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 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.

Weak solutions for a second order impulsive boundary value problem

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6431-6439
Author(s):  
Keyu Zhang ◽  
Jiafa Xu ◽  
Donal O’Regan
Keyword(s):  
Boundary Value Problem ◽  
Weak Solutions ◽  
Boundary Value ◽  
Degree Theory ◽  
Point Theory ◽  
Second Order ◽  
Positive Parameter ◽  
Topological Degree Theory ◽  
Existence Of Weak Solutions ◽  
Impulsive Boundary Value Problem

In this paper we use topological degree theory and critical point theory to investigate the existence of weak solutions for the second order impulsive boundary value problem {-x??(t)- ?x(t) = f (t), t ? tj, t ? (0,?), ?x?(tj) = x?(t+j)- x?(t-j) = Ij(x(tj)), j=1,2,..., p, x(0) = x(?) = 0, where ? is a positive parameter, 0 = t0 < t1 < t2 < ... < tp < tp+1 = ?, f ? L2(0,?) is a given function and Ij ? C(R,R) for j = 1,2,..., p.

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