Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.22
Zhengxin Zhou
In this paper, we give the necessary and sufficient conditions for a class of higher degree polynomial systems to have a weak center. As corollaries, we prove the correctness of the two conjectures about the weak center problem for the Λ – Ω differential systems.
{"title":"Weak center for a class of Λ– ","authors":"Zhengxin Zhou","doi":"10.14232/ejqtde.2022.1.22","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.22","url":null,"abstract":"In this paper, we give the necessary and sufficient conditions for a class of higher degree polynomial systems to have a weak center. As corollaries, we prove the correctness of the two conjectures about the weak center problem for the Λ – Ω differential systems.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66584084","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 : 2022-01-01DOI: 10.14232/ejqtde.2022.1.20
Xiaoyan Gao
It is our purpose in this paper to make a detailed description for the structure of the set of the nonconstant steady states for the two-dimensional epidemic S-I model with diffusion incorporating demographic and epidemiological processes with zero-flux boundary conditions. We first study the conditions of diffusion-driven instability occurrence, which induces spatial inhomogeneous patterns. The results will extend to the derivative of prey's functional response with prey is positive. Moreover, we establish the local and global structure of nonconstant positive steady state solutions. A priori estimates for steady state solutions will play a key role in the proof. Our results indicate that the diffusion has a great influence on the spread of the epidemic and extend well the finding of spatiotemporal dynamics in the epidemic model.
{"title":"Nonconstant positive steady states and pattern formation of a diffusive epidemic model","authors":"Xiaoyan Gao","doi":"10.14232/ejqtde.2022.1.20","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.20","url":null,"abstract":"It is our purpose in this paper to make a detailed description for the structure of the set of the nonconstant steady states for the two-dimensional epidemic S-I model with diffusion incorporating demographic and epidemiological processes with zero-flux boundary conditions. We first study the conditions of diffusion-driven instability occurrence, which induces spatial inhomogeneous patterns. The results will extend to the derivative of prey's functional response with prey is positive. Moreover, we establish the local and global structure of nonconstant positive steady state solutions. A priori estimates for steady state solutions will play a key role in the proof. Our results indicate that the diffusion has a great influence on the spread of the epidemic and extend well the finding of spatiotemporal dynamics in the epidemic model.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66584412","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 : 2022-01-01DOI: 10.14232/ejqtde.2022.1.37
S. Stević, B. Iricanin, W. Kosmala, Z. Šmarda
We consider a class of nonlinear difference equations of the fourth order, which extends some equations in the literature. It is shown that the class of equations is solvable in closed form explaining theoretically, among other things, solvability of some previously considered very special cases. We also present some applications of the main theorem through two examples, which show that some results in the literature are not correct.
{"title":"On a solvable class of nonlinear difference equations of fourth order","authors":"S. Stević, B. Iricanin, W. Kosmala, Z. Šmarda","doi":"10.14232/ejqtde.2022.1.37","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.37","url":null,"abstract":"We consider a class of nonlinear difference equations of the fourth order, which extends some equations in the literature. It is shown that the class of equations is solvable in closed form explaining theoretically, among other things, solvability of some previously considered very special cases. We also present some applications of the main theorem through two examples, which show that some results in the literature are not correct.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66585364","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 : 2022-01-01DOI: 10.14232/ejqtde.2022.1.4
Qinlong Wang, Yu'e Xiong, Wentao Huang, V. Romanovski
We study the isolated periodic wave trains in a class of modified generalized Burgers–Huxley equation. The planar systems with a degenerate equilibrium arising after the traveling transformation are investigated. By finding certain positive definite Lyapunov functions in the neighborhood of the degenerate singular points and the Hopf bifurcation points, the number of possible limit cycles in the corresponding planar systems is determined. The existence of isolated periodic wave trains in the equation is established, which is universal for any positive integer n in this model. Within the process, one interesting example is obtained, namely a series of limit cycles bifurcating from a semi-hyperbolic singular point with one zero eigenvalue and one non-zero eigenvalue for its Jacobi matrix.
{"title":"Isolated periodic wave trains in a generalized Burgers–Huxley equation","authors":"Qinlong Wang, Yu'e Xiong, Wentao Huang, V. Romanovski","doi":"10.14232/ejqtde.2022.1.4","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.4","url":null,"abstract":"We study the isolated periodic wave trains in a class of modified generalized Burgers–Huxley equation. The planar systems with a degenerate equilibrium arising after the traveling transformation are investigated. By finding certain positive definite Lyapunov functions in the neighborhood of the degenerate singular points and the Hopf bifurcation points, the number of possible limit cycles in the corresponding planar systems is determined. The existence of isolated periodic wave trains in the equation is established, which is universal for any positive integer n in this model. Within the process, one interesting example is obtained, namely a series of limit cycles bifurcating from a semi-hyperbolic singular point with one zero eigenvalue and one non-zero eigenvalue for its Jacobi matrix.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66585515","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 : 2022-01-01DOI: 10.14232/ejqtde.2022.1.41
Zhenhai Liu, N. Papageorgiou
We consider a double phase Dirichlet problem with a gradient dependent reaction term (convection). Using the theory of nonlinear operators of monotone type, we show the existence of a bounded strictly positive solution. Moreover, we show that the set of these solutions is compact in the corresponding generalized Sobolev–Orlicz space.
{"title":"Dirichlet Problems with Unbalanced Growth and Convection","authors":"Zhenhai Liu, N. Papageorgiou","doi":"10.14232/ejqtde.2022.1.41","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.41","url":null,"abstract":"We consider a double phase Dirichlet problem with a gradient dependent reaction term (convection). Using the theory of nonlinear operators of monotone type, we show the existence of a bounded strictly positive solution. Moreover, we show that the set of these solutions is compact in the corresponding generalized Sobolev–Orlicz space.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66585080","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 : 2022-01-01DOI: 10.14232/ejqtde.2022.1.8
R. Ma, Zhongzi Zhao, Mantang Ma
We prove existence of weak solutions of Neumann problem of nonhomogeneous elliptic system with asymmetric nonlinearities that may resonant at − ∞ and superlinear at + ∞ . The proof is based on Mawhin's coincidence theory and the product formula of Brouwer degree.
{"title":"Neumann problems of superlinear elliptic systems at resonance","authors":"R. Ma, Zhongzi Zhao, Mantang Ma","doi":"10.14232/ejqtde.2022.1.8","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.8","url":null,"abstract":"We prove existence of weak solutions of Neumann problem of nonhomogeneous elliptic system with asymmetric nonlinearities that may resonant at − ∞ and superlinear at + ∞ . The proof is based on Mawhin's coincidence theory and the product formula of Brouwer degree.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66585435","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 : 2022-01-01DOI: 10.14232/ejqtde.2022.1.66
Equations L. Ding, Jiu Liu, Yan-Xiang Yuan
With the help of the monotonicity trick, a nonautonomous Choquard equations with general nonlinearity is studied and a nontrivial solution is obtained.
利用单调性技巧，研究了一类具有一般非线性的非自治Choquard方程的非平凡解。
{"title":"A nontrivial solution for a nonautonomous Choquard equation with\u0000 general nonlinearity","authors":"\t\tEquations\t\t\tL. Ding, Jiu Liu, Yan-Xiang Yuan","doi":"10.14232/ejqtde.2022.1.66","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.66","url":null,"abstract":"With the help of the monotonicity trick, a nonautonomous Choquard equations with general nonlinearity is studied and a nontrivial solution is obtained.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"4 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75481165","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 : 2022-01-01DOI: 10.14232/ejqtde.2022.1.14
Xiucai Ding
This paper is considered with a scalar delay Nicholson's blowflies equation in periodic environments. By taking advantage of some novel differential inequality techniques and the fluctuation lemma, we set up the sharp condition to characterize the global asymptotic stability of positive periodic solutions on the addressed equation. The obtained results improve and supplement some existing ones in recent literature, and then give a more perfect answer to an open problem proposed by Berezansky et al. in [Appl. Math. Model. 34(2010) 1405–1417]. In particular, two numerical examples are provided to verify the reliability and feasibility of the theoretical findings.
{"title":"Global asymptotic stability of a scalar delay Nicholson's blowflies equation in periodic environment","authors":"Xiucai Ding","doi":"10.14232/ejqtde.2022.1.14","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.14","url":null,"abstract":"This paper is considered with a scalar delay Nicholson's blowflies equation in periodic environments. By taking advantage of some novel differential inequality techniques and the fluctuation lemma, we set up the sharp condition to characterize the global asymptotic stability of positive periodic solutions on the addressed equation. The obtained results improve and supplement some existing ones in recent literature, and then give a more perfect answer to an open problem proposed by Berezansky et al. in [Appl. Math. Model. 34(2010) 1405–1417]. In particular, two numerical examples are provided to verify the reliability and feasibility of the theoretical findings.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66583887","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 : 2022-01-01DOI: 10.14232/ejqtde.2022.1.49
Equations Guangjie Li, Zhipei Hu, F. Deng, Huiyan Zhang
Little work seems to be known about stabilization results of highly nonlinear stochastic time-varying delay systems (STVDSs) with Markovian switching and Poisson jump. This paper is concerned with the stabilization problem for a class of STVDSs with Markovian switching and Poisson jump. The coefficients of such systems do not satisfy the conventional linear growth conditions, but are subject to high nonlinearity. The aim of this paper is to design a delay feedback controller to make an unstable highly nonlinear STVDSs with Markovian switching and Poisson jump H ∞ -stable and asymptotically stable. Besides, an illustrative example is provided to support the theoretical results.
{"title":"Stabilization via delay feedback for highly nonlinear stochastic\u0000 time-varying delay systems with Markovian switching and Poisson jump","authors":"\t\tEquations\t\t\tGuangjie Li, Zhipei Hu, F. Deng, Huiyan Zhang","doi":"10.14232/ejqtde.2022.1.49","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.49","url":null,"abstract":"Little work seems to be known about stabilization results of highly nonlinear stochastic time-varying delay systems (STVDSs) with Markovian switching and Poisson jump. This paper is concerned with the stabilization problem for a class of STVDSs with Markovian switching and Poisson jump. The coefficients of such systems do not satisfy the conventional linear growth conditions, but are subject to high nonlinearity. The aim of this paper is to design a delay feedback controller to make an unstable highly nonlinear STVDSs with Markovian switching and Poisson jump H ∞ -stable and asymptotically stable. Besides, an illustrative example is provided to support the theoretical results.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"6 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83183998","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 : 2021-10-03DOI: 10.14232/ejqtde.2021.1.3
J. Schlage-Puchta
Consider the differential equation y = F (x, y). We determine the weakest possible upper bound on |F (x, y)−F (x, z)| which guarantees that this equation has for all initial values a unique solution, which exists globally. Let F : R → R be a continuous function. The well known global Picard-Lindelöf theorem states that if F is Lipschitz continuous with respect to the second variable, then for every real number y0, the initial value problem y ′ = F (x, y), y(0) = y0 has a unique solution, which exists globally. On the other hand the initial value problem y′ = 2 √ |y|, y(0) = 0 has infinitely many solutions, which can be parametrized by real numbers −∞ ≤ a ≤ b ≤ ∞ as
{"title":"Optimal version of the Picard-Lindel\"of theorem","authors":"J. Schlage-Puchta","doi":"10.14232/ejqtde.2021.1.3","DOIUrl":"https://doi.org/10.14232/ejqtde.2021.1.3","url":null,"abstract":"Consider the differential equation y = F (x, y). We determine the weakest possible upper bound on |F (x, y)−F (x, z)| which guarantees that this equation has for all initial values a unique solution, which exists globally. Let F : R → R be a continuous function. The well known global Picard-Lindelöf theorem states that if F is Lipschitz continuous with respect to the second variable, then for every real number y0, the initial value problem y ′ = F (x, y), y(0) = y0 has a unique solution, which exists globally. On the other hand the initial value problem y′ = 2 √ |y|, y(0) = 0 has infinitely many solutions, which can be parametrized by real numbers −∞ ≤ a ≤ b ≤ ∞ as","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2021-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66581403","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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