scholarly journals Existence and uniqueness of positive solution for singular boundary value problem

2005 ◽  
Vol 50 (1-2) ◽  
pp. 133-143 ◽  
Author(s):  
Yansheng Liu ◽  
Huimin Yu
Keyword(s):  
Boundary Value Problem ◽  
Positive Solution ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Singular Boundary ◽  
Singular Boundary Value Problem

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR SINGULAR BOUNDARY VALUE PROBLEM ON TIME SCALES

2012 ◽  
Vol 23 (07) ◽  
pp. 1250070 ◽  
Author(s):  
ZHENJIE LIU
Keyword(s):  
Boundary Value Problem ◽  
Time Scales ◽  
Time Scale ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Second Order ◽  
Singular Boundary ◽  
Singular Boundary Value Problem ◽  
Uniqueness Of Solutions ◽  
Monotone Method

This paper investigates the existence and uniqueness of solutions for singular second-order boundary value problem on time scales by using mixed monotone method. The theorems obtained are very general and complement the previous known results. When the time scale 𝕋 is chosen as ℝ or ℤ, the problem will be the corresponding continuous or discrete boundary value problem.

scholarly journals On a Constructive Approach for Derivative-Dependent Singular Boundary Value Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
R. K. Pandey ◽  
Amit K. Verma
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Uniqueness Of Solution ◽  
Singular Boundary ◽  
Singular Boundary Value Problem ◽  
Singular Boundary Value Problems ◽  
Constructive Approach

We present a constructive approach to establish existence and uniqueness of solution of singular boundary value problem-(p(x)y′(x))′=q(x)f(x,y,py′)for0<x≤b,y(0)=a,α1y(b)+β1p(b)y′(b)=γ1.Herep(x)>0on(0,b)allowingp(0)=0. Furtherq(x)may be allowed to have integrable discontinuity atx=0, so the problem may be doubly singular.

scholarly journals EXISTENCE OF POSITIVE SOLUTIONS FOR BOUNDARY-VALUE PROBLEMS WITH SINGULARITIES IN PHASE VARIABLES

2004 ◽  
Vol 47 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Donal O’Regan ◽  
Svatoslav Staněk
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Boundary Value ◽  
Singular Boundary ◽  
Singular Boundary Value Problem ◽  
Existence Of Positive Solutions ◽  
Carathéodory Conditions ◽  
Primary 34B16

AbstractThe singular boundary-value problem $(g(x'))'=\mu f(t,x,x')$, $x'(0)=0$, $x(T)=b>0$ is considered. Here $\mu$ is the parameter and $f(t,x,y)$, which satisfies local Carathéodory conditions on $[0,T]\times(\mathbb{R}\setminus\{b\})\times(\mathbb{R}\setminus\{0\})$, may be singular at the values $x=b$ and $y=0$ of the phase variables $x$ and $y$, respectively. Conditions guaranteeing the existence of a positive solution to the above problem for suitable positive values of $\mu$ are given. The proofs are based on regularization and sequential techniques and use the topological transversality theorem.AMS 2000 Mathematics subject classification: Primary 34B16; 34B18

scholarly journals SOLVABILITY OF SINGULAR DIRICHLET BOUNDARY-VALUE PROBLEMS WITH GIVEN MAXIMAL VALUES FOR POSITIVE SOLUTIONS

2005 ◽  
Vol 48 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Donal O’Regan ◽  
Svatoslav Staněk
Keyword(s):  
Boundary Value Problem ◽  
Positive Solution ◽  
Convergence Theorem ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Singular Boundary ◽  
Singular Boundary Value Problem ◽  
Negative Function ◽  
Carathéodory Conditions ◽  
Primary 34B16

AbstractThe singular boundary-value problem $(g(x'(t)))'=\mu f(t,x(t),x'(t))$, $x(0)=x(T)=0$ and $\max\{x(t):0\le t\le T\}=A$ is considered. Here $\mu$ is the parameter and the negative function $f(t,u,v)$ satisfying local Carathéodory conditions on $[0,T]\times(0,\infty)\times(\mathbb{R}\setminus\{0\})$ may be singular at the values $u=0$ and $v=0$ of the phase variables $u$ and $v$. The paper presents conditions which guarantee that for any $A>0$ there exists $\mu_A>0$ such that the above problem with $\mu=\mu_A$ has a positive solution on $(0,T)$. The proofs are based on the regularization and sequential techniques and use the Leray–Schauder degree and Vitali’s convergence theorem.AMS 2000 Mathematics subject classification: Primary 34B16; 34B18

Existence of solutions in a singular biharmonic nonlinear problem

1993 ◽  
Vol 36 (3) ◽  
pp. 537-546 ◽  
Author(s):  
Gaston L. Hernandez ◽  
Y. Choi
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Nonlinear Problem ◽  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Singular Boundary ◽  
Singular Boundary Value Problem ◽  
Image Position

In this work we prove the existence and uniqueness of positive solutions of the nonlinear singular boundary value problemwhere 0<σ<1.Extensions of the above results to the case of Δ2u−f(x, u) = 0 with appropriate singularity built into f are also given.

scholarly journals The Existence of Positive Solution to Three-Point Singular Boundary Value Problem of Fractional Differential Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
Yuansheng Tian ◽  
Anping Chen
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Boundary Value ◽  
Singular Boundary ◽  
Singular Boundary Value Problem ◽  
Nonlinear Fractional Differential Equation ◽  
Fixed Point Index Theory ◽  
Fractional Differential

We investigate the existence of positive solution to nonlinear fractional differential equation three-point singular boundary value problem:Dqu(t)+f(t,u(t))=0,0<t<1,u(0)=0,u(1)=αD(q−1)/2u(t)|t=ξ, where1<q≤2is a real number,ξ∈(0,1/2],α∈(0,+∞)andαΓ(q)ξ(q−1)/2<Γ((q+1)/2),Dqis the standard Riemann-Liouville fractional derivative, andf∈C((0,1]×[0,+∞),[0,+∞)),lim⁡t→+0f(t,⋅)=+∞(i.e.,fis singular att=0). By using the fixed-point index theory, the existence result of positive solutions is obtained.

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