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 SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR SECOND-ORDER SINGULAR NONLINEAR DIFFERENTIAL EQUATIONS

New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the conditions 0 less than or equal to f(0)(+) < M-1, m(1) < f(infinity)(-)less than or equal to infinity or 0 less than or equal to f(infinity)(+)< M-1, m(1) < f (-)(0)less than or equal to infinity where f(0)(+) = lim(u -->0)f(u)/u, f(infinity)(-)= lim(u --> infinity)f(u)/u, f(0)(-)= lim(u -->0)f(u)/u, f(infinity)(+) = lim(u --> infinity)f(u)/u, g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.

EXISTENCE OF POSITIVE SOLUTIONS FOR A CLASS OF SINGULAR TWO POINT BOUNDARY VALUE PROBLEMS OF SECOND ORDER NONLINEAR EQUATION

In this paper, we establish the existence of positive solutions of (|y'| p-2g' )'+f(t,y)= 0 (P>1 ). y (0)=y (1) = 0. The function f is allowed to be singular when y= 0.

MULTIPLE POSITIVE SOLUTIONS OF SINGULAR THIRD-ORDER PERIODIC BOUNDARY VALUE PROBLEM

This paper deals with the singular nonlinear third-order periodic boundary value problem u'' + p(3)u = f (t, u), 0 less than or equal to t less than or equal to 2pi, with u((i)) (0) = u((i)) (2pi), i = 0, 1, 2, where p is an element of (Graphics) and f is singular at t = 0, t = 1 and u = 0. Under suitable weaker conditions than those of [1], it is proved by constructing a special cone in C[0, 2pi] and employing the fixed point index theory that the problem has at least one or at least two positive solutions.

Positive Solutions of Boundary Value Problems for Systems of Nonlinear Second-order Differential Equations

By using fixed-point index theory,we study boundary value problems for systems of nonlinear second-order differential equation,and a result on existence and multiplicity of positive solutions is obtained.

Symmetric Positive Solutions of Nonlinear Singular Second-order Three-point Boundary Value Problem

In this paper, the second-order three-point boundary value problem u(t) + λa(t)f(t, u(t)) = 0, 0 < t < 1,u(t) = u(1- t), u(0)- u(1) = u(12)is studied, where λ is a positive parameter, under various assumption on a and f, we establish intervals of the parameter λ, which yield the existence of positive solution, our proof based on Krasnosel'skii fixed-point theorem in cone.{u"(t)+λa(t)f(t,u(t))=0,0<t<1,u(t)=u(1-t),u′(0)-u′(1)=u(1/2)is studied,where A is a positive parameter,under various assumption on a and f,we establish intervals of the parameter A,which yield the existence of positive solution,our proof based on Krasnosel'skii fixed-point theorem in cone.

Existence and Multiplicity of Positive Solutions for a Singular Third-Order Three-Point Boundary Value Problem with a Parameter

In this paper, we investigate the existence of positive solutions for a singular third-order three-point boundary value problem with a parameter. By using fixed point index theory, some existence, multiplicity and nonexistence results for positive solutions are derived in terms of different values of λ.

Multiple positive solutions for a class of nonlinear four-point boundary value problem with p-Laplacian

This paper deals with the existence of multiple positive solutions for a class of nonlinear singular four-point boundary value problem with p-Laplacian：{（φ（u′））′＋a（t）f（u（t））=0, 0〈t〈1, αφ（u（0））-βφ（u′（ξ））=0,γφ（u（1））＋δφ（u′（η））0,where φ（x） = ｜x｜^p-2x,p 〉 1, a（t） may be singular at t = 0 and/or t = 1. By applying Leggett-Williams fixed point theorem and Schauder fixed point theorem, the sufficient conditions for the existence of multiple （at least three） positive solutions to the above four-point boundary value problem are provided. An example to illustrate the importance of the results obtained is also given.

NONEXISTENCE AND EXISTENCE OF POSITIVE SOLUTIONS FOR 2nTH-ORDER SINGULAR SUPERLINEAR PROBLEMS WITH STRUM-LIOUVILLE BOUNDARY CONDITIONS

This paper investigates a class of 2nth-order singular superlinear problems with Strum-Liouville boundary conditions. We obtain a necessary and sufficient condition for the existence of C 2 n- 2 [0, 1] positive solutions, and a sufficient condition, a necessary condition for the existence of C 2 n-1 [0, 1] positive solutions. Relations between the positive solutions and the Green’s functions are depicted. The results are used to judge nonexistence or existence of positive solutions for given boundary value problems.

Existence of Monotone Positive Solution for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function

This paper is concerned with the following fourth-order three-point boundary value problem , where , we discuss the existence of positive solutions to the above problem by applying to the fixed point theory in cones and iterative technique.

Positive Solutions for Nonlineary nth-order Boundary Value Problem

The existence of positive solutions for the following nonlineary nth-oreder boundary value problem was studied. u^（n）（t） ＋h（t）f（t,u（t）） =0 0 〈 t 〈 1, u（O） = ∫1 0u（t）dα（t）,u（1） = ∫1 0u（t）dβ（t） u＇（O）=……u^（n-3）（0） = u^（n-2）（0） = 0 Whereh ∈ C（O,1） ∩L（O,1） is nonnegative and may be singular att = Oandt = 1 ,f∈C（[O,1] ×R^＋,R^＋）（R^＋ = signed , ∫1 0 u（t）dα（t） and ∫1 0 u（t）dβ（t） denote the Riemann-Stiehjes integral with a measure, that is, α（t） andβ（t） have bounded variation.

The Existence and Multiplicity of Solutions for Singular Boundary Value Systems with p-Laplacian

This paper presents sufficient conditions for the existence of positive solutions for the fourth-order boundary value problem system with p-Laplacian operator. The existence of single or multiple positive solutions for the system is showed through the fixed point index theory in cones under some assumptions.

Positive Solution for a Singular Fourth-Order Differential System

In this paper, we investigate the existence of positive solutions for the singular fourth-order differential system u(4) = φu + f (t, u, u”, φ), 0 < t < 1, -φ” = μg (t, u, u”), 0 < t < 1, u (0) = u (1) = u”(0) = u”(1) = 0, φ (0) = φ (1) = 0;where μ > 0 is a constant, and the nonlinear terms f, g may be singular with respect to both the time and space variables. The results obtained herein generalize and improve some known results including singular and non-singular cases.

Multiple Positive Solutions of Nonlinear Singular Two-point Boundary Value Problems in Banach Spaces

In this paper, we study the nonlinear singular boundary value problems in Banach spaces：-x=f（t,x）,t∈（0,1）,a1x（0）-a2x＇（0）=θ,b1x（1）＋b2x＇（1）=θ.where θ denotes the zero element of E, E is a real Banach space, and f （t, x） is allowed to be singular at both end point t = 0 and t = 1. We show the existence of at least two positive solutions of this problem.

Existence of Positive Solutions for Singular Second-order m-Point Boundary Value Problems

The singular second-order m-point boundary value problem , is considered under some conditions concerning the first eigenvalue of the relevant linear operators, where (Lϕ)(x) = (p(x)ϕ′(x))′ + q(x)ϕ(x) and ξ i ∈ (0, 1) with 0 【 ξ1 【 ξ2 【 · · · 【 ξ m−2 【 1, a i ∈ [0, ∞). h(x) is allowed to be singular at x = 0 and x = 1. The existence of positive solutions is obtained by means of fixed point index theory. Similar conclusions hold for some other m-point boundary value conditions.

POSITIVE SOLUTIONS OF SINGULAR SUBLINEAR SECOND ORDER BOUNDARY VALUE PROBLEMS

This paper mainly studies the existence of positive solutions of singular sub-linear boundary value problems concerning the generalized Emden-Fowler equations. Anecessary and sufficient condition for the existence of positive solutions to this problemhas been obtained by using the method of lower and upper solutions with the fixed poilltt heorems.

Positive Solution of Singular Boundary Value Problems on a Half-Line

This paper investigates existence of positive solutions of singular sub-linear boundary value problems on a half-line. Necessary and sufficient conditions for the existence of positive continuous solutions or smooth solutions on [0, ∞] are given by constructing new lower and upper solutions.

Positive Solutions of Singular Boundary Value Problems of Fourth-Order Differential Equations

The singular boundary value problem{φ^（4）（x）-h（x）f（φ（x））=0,0〈x〈1, φ（0）=φ（1）=φ′（0）=φ′（1）=0.is considered under some conditions concerning the first eigenvaiues corresponding to the relevant linear operators, where h（x） is allowed to be singular at both x = 0 and x = 1. The existence results of positive solutions are obtained by means of the cone theory and the fixed point index.

EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS TO SINGULAR SECOND ORDER NEUMANN BOUNDARY VALUE PROBLEMS

Using the fixed point index in cones, we study the existence of one and two positive solutions to singular second order Neumann boundary value problems under some conditions, which concerns the first eigenvalue of the relevant linear problem. Our results improve and extend some known ones in the previous literature.

Triple Positive Periodic Solutions of Nonlinear Singular Second-order Boundary Value Problems

This paper studies the positive solutions of the nonlinear second-order periodic boundary value problem u″（t） ＋ λ（t）u（t） = f（t,u（t））,a.e.t ∈ [0,2π],u（0） = u（2π）,u′（0） = u′（2π）,where f（t,u） is a local Carath′eodory function.This shows that the problem is singular with respect to both the time variable t and space variable u.By applying the Leggett-Williams and Krasnosel＇skii fixed point theorems on cones,an existence theorem of triple positive solutions is established.In order to use these theorems,the exact a priori estimations for the bound of solution are given,and some proper height functions are introduced by the estimations.