scholarly journals Triple positive solutions for the one-dimensional $p$-laplacian

2006 ◽  
Vol 37 (1) ◽  
pp. 15-25
Author(s):  
Zhanbing Bai ◽  
Mingfu Ma ◽  
Xiangqian Liang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Nonlinear Term ◽  
Sufficient Conditions ◽  
One Dimensional ◽  
First Order ◽  
Triple Positive Solutions ◽  
The One

We consider the boundary value problem: $ \left(\varphi_p(x'(t))\right)'+ q(t)f(t, x(t), x'(t))=0, p>1, t \in [0, 1] $, with $ x(0)=x(1)=0 $, or $ x(0)=x'(1)=0 $. Using a fixed point theorem due to Avery and Peterson, sufficient conditions are obtained that guarantee the existence of at least three positive solutions. The emphasis here is the nonlinear term $ f $ is involved with the first order derivative. An example is also included to illustrate the importance of the results obtained

scholarly journals Existence of Positive Solutions for Fourth-Order Boundary Value Problems with Sign-Changing Nonlinear Terms

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Xingfang Feng ◽  
Hanying Feng
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fourth Order ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Existence Of Positive Solutions

the existence of positive solutions for a fourth-order boundary value problem with a sign-changing nonlinear term is investigated. By using Krasnoselskii’s fixed point theorem, sufficient conditions that guarantee the existence of at least one positive solution are obtained. An example is presented to illustrate the application of our main results.

scholarly journals Existence of Multiple Positive Solutions for Even Order Multi-Point Boundary Value Problems

2007 ◽  
Vol 14 (4) ◽  
pp. 775-792
Author(s):  
Youyu Wang ◽  
Weigao Ge
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Even Order

Abstract In this paper, we consider the existence of multiple positive solutions for the 2𝑛th order 𝑚-point boundary value problem: where (0,1), 0 < ξ 1 < ξ 2 < ⋯ < ξ 𝑚–2 < 1. Using the Leggett–Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The associated Green's function for the above problem is also given.

scholarly journals POSITIVE SOLUTIONS OF NTH-ORDER BOUNDARY VALUE PROBLEMS WITH INTEGRAL BOUNDARY CONDITIONS

2015 ◽  
Vol 20 (2) ◽  
pp. 188-204 ◽  
Author(s):  
Ilkay Yaslan Karaca ◽  
Fatma Tokmak Fen
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Point Theorem ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Integral Boundary Conditions ◽  
Integral Boundary

In this paper, by using double fixed point theorem and a new fixed point theorem, some sufficient conditions for the existence of at least two and at least three positive solutions of an nth-order boundary value problem with integral boundary conditions are established, respectively. We also give two examples to illustrate our main results.

scholarly journals Positive Solutions to a Second-Order Discrete Boundary Value Problem

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaojie Lin ◽  
Wenbin Liu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Second Order ◽  
The Fixed Point Theorem

We are concerned with second-order discrete boundary value problems and obtain some sufficient conditions for the existence of at least one positive solution by using the fixed point theorem due to Krasnosel'skii on a cone.

scholarly journals Existence of Three Positive Solutions form-Point Discrete Boundary Value Problems withp-Laplacian

2009 ◽  
Vol 2009 ◽  
pp. 1-15 ◽  
Author(s):  
Yanping Guo ◽  
Wenying Wei ◽  
Yuerong Chen
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Point Theorem ◽  
Boundary Value ◽  
Laplacian Operator ◽  
One Dimensional

We consider the multi-point discrete boundary value problem with one-dimensionalp-Laplacian operatorΔ(ϕp(Δu(t−1))+q(t)f(t,u(t),Δu(t))=0,t∈{1,…,n−1}subject to the boundary conditions:u(0)=0,u(n)=∑i=1m−2aiu(ξi), whereϕp(s)=|s|p−2s,p>1,ξi∈{2,…,n−2}with1<ξ1<⋯<ξm−2<n−1andai∈(0,1),0<∑i=1m−2ai<1. Using a new fixed point theorem due to Avery and Peterson, we study the existence of at least three positive solutions to the above boundary value problem.

scholarly journals Upper and Lower Solutions form-Point Impulsive BVP with One-Dimensionalp-Laplacian

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Junfang Zhao ◽  
Jianfeng Zhao ◽  
Weigao Ge
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Point Theorem ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Upper And Lower Solutions ◽  
First Order ◽  
Impulsive Boundary Value Problems

Upper and lower solutions theories are established for a kind ofm-point impulsive boundary value problems withp-Laplacian. By using such techniques and the Schauder fixed point theorem, the existence of solutions and positive solutions is obtained. Nagumo conditions play an important role in the nonlinear term involved with the first-order derivatives.

scholarly journals Existence of Triple Positive Solutions for Second-Order Discrete Boundary Value Problems

2007 ◽  
Vol 2007 ◽  
pp. 1-10 ◽  
Author(s):  
Yanping Guo ◽  
Jiehua Zhang ◽  
Yude Ji
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Point Theorem ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Second Order ◽  
Triple Positive Solutions

By using a new fixed-point theorem introduced by Avery and Peterson (2001), we obtain sufficient conditions for the existence of at least three positive solutions for the equationΔ2x(k−1)+q(k)f(k,x(k),Δx(k))=0, fork∈{1,2,…,n−1}, subject to the following two boundary conditions:x(0)=x(n)=0orx(0)=Δx(n−1)=0, wheren≥3.

scholarly journals Existence of positive solutions of PBVPs for first-order difference equations

2006 ◽  
Vol 2006 ◽  
pp. 1-8 ◽  
Author(s):  
Yinggao Zhou
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Periodic Boundary ◽  
Multiplicity Results ◽  
Order Difference ◽  
First Order ◽  
Existence Of Positive Solutions ◽  
Krasnosel’Skii’S Fixed Point Theorem

We consider the existence of positive solutions for the following first-order periodic boundary value problem:x(n+1)=x(n)−f(n,x(n)),0≤n≤ω−1,x(0)=x(ω). Some criteria for existence of positive solutions of the above difference boundary problem are established by using Krasnosel'skiĭ's fixed point theorem, and some multiplicity results of positive solutions are also derived.

scholarly journals Exact Multiplicity of Solutions for a Class of Singular Generalized One-Dimensionalp-Laplacian Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Youwei Zhang
Keyword(s):  
Fixed Point ◽  
Positive Solutions ◽  
Nonlinear Term ◽  
Fixed Point Theory ◽  
Point Theory ◽  
One Dimensional ◽  
First Order ◽  
Existence Of Positive Solutions ◽  
Exact Multiplicity Of Solutions ◽  
Exact Multiplicity

We describe the existence of positive solutions for a class of singular generalized one-dimensionalp-Laplacian problem. By applying the related fixed point theory in cone, some new and general results on the existence of positive solutions to the singular generalizedp-Laplacian problem are obtained. Note that the nonlinear termfinvolves the first-order derivative explicitly.

scholarly journals Existence of Positive Solutions form-Point Boundary Value Problems on Time Scales

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Ying Zhang ◽  
ShiDong Qiao
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Time Scales ◽  
Point Theorem ◽  
Boundary Value ◽  
Point Boundary ◽  
One Dimensional ◽  
Existence Of Positive Solutions

We study the one-dimensionalp-Laplacianm-point boundary value problem(φp(uΔ(t)))Δ+a(t)f(t,u(t))=0,t∈[0,1]T,u(0)=0,u(1)=∑i=1m−2aiu(ξi), whereTis a time scale,φp(s)=|s|p−2s,p>1, some new results are obtained for the existence of at least one, two, and three positive solution/solutions of the above problem by usingKrasnosel′skll′sfixed point theorem, new fixed point theorem due to Avery and Henderson, as well as Leggett-Williams fixed point theorem. This is probably the first time the existence of positive solutions of one-dimensionalp-Laplacianm-point boundary value problem on time scales has been studied.

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