Oblique derivative problems for nonlinear elliptic systems with measurable coefficients

1 Problem formulation Let Q be a bounded multiply connected domain in RN and the boundary OQ E C2. W6consider the nonlinear elliptic system of second order equationsUnder certain conditions, system (1) can be reduced to the formwhere u = (ul,' t u.), Da = (u..), DZu = (u:..,), andSuppose that (1) (or (2)) satisfiesCondition C For arbitrary functions u'(x), u'(z) E Cd(~) n W::(Q), Fk(x,u,Du,DZu)(k = 1,' I m) satisfy the conditionswhere 0 < g < 1, u == al - u2, and al:), bit), of*),…     

奇异三阶两点边值问题解的一个存在定理

By constructing suitable Banach space.an existence theorem is established under a condition of linear growth for the third-order boundary value problem u'"(t) f(t,u(t),t'(t),u"(t))=0,0＜t＜1,u(0)=u'(0)=u'(1)=0,where the nonlinear term contains first and second derivatives of unknown function.In this theorem the nonlinear term f(t,u,v,w)may be singular at t=0 and t=1.The main ingredient is Leray-Schauder nonlinear alternative.     

Periodic Solutions of Third-order Differential Equations with Finite Delay in Vector-valued Functional Spaces

In this paper, we study the well-posedness of the third-order differential equation with finite delay(P_3): αu'"(t) + u"(t) = Au(t) + Bu'(t) + Fut +f(t)(t ∈ T := [0,2π]) with periodic boundary conditions u(0) = u(2π), u'(0) = u"(2π),u"(0)=u"(2π) in periodic Lebesgue-Bochner spaces Lp(T;X) and periodic Besov spaces B_(p,q)~s(T;X), where A and B are closed linear operators on a Banach space X satisfying D(A) ∩ D(B) ≠ {0}, α≠ 0 is a fixed constant and F is a bounded linear operator from Lp([-2π, 0]; X)(resp. Bp,qs([-2π, 0]; X)) into X, ut is given by ut(s) = u(t + s) when s ∈ [-2π,0]. Necessary and sufficient conditions for the Lp-well-posedness(resp. B_(p,q)~s-well-posedness)of(P_3) are given in the above two function spaces. We also give concrete examples that our abstract results may be applied.     

一阶离散周期问题的多解性

Let T 1 be an integer, T = {0, 1, 2,..., T- 1}. This paper is concerned with the existence of periodic solutions of the discrete first-order periodic boundary value problems△u(t)- a(t)u(t) = λu(t) + f(u(t- τ(t)))- h(t), t ∈ T,u(0) = u(T),where △u(t) = u(t + 1)- u(t), a : T → R and satisfies∏T-1t=0(1 + a(t)) = 1, τ : T → Z t- τ(t) ∈ T for t ∈ T, f : R → R is continuous and satisfies Landesman-Lazer type condition and h : T → R. The proofs of our main results are based on the Rabinowitz's global bifurcation theorem and Leray-Schauder degree.     

POSITIVE SOLUTION FOR NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS

Under suitable conditions on f(·,u), it is shown that the two-point boundaryvalue problem((u'))' + λq(t)f(u) = 0in (0, 1),u(0) = u(1) = 0,has two positive solution or at least one positive solution for λ in a compatibleinterval.     

POSITIVE SOLUTION TO FOURTH-ORDER IMPULSIVE DIFFERENTIAL EQUATIONS WITH P-LAPLACIAN

In order to study three-point BVPs for fourth-order impulsive differential equation of the form(\phi_p(u'(t)))'- f(t,u(t))=0, t≠ t_i,△ u(t_i)=-I_i(u(t_i)), i=1, 2, ..., k,△u'(t_i)=-L_i(u(t_i)), i=1, 2, ..., k,(\star)with the following boundary conditionsu'(0)=u(1)=0, u'(0)=0=u'(1)-\phi_q(α)u'(η),the authors translate the fourth-order impulsive differential equations with p-Laplacian (\star) into three-point BVPs for second-order differential equation without impulses and two-point BVPs for second-order impulsive differential equation by a variable transform. Based on it, existence theorems of positive solutions for the boundary value problems (\star) are obtained.     

Extreme solutions of initial value problems for nonlinear second order integrodifferential equations in Banach spaces

1. IntroductionIn paPer l1], Guo Da jun established the eristence of extreme solutions of initnd Vaueproblems fOr first order illtegrodmerelltial eqllations of VOlterra type in Banach spaces.Now, in this paPer, we consider the IVP for second order illtegrodifferelltial equatinns onthe infinite iliterVa R in BanaCh space E:U" = F(t, u,u', Tu), Vt E R , u(0) = xo, u'(0) = x1) (1)where xo,x1 E E,F E C(R x E x E x E,E), and(Tx)(t) = l'k(,,.).(.)d., Vt E R , (2)jok E C(fl, R ), fl…     

THREE POINT BOUNDARY VALUE PROBLEMS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, we study existence and uniqueness of solutions to nonlinear three point boundary value problems for fractional differential equation of the type c D δ 0+ u(t) = f (t, u(t), c D σ 0+ u(t)), t ∈ [0, T ], u(0) = αu(η), u(T ) = βu(η), where 1 < δ < 2, 0 < σ < 1, α, β∈ R, η∈ (0, T ), αη(1 -β) + (1-α)(T βη) = 0 and c D δ 0+ , c D σ 0+ are the Caputo fractional derivatives. We use Schauder fixed point theorem and contraction mapping principle to obtain existence and uniqueness results. Examples are also included to show the applicability of our results.     

REGULARITY OF WEAK SOLUTIONS TO MAGNETO-MICROPOLAR FLUID EQUATIONS

In this article,we study the regularity of weak solutions and the blow-up criteria for smooth solutions to the magneto-micropolar fluid equations in R~3.We obtain the classical blow-up criteria for smooth solutions(u,ω,b),i.e.,u ∈ L q(0,T;L p(R 3)) for 2 q + 3 p ≤ 1 with 3p≤∞,u ∈ C([0,T);L 3(R 3)) or u ∈L q(0,T;L p) for 3 2p≤∞ satisfying 2 q + 3p≤2.Moreover,our results indicate that the regularity of weak solutions is dominated by the velocity u of the fluid.In the end-point case p = ∞,the blow-up criteria can be extended to more general spaces u∈ L~1(0,T;B_(∞,∞)~0(R~3)).     

二阶微分方程多点边值问题的三重正解

We consider the second-order differential equation u(t) + q(t)f(t,u(t),u (t)) = 0,0 t 1,subject to three-point boundry condition u(0) = 0,u(1) = a 0 u(ξ 0 ),or to m-point boundary conditionu (0) = m2 i=1 b i u (ξ i ),u(1) = m2 i=1 a i u(ξ i ).We show the existence of at least three positive solutions of the above multi-point boundary-value problem by applying a new fixed-point theorem introduced by Avery and Peterson.     

四阶微分方程的迭代解

利用一个构造性的方法,在假设边值问题存在上解α和下解β,满足β≤α的前提下,给出了两个单调序列它们一致收敛于如下两类边值问题的极值解u(4)(x)-Mu″(x)=f(x,u(x),u'(x),u″(x),u″'(x)),0＜x＜1,u(0)=u'(1)=u″(0)=u″'(1)=0;u(4)(x)-Mu″(x)=g(x,u(x),u'(x),u″(x)),0＜x＜1,u(0)=u'(1)=u″(0)=u″'(1)=0.     

Existence of positive solutions for singular four-point boundary value problem with a <Emphasis Type="Italic">p</Emphasis>-Laplacian

In this paper we deal with the four-point singular boundary value problem

Existence and iteration of monotone positive solutions for a third-order two-point boundary value problem

The existence of nondecreasing positive solutions for the nonlinear third-order twopoint boundary value problem u′″（t） ＋ q（t）f（t,u（t）,u′（t）） = 0, 0 〈 t 〈 1, u（0） = u″（0） = u′（1） = 0 is studied. The iterative schemes for approximating the solutions are obtained by applying a monotone iterative method.     

一维p-Laplacian方程正解的存在性

本文考虑一维p-Laplacian非线性边值问题(ψ(u′))′+f(t,u)=0,αψ(u(0))—βψ(u′(0))=O,γψ(u(1))+δψ(u′(1))=0,其中ψ(s):=|s|~(p-2)s,P>1.通过应用Krasnoselskii锥不动点定理,建立了该问题存在多个正解的充分条件,推广并丰富了以往文献的一些结论.     

带p-Laplacian算子的四点边值问题拟对称正解的存在性

利用范数形式的锥拉伸与压缩不动点定理,研究了一类p-Laplacian方程四点边值问题(φp(u′(t)))′(t)+λf(t,u(t))=0,t∈(0,1),u(0)-βu′(ξ)=0,u(ξ)-δu′(η)=u(1)+δu′(1+ξ-η),其中φp(s)=sp-2·s,p>1.获得了其拟对称正解的存在性定理.     

二阶微分方程Neumann边值问题正解存在性

本文利用锥不动点定理证明了－ｕ"＋Ｍｕ＝ｆ（ｔ,ｕ）,ｕ′（０）＝ｕ′（１）＝０和ｕ"＋Ｍｕ＝ ｆ（ｔ,ｕ）,ｕ′（０）＝ｕ′（１）＝ ０两个二阶微分方程 Ｎｅｕｍａｎｎ边值问题正解的存在性。     

一类二阶非线性方程奇异边值问题

本文证明了方程div（|Du|p-2Du） f(r.u（r），u'(r))=0（R1＜r＜R0）正对称解的存在性．这里f允许在u=0或。u’=0处奇异．     

一维 p-Laplacian方程的两点奇异边值问题正解的存在性

该文讨论了如下一维 p-Laplacian 方程-(|u'(t)|^p-2u'(t))'=a(t)f(u(t)), t∈(0,1) u(0)=u(1)=0 的两点奇异边值问题正解的存在性,其中f可能在t=0,1都有奇点.     

Solvability of a class of elastic beam equations with strong Carathéodory nonlinearity

We study the existence of a solution to the nonlinear fourth-order elastic beam equation with nonhomogeneous boundary conditions

一维奇异p-Laplacian方程多解的存在性

该文通过利用Leggett-Williams定理,建立了一维奇异p-Laplacian非线性边值问题(\varphi(u'))'+a(t)f(u)=0,\u'(0)=u(1)=0 (或者u(0)=u'(1)=0),其中\varphi(s)=|s|^{p-2}s, p>1三解的存在性定理,推广并丰富了以往文献的一些结论.