Existence of positive solutions for a general nonlinear eigenvalue problem

Let Ω R n be a bounded domain, H = L 2 (Ω), L : D(L) H → H be an unbounded linear operator, f ∈ C(■× R, R) and λ∈ R. The paper is concerned with the existence of positive solutions for the following nonlinear eigenvalue problem Lu = λf (x, u), u ∈ D(L), which is the general form of nonlinear eigenvalue problems for differential equations. We obtain the global structure of positive solutions, then we apply the results to some nonlinear eigenvalue problems for a second-order ordinary differential equation and a fourth-order beam equation, respectively. The discussion is based on the fixed point index theory in cones.     

Stability of planar diffusion wave for nonlinear evolution equation

It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f’(u) > 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.     

非线性二阶阻尼微分方程的振动定理

Several oscillation criteria are given for the second order nonlinear differential equation with damped term of the form [α(t)(y'(t))σ]' p(t)(y'(t))σ q(t)f(y(t)) = 0, where α∈C(R, (0,∞)), p(t) and q(t) are allowed to change sign on [t0, ∞), and f∈C1 (R, R) such that xf(x) > 0 for x ≠0. Our results improve and extend some known oscillation criteria. Examples are inserted to illustrate our results.     

EXISTENCE AND ASYMPTOTIC ESTIMATE OF SOLUTION FOR THIRD ORDER BOUNDARY VALUE PROBLEM

This article shows the existence and asymptotic estimates of solutions of singularly perturbed boundary value problems for a class of third order nonlinear differential equations εx'" = f(t,x,x',ε), x(0) = A, x'(0) = x'(1), x"(0) = x"(1).     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF NONLINEAR NEUTRAL DIFFERENTIAL EQUATION

The boundedness of the every solution and the asymptotic behavior of all solutions of the nonlinear neutral delay differential equation [x(t) - P(t)x(t - t)]' Q1 (t)f(x{t-σ1))-Q2(t)f(x(t -σ2))=0,t≥t0 are investigated, whereτ,σ1,σ2∈(0,∞), P∈C([t0,∞),R), and Q1,Q2∈C([t0,∞),R), f∈C(R,R). The sufficient conditions obtained improve the existing results in the literatures.     

TWO POSITIVE SOLUTIONS TO THREE-POINT SINGULAR BOUNDARY VALUE PROBLEMS

In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem: -u′′(t) = λf(t, u(t)) for all t ∈ (0, 1) subjecting to u(0) = 0 and αu(η) = u(1), where η∈ (0, 1), α∈ [0, 1), and λ is a positive parameter. The nonlinear term f(t, u) is nonnegative, and may be singular at t = 0, t = 1, and u = 0. By the fixed point index theory and approximation method, we establish that there exists λ* ∈ (0, +∞], such that the above problem has at least two positive solutions for any λ∈ (0, λ*) under certain conditions on the nonlinear term f.     

Nonlinear wavelet methods for high-dimensional backward heat equation

The backward heat equation is a typical ill-posed problem. In this paper, we shall apply a dual least squares method connecting Shannon wavelet to the following equation ut (x, y, t) = u xx (x, y, t) + uyy (x, y, t), x ∈ R, y ∈ R, 0 ≤ t 1, u(x, y, 1) = (x, y), x ∈ R, y ∈ R. Motivated by Regińska's work, we shall give two nonlinear approximate methods to regularize the approximate solutions for high-dimensional backward heat equation, and prove that our methods are convergent.     

Positive solutions for some 1-dimensional boundary value problems of Laplace-type

This paper deals with the existence of triple positive solutions for the 1-dimensional equation of Laplace-type(φ(x'(t)))' q(t)f(t,x(t),x'(t)) = 0, t ∈ (0, 1),subject to the following boundary condition:a1φ(x(0)) - a2φ(x'(0)) = 0, a3φ(x(1)) a4φ(x'(1)) = 0,where φ is an odd increasing homogeneous homeomorphism. By using a new fixed point theorem,sufficient conditions are obtained that guarantee the existence of at least three positive solutions. The emphasis here is that the nonlinear term f is involved with the first order derivative explicitly.     

ON THE METHOD OF SOLUTION FOR A KIND OFNONLINEAR SINGULAR INTEGRAL EQUATION

The solutions of the nonlinear singular integral equation ψo(t)2 2b/πi ∫L ψ(τ)/T-t dr =f(t), t ∈ L, are considered, where L is a closed contour in the complex plane, b ≠- 0 is a constant and f(t) is a polynomial. It is an extension of the results obtained in [1] when f(t) is a constant. Certain special cases are illustrated.     

Existence of positive solutions to some nonlinear equations on locally finite graphs

Let G =(V,E) be a locally finite graph,whose measure μ(x) has positive lower bound,and A be the usual graph Laplacian.Applying the mountain-pass theorem due to Ambrosetti and Rabinowitz(1973),we establish existence results for some nonlinear equations,namely △u+hu=f(x,u),x∈V.In particular,we prove that if h and f satisfy certain assumptions,then the above-mentioned equation has strictly positive solutions.Also,we consider existence of positive solutions of the perturbed equation △u+hu=f(x,u)+∈g.Similar problems have been extensively studied on the Euclidean space as well as on Riemannian manifolds.     

依赖于导数项的非对称振子解的无界性（英文）

In this paper, we consider the unboundedness of solutions for the asymmetric equation x'+ax~+-bx~-+(x)ψ(x')+f(x)+g(x')=p(t),where x~+= max{x, 0}, x~-= max{-x, 0}, a and b are two different positive constants,f(x) is locally Lipschitz continuous and bounded, (x), ψ(x), g(x) and p(t) are continuous functions, p(t) is a 2π-periodic function. We discuss the existence of unbounded solutions under two classes of conditions: the resonance case 1/a~(1/2)+1/b~(1/2)∈Q and the nonresonance case 1/a~(1/2)+1/b~(1/2)?Q     

Forced oscillation of second order differential equations with mixed nonlinearities

This article is concerned with the oscillation of the forced second order differential equation with mixed nonlinearities a(t) x ′ (t) γ′ + p 0 (t) x γ (g 0 (t)) + n i =1 p i (t) | x (g i (t)) | α i sgn x (g i (t)) = e(t), where γ is a quotient of odd positive integers, α i > 0, i = 1, 2, ··· , n, a, e, and p i ∈ C ([0, ∞ ) , R), a (t) > 0, gi : R → R are positive continuous functions on R with lim t →∞ g i (t) = ∞ , i = 0, 1, ··· , n. Our results generalize and improve the results in a recent article by Sun and Wong [29].     

Quasi-periodic Solutions of Nonlinear SchrSdinger Equations of Higher Dimension

We are concerned with the existence of the quasi-periodic solutions of the nonlinear Schrodinger(NLS) equation + (-△ + Mσ)u + ε|u|2u = 0, x ∈Td where △ is the d-Laplace and Mσ is a Fourier multiplier, i.e.,Mσe -1<,x> = σne -1<,x>, σn ∈ R. Regarding (1) as a Hamiltonian system and using the well-known infinite dimensional KAM theorem developed by them, Kuksin and Poschel[4] showed that there are invariant tori (thus quasi-periodic solutions) for Eq.(1) subject to Dirichlet boundary with d = 1.     

一类具偏差变元的二阶微分方程周期解

By using the theory of coincidence degree,we study a kind of periodic solutions to second order differential equation with a deviating argument such as x"(t) f(x'(t)) h(x(t))x'(t) g(x(t-τ(t)))=p(t),some sufficient conditions on the existence of periodic solutions are obtained.     

GLOBAL EXISTENCE AND EXPONENTIAL DECAY OF SOLUTIONS OF GENERALIZED KURAMOTO-SIVASHINSKY EQUATIONS

We study the Dirichlet initial-boundary value problem of the generalized Kuramoto-Sivashinsky equation ut+uxxxx+λuxx+f(u)x=0 on the interval [0,l],The nonlinear function f satisfies the conditon |f′(u)|≤c|u|^α-1 for some α&gt;1. We prove that if λ4π^2/t^2,then the strong solution is global and exponentially decays to zero for and initial datum uo∈H0^2(0,l) if 1&lt;α≤7,and for small u0∈H0^2(0,l)if α&gt;7,We the consider the equation ut+uxxxx+λuzz+μu+auxxx+bux=F(u,ux,uxx,uxxx),We prove that if F is twice differentiable,Δ↓F is Lipschitz continuous,and F(0)=Δ↓F（0)=0,and if λand μsatisfu μ+σ(λ）&gt;0(σ（λ）=the first eigenvalue of the operator d^4/dx^4+λd^2/dx^2),then the solution for small initial datum is global and exponentially decays to zero.     

On Reducibility of Beam Equation with Quasi-p erio dic Forcing Potential

In this paper, the Dirichlet boundary value problems of the nonlinear beam equation u_(tt) + ?_u~2 + αu + ∈Φ(t)(u + u~3) = 0, α 0 in the dimension one is considered, where u(t, x) and Φ(t) are analytic quasi-periodic functions in t, and∈ is a small positive real-number parameter. It is proved that the above equation admits a small-amplitude quasi-periodic solution. The proof is based on an infinite dimensional KAM iteration procedure.     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR THE CHAFEE-INFANTE EQUATION

In higher dimension, there are many interesting and challenging problems about the dynamics of non-autonomous Chafee-Infante equation. This article is concerned with the asymptotic behavior of solutions for the non–autonomous Chafee-Infante equation (?u)/(?t)-?u =λ(t)(u-u~3) in higher dimension, where λ(t) ∈ C~1[0, T ] and λ(t) is a positive, periodic function.We denote λ_1 as the first eigenvalue of-?? = λ?, x ∈ ?; ? = 0, x ∈ ??. For any spatial dimension N ≥ 1, we prove that if λ(t) ≤λ_1, then the nontrivial solutions converge to zero,namely, ■ u(x, t) = 0, x ∈ ?; if λ(t) λ_1 as t → +∞, then the positive solutions t→+∞are "attracted" by positive periodic solutions. Specially, if λ(t) is independent of t, then the positive solutions converge to positive solutions of-?U = λ(U-U~3). Furthermore,numerical simulations are presented to verify our results.     

SOLUTIONS AND STABILITY OF A GENERALIZATION OF WILSON'S EQUATION

In this paper we study the solutions and stability of the generalized Wilson's functional equation ∫_Gf(xty)dμ(t) + ∫_Gf(xtσ(y))dμ(t) = 2f(x)g(y),x,y ∈ G,where G is a locally compact group,σ is a continuous involution of G and μ is an idempotent complex measure with compact support and which is σ-invariant.We show that ∫_Gg(xty)dμ(t) + ∫_Gg(xtσ(y))dμ(t) = 2g(x)g(y) if f ≠0 and ∫_Gf(t.)dμ(t)≠0,where [ ∫_Gf(t.)dμ(t)](x) = ∫_Gf(tx)dμ(t).We also study some stability theorems of that equation and we establish the stability on noncommutative groups of the classical Wilson's functional equation f(xy) +χ(y)f(xσ(y)) = 2f(x)g(y) x,y ∈ G,where χ is a unitary character of G.     

Boundedness in Asymmetric Quasi-periodic Oscillations

In the paper, by applying the method of main integration, we show the boundedness of the quasi-periodic second order differential equation x′′+ ax~+-bx~-+φ(x) = p(t), where a≠b are two positive constants and φ(s), p(t) are real analytic functions. Moreover, the p(t) is quasi-periodic coefficient, whose frequency vectors are Diophantine. The results we obtained also imply that, under some conditions,the quasi-periodic oscillator has the Lagrange stability.     

BOUNDEDNESS AND CONVERGENCE FOR THE NON-LINARD TYPE DIFFERENTIAL EQUATION

In this article, the author studies the boundedness and convergence for the non-Lienard type differential equation (x|·)=a(y)-f(x) (y|·)=b(y)β(x)-g(x) e(t) where a(y),b(y),f(x),g(x),β(x) are real continuous functions in y∈R or x∈R,β(x)≥0 for all x and e(t) is a real continuous function on R = {t: t≥0} such that the equation has a unique solution for the initial value problem. The necessary and sufficient conditions are obtained and some of the results in the literatures are improved and extended.