scholarly journals Global Structure of Positive Solutions for Some Second-Order Multipoint Boundary Value Problems

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Hongyu Li ◽  
Junting Zhang
Keyword(s):  
Boundary Value Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Boundary Value ◽  
Global Bifurcation ◽  
Global Structure ◽  
Second Order ◽  
Bifurcation Method ◽  
Multipoint Boundary ◽  
Multipoint Boundary Value Problems

We investigate in this paper the following second-order multipoint boundary value problem:-(Lφ)(t)=λf(t,φ(t)),0≤t≤1,φ′0=0,φ1=∑i=1m-2βiφηi. Under some conditions, we obtain global structure of positive solution set of this boundary value problem and the behavior of positive solutions with respect to parameterλby using global bifurcation method. We also obtain the infinite interval of parameterλabout the existence of positive solution.

scholarly journals Positive Solutions for Resonant and Nonresonant Nonlinear Third-Order Multipoint Boundary Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Liu Yang ◽  
Chunfang Shen ◽  
Dapeng Xie
Keyword(s):  
Boundary Value Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Boundary Value ◽  
Type Theorem ◽  
Third Order ◽  
Multipoint Boundary ◽  
Multipoint Boundary Value Problems ◽  
First Time ◽  
Multipoint Boundary Value Problem

Positive solutions for a kind of third-order multipoint boundary value problem under the nonresonant conditions and the resonant conditions are considered. In the nonresonant case, by using the Leggett-Williams fixed point theorem, the existence of at least three positive solutions is obtained. In the resonant case, by using the Leggett-Williams norm-type theorem due to O’Regan and Zima, the existence result of at least one positive solution is established. It is remarkable to point out that it is the first time that the positive solution is considered for the third-order boundary value problem at resonance. Some examples are given to demonstrate the main results of the paper.

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 Bifurcation from Interval and Positive Solutions of a Nonlinear Second-Order Dynamic Boundary Value Problem on Time Scales

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Hua Luo
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Time Scale ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Global Bifurcation ◽  
Degree Theory ◽  
Second Order ◽  
Topological Degree Theory ◽  
Bifurcation From Interval

Let𝕋be a time scale with0,T∈𝕋. We give a global description of the branches of positive solutions to the nonlinear boundary value problem of second-order dynamic equation on a time scale𝕋,uΔΔ(t)+f(t,uσ(t))=0,  t∈[0,T]𝕋,  u(0)=u(σ2(T))=0, which is not necessarily linearizable. Our approaches are based on topological degree theory and global bifurcation techniques.

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 Multiple Positive Solutions to Multipoint Boundary Value Problem for a System of Second-Order Nonlinear Semipositone Differential Equations on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Gang Wu ◽  
Longsuo Li ◽  
Xinrong Cong ◽  
Xiufeng Miao
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Time Scales ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Second Order ◽  
Dynamic Equations ◽  
Multiple Positive Solutions ◽  
Multipoint Boundary ◽  
Multipoint Boundary Value Problem

We study a system of second-order dynamic equations on time scales(p1u1∇)Δ(t)-q1(t)u1(t)+λf1(t,u1(t),u2(t))=0,t∈(t1,tn),(p2u2∇)Δ(t)-q2(t)u2(t)+λf2(t,u1(t),u2(t))=0, satisfying four kinds of different multipoint boundary value conditions,fiis continuous and semipositone. We derive an interval ofλsuch that anyλlying in this interval, the semipositone coupled boundary value problem has multiple positive solutions. The arguments are based upon fixed-point theorems in a cone.

scholarly journals Global Structure of Positive Solutions for a Singular Fourth-Order Integral Boundary Value Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Wenguo Shen ◽  
Tao He
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fourth Order ◽  
Boundary Value ◽  
Global Bifurcation ◽  
Global Structure ◽  
Stieltjes Integral ◽  
Integral Boundary ◽  
Integral Boundary Value Problem

We consider fourth-order boundary value problemsu′′′′(t)=λh(t)f(u(t)),  0<t<1,  u(0)=∫01‍u(s)dα(s),  u′(0)=u(1)=u′(1)=0, where∫01‍u(s)dα(s)is a Stieltjes integral withα(t)being nondecreasing andα(t)being not a constant on[0,1];h(t)may be singular att=0andt=1,h∈C((0,1),[0,∞))withh(t)≢0on any subinterval of(0,1);f∈C([0,∞),[0,∞))andf(s)>0for alls>0, andf0=∞,  f∞=0,  f0=lims→0+f(s)/s,  f∞=lims→+∞f(s)/s.We investigate the global structure of positive solutions by using global bifurcation techniques.

scholarly journals S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems

2020 ◽  
Vol 18 (1) ◽  
pp. 1658-1666
Author(s):  
Liangying Miao ◽  
Jing Liu ◽  
Zhiqian He
Keyword(s):  
Boundary Value Problem ◽  
Weight Function ◽  
Positive Solutions ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Second Order ◽  
Connected Component ◽  
Bifurcation Method ◽  
Neumann Boundary Value Problem ◽  
Neumann Boundary Value Problems

Abstract By using the bifurcation method, we study the existence of an S-shaped connected component in the set of positive solutions for discrete second-order Neumann boundary value problem. By figuring the shape of unbounded connected component of positive solutions, we show that the Neumann boundary value problem has three positive solutions suggesting suitable conditions on the weight function and nonlinearity.

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