Nonexistence and existence of positive solutions for?second order singular three-point boundary value problems with derivative dependent and sign-changing nonlinearities

This paper discusses the nonexistence and the existence of positive solutions for second order singular three-point boundary value problems when the nonlinear term f(t,x,y) is sign-changing and may be singular at t=0, x=0, y=0.     

Nonlinear Boundary Value Problems for Second Order Differential Inclusions

In this paper we study two boundary value problems for second order strongly nonlinear differential inclusions involving a maximal monotone term. The first is a vector problem with Dirichlet boundary conditions and a nonlinear differential operator of the form x ↦ a(x, x′)′. In this problem the maximal monotone term is required to be defined everywhere in the state space ℝ^N. The second problem is a scalar problem with periodic boundary conditions and a differential operator of the form x ↦ (a(x)x′)′. In this case the maximal monotone term need not be defined everywhere, incorporating into our framework differential variational inequalities. Using techniques from multivalued analysis and from nonlinear analysis, we prove the existence of solutions for both problems under convexity and nonconvexity conditions on the multivalued right-hand side.     

Barrier strips and positive solutions of first order singular and regular initial value problems

We use the barrier strip method to prove sufficient conditions for the global solvability of the initial value problem f(t, x, x′) = 0, x(0) = A, including the case in which the function (t, x, y) → f(t, x, y) has a singularity at x = A.     

Existence and uniqueness of positive solutions for some singular boundary value problems with linear functional boundary conditions

By using the upper and lower solution method and fixed point theory, we investigate some nonlinear singular second-order differential equations with linear functional boundary conditions. The nonlinear term f(t, u) is nonincreasing with respect to u, and only possesses some integrability. We obtain the existence and uniqueness of the C[0,1] positive solutions as well as the C ¹[0, 1] positive solutions.     

On Sturm–Liouville boundary value problems for second-order nonlinear functional finite difference equations

Motivated by the interesting paper [I. Karaca, Discrete third-order three-point boundary value problem, J. Comput. Appl. Math. 205 (2007) 458–468], this paper is concerned with a class of boundary value problems for second-order functional difference equations. Sufficient conditions for the existence of at least one solution of a Sturm–Liouville boundary value problem for second-order nonlinear functional difference equations are established. We allow f to be at most linear, superlinear or sublinear in obtained results.     

Oscillatory and asymptotic behaviour of fourth order nonlinear difference equations

This paper considers a class of fourth order nonlinear difference equations Δ²(r _n Δ²(y _n ) + Δ²(r _n ,f(n,n _n )=0,n ∈N(n ₀) wheref(n, y) may be classified as superlinear, sublinear, strongly superlinear and strongly sublinear. In superlinear and sublinear cases, necessary and sufficient conditions are obtained for the difference equation to admit the existence of nonoscillatory solutions with special asymptotic properties. In strongly superlinear and strongly sublinear cases, sufficient conditions are given for all solutions to be oscillatory. Partially Supported by the National Science Foundation of China     

Boundary Value Problems for Singular Second-Order Functional Differential Equations

Abstract Positive solutions to the boundary value problem, y'=-f(x,y(w(x)) 0相似文献   

Positive solutions for higher order multi-point boundary value problems with nonhomogeneous boundary conditions

We study nth order boundary value problems with a nonlinear term f(t,x) subject to nonhomogeneous multi-point boundary conditions. Criteria for the existence of positive solutions of such problems are established. Conditions are determined by the relationship between the behavior of f(t,x)/x near 0 and ∞ when compared with the smallest positive characteristic value of an associated linear integral operator. This work improves and extends some recent results in the literature for the second order problems. The results are illustrated with examples.     

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.     

Positive and maximal positive solutions of singular mixed boundary value problem

The paper is concerned with existence results for positive solutions and maximal positive solutions of singular mixed boundary value problems. Nonlinearities h(t;x;y) in differential equations admit a time singularity at t=0 and/or at t=T and a strong singularity at x=0.     

Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order boundary value problem

We consider the classical nonlinear fourth-order two-point boundary value problem . In this problem, the nonlinear term h(t)f(t, u(t), u′(t), u″(t)) contains the first and second derivatives of the unknown function, and the function h(t)f(t, x, y, z) may be singular at t = 0, t = 1 and at x = 0, y = 0, z = 0. By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals.     

Uniqueness results and asymptotic behavior of solutions with boundary blow-up for logistic-type porous media equations

Under the proper structure conditions on the nonlinear term f(u) and weight function b(x), the paper shows the uniqueness and asymptotic behavior near the boundary of boundary blow-up solutions to the porous media equations of logistic type −Δu = a(x)u ^1/m − b(x)f(u) with m > 1.     

Weighted mean convergence of Hakopian interpolation on the disk

In this paper, we study weighted mean integral convergence of Hakopian interpolation on the unit disk D. We show that the inner product between Hakopian interpolation polynomial Hn(f;x,y) and a smooth function g(x,y) on D converges to that of f(x,y) and g(x,y) on D when n →∞, provided f(x,y) belongs to C(D) and all first partial derivatives of g(x,y) belong to the space LipαM(0 ＜α≤ 1). We further show that provided all second partial derivatives of g(x,y) also belong to the space LipαM and f(x,y) belongs to C1 (D), the inner product between the partial derivative of Hakopian interpolation polynomial (6)/(6)xHn(f;x,y) and g(x,y) on D converges to that between (6)/(6)xf(x,y) and g(x,y) on D when n →∞.     

A fourth-order difference method for elliptic equations with nonlinear first derivative terms

We present a nine-point fourth-order finite difference method for the nonlinear second-order elliptic differential equation Au_xx + Bu_yy = f(x, y, u, u_x, u_y) on a rectangular region R subject to Dirichlet boundary conditions u(x, y) = g(x, y) on ?R. We establish, under appropriate conditions O(h⁴)-convergence of the finite difference scheme. Numerical examples are given to illustrate the method and its fourth-order convergence.     

TWIN POSITIVE SOLUTIONS TO A CLASS OF NONLINEAR SECOND-ORDER THREE-POINTS BOUNDARY VALUE PROBLEMS

In this paper, we establish two existence theorems of twin positive solutions for a class of nonlinear second-order three-point boundary value problems, and concentrate on the case when nonlinear term does not satisfy usual conditions.     

Monotone positive solution for three-point boundary value problem

In this paper, the existence of monotone positive solution for the following secondorder three-point boundary value problem is studied：
x″（t）＋f（t,x（t））=0,0〈t〈1,
x′（0）=0,x（1）＋δx′（η）=0,
where η ∈ （0, 1）, δ∈ [0, ∞）, f ∈ C（[0, 1] × [0, ∞）, [0, ∞））. Under certain growth conditions on the nonlinear term f and by using a fixed point theorem of cone expansion and compression of functional type due to Avery, Anderson and Krueger, sufficient conditions for the existence of monotone positive solution are obtained and the bounds of solution are given. At last, an example is given to illustrate the result of the paper.     

The 2-circle and 2-disk problems on trees

Our purpose here is to consider on a homogeneous tree two Pompeiutype problems which classically have been studied on the plane and on other geometric manifolds. We obtain results which have remarkably the same flavor as classical theorems. Given a homogeneous tree, letd(x, y) be the distance between verticesx andy, and letf be a function on the vertices. For each vertexx and nonnegative integern let Σ _n f(x) be the sum Σ _{d(x, y)=n} f(y) and letB _n f(x)=Σ _{d(x, y)≦n} f(y). The purpose is to study to what extent Σ _n f andB _n f determinef. Since these operators are linear, this is really the study of their kernels. It is easy to find nonzero examples for which Σ _n f orB _n f vanish for one value ofn. What we do here is to study the problem for two values ofn, the 2-circle and the 2-disk problems (in the cases of Σ _n andB _n respectively). We show for which pairs of values there can exist non-zero examples and we classify these examples. We employ the combinatorial techniques useful for studying trees and free groups together with some number theory.     

Remark on the phase shift in the Kuzmak-Whitham ansatz

We consider one-phase (formal) asymptotic solutions in the Kuzmak-Whitham form for the nonlinear Klein-Gordon equation and for the Korteweg-de Vries equation. In this case, the leading asymptotic expansion term has the form X(S(x, t)/h+Φ(x, t), I(x, t), x, t) +O(h), where h ≪ 1 is a small parameter and the phase S}(x, t) and slowly changing parameters I(x, t) are to be found from the system of “averaged” Whitham equations. We obtain the equations for the phase shift Φ(x, t) by studying the second-order correction to the leading term. The corresponding procedure for finding the phase shift is then nonuniform with respect to the transition to a linear (and weakly nonlinear) case. Our observation, which essentially follows from papers by Haberman and collaborators, is that if we incorporate the phase shift Φ into the phase and adjust the parameter Ĩ by setting $ \tilde S $ \tilde S = S +hΦ+O(h ²),Ĩ = I + hI ₁ + O(h ²), then the functions $ \tilde S $ \tilde S (x, t, h) and Ĩ(x, t, h) become solutions of the Cauchy problem for the same Whitham system but with modified initial conditions. These functions completely determine the leading asymptotic term, which is X($ \tilde S $ \tilde S (x, t, h)/h, Ĩ(x, t, h), x, t) + O(h).     

Existence of Single and Multiple Solutions for First Order Periodic Boundary Value Problems

This paper is devoted to the study ofthe existence of single and multiple positive solutions forthe first order boundary value problem x′= f(t,x),x(0)=x(T),where f ∈ C([0,T]×R).In addition,weapply our existence theorems to a class of nonlinear periodic boundary value problems with a singularity at theorigin.Our proofs are based on a fixed point theorem in cones.Our results improve some recent results in theliteratures.     

Diameter preserving linear maps and isometries, II

We study linear bijections of simplex spacesA(S) which preserve the diameter of the range, that is, the seminorm ϱ(f)=sup{|f(x)−f(y)|:x,yεS}.