Nash equilibrium correspondence of strategic-form games to characterize a class of fixed-point correspondences, that is, correspondences assigning, for a given parametrized function, the fixed-points associated with each value of the parameter. After generalizing recent results from the game-theoretic literature, we deduce that every fixed-point correspondence associated with a semi-algebraic function is the projection of a Nash equilibrium correspondence, and hence its graph is a slice of a projection, as well as a projection of a slice, of a manifold that is homeomorphic, even isotopic, to a Euclidean space. As a result, we derive an illustrative proof of Browder’s theorem for fixed-point correspondences.","PeriodicalId":54835,"journal":{"name":"Journal of Fixed Point Theory and Applications","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136283098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-25DOI: 10.1007/s11784-023-01087-3
C. Deconinck, K. Dekimpe
{"title":"Nielsen numbers of affine n-valued maps on nilmanifolds","authors":"C. Deconinck, K. Dekimpe","doi":"10.1007/s11784-023-01087-3","DOIUrl":"https://doi.org/10.1007/s11784-023-01087-3","url":null,"abstract":"","PeriodicalId":54835,"journal":{"name":"Journal of Fixed Point Theory and Applications","volume":"95 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135218767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-16DOI: 10.1007/s11784-023-01085-5
Bartosz Bieganowski, Adam Konysz
Abstract We are interested in the following Dirichlet problem: $$begin{aligned} left{ begin{array}{ll} -Delta u + lambda u - mu frac{u}{|x|^2} - nu frac{u}{textrm{dist}(x,mathbb {R}^N setminus Omega )^2} = f(x,u) &{} quad text{ in } Omega u = 0 &{} quad text{ on } partial Omega , end{array} right. end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mfenced> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>-</mml:mo> <mml:mi>μ</mml:mi> <mml:mfrac> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>x</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mfrac> <mml:mo>-</mml:mo> <mml:mi>ν</mml:mi> <mml:mfrac> <mml:mi>u</mml:mi> <mml:mrow> <mml:mtext>dist</mml:mtext> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> <mml:mo></mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:mtd> <mml:mtd> <mml:mrow> <mml:mrow /> <mml:mspace /> <mml:mspace /> <mml:mtext>in</mml:mtext> <mml:mspace /> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mrow /> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:mtd> <mml:mtd> <mml:mrow> <mml:mrow /> <mml:mspace /> <mml:mspace /> <mml:mtext>on</mml:mtext> <mml:mspace /> <mml:mi>∂</mml:mi> <mml:mi>Ω</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:mfenced> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> on a bounded domain $$Omega subset mathbb {R}^N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> </mml:math> with $$0 in Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>∈</mml:mo> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> . We assume that the nonlinear part is superlinear on some closed subset $$K subset Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> and asymptotically linear on $$Omega setminus K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo></mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> . We find a solution with the energy bounded by a certain min–max level, and infinitely, many solutions provided that f is odd in u . Moreover, we study also
我们对以下Dirichlet问题感兴趣: $$begin{aligned} left{ begin{array}{ll} -Delta u + lambda u - mu frac{u}{|x|^2} - nu frac{u}{textrm{dist}(x,mathbb {R}^N setminus Omega )^2} = f(x,u) &{} quad text{ in } Omega u = 0 &{} quad text{ on } partial Omega , end{array} right. end{aligned}$$ - Δ u + λ u - μ u | x | 2 - ν u dist (x, R N Ω) 2 = f (x, u) in Ω u = 0 on∂Ω，在有界域中 $$Omega subset mathbb {R}^N$$ Ω∧R N with $$0 in Omega $$ 0∈Ω。我们假设非线性部分在某个闭子集上是超线性的 $$K subset Omega $$ K∧Ω并在上渐近线性 $$Omega setminus K$$ Ω我们找到一个解，它的能量有一个最小-最大能级的边界，并且有无穷多个解，只要f在u中是奇数。此外，我们还研究了相关归一化问题解的多重性。
{"title":"Elliptic problems with mixed nonlinearities and potentials singular at the origin and at the boundary of the domain","authors":"Bartosz Bieganowski, Adam Konysz","doi":"10.1007/s11784-023-01085-5","DOIUrl":"https://doi.org/10.1007/s11784-023-01085-5","url":null,"abstract":"Abstract We are interested in the following Dirichlet problem: $$begin{aligned} left{ begin{array}{ll} -Delta u + lambda u - mu frac{u}{|x|^2} - nu frac{u}{textrm{dist}(x,mathbb {R}^N setminus Omega )^2} = f(x,u) &{} quad text{ in } Omega u = 0 &{} quad text{ on } partial Omega , end{array} right. end{aligned}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mfenced> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>-</mml:mo> <mml:mi>μ</mml:mi> <mml:mfrac> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>x</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mfrac> <mml:mo>-</mml:mo> <mml:mi>ν</mml:mi> <mml:mfrac> <mml:mi>u</mml:mi> <mml:mrow> <mml:mtext>dist</mml:mtext> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> <mml:mo></mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:mtd> <mml:mtd> <mml:mrow> <mml:mrow /> <mml:mspace /> <mml:mspace /> <mml:mtext>in</mml:mtext> <mml:mspace /> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mrow /> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:mtd> <mml:mtd> <mml:mrow> <mml:mrow /> <mml:mspace /> <mml:mspace /> <mml:mtext>on</mml:mtext> <mml:mspace /> <mml:mi>∂</mml:mi> <mml:mi>Ω</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:mfenced> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> on a bounded domain $$Omega subset mathbb {R}^N$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> </mml:math> with $$0 in Omega $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>∈</mml:mo> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> . We assume that the nonlinear part is superlinear on some closed subset $$K subset Omega $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> and asymptotically linear on $$Omega setminus K$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo></mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> . We find a solution with the energy bounded by a certain min–max level, and infinitely, many solutions provided that f is odd in u . Moreover, we study also","PeriodicalId":54835,"journal":{"name":"Journal of Fixed Point Theory and Applications","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136079833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1007/s11784-023-01084-6
Shengbing Deng, Fang Yu
{"title":"On a biharmonic elliptic problem with slightly subcritical non-power nonlinearity","authors":"Shengbing Deng, Fang Yu","doi":"10.1007/s11784-023-01084-6","DOIUrl":"https://doi.org/10.1007/s11784-023-01084-6","url":null,"abstract":"","PeriodicalId":54835,"journal":{"name":"Journal of Fixed Point Theory and Applications","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135858378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-05DOI: 10.1007/s11784-023-01083-7
Alberto Cabada, Lucía López-Somoza, Mouhcine Yousfi
Abstract In this paper, we obtain an explicit expression for the Green’s function of a certain type of systems of differential equations subject to non-local linear boundary conditions. In such boundary conditions, the dependence on certain parameters is considered. The idea of the study is to transform the given system into another first-order differential linear system together with the two-point boundary value conditions. To obtain the explicit expression of the Green’s function of the considered linear system with non-local boundary conditions, it is assumed that the Green’s function of the homogeneous problem, that is, when all the parameters involved in the non-local boundary conditions take the value zero, exists and is unique. In such a case, the homogeneous problem has a unique solution that is characterized by the corresponding Green’s function g . The expression of the Green’s function of the given system is obtained as the sum of the function g and a part that depends on the parameters involved in the boundary conditions and the expression of function g . The novelty of our work is that in the system to be studied, the unknown functions do not appear separated neither in the equations nor in the boundary conditions. The existence of solutions of nonlinear systems with linear non-local boundary conditions is also studied. We illustrate the obtained results in this paper with examples.
{"title":"Existence of solutions of nonlinear systems subject to arbitrary linear non-local boundary conditions","authors":"Alberto Cabada, Lucía López-Somoza, Mouhcine Yousfi","doi":"10.1007/s11784-023-01083-7","DOIUrl":"https://doi.org/10.1007/s11784-023-01083-7","url":null,"abstract":"Abstract In this paper, we obtain an explicit expression for the Green’s function of a certain type of systems of differential equations subject to non-local linear boundary conditions. In such boundary conditions, the dependence on certain parameters is considered. The idea of the study is to transform the given system into another first-order differential linear system together with the two-point boundary value conditions. To obtain the explicit expression of the Green’s function of the considered linear system with non-local boundary conditions, it is assumed that the Green’s function of the homogeneous problem, that is, when all the parameters involved in the non-local boundary conditions take the value zero, exists and is unique. In such a case, the homogeneous problem has a unique solution that is characterized by the corresponding Green’s function g . The expression of the Green’s function of the given system is obtained as the sum of the function g and a part that depends on the parameters involved in the boundary conditions and the expression of function g . The novelty of our work is that in the system to be studied, the unknown functions do not appear separated neither in the equations nor in the boundary conditions. The existence of solutions of nonlinear systems with linear non-local boundary conditions is also studied. We illustrate the obtained results in this paper with examples.","PeriodicalId":54835,"journal":{"name":"Journal of Fixed Point Theory and Applications","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135481236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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