Nonlinear Analysis-Real World Applications最新文献

{"title":"On the limit cycles of a quartic model for Evolutionary Stable Strategies","authors":"Armengol Gasull ,&nbsp;Luiz F.S. Gouveia ,&nbsp;Paulo Santana","doi":"10.1016/j.nonrwa.2024.104313","DOIUrl":"10.1016/j.nonrwa.2024.104313","url":null,"abstract":"<div><div>This paper studies the number of centers and limit cycles of the family of planar quartic polynomial vector fields that has the invariant algebraic curve <span><math><mrow><mrow><mo>(</mo><mn>4</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>4</mn><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>.</mo></mrow></math></span> The main interest for this type of vector fields comes from their appearance in some mathematical models in Game Theory composed by two players. In particular, we find examples with five nested limit cycles surrounding the same singularity, as well as examples with four limit cycles formed by two disjoint nests, each one of them with two limit cycles. We also prove a Berlinskiĭ’s type result for this family of vector fields.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104313"},"PeriodicalIF":1.8,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143179292","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":"Stability and Hopf Bifurcation for a delayed hand–foot–mouth disease model with continuous age-structure in the exposed class","authors":"Dongxue Yan ,&nbsp;Yongxian Jin ,&nbsp;Hui Cao ,&nbsp;Yu Cao","doi":"10.1016/j.nonrwa.2024.104310","DOIUrl":"10.1016/j.nonrwa.2024.104310","url":null,"abstract":"<div><div>Hand–foot–mouth disease (HFMD) is a mild and highly contagious viral infectious disease common in young children, but anyone can get it. In order to reveal the transmission phenomena of HFMD, we formulate a HFMD model with age structure for latently infected individuals and atime delay. The time delay occurs during the transition from latent to infectious individuals. We reformulate the model as an abstract Cauchy problem and show the presence of equilibria. We specify the basic reproduction number <span><math><msub><mrow><mi>ℛ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> which determines the threshold dynamics of the HFMD model. For <span><math><mrow><msub><mrow><mi>ℛ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>&lt;</mo><mn>1</mn></mrow></math></span>, the disease-free equilibrium <span><math><msub><mrow><mover><mrow><mi>E</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow><mrow><mn>0</mn></mrow></msub></math></span> is globally asymptotically stable. For <span><math><mrow><msub><mrow><mi>ℛ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>&gt;</mo><mn>1</mn></mrow></math></span>, we derive that the endemic equilibrium <span><math><msub><mrow><mover><mrow><mi>E</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow><mrow><mo>∗</mo></mrow></msub></math></span> is unstable, which is the criteria for the occurrence of Hopf bifurcation. Finally, some numerical simulations demonstrate the obtained theoretical results and shed light on the impact of time delay on the evolution of HFMD spread.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104310"},"PeriodicalIF":1.8,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143180533","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":"Well-posedness of the growth-coagulation equation with singular kernels","authors":"Ankik Kumar Giri ,&nbsp;Philippe Laurençot ,&nbsp;Saroj Si","doi":"10.1016/j.nonrwa.2024.104300","DOIUrl":"10.1016/j.nonrwa.2024.104300","url":null,"abstract":"<div><div>The well-posedness of the growth-coagulation equation is established for coagulation kernels having singularity near the origin and growing at most linearly at infinity. The existence of weak solutions is shown by means of the method of the characteristics and a weak <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-compactness argument. For the existence result, we also show our gratitude to Banach fixed point theorem and a refined version of the Arzelà-Ascoli theorem. In addition, the continuous dependence of solutions upon the initial data is shown with the help of the DiPerna-Lions theory, Gronwall’s inequality and moment estimates. Moreover, the uniqueness of solution follows from the continuous dependence. The results presented in this article extend the contributions made in earlier literature.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104300"},"PeriodicalIF":1.8,"publicationDate":"2025-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143179297","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":"Theoretical investigation on fractal–fractional nonlinear ordinary differential equations","authors":"Abdon Atangana ,&nbsp;Seda İğret Araz","doi":"10.1016/j.nonrwa.2024.104296","DOIUrl":"10.1016/j.nonrwa.2024.104296","url":null,"abstract":"<div><div>In this study, we examine the existence and uniqueness conditions of the solutions of the nonlinear fractal-fractional differential equations. Particular emphasis is placed on four cases: exponential decay, power law, generalized Mittag-Leffler kernels and the Delta-Dirac function. Our first contribution is the formulation of some basic inequalities inspired from Gronwall inequality setting up a solid foundation for our analysis to follow. We subsequently carefully obtain the maximal and minimal solutions in each scenario, providing a complete picture of their structure. Finally we show convergence of four different successive approximation schemes, validating their applicability in the various contexts. This is an important finding that adds to the growing literature on the use of fractional calculus in complex dynamical systems.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104296"},"PeriodicalIF":1.8,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143179296","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":"Traveling wave solutions of a cholera transmission model with nonlocal diffusion and spatio-temporal delay","authors":"Chenwei Song ,&nbsp;Rui Xu","doi":"10.1016/j.nonrwa.2024.104304","DOIUrl":"10.1016/j.nonrwa.2024.104304","url":null,"abstract":"<div><div>In this paper, we consider the traveling wave solutions of a cholera transmission model with nonlocal diffusion and spatio-temporal delay, in which the discrete delay <span><math><mi>τ</mi></math></span> represents the latent period of cholera and a nonlocal infection term is introduced to describe the impact of infections at all possible locations at time <span><math><mrow><mi>t</mi><mo>−</mo><mi>τ</mi></mrow></math></span> on the current location at time <span><math><mi>t</mi></math></span>. The basic reproduction number <span><math><msub><mrow><mi>R</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is calculated by using the method of next generation matrix. In addition, the critical wave speed <span><math><msup><mrow><mi>c</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span> is established. Firstly, when <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>&gt;</mo><mn>1</mn></mrow></math></span> and the wave speed <span><math><mrow><mi>c</mi><mo>&gt;</mo><msup><mrow><mi>c</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span>, the existence of traveling waves connecting the disease-free steady state and endemic steady state is obtained by using Schauder’s fixed point theorem, the prior estimate, limit theory and suitable Lyapunov functional. By employing a limiting argument, the existence of traveling waves is established when <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>&gt;</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>c</mi><mo>=</mo><msup><mrow><mi>c</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span>. Secondly, when <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>&gt;</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mn>0</mn><mo>&lt;</mo><mi>c</mi><mo>&lt;</mo><msup><mrow><mi>c</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span>, the nonexistence of traveling wave solution is proved by means of two-sided Laplace transform. It is shown that <span><math><msup><mrow><mi>c</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span> is indeed the minimal wave speed. Numerical simulations are carried out to illustrate the theoretical results. Finally, the impacts of nonlocal diffusion and latent period on minimal wave speed are addressed.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104304"},"PeriodicalIF":1.8,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143179295","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":"Riemann problem for a nonsymmetric Keyfitz–Kranzer and pressureless gas systems with a time-dependent Coulomb-like friction term","authors":"Richard De la cruz ,&nbsp;Wladimir Neves","doi":"10.1016/j.nonrwa.2024.104301","DOIUrl":"10.1016/j.nonrwa.2024.104301","url":null,"abstract":"<div><div>In this paper, we study the Riemann solutions for two systems: the nonsymmetric Keyfitz–Kranzer system and the pressureless system, both characterized by a time-dependent Coulomb-like friction term. Our analysis identifies two types of Riemann solutions: contact discontinuities and delta-shock solutions. We obtain generalized Rankine–Hugoniot conditions, which support the construction of the delta-shock solution for the nonsymmetric Keyfitz–Kranzer system with a time-dependent Coulomb-like friction term. Furthermore, we demonstrate that as the pressure tends to zero, the Riemann solutions of the nonsymmetric Keyfitz–Kranzer system converge to those of the pressureless system, both incorporating a time-dependent Coulomb-like friction term.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104301"},"PeriodicalIF":1.8,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143179294","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":"Global attractors for Moore–Gibson–Thompson thermoelastic extensible beams and Berger plates","authors":"Filippo Dell’Oro ,&nbsp;Lorenzo Liverani ,&nbsp;Vittorino Pata ,&nbsp;Ramon Quintanilla","doi":"10.1016/j.nonrwa.2024.104298","DOIUrl":"10.1016/j.nonrwa.2024.104298","url":null,"abstract":"<div><div>For <span><math><mrow><mi>p</mi><mo>∈</mo><mi>R</mi></mrow></math></span>, and <span><math><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>η</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>, we consider an abstract version of the system <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>+</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>−</mo><mrow><mo>(</mo><mrow><mi>p</mi><mo>+</mo><msup><mrow><mo>‖</mo><mo>∇</mo><mi>u</mi><mo>‖</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow><mi>Δ</mi><mi>u</mi><mo>=</mo><mi>η</mi><mi>Δ</mi><mrow><mo>(</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>+</mo><mi>α</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mspace></mspace></mtd></mtr><mtr><mtd><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>t</mi><mi>t</mi><mi>t</mi></mrow></msub><mo>+</mo><mi>α</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>−</mo><mi>β</mi><mi>Δ</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><mi>γ</mi><mi>Δ</mi><mi>ϕ</mi><mo>−</mo><mi>δ</mi><mi>Δ</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>=</mo><mo>−</mo><mi>η</mi><mi>Δ</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mspace></mspace></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>describing the dynamics of thermoelastic extensible beams or Berger plates, where the evolution of the temperature is ruled by a regularized Moore–Gibson–Thompson type equation. The existence of a global attractor of optimal regularity is proved.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104298"},"PeriodicalIF":1.8,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143178796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Mathematical analysis of steady non-isothermal flows of a micropolar fluid","authors":"E.S. Baranovskii ,&nbsp;E.Yu. Prosviryakov ,&nbsp;S.V. Ershkov","doi":"10.1016/j.nonrwa.2024.104294","DOIUrl":"10.1016/j.nonrwa.2024.104294","url":null,"abstract":"<div><div>This paper deals with a boundary value problem (BVP) describing the 3D steady non-isothermal flow of a micropolar fluid (with couple stresses) in a bounded vessel. The presence of couple shear stresses is a consequence of taking into account the rotational degrees of freedom for an elementary volume of a fluid. Since the governing equations of the couple stress fluid are of order 4, for a well-posed setting of a BVP modeling internal flows, it is not enough to prescribe the no-slip (stick) condition on solid walls on a vessel. Therefore, we come across the non-trivial issue of specifying extra boundary conditions for the velocity field that are reasonable from both physical and mathematical points of view. As one of approaches to solving this problem, we suggest introducing a vorticity-type boundary condition with a parameter, the choice of which determines one of two scenarios: either the no-slip regime together with the vanishing of the vorticity on the boundary (the “super-stick” regime) or the no-slip regime under the vanishing of the couple stresses on the boundary. The interpretation of both boundary conditions is proposed in the terms of the normal and tangential components of the couple stress vector. Another important feature of our work is that we take into account the viscous dissipation effect in the energy balance equation unlike conventional approaches that overlook this effect. We introduce both weak and strong formulations of the considered BVP and study the relationship between the ones. Applying a generalized version of the Leray–Schauder fixed-point theorem, we prove the existence of a weak solution and, under additional assumptions for the model data, the uniqueness of this solution. Moreover, some qualitative and quantitative properties of solutions are established. In particular, we analyze the convergence of the constructed solutions to the solutions of the stationary Navier–Stokes system as the couple stress viscosity coefficient tends to zero.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104294"},"PeriodicalIF":1.8,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143179293","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":"Well-posedness and stability of a nonlinear plate model with energy damping","authors":"Eduardo H. Gomes Tavares ,&nbsp;Linfang Liu ,&nbsp;Vando Narciso ,&nbsp;JinYun Yuan","doi":"10.1016/j.nonrwa.2024.104291","DOIUrl":"10.1016/j.nonrwa.2024.104291","url":null,"abstract":"<div><div>A plate model with nonlinear damping is considered. This model presents a new damping mechanism which is inspired by a combination between an energy damping mechanism and a nonlinear monotonic damping mechanism. The focus of this work is to present the well-posedness and stability of solutions for this model. In particular, the solutions will decay at a polynomial rate, which unifies, in some sense, the decay rates obtained in previous models.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104291"},"PeriodicalIF":1.8,"publicationDate":"2024-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143178795","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 shooting approach for some semilinear scalar field equation with a Dirac-like potential in one-dimension","authors":"Yohei Sato","doi":"10.1016/j.nonrwa.2024.104297","DOIUrl":"10.1016/j.nonrwa.2024.104297","url":null,"abstract":"<div><div>We study the following semilinear scalar field equation in one-dimension <span><span><span><math><mrow><mo>−</mo><msup><mrow><mi>u</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mo>+</mo><mrow><mo>(</mo><msup><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>b</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mi>u</mi><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mspace></mspace><mspace></mspace><mtext>in</mtext><mi>R</mi><mo>,</mo><mspace></mspace><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>→</mo><mn>0</mn><mspace></mspace><mspace></mspace><mtext>as</mtext><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>→</mo><mi>∞</mi><mo>.</mo></mrow></math></span></span></span>Here, <span><math><mrow><mi>λ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>, <span><math><mrow><mi>b</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> satisfies <span><math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>μ</mi><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>μ</mi><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow></msup><mo>≤</mo><mrow><mo>|</mo><mi>b</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>≤</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>μ</mi><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>μ</mi><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow></msup></mrow></math></span>, and <span><math><mi>f</mi></math></span> is a locally Lipschitz function with <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span> that is supposed as general condition as possible. Then there exists <span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mo>≥</mo><mi>λ</mi><mo>)</mo></mrow></mrow></math></span> that is explicitly determined from <span><math><mi>f</mi></math></span>, and we prove the following. If <span><math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>&gt;</mo><mi>γ</mi></mrow></math></span>, then there exist no non-trivial solutions for large <span><math><mi>μ</mi></math></span>. If <span><math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&lt;</mo><mi>λ</mi></mrow></math></span>, then there exists at least a positive solution for large <span><math><mi>μ</mi></math></span>. If <span><math><mrow><mi>γ</mi><mo>&lt;</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>&lt;</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&lt;</mo><mi>λ</mi></mrow></math></span> and <span><math><mrow><mi>b</mi><mrow><mo>(</mo><mo>−</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>b</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>, then there exist at least two positive solutions for large <span><math><mi>μ</mi></math></span>. In the proofs, we use a shooting method from <span><math><mrow><mo>±</mo><mi>∞</mi></mrow></math></span>.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104297"},"PeriodicalIF":1.8,"publicationDate":"2024-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143178794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}