DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

POSITIVE SOLUTIONS OF A CLASS OF SINGULAR
FUNCTIONAL BOUNDARY VALUE PROBLEMS
WITH φ-LAPLACIAN
SVATOSLAV STANĚK
The paper discusses the existence of positive solutions (in C 1 [0,T]) to the functional
differential equations of the form (φ(x ′ )) ′ = F(t,x,x ′ ,x ′ (0),x ′ (T)) satisfying the Dirichlet
boundary conditions x(0) = x(T) = 0. The nonlinearity F may be singular at x = 0and
changes its sign.
Copyright © 2006 Svatoslav Staněk. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let T be a positive number. Consider the functional-differential equation
( φ
( x ′ (t) )) ′
= f
( x(t)
) ω
( x ′ (t) ) − p 1
( t,x(t),x ′ (t) ) x ′ (0) + p 2
( t,x(t),x ′ (t) ) x ′ (T) (1.1)
together with the Dirichlet boundary conditions
x(0) = 0, x(T) = 0. (1.2)
A function x ∈ C 1 [0,T] issaidtobea positive solution of the boundary value problem
(BVP) (1.1), (1.2)ifφ(x ′ ) ∈ C 1 (0,T), x>0on(0,T), x satisfies the boundary conditions
(1.2), and (1.1)holdsfort ∈ (0,T).
The aim of this paper is to give conditions for the existence of a positive solution of
the BVP (1.1), (1.2). Our results generalize those in [1] where the equation (φ(x ′ (t))) ′ =
f (x(t)) − q(t)h(x(t))x ′ (0) + r(t)p(x(t))x ′ (T) was discussed. The form of our equation
(1.1) is motivated by a regular functional-differential equation considered in [2]together
with (1.2). This problem is a mathematical model for a biological population.
Throughout this paper, we will use the following assumptions on the functions φ, f ,
ω, p 1 ,andp 2 .
(H 1 ) φ ∈ C 0 (R) is increasing and odd on R and lim u→∞ φ(u) =∞.
(H 2 ) f ∈ C 0 (0,∞), lim x→0 + f (x) =−∞, there is a χ>0suchthat f0
on (χ,∞), and ∫ χ
0 f (s)ds > −∞, ∫ ∞
χ f (s)ds =∞.
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 1029–1039