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1.
Considered is the periodic functional differential system with a parameter, x(t)=A(t,x(t))x(t)+λf(t,xt). Using the eigenvalue problems of completely continuous operators, we establish some criteria on the existence of positive periodic solutions. Moreover, we apply the results to a couple of population models and obtain sufficient conditions for the existence of positive periodic solutions, which are compared with existing ones.  相似文献   

2.
3/2-criterion is built, which guarantees the global attractivity of positive solution for equation having the form x(t)=a(t)x(t)(1−L(t,xt)), where a(t)?0 and the linear functional L(t,⋅) is positive. Moreover, when the equation is almost periodic, the similar conditions can also guarantee the existence and uniqueness of almost periodic solution that is globally attractive. Our results improve those in literature.  相似文献   

3.
This paper is concerned with the existence and nonexistence of positive solutions of the second-order nonlinear dynamic equation uΔΔ(t)+λa(t)f(u(σ(t)))=0, t∈[0,1], satisfying either the conjugate boundary conditions u(0)=u(σ(1))=0 or the right focal boundary conditions u(0)=uΔ(σ(1))=0, where a and f are positive. We show that there exists a λ>0 such that the above boundary value problem has at least two, one and no positive solutions for 0<λ<λ, λ=λ and λ>λ, respectively. Furthermore, by using the semiorder method on cones of the Banach space, we establish an existence and uniqueness criterion for positive solution of the problem. In particular, such a positive solution uλ(t) of the problem depends continuously on the parameter λ, i.e., uλ(t) is nondecreasing in λ, limλ0+uλ‖=0 and limλ→+∞‖uλ‖=+∞.  相似文献   

4.
This paper is concerned with the linear ODE in the form y′(t) = λρ(t)y(t) + b(t), λ < 0 which represents a simplified storage model of the carbon in the soil. In the first part, we show that, for a periodic function ρ(t), a linear drift in the coefficient b(t) involves a linear drift for the solution of this ODE. In the second part, we extend the previous results to a classical heat non-homogeneous equation. The connection with an analytic semi-group associated to the ODE equation is considered in the third part. Numerical examples are given.  相似文献   

5.
In this paper, applying the theory of semigroups of operators to evolution families and Banach fixed point theorem, we prove the existence and uniqueness of the weighted pseudo almost periodic mild solution of the semilinear evolution equation x(t)=A(t)x(t)+f(t,x(t)) with nonlocal conditions x(0)=x0+g(x) in Banach space X under some suitable hypotheses.  相似文献   

6.
This paper is concerned with the existence and nonexistence of positive solutions of the nonlinear fourth-order beam equation u(4)(t)+ηu(t)−ζu(t)=λf(t,u(t)), 0<t<1, u(0)=u(1)=u(0)=u(1)=0, where is continuous and ζ, η and λ are parameters. We show that there exists a such that the above boundary value problem (BVP) has at least two, one and no positive solutions for 0<λ<λ*, λ=λ* and λ>λ*, respectively. Furthermore, by using the semiorder method on cones of Banach space, we establish a uniqueness criterion for positive solution of the BVP. In particular such a positive solution uλ(t) of the BVP depends continuously on the parameter λ, i.e., uλ(t) is nondecreasing in λ, limλ0+uλ(t)‖=0 and limλ→+∞‖uλ(t)‖=+∞ for any t∈[0,1].  相似文献   

7.
This paper is concerned with the existence and asymptotic behavior of solutions of a nonlocal dispersal equation. By means of super-subsolution method and monotone iteration, we first study the existence and asymptotic behavior of solutions for a general nonlocal dispersal equation. Then, we apply these results to our equation and show that the nonnegative solution is unique, and the behavior of this solution depends on parameter λ in equation. For λλ1(Ω), the solution decays to zero as t; while for λ>λ1(Ω), the solution converges to the unique positive stationary solution as t. In addition, we show that the solution blows up under some conditions.  相似文献   

8.
This paper is concerned with the boundary value problems y″+λ(ypyq)=0 and y(−1)=y(1)=0, where p>q>−1 and λ>0 is a positive parameter. We discuss the existence of positive solutions and give a complete study.  相似文献   

9.
We consider a nonlinear periodic problem, driven by the scalar p-Laplacian with a concave term and a Caratheodory perturbation. We assume that this perturbation f(t,x) is (p−1)-linear at ±∞, and resonance can occur with respect to an eigenvalue λm+1, m?2, of the negative periodic scalar p-Laplacian. Using a combination of variational techniques, based on the critical point theory, with Morse theory, we establish the existence of at least three nontrivial solutions. Useful in our considerations is an alternative minimax characterization of λ1>0 (the first nonzero eigenvalue) that we prove in this work.  相似文献   

10.
Starting with the famous article [A. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209-243], many papers have been devoted to the uniqueness question for positive solutions of −Δu=λu+up in Ω, u=0 on ∂Ω, where p>1 and λ ranges between 0 and the first Dirichlet eigenvalue λ1(Ω) of −Δ. For the case when Ω is a ball, uniqueness could be proved, mainly by ODE techniques. But very little is known when Ω is not a ball, and then only for λ=0. In this article, we prove uniqueness, for all λ∈[0,λ1(Ω)), in the case Ω=2(0,1) and p=2. This constitutes the first positive answer to the uniqueness question in a domain different from a ball. Our proof makes heavy use of computer assistance: we compute a branch of approximate solutions and prove existence of a true solution branch close to it, using fixed point techniques. By eigenvalue enclosure methods, and an additional analytical argument for λ close to λ1(Ω), we deduce the non-degeneracy of all solutions along this branch, whence uniqueness follows from the known bifurcation structure of the problem.  相似文献   

11.
The paper discusses the existence of positive and dead core solutions of the singular differential equation (?(u))=λf(t,u,u,u) satisfying the boundary conditions u(0)=A, u(T)=A, min{u(t):t∈[0,T]}=0. Here λ is a nonnegative parameter, A is a positive constant and the Carathéodory function f(t,x,y,z) is singular at the value 0 of its space variable y.  相似文献   

12.
A multiplicity result for the singular ordinary differential equation y+λx−2yσ=0, posed in the interval (0,1), with the boundary conditions y(0)=0 and y(1)=γ, where σ>1, λ>0 and γ?0 are real parameters, is presented. Using a logarithmic transformation and an integral equation method, we show that there exists Σ?∈(0,σ/2] such that a solution to the above problem is possible if and only if λγσ−1?Σ?. For 0<λγσ−1<Σ?, there are multiple positive solutions, while if γ=(λ−1Σ?)1/(σ−1) the problem has a unique positive solution which is monotonic increasing. The asymptotic behavior of y(x) as x0+ is also given, which allows us to establish the absence of positive solution to the singular Dirichlet elliptic problem −Δu=d−2(x)uσ in Ω, where ΩRN, N?2, is a smooth bounded domain and d(x)=dist(x,∂Ω).  相似文献   

13.
In this paper, the authors study the existence of periodic solutions to a p-Laplacian Rayleigh differential equation with a delay as follows:
(φp(y(t)))+f(y(t))+g(y(tτ(t)))=e(t),  相似文献   

14.
This paper is concerned with the exact number of positive solutions for the boundary value problem (|y|p−2y)+λf(y)=0 and y(−1)=y(1)=0, where p>1 and λ>0 is a positive parameter. We consider the case in which both f(u) and g(u)=(p−1)f(u)−uf(u) change sign exactly once from negative to positive on (0,∞).  相似文献   

15.
In this paper, applying the theory of semigroups of operators to evolution family and Banach fixed point theorem, we prove the existence and uniqueness of an (a) almost automorphic (weighted pseudo almost automorphic) mild solution of the semilinear evolution equation x(t)=A(t)x(t)+f(t,x(t)) in Banach space under conditions.  相似文献   

16.
In this paper, we investigate the existence of positive solutions for the singular fractional boundary value problem: Dαu(t)+f(t,u(t),Dμu(t))=0, u(0)=u(1)=0, where 1<α<2, 0<μ?α−1, Dα is the standard Riemann-Liouville fractional derivative, f is a positive Carathéodory function and f(t,x,y) is singular at x=0. By means of a fixed point theorem on a cone, the existence of positive solutions is obtained. The proofs are based on regularization and sequential techniques.  相似文献   

17.
In this paper we consider the wave equation with nonlinear damping and source terms. We are interested in the interaction between the boundary damping −|yt(L,t)|m−1yt(L,t) and the interior source |y(t)|p−1y(t). We find a sufficient condition for obtaining the blow-up solution of the problem. Furthermore, we also obtain that the solution may blow up even if mp.  相似文献   

18.
We consider an inverse problem for identifying a leading coefficient α(x) in −(α(x)y′(x))′ + q(x)y(x) = H(x), which is known as an inverse coefficient problem for the Sturm-Liouville operator. We transform y(x) to u(xt) =  (1 + t)y(x) and derive a parabolic type PDE in a fictitious time domain of t. Then we develop a Lie-group adaptive method (LGAM) to find the coefficient function α(x). When α(x) is a continuous function of x, we can identify it very well, by giving boundary data of y, y′ and α. The efficiency of LGAM is confirmed by comparing the numerical results with exact solutions. Although the data used in the identification are limited, we can provide a rather accurate solution of α(x).  相似文献   

19.
This paper is concerned with the exact number of positive solutions for boundary value problems (|y|p−2y)+λf(y)=0 and y(−1)=y(1)=0, where p>1 and λ>0 is a positive parameter. We consider the case in which the nonlinearity f is positive on (0,∞) and (p−1)f(u)−uf(u) changes sign from negative to positive.  相似文献   

20.
By using the theory of coincidence degree, we study a kind of periodic solutions to p-Laplacian neutral functional differential equation with deviating arguments such as (φp(x(t)−cx(tσ)))+g(t,x(tτ(t)))=p(t), a result on the existence of periodic solutions is obtained.  相似文献   

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