Nonlinear Analysis-Real World Applications最新文献

{"title":"Steklov vs. Steklov: A fourth-order affair related to the Babuška paradox","authors":"Francesco Ferraresso ,&nbsp;Pier Domenico Lamberti","doi":"10.1016/j.nonrwa.2025.104464","DOIUrl":"10.1016/j.nonrwa.2025.104464","url":null,"abstract":"<div><div>We discuss two fourth-order Steklov problems and highlight a Babuška paradox appearing in their approximations on convex domains via sequences of convex polygons. To do so, we prove that the eigenvalues of one of the two problems depend with continuity upon domain perturbation in the class of convex domains, extending a result known in the literature for the first eigenvalue. This is obtained by examining in detail a nonlocal second order problem for harmonic functions introduced by Ferrero, Gazzola, and Weth. We further review how this result is connected to diverse variants of the classical Babuška paradox for the hinged plate and to a degeneration result by Maz’ya and Nazarov.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"87 ","pages":"Article 104464"},"PeriodicalIF":1.8,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144695371","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}

{"title":"A double step Rothe scheme for hyperbolic Clarke subdifferential inclusions controlled by evolution equations","authors":"Jinxia Cen ,&nbsp;Krzysztof Bartosz ,&nbsp;Jen-Chih Yao ,&nbsp;Shengda Zeng","doi":"10.1016/j.nonrwa.2025.104463","DOIUrl":"10.1016/j.nonrwa.2025.104463","url":null,"abstract":"<div><div>In this paper we deal with a coupled system which consists of a hyperbolic Clarke subdifferential inclusion and an evolution equation in Banach spaces. Using temporally semidiscrete method based on the double step scheme, we construct a discrete approximate system. The existence of solutions and its a-priori estimates for the discrete approximate system are provided by the surjectivity of multivalued pesudomonotone operators and discrete Gronwall’s inequality. Finally, we show that the solution sequence of the discrete approximate system converges weakly to a limit element, which is a solution of the coupled original system.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104463"},"PeriodicalIF":1.8,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144694866","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}

{"title":"Maximum principle for higher order elliptic operators with inertia in general domains and any dimension","authors":"Daniele Cassani ,&nbsp;Camilla Chiara Polvara ,&nbsp;Antonio Tarsia","doi":"10.1016/j.nonrwa.2025.104465","DOIUrl":"10.1016/j.nonrwa.2025.104465","url":null,"abstract":"<div><div>It is well known how the Maximum Principle (MP) in general fails to hold for uniformly elliptic operators of order higher than two, even in smooth convex domains. In D. Cassani and A. Tarsia (2022) it was shown in dimension <span><math><mrow><mi>N</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></math></span>, by establishing a new Harnack type inequality, that the validity of the positivity preserving property can be restored when lower order derivatives are taken into account as a perturbation of the higher order differential operator. The restriction to the dimension was due to regularity issues which we develop here, extending the validity of the MP to any dimension and fairly general domains. Moreover, we show that the presence of inertial terms affects the range of the perturbation parameter, providing a balance between the positivity restoring effect of lower order derivatives and the mass energy. The method provided here is flexible with respect to the form of differential operators involved and thus suitable to be further extended to other classes of operators than just elliptic.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"87 ","pages":"Article 104465"},"PeriodicalIF":1.8,"publicationDate":"2025-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144686227","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}

{"title":"Approximate controllability for the μ-version of the b-family Camassa–Holm equations with a finite-dimensional force","authors":"Yanpeng Jin ,&nbsp;Xiaoping Wu","doi":"10.1016/j.nonrwa.2025.104462","DOIUrl":"10.1016/j.nonrwa.2025.104462","url":null,"abstract":"<div><div>Considered herein is the approximate controllability of the <span><math><mi>μ</mi></math></span>-<span><math><mi>b</mi></math></span>-Camassa–Holm equation on the circle. We first prove the well-posedness and stability of the <span><math><mi>μ</mi></math></span>-<span><math><mi>b</mi></math></span>-family Camassa–Holm equations with a source term. Then we establish the asymptotic property of this equation. Finally, based on the Agrachev–Sarychev approach in geometric control theory, the approximate controllability for the <span><math><mi>μ</mi></math></span>-version of the <span><math><mi>b</mi></math></span>-family Camassa–Holm equations with two dimensional external force is shown.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"87 ","pages":"Article 104462"},"PeriodicalIF":1.8,"publicationDate":"2025-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144665507","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}

{"title":"Non-convex sweeping processes in contact mechanics","authors":"Erich Bauer ,&nbsp;Victor A. Kovtunenko ,&nbsp;Pavel Krejčí ,&nbsp;Giselle A. Monteiro ,&nbsp;Laetitia Paoli ,&nbsp;Adrien Petrov","doi":"10.1016/j.nonrwa.2025.104456","DOIUrl":"10.1016/j.nonrwa.2025.104456","url":null,"abstract":"<div><div>We propose a model for irreversible dynamics of the rail foundation under the effects of rail traffic, taking into account the granular structure of the ballast subject to changing void ratio and to mechanical degradation. The rail is modeled as an Euler–Bernoulli beam with distributed forcing terms representing the moving traffic load as well as the interaction with the foundation. This interaction is described by an implicit variational inequality with non-convex constraint depending in turn on the solution of the underlying PDE. The problem is reduced to a fixed point problem in a suitable Banach space, and its unique solvability is proved using the contraction principle.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"87 ","pages":"Article 104456"},"PeriodicalIF":1.8,"publicationDate":"2025-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144631493","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}

{"title":"Fractional Orlicz–Sobolev embeddings into Campanato type spaces","authors":"Angela Alberico ,&nbsp;Andrea Cianchi ,&nbsp;Luboš Pick ,&nbsp;Lenka Slavíková","doi":"10.1016/j.nonrwa.2025.104455","DOIUrl":"10.1016/j.nonrwa.2025.104455","url":null,"abstract":"<div><div>Optimal embeddings for fractional Orlicz–Sobolev spaces into (generalized) Campanato spaces on the Euclidean space are exhibited. Embeddings into vanishing Campanato spaces are also characterized. Sharp embeddings into <span><math><mrow><mo>BMO</mo><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mo>VMO</mo><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> are derived as special instances. Dissimilarities to corresponding embeddings for classical fractional Sobolev spaces are pointed out.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"87 ","pages":"Article 104455"},"PeriodicalIF":1.8,"publicationDate":"2025-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144605282","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}

{"title":"Optimal control and dynamic analysis of a new Atangana–Baleanu fractional rumor dissemination model involving media","authors":"Guotao Wang,&nbsp;Ningning Huang,&nbsp;Lihong Zhang","doi":"10.1016/j.nonrwa.2025.104457","DOIUrl":"10.1016/j.nonrwa.2025.104457","url":null,"abstract":"<div><div>In recent years, how to control the spreading of rumors effectively has been one of the public hot issues. Mathematical models can provide quantitative analysis and decision support for this problem. In this paper, a rumor dissemination model with the Atangana–Baleanu fractional derivative in social networks is established, which includes the influence of authoritative media reports and the transformation mechanism of media. The basic properties of the solution for the considered system and the Ulam stability of the system are verified. In order to minimize the cost of controlling rumors, a real-time optimal control model is also proposed by using the Pontryagins maximum principle. At last, numerical simulation results are given to analyze the impacts of different parameters on the Atangana–Baleanu fractional rumor dissemination model, which indicates that our model is realistic and effective.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"87 ","pages":"Article 104457"},"PeriodicalIF":1.8,"publicationDate":"2025-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144605287","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}

{"title":"Multiple solutions for a nonlocal Neumann p-Laplacian problem","authors":"Ángel Crespo-Blanco ,&nbsp;Giuseppe Failla ,&nbsp;Bruno Vassallo","doi":"10.1016/j.nonrwa.2025.104449","DOIUrl":"10.1016/j.nonrwa.2025.104449","url":null,"abstract":"<div><div>We prove the existence of multiple pairs of positive smooth solutions for a nonlocal <span><math><mi>p</mi></math></span>-Laplacian problem with a non-homogeneous Neumann boundary condition. A fully variational approach is used. Moreover, we move from a variational problem to a one-dimensional fixed-point map. Finally, our solutions are ordered in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span>-norm and a conclusive example is furnished.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"87 ","pages":"Article 104449"},"PeriodicalIF":1.8,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144595597","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}

{"title":"Higher differentiability of minimizers for non-autonomous orthotropic functionals","authors":"Stefania Russo","doi":"10.1016/j.nonrwa.2025.104450","DOIUrl":"10.1016/j.nonrwa.2025.104450","url":null,"abstract":"<div><div>We establish the higher differentiability for the minimizers of the following non-autonomous integral functionals <span><span><span><math><mrow><mi>F</mi><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>Ω</mi><mo>)</mo></mrow><mo>≔</mo><mspace></mspace><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><munderover><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mspace></mspace><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><msub><mrow><mi>u</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>|</mo></mrow></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup><mi>d</mi><mi>x</mi><mo>,</mo></mrow></math></span></span></span>with exponents <span><math><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≥</mo><mn>2</mn></mrow></math></span> and with coefficients <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> that satisfy a suitable Sobolev regularity. The main result is obtained, as usual, by imposing a gap bound on the exponents <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, which depends on the dimension and on the degree of regularity of the coefficients <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"87 ","pages":"Article 104450"},"PeriodicalIF":1.8,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144549340","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}

{"title":"A Poincaré–Hopf formula for functionals associated to quasilinear elliptic systems","authors":"Natalino Borgia ,&nbsp;Silvia Cingolani ,&nbsp;Giuseppina Vannella","doi":"10.1016/j.nonrwa.2025.104443","DOIUrl":"10.1016/j.nonrwa.2025.104443","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We consider the functional &lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∫&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;∇&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∫&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;∇&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∫&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is a smooth bounded domain of &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Here &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;≔&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; denotes the product space, endowed with the norm &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, for any &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, being &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;mi&gt;⋅&lt;/mi&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; the usual norm in &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"87 ","pages":"Article 104443"},"PeriodicalIF":1.8,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144549342","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}