A link-type-independent adaptive network model of SIS epidemic propagation is considered. In the model links can be activated or deleted randomly regardless to the type of nodes. A four-variable pairwise ODE approximation is used to describe how the number of quantities such as number of infected nodes evolves in time. In order to investigate bifurcations in the model an invariant manifold is defined. Using the theory of asymptotically autonomous systems, results obtained for the reduced system on the manifold are extended to the full pairwise model and a non-oscillating behaviour is proven.
{"title":"Dynamics of a link-type independent adaptive epidemic model","authors":"A. Szabó","doi":"10.7153/DEA-09-09","DOIUrl":"https://doi.org/10.7153/DEA-09-09","url":null,"abstract":"A link-type-independent adaptive network model of SIS epidemic propagation is considered. In the model links can be activated or deleted randomly regardless to the type of nodes. A four-variable pairwise ODE approximation is used to describe how the number of quantities such as number of infected nodes evolves in time. In order to investigate bifurcations in the model an invariant manifold is defined. Using the theory of asymptotically autonomous systems, results obtained for the reduced system on the manifold are extended to the full pairwise model and a non-oscillating behaviour is proven.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"10 1","pages":"105-122"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73275351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider the following nonlinear delay differential equation with a forcing term r(t) : x′(t)+a(t)x(t)+b(t) f (x(t − τ(t))) = r(t), t 0, where a ∈ C[[0,∞), [0,∞)] , b,τ ∈C[[0,∞),(0,∞)] , r ∈C[[0,∞),R] , f ∈ C[(L,∞),(L,∞)] with −∞ L 0 , and limt→∞(t − τ(t)) = ∞ . We establish a sufficient condition for every solution of the equation to converge to zero. By applying the result to some special cases and differential equation models from applications, we obtain several new criteria on the global convergence of solutions.
考虑下面的非线性时滞微分方程与强迫项r (t): x (t) + (t) x (t) + b (t) f (x (t−τ(t))) = r (t) t 0,在∈C[[0,∞),[0,∞)],b,τ∈C[[0,∞),(0,∞)],C r∈([0,∞),r], f∈C [(L,∞)(L,∞)]与−∞L 0,和limt→∞(t−τ(t)) =∞。我们建立了方程的每一个解收敛于零的充分条件。将结果应用于一些特殊情况和应用中的微分方程模型，得到了解全局收敛的几个新判据。
{"title":"On global convergence of forced nonlinear delay differential equations and applications","authors":"D. Hai, C. Qian","doi":"10.7153/DEA-09-02","DOIUrl":"https://doi.org/10.7153/DEA-09-02","url":null,"abstract":"Consider the following nonlinear delay differential equation with a forcing term r(t) : x′(t)+a(t)x(t)+b(t) f (x(t − τ(t))) = r(t), t 0, where a ∈ C[[0,∞), [0,∞)] , b,τ ∈C[[0,∞),(0,∞)] , r ∈C[[0,∞),R] , f ∈ C[(L,∞),(L,∞)] with −∞ L 0 , and limt→∞(t − τ(t)) = ∞ . We establish a sufficient condition for every solution of the equation to converge to zero. By applying the result to some special cases and differential equation models from applications, we obtain several new criteria on the global convergence of solutions.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"17 1","pages":"13-28"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74587943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the Non homogeneous Dirichlet problem with large initial data for the KdVB equation on the interval x ∈ (0,1) ⎪⎪⎨ ⎪⎪⎩ ut +uxu−uxx +uxxx = 0, t > 0, x ∈ (0,1) u(x,0) = u0(x), x ∈ (0,1) u(0,t) = u(1,t) = 0, t > 0 ux(1,t) = h(t), t > 0. (1) We prove that if the initial data u0 ∈ L2 and boundary data h(t) ∈ H∞(0,∞) then there exist a unique solution u ∈ C([0,∞) ;L2)∪C((0,∞) ;H1) of the initial-boundary value problem (1). We also obtain the large time asymptotic of solution uniformly with respect to x ∈ (0,1) as t → ∞. Mathematics subject classification (2010): 35Q35.
{"title":"Non homogeneous Dirichlet problem for the KdVB equation on a segment","authors":"Isahi Sánchez Suárez, Gerardo Loreto Gómez, Marcela Morales Morfín","doi":"10.7153/DEA-09-21","DOIUrl":"https://doi.org/10.7153/DEA-09-21","url":null,"abstract":"We study the Non homogeneous Dirichlet problem with large initial data for the KdVB equation on the interval x ∈ (0,1) ⎪⎪⎨ ⎪⎪⎩ ut +uxu−uxx +uxxx = 0, t > 0, x ∈ (0,1) u(x,0) = u0(x), x ∈ (0,1) u(0,t) = u(1,t) = 0, t > 0 ux(1,t) = h(t), t > 0. (1) We prove that if the initial data u0 ∈ L2 and boundary data h(t) ∈ H∞(0,∞) then there exist a unique solution u ∈ C([0,∞) ;L2)∪C((0,∞) ;H1) of the initial-boundary value problem (1). We also obtain the large time asymptotic of solution uniformly with respect to x ∈ (0,1) as t → ∞. Mathematics subject classification (2010): 35Q35.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"14 1","pages":"265-283"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88682018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper, deals with Lyapunov inequalities of conformable fractional boundary value problems on an N-dimensional spherical shell. Applicability of these Lyapunov inequalities will be examined by establishing the disconjugacy as a nonexistence criterion for nontrivial solutions, lower bound estimation for eigenvalues of the corresponding fractional eigenvalue problem, upper bound estimation for maximum number of zeros of the nontrivial solutions and distance between consecutive zeros of an oscillatory solution.
{"title":"Fractional Lyapunov inequalities on spherical shells","authors":"Yousef Gholami, K. Ghanbari","doi":"10.7153/DEA-2017-09-25","DOIUrl":"https://doi.org/10.7153/DEA-2017-09-25","url":null,"abstract":"This paper, deals with Lyapunov inequalities of conformable fractional boundary value problems on an N-dimensional spherical shell. Applicability of these Lyapunov inequalities will be examined by establishing the disconjugacy as a nonexistence criterion for nontrivial solutions, lower bound estimation for eigenvalues of the corresponding fractional eigenvalue problem, upper bound estimation for maximum number of zeros of the nontrivial solutions and distance between consecutive zeros of an oscillatory solution.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"25 1","pages":"353-368"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90634496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the question of global existence and asymptotics of small, smooth, and localized solutions of a certain pseudoparabolic equation in one dimension, posed on half-line x > 0 , ⎪⎨ ⎪⎩ ( 1−∂ 2 x ) ut = ∂ 2 x (u+α2 (|u|2 u))+α1 |u|1 u, x ∈ R+, t > 0, u(0,x) = u0 (x) , x ∈ R+, u(0,t) = h(t), (0.1) where αi ∈ R,qi > 0, i = 1,2,u : Rx × R+ t ∈ C. This model is motivated by the a wave equation for media with a strong spatial dispersion, which appear in the nonlinear theory of the quasy-stationary processes in the electric media. We show that the problem (0.1) admits global solutions whose long-time behavior depend on boundary data. More precisely, we prove global existence and modified by boundary scattering of solutions. Mathematics subject classification (2010): 35Q35, 35B40.
考虑半直线x >，⎪⎪(1−∂2 x) ut =∂2 x (u+α2 (|u|2 u))+α1 |u|1 u, x∈R+， t > 0, u(0,x) = u0 (x)，x∈R+， u(0,t) = h(t)，(0.1)其中αi∈R,qi > 0, i = 1,2,u:Rx × R+ t∈c，该模型是由电介质中准平稳过程非线性理论中出现的具有强空间色散的介质的波动方程驱动的。我们证明了问题(0.1)允许其长期行为依赖于边界数据的全局解。更准确地说，我们证明了解的整体存在性，并通过边界散射修正了它。数学学科分类(2010):35Q35, 35B40。
{"title":"Nonlinear model of quasi-stationary process in crystalline semiconductor","authors":"B. Juárez-Campos, E. Kaikina, H. Ruiz-Paredes","doi":"10.7153/DEA-09-04","DOIUrl":"https://doi.org/10.7153/DEA-09-04","url":null,"abstract":"We consider the question of global existence and asymptotics of small, smooth, and localized solutions of a certain pseudoparabolic equation in one dimension, posed on half-line x > 0 , ⎪⎨ ⎪⎩ ( 1−∂ 2 x ) ut = ∂ 2 x (u+α2 (|u|2 u))+α1 |u|1 u, x ∈ R+, t > 0, u(0,x) = u0 (x) , x ∈ R+, u(0,t) = h(t), (0.1) where αi ∈ R,qi > 0, i = 1,2,u : Rx × R+ t ∈ C. This model is motivated by the a wave equation for media with a strong spatial dispersion, which appear in the nonlinear theory of the quasy-stationary processes in the electric media. We show that the problem (0.1) admits global solutions whose long-time behavior depend on boundary data. More precisely, we prove global existence and modified by boundary scattering of solutions. Mathematics subject classification (2010): 35Q35, 35B40.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"5 1","pages":"37-55"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87018747","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Virial identities for nonlinear Schrödinger equations with some strongly singular potential (a|x|−2 ) are established. Here if a = a(N) :=−(N−2)2/4 , then Pa(N) :=−Δ+a(N)|x|−2 is nonnegative selfadjoint in the sense of Friedrichs extension. But the energy class D((1 + Pa(N))) does not coincide with H1(RN ) . Thus justification of the virial identities has a lot of difficulties. The identities can be applicable for showing blow-up in finite time and for proving the existence of scattering states. Mathematics subject classification (2010): 35Q55, 35Q40, 81Q15.
{"title":"Virial identities for nonlinear Schrödinger equations with a critical coefficient inverse-square potential","authors":"Toshiyuki Suzuki","doi":"10.7153/DEA-2017-09-24","DOIUrl":"https://doi.org/10.7153/DEA-2017-09-24","url":null,"abstract":"Virial identities for nonlinear Schrödinger equations with some strongly singular potential (a|x|−2 ) are established. Here if a = a(N) :=−(N−2)2/4 , then Pa(N) :=−Δ+a(N)|x|−2 is nonnegative selfadjoint in the sense of Friedrichs extension. But the energy class D((1 + Pa(N))) does not coincide with H1(RN ) . Thus justification of the virial identities has a lot of difficulties. The identities can be applicable for showing blow-up in finite time and for proving the existence of scattering states. Mathematics subject classification (2010): 35Q55, 35Q40, 81Q15.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"1 1","pages":"327-352"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91333543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper using the fixed point index and the Leggett-Williams fixed point theorem we establish the existence and multiplicity of positive solutions for a class of fractional difference boundary value problems.
{"title":"Positive solutions for a class of fractional difference boundary value problems","authors":"Ji Xu, D. Regan, Chengmin Hou, Yuanda Cui","doi":"10.7153/DEA-2017-09-32","DOIUrl":"https://doi.org/10.7153/DEA-2017-09-32","url":null,"abstract":"In this paper using the fixed point index and the Leggett-Williams fixed point theorem we establish the existence and multiplicity of positive solutions for a class of fractional difference boundary value problems.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"4 1","pages":"479-493"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74423124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we provide sufficient conditions for the existence of solutions to nonlinear boundary value problems. We do so by applying a general abstract strategy for solving nonlinear equations with a linear component. We apply this to general systems by first isolating a linear periodic system and using the general theory of periodic solutions to find conditions on the additional nonlinear components to guarantee solutions.
{"title":"Existence of solutions to nonlinear boundary value problems","authors":"Jesús F. Rodríguez, Adam J. Suarez","doi":"10.7153/DEA-09-01","DOIUrl":"https://doi.org/10.7153/DEA-09-01","url":null,"abstract":"In this paper we provide sufficient conditions for the existence of solutions to nonlinear boundary value problems. We do so by applying a general abstract strategy for solving nonlinear equations with a linear component. We apply this to general systems by first isolating a linear periodic system and using the general theory of periodic solutions to find conditions on the additional nonlinear components to guarantee solutions.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"24 1","pages":"1-11"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79190695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we study the oscillation and nonoscillation properties of a class of second order neutral impulsive differential equations with constant coefficients and constant delays by using pulsatile constant. Also, an attempt is made to extend the constant coefficient results to variable coefficient equations.
{"title":"Characterization of a class of second order neutral impulsive systems via pulsatile constant","authors":"A. Tripathy, S. Santra","doi":"10.7153/DEA-09-07","DOIUrl":"https://doi.org/10.7153/DEA-09-07","url":null,"abstract":"In this work, we study the oscillation and nonoscillation properties of a class of second order neutral impulsive differential equations with constant coefficients and constant delays by using pulsatile constant. Also, an attempt is made to extend the constant coefficient results to variable coefficient equations.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"73 1","pages":"87-98"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86605790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a boundary value problem for the beam equation. Some upper and lower bounds for positive solutions of the boundary value problem are obtained. As an application, some new sufficient conditions for the existence and nonexistence of positive solutions for the boundary value problem are established.
{"title":"Maximum Principle for a fourth order boundary value problem","authors":"Bo Yang","doi":"10.7153/DEA-2017-09-33","DOIUrl":"https://doi.org/10.7153/DEA-2017-09-33","url":null,"abstract":"We consider a boundary value problem for the beam equation. Some upper and lower bounds for positive solutions of the boundary value problem are obtained. As an application, some new sufficient conditions for the existence and nonexistence of positive solutions for the boundary value problem are established.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"234 1","pages":"495-504"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78523414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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