In this paper, we study the existence of multiple positive solutions for the following equation: −Δu+u = (Kα (x)∗ |u|p)|u|p−2u +λ f (x), x ∈ R , where N 3, α ∈ (0,N), p ∈ (1+ α/N,(N + α)/(N− 2)), Kα (x) is the Riesz potential, and f (x) ∈ H−1(RN) , f (x) 0 , f (x) ≡ 0. We prove that there exists a constant λ ∗ > 0 such that the equation above possesses at least two positive solutions for all λ ∈ (0,λ ∗) . Furthermore, we can obtain the existence of the ground state solution.
本文研究了下列方程的多个正解的存在性:−Δu+u = (Kα (x)∗|u|p)|u|p−2u +λ f (x)， x∈R，其中N 3， α∈(0,N)， p∈(1+ α/N，(N + α)/(N−2))，Kα (x)是Riesz势，f (x)∈H−1(RN)， f (x) 0, f (x)≡0。我们证明了存在一个常数λ∗> 0，使得上述方程对所有λ∈(0，λ∗)至少有两个正解。进一步，我们可以得到基态解的存在性。
{"title":"Multiple positive solutions for a nonlinear Choquard equation with nonhomogeneous","authors":"Haiyang Li, Chunlei Tang, Xing-Ping Wu","doi":"10.7153/dea-2017-09-38","DOIUrl":"https://doi.org/10.7153/dea-2017-09-38","url":null,"abstract":"In this paper, we study the existence of multiple positive solutions for the following equation: −Δu+u = (Kα (x)∗ |u|p)|u|p−2u +λ f (x), x ∈ R , where N 3, α ∈ (0,N), p ∈ (1+ α/N,(N + α)/(N− 2)), Kα (x) is the Riesz potential, and f (x) ∈ H−1(RN) , f (x) 0 , f (x) ≡ 0. We prove that there exists a constant λ ∗ > 0 such that the equation above possesses at least two positive solutions for all λ ∈ (0,λ ∗) . Furthermore, we can obtain the existence of the ground state solution.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"119 1","pages":"553-563"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91049292","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}
Given T > 0 , the Abel-like equation θ ′ = f0 + ∑ j∈N f jθ j is generalized to the case where θ and θ ′ are real functions on [0,T ] subject to given state dependent discontinuities. Each f j is a real function of bounded variation for which f j(0) = (−1) j+1 f j(T ) . Under appropriate conditions, this equation is shown to admit a solution of bounded variation on [0,T ] which is T -anti-periodic in the sense that θ (0) = −θ (T) . The contraction principle yields a bound for the rate of uniform convergence to the solution of a sequence of iterates.
{"title":"Anti-periodic solutions of Abel differential equations with state dependent discontinuities","authors":"J. Belley, A. Gueye","doi":"10.7153/DEA-09-18","DOIUrl":"https://doi.org/10.7153/DEA-09-18","url":null,"abstract":"Given T > 0 , the Abel-like equation θ ′ = f0 + ∑ j∈N f jθ j is generalized to the case where θ and θ ′ are real functions on [0,T ] subject to given state dependent discontinuities. Each f j is a real function of bounded variation for which f j(0) = (−1) j+1 f j(T ) . Under appropriate conditions, this equation is shown to admit a solution of bounded variation on [0,T ] which is T -anti-periodic in the sense that θ (0) = −θ (T) . The contraction principle yields a bound for the rate of uniform convergence to the solution of a sequence of iterates.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"48 1","pages":"219-239"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78800661","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 presents new sufficient conditions, involving limsup and lim inf , for the oscillation of all solutions of differential equations with several non-monotone deviating arguments and nonnegative coefficients. Corresponding differential equations of both delay and advanced type are studied. We illustrate the results and the improvement over other known oscillation criteria by examples, numerically solved in MATLAB.
{"title":"Oscillations caused by several non-monotone deviating arguments","authors":"G. Chatzarakis","doi":"10.7153/DEA-2017-09-22","DOIUrl":"https://doi.org/10.7153/DEA-2017-09-22","url":null,"abstract":"This paper presents new sufficient conditions, involving limsup and lim inf , for the oscillation of all solutions of differential equations with several non-monotone deviating arguments and nonnegative coefficients. Corresponding differential equations of both delay and advanced type are studied. We illustrate the results and the improvement over other known oscillation criteria by examples, numerically solved in MATLAB.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"1 1","pages":"285-310"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75300883","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 nonexistence of solutions for second-order initial value problems. Two results are given: one in which the problems are singular in the time variable, and one in which the problems are singular in both the time and state variables. We consider nonexistence of solutions to singular second-order initial value problems. The results and proofs were originally motivated by Proposition 3.2 in [6]. Existence of solutions to singular differential equations has received a great deal of attention – see, for example, the monograph [1]. For more recent results regarding second-order problems, see [2], [4], [7], [9], [10], [12], [13], [16] and [17]. On the other hand, sometimes nonexistence can be trivial: For example, if f is not Lebesgue integrable in a neighborhood of 0, then clearly x′′(t) = f (t) , x(0) = x0 , x′(0) = x1 has no Carathéodory solution. Results in the literature for nonexistence for singular second-order differential equations typically involve boundary conditions, see for example, [3], [5], [11], [14] and [15]. In [8], existence and nonexistence of positive solutions are studied for the problem x′′ = f (t,x,x′) , x(0) = 0, x′(0) = 0. We begin with the following definition. DEFINITION 1. u is a solution to the initial value problem p(t)u′′(t) = g(t,u(t),u′(t)) u(0) = α, u′(0) = β if there exists a T > 0 such that all of the following are satisfied: i) u , u′ are absolutely continuous on [0,T ] , ii) p(t)u′′(t) = g(t,u(t),u′(t)) a.e. on [0,T ] , iii) u(0) = α , u′(0) = β . We define solution for the problem in Theorem 2 below similarly. Throughout the paper, we assume a,b, f , p,q and u are real-valued. Our first result is the following: Mathematics subject classification (2010): 34A12, 34A34, 34A36.
{"title":"Nonexistence of solutions for second-order initial value problems","authors":"D. Biles","doi":"10.7153/DEA-09-11","DOIUrl":"https://doi.org/10.7153/DEA-09-11","url":null,"abstract":"We consider nonexistence of solutions for second-order initial value problems. Two results are given: one in which the problems are singular in the time variable, and one in which the problems are singular in both the time and state variables. We consider nonexistence of solutions to singular second-order initial value problems. The results and proofs were originally motivated by Proposition 3.2 in [6]. Existence of solutions to singular differential equations has received a great deal of attention – see, for example, the monograph [1]. For more recent results regarding second-order problems, see [2], [4], [7], [9], [10], [12], [13], [16] and [17]. On the other hand, sometimes nonexistence can be trivial: For example, if f is not Lebesgue integrable in a neighborhood of 0, then clearly x′′(t) = f (t) , x(0) = x0 , x′(0) = x1 has no Carathéodory solution. Results in the literature for nonexistence for singular second-order differential equations typically involve boundary conditions, see for example, [3], [5], [11], [14] and [15]. In [8], existence and nonexistence of positive solutions are studied for the problem x′′ = f (t,x,x′) , x(0) = 0, x′(0) = 0. We begin with the following definition. DEFINITION 1. u is a solution to the initial value problem p(t)u′′(t) = g(t,u(t),u′(t)) u(0) = α, u′(0) = β if there exists a T > 0 such that all of the following are satisfied: i) u , u′ are absolutely continuous on [0,T ] , ii) p(t)u′′(t) = g(t,u(t),u′(t)) a.e. on [0,T ] , iii) u(0) = α , u′(0) = β . We define solution for the problem in Theorem 2 below similarly. Throughout the paper, we assume a,b, f , p,q and u are real-valued. Our first result is the following: Mathematics subject classification (2010): 34A12, 34A34, 34A36.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"66 1","pages":"141-146"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72851470","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 explore P-type and D-type learning laws for two classes of RiemannLiouville fractional-order controlled systems to track the varying reference accurately by adopting a few iterations in a finite time interval. Firstly, we establish open and closed-loop P-type convergence results in the sense of (1−α ,λ) -weighted norm ‖ ·‖1−α,λ for Riemann-Liouville fractional-order system of order 0 < α < 1 with initial state learning. Secondly, we establish open and closed-loop D-type convergence results in the sense of λ -weighted norm ‖ · ‖λ for Riemann-Liouville fractional-order system of order 1 < α < 2 with initial state learning. Finally, two numerical examples are given to illustrate our theoretical results. Mathematics subject classification (2010): 34A37, 93C15, 93C40.
{"title":"Study on iterative learning control for Riemann-Liouville type fractional-order systems","authors":"Zijian Luo, Jin Rong Wang","doi":"10.7153/dea-09-10","DOIUrl":"https://doi.org/10.7153/dea-09-10","url":null,"abstract":"In this paper, we explore P-type and D-type learning laws for two classes of RiemannLiouville fractional-order controlled systems to track the varying reference accurately by adopting a few iterations in a finite time interval. Firstly, we establish open and closed-loop P-type convergence results in the sense of (1−α ,λ) -weighted norm ‖ ·‖1−α,λ for Riemann-Liouville fractional-order system of order 0 < α < 1 with initial state learning. Secondly, we establish open and closed-loop D-type convergence results in the sense of λ -weighted norm ‖ · ‖λ for Riemann-Liouville fractional-order system of order 1 < α < 2 with initial state learning. Finally, two numerical examples are given to illustrate our theoretical results. Mathematics subject classification (2010): 34A37, 93C15, 93C40.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"20 1","pages":"123-139"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88775231","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 the authors established sufficient conditions for the oscillation of all solutions of a nonlinear differential equation ( a(t) ( x(t)+ p(t)xα (τ(t)) )′)′ +q(t)xβ ( σ(t) ) = 0, t t0, where α and β are ratio of odd positive integers. The results obtained here extend and improve some of the existing results. Examples are included to illustrate the importance of the results. Mathematics subject classification (2010): 34C10, 34K11.
{"title":"Oscillation of second order nonlinear differential equation with sub-linear neutral term","authors":"S. Tamilvanan, E. Thandapani, J. Džurina","doi":"10.7153/DEA-09-03","DOIUrl":"https://doi.org/10.7153/DEA-09-03","url":null,"abstract":"In this paper the authors established sufficient conditions for the oscillation of all solutions of a nonlinear differential equation ( a(t) ( x(t)+ p(t)xα (τ(t)) )′)′ +q(t)xβ ( σ(t) ) = 0, t t0, where α and β are ratio of odd positive integers. The results obtained here extend and improve some of the existing results. Examples are included to illustrate the importance of the results. Mathematics subject classification (2010): 34C10, 34K11.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"1 1","pages":"29-35"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85309912","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 consider some boundary value problems composed by coupled systems of second order differential equations with full nonlinearities and general functional boundary conditions verifying some monotone assumptions. The arguments apply lower and upper solutions method and fixed point theory. Due to an adequate auxiliary problem, including a convenient truncature, there is no need of sign, bound, monotonicity or other growth assumptions on the nonlinearities, besides the Nagumo condition. An application to a coupled mass-spring system with functional behavior at the final instant is shown.
{"title":"Existence of solution for functional coupled systems with full nonlinear terms and applications to a coupled mass-spring model","authors":"F. Minhós, R. Sousa","doi":"10.7153/DEA-2017-09-30","DOIUrl":"https://doi.org/10.7153/DEA-2017-09-30","url":null,"abstract":"In this paper we consider some boundary value problems composed by coupled systems of second order differential equations with full nonlinearities and general functional boundary conditions verifying some monotone assumptions. The arguments apply lower and upper solutions method and fixed point theory. Due to an adequate auxiliary problem, including a convenient truncature, there is no need of sign, bound, monotonicity or other growth assumptions on the nonlinearities, besides the Nagumo condition. An application to a coupled mass-spring system with functional behavior at the final instant is shown.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"61 1","pages":"433-452"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86608718","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}
The concept of practical stability is generalized to nonlinear differential equations with non-instantaneous impulses. These type of impulses start their action abruptly at some points and then continue on given finite intervals. The practical stability and strict practical stability is studied using Lyapunov like functions and comparison results for scalar differential equations with non-instantaneous impulses. Several sufficient conditions for various types of practical stability, practical quasi stability and strict practical stability are established. Some examples are included to illustrate our theoretical results.
{"title":"Practical stability of differential equations with non-instantaneous impulses","authors":"R. Agarwal, S. Hristova, D. Regan","doi":"10.7153/DEA-2017-09-29","DOIUrl":"https://doi.org/10.7153/DEA-2017-09-29","url":null,"abstract":"The concept of practical stability is generalized to nonlinear differential equations with non-instantaneous impulses. These type of impulses start their action abruptly at some points and then continue on given finite intervals. The practical stability and strict practical stability is studied using Lyapunov like functions and comparison results for scalar differential equations with non-instantaneous impulses. Several sufficient conditions for various types of practical stability, practical quasi stability and strict practical stability are established. Some examples are included to illustrate our theoretical results.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"196 1","pages":"413-432"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75769087","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 interpret the global stability properties of the delayed single species chemostat in terms of monotone dynamics on an asymptotically invariant hyperplane in the state space. The co ...
本文从状态空间中渐近不变超平面上的单调动力学角度解释了时滞单种恒化器的全局稳定性。公司……
{"title":"Monotone dynamics or not? : Dynamical consequences of various mechanisms for delayed logistic growth","authors":"T. Lindström","doi":"10.7153/DEA-2017-09-27","DOIUrl":"https://doi.org/10.7153/DEA-2017-09-27","url":null,"abstract":"In this paper we interpret the global stability properties of the delayed single species chemostat in terms of monotone dynamics on an asymptotically invariant hyperplane in the state space. The co ...","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"7 1","pages":"379-382"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79960113","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 investigate nonlinear boundary value problems for impulsive differential equations with causal operators. Our boundary condition is given by a nonlinear function, and more general than ones given before. To begin with, we prove a comparison theorem. Then by using this theorem, we show the existence of solutions for linear problems. Finally, by using the monotone iterative technique, we obtain the existence of extremal solutions for nonlinear boundary value problems with causal operators. An example satisfying the assumptions is presented.
{"title":"Nonlinear boundary value problems for impulsive differential equations with causal operators","authors":"Wen-Li Wang, Jingfeng Tian","doi":"10.7153/DEA-09-13","DOIUrl":"https://doi.org/10.7153/DEA-09-13","url":null,"abstract":"In this work, we investigate nonlinear boundary value problems for impulsive differential equations with causal operators. Our boundary condition is given by a nonlinear function, and more general than ones given before. To begin with, we prove a comparison theorem. Then by using this theorem, we show the existence of solutions for linear problems. Finally, by using the monotone iterative technique, we obtain the existence of extremal solutions for nonlinear boundary value problems with causal operators. An example satisfying the assumptions is presented.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"31 1","pages":"161-170"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90602535","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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