scholarly journals Solutions of two-point boundary value problems via phase-plane analysis

2016 ◽  
Author(s):  
Svetlana Atslega ◽  
F. Sadyrbaev
Keyword(s):  
Boundary Value Problems ◽  
Phase Plane ◽  
Boundary Value ◽  
Phase Plane Analysis ◽  
Point Boundary ◽  
Plane Analysis

scholarly journals Multiple Positive Solutions for the Dirichlet Boundary Value Problems by Phase Plane Analysis

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
A. Kirichuka ◽  
F. Sadyrbaev
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Phase Plane ◽  
Boundary Value ◽  
Phase Plane Analysis ◽  
Multiple Positive Solutions ◽  
Degree Polynomial ◽  
Plane Analysis ◽  
Phase Plane Method ◽  
Scalar Differential Equation

We consider boundary value problems for scalar differential equationx′′+λfx=0,x(0)=0,x(1)=0, wheref(x)is a seventh-degree polynomial andλis a parameter. We use the phase plane method combined with evaluations of time-map functions and make conclusions on the number of positive solutions. Bifurcation diagrams are constructed and examples are considered illustrating the bifurcation processes.

Infinitely Many Rotationally Symmetric Solutions to a Class of Semilinear Laplace–Beltrami Equations on Spheres

2015 ◽  
Vol 58 (4) ◽  
pp. 723-729 ◽  
Author(s):  
Alfonso Castro ◽  
Emily M. Fischer
Keyword(s):  
Differential Equation ◽  
Phase Plane ◽  
Phase Plane Analysis ◽  
Point Boundary ◽  
Initial Value ◽  
Beltrami Equations ◽  
Plane Analysis ◽  
Singular Ordinary Differential Equation ◽  
Rotationally Symmetric ◽  
Pohozaev Type Identity

AbstractWe show that a class of semilinear Laplace–Beltrami equations on the unit sphere in ℝn has inûnitely many rotationally symmetric solutions. The solutions to these equations are the solutions to a two point boundary value problem for a singular ordinary differential equation. We prove the existence of such solutions using energy and phase plane analysis. We derive a Pohozaev-type identity in order to prove that the energy to an associated initial value problem tends to infinity as the energy at the singularity tends to infinity. The nonlinearity is allowed to grow as fast as |s|p-1s for |s| large with 1 < p < (n + 5)/(n − 3).

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.

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