scholarly journals Existence of Positive Solutions for a Discrete Three-Point Boundary Value Problem

2007 ◽  
Vol 2007 ◽  
pp. 1-14
Author(s):  
Huting Yuan ◽  
Guang Zhang ◽  
Hongliang Zhao
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Positive Parameter ◽  
Point Boundary ◽  
Constant Number ◽  
Fixed Integer ◽  
Negative Numbers ◽  
Existence Of Positive Solutions

A discrete three-point boundary value problemΔ2xk−1+λfk(xk)=0,k=1,2,…,n, x0=0,axl=xn+1, is considered, where1≤l≤nis a fixed integer,ais a real constant number, andλis a positive parameter. A characterization of the values ofλis carried out so that the boundary value problem has the positive solutions. Particularly, in this paper the constantacan be negative numbers. The similar results are not valid for the three-point boundary value problem of differential equations.

scholarly journals Positive Solutions for a Class of Third-Order Three-Point Boundary Value Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Xiaojie Lin ◽  
Zhengmin Fu
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Positive Parameter ◽  
Point Boundary ◽  
Point Index ◽  
Third Order ◽  
Index Theorems ◽  
Existence Of Positive Solutions

We investigate the problem of existence of positive solutions for the nonlinear third-order three-point boundary value problemu‴(t)+λa(t)f(u(t))=0,0<t<1,u(0)=u′(0)=0,u″(1)=∝u″(η), whereλis a positive parameter,∝∈(0,1),η∈(0,1),f:(0,∞)→(0,∞),a:(0,1)→(0,∞)are continuous. Using a specially constructed cone, the fixed point index theorems and Leray-Schauder degree, this work shows the existence and multiplicities of positive solutions for the nonlinear third-order boundary value problem. Some examples are given to demonstrate the main results.

scholarly journals Positive Solutions for a Fractional Boundary Value Problem with Changing Sign Nonlinearity

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Yongqing Wang ◽  
Lishan Liu ◽  
Yonghong Wu
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Positive Parameter ◽  
Fractional Boundary Value Problem ◽  
Point Boundary ◽  
Liouville Derivative ◽  
Existence Of Positive Solutions

We discuss the existence of positive solutions to the following fractionalm-point boundary value problem with changing sign nonlinearityD0+αu(t)+λf(t,u(t))=0,0<t<1,u(0)=0,D0+βu(1)=∑i=1m-2ηiD0+βu(ξi), whereλis a positive parameter,1<α≤2,0<β<α-1,0<ξ1<⋯<ξm-2<1with∑i=1m-2ηiξiα-β-1<1,D0+αis the standard Riemann-Liouville derivative,fand may be singular att=0and/ort=1and also may change sign. The work improves and generalizes some previous results.

Existence of Positive Solutions of Nonlinear Second-Order M-Point Boundary Value Problem

2011 ◽  
Vol 2 (1) ◽  
pp. 28-33
Author(s):  
F. H. Wong ◽  
C. J. Chyan ◽  
S. W. Lin
Keyword(s):  
Boundary Value Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Point Boundary ◽  
Existence Of Positive Solutions

Under suitable conditions on, the nonlinear second-order m-point boundary value problem has at least one positive solution. In this paper, the authors examine the positive solutions of nonlinear second-order m-point boundary value problem.

scholarly journals EXISTENCE OF POSITIVE SOLUTIONS FOR SUPERLINEAR SEMIPOSITONE $m$-POINT BOUNDARY-VALUE PROBLEMS

2003 ◽  
Vol 46 (2) ◽  
pp. 279-292 ◽  
Author(s):  
Ruyun Ma
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Point Theorem ◽  
Boundary Value ◽  
Positive Parameter ◽  
Point Boundary ◽  
Existence Of Positive Solutions ◽  
The Fixed Point Theorem

AbstractIn this paper we consider the existence of positive solutions to the boundary-value problems\begin{align*} (p(t)u')'-q(t)u+\lambda f(t,u)\amp=0,\quad r\ltt\ltR, \\[2pt] au(r)-bp(r)u'(r)\amp=\sum^{m-2}_{i=1}\alpha_iu(\xi_i), \\ cu(R)+dp(R)u'(R)\amp=\sum^{m-2}_{i=1}\beta_iu(\xi_i), \end{align*}where $\lambda$ is a positive parameter, $a,b,c,d\in[0,\infty)$, $\xi_i\in(r,R)$, $\alpha_i,\beta_i\in[0,\infty)$ (for $i\in\{1,\dots m-2\}$) are given constants satisfying some suitable conditions. Our results extend some of the existing literature on superlinear semipositone problems. The proofs are based on the fixed-point theorem in cones.AMS 2000 Mathematics subject classification: Primary 34B10, 34B18, 34B15

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