Multiple solutions of three-point boundary value problems for second-order impulsive differential equation at resonance
Zhao et al. Boundary Value Problems 2014, 2014:103
http://www.boundaryvalueproblems.com/content/2014/1/103
RESEARCH
Open Access
Multiple solutions of three-point boundary
value problems for second-order impulsive
differential equation at resonance
Yulin Zhao1* , Haibo Chen2 and Qiming Zhang1
*
Correspondence:
1
School of Science, Hunan
University of Technology, Zhuzhou,
Hunan 412007, PR China
Full list of author information is
available at the end of the article
Abstract
In this paper, by using the coincidence degree theory and the upper and lower
solutions method, we deal with the existence of multiple solutions to three-point
boundary value problems for second-order differential equation with impulses at
resonance. An example is given to show the validity of our results.
Keywords: resonance; coincidence degree; upper and lower solutions; impulsive;
three-point boundary value problems
1 Introduction
The purpose of the present paper is to investigate the following second-order impulsive
differential equations:
⎧
⎪
⎪
⎨(ρ(t)u (t)) = f (t, u(t), u (t)), t ∈ J, t = tk ,
u(tk ) = Ik (tk , u(tk )), k = , , . . . , m,
⎪
⎪
⎩
u (tk ) = Jk (tk , u(tk )),
(.)
together with the boundary conditions:
u () = ,
u() = u(η),
(.)
where J = [, ], ρ : J → (, +∞) is a continuous differentiable function, f : J × R → R is
continuous, < η < , Ik , Jk ∈ C(J, R) for ≤ k ≤ m, m is a fixed positive integer, = t <
t < t < · · · < tm < tm+ = , η = tk , u(tk ) = u(tk+ ) – u(tk– ) denotes the jump of u(t) at t = tk ,
u (tk ) = u (tk+ ) – u (tk– ). u (tk+ ), u(tk+ ) (u (tk– ), u(tk– )) represent the right limit (left limit) of
u (t) and u(t) at t = tk , respectively.
Impulsive differential equations describe processes which experience a sudden change
of their state at certain moments. The theory of impulse differential equations has been
a significant development in recent years and played a very important role in modern
applied mathematical models of real processes rising in phenomena studied in physics,
population dynamics, chemical technology, biotechnology, and economics; see [–] and
the references therein.
©2014 Zhao et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Zhao et al. Boundary Value Problems 2014, 2014:103
http://www.boundaryvalueproblems.com/content/2014/1/103
Page 2 of 17
Recently, several authors (see [, –] and the references therein) have studied the
existence of nontrivial or positive solutions for second-order three-point boundary value
problem of the type
⎧
⎨u = f (t, u, u ),
⎩u () = ,
(.)
u() = au(η).
Note that the nonlinear term f depends on u and its derivative u , then the relative problem becomes more complicated. A general method to deal with this difficulty is to add
some conditions to restrict the growth of the u term. One condition is the Caratheodory
nonlinearity, the other usual condition is Nagumo condition or Nagumo-Winter condition (see [, , , –]). When a = , the linear operator Lu = u is invertible, this is the
so-called non-resonance case. Gupta et al. made use of the Leray-Schauder continuation
theorem to get the results on the existence of the solution for the problems (.) when
a = in []. By using the Leray-Schauder continuation theorem and in the presence of
two pairs of upper and lower solutions, Khan and Webb [] established the existence of
at least three solutions for the problem (.) when a = . The linear operator Lu = u is
non-invertible when a = , this is the so-called resonance case, and the Leray-Schauder
continuation theorem cannot be applied. In [], by using the coincidence degree theory
of Mawhin [] and some linear or non-linear growth assumptions on f , Feng and Webb
obtained the existence of the solution of the problem (.) when a = . By applying the
nonlinear alternative of Leray-Schauder, Ma [] have showed the existence of at least one
solution for the problem (.) when a = .
Recently, using the coincidence degree theory and the concept of autonomous curvature
bound set, Liu and Yu [] have studied the existence of at least one solution for the problem
(.)-(.) when u(tk ) = Ik (u(tk ), u (tk )), u (tk ) = Jk (u(tk ), u (tk )).
In the present paper, we assume that there exist n (n ∈ N and n ≥ ) pairs of upper and
lower solutions for problem (.)-(.) and the nonlinear f satisfies a Nagumo-like growth
condition with respect to u . By considering a suitably modified nonlinearity and applying
the coincidence degree method of Mawhin [], the existence of multiple solutions for the
problem (.)-(.) is given.
2 Preliminaries
Let
X = PC (J) ∩ u () = , u() = u(η) ,
Z = PC(J) × Rm ,
where
PC(J) = u ∈ C J * , u t – and u t + exist, and u tk– = u(tk ) .
PC (J) = u : J → R : u(t) is continuously differentiable for t = , , tk ; u t –
and u t + exist, and u tk– = u (tk ) , J * = J\{t , t , . . . , tm }.
Obviously, X is a Banach space with the following norm:
u X = max sup u(t) , sup u (t) .
t∈J
t∈J
Zhao et al. Boundary Value Problems 2014, 2014:103
http://www.boundaryvalueproblems.com/content/2014/1/103
Page 3 of 17
In the following, we recall the concept of strict upper and lower solutions for problem
(.)-(.).
Definition . A function α(t) ∈ PC (J) ∩ C (J * ) is said to be a strict lower solution of the
problem (.)-(.) if
ρ(t)α (t) > f t, α(t), α (t) , t ∈ J * ,
α (tk ) ≥ Jk tk , α(tk ) ,
α(tk ) = Ik tk , α(tk ) ,
α () ≥ ,
(.)
k = , , . . . , m,
α() – α(η) ≤ .
(.)
(.)
Similarly, a function β(t) ∈ PC (J) ∩ C (J * ) is said to be a strict upper solution of the
problem (.)-(.) if
ρ(t)β (t) < f t, β(t), β (t) , t ∈ J * ,
β (tk ) ≤ Jk tk , β(tk ) ,
β(tk ) = Ik tk , β(tk ) ,
β () ≤ ,
(.)
k = , , . . . , m,
β() – β(η) ≥ .
(.)
(.)
Remark . Let f : J × R → R be continuous, Ik , Jk ∈ C(J, R), and u ∈ PC (J) ∩ C (J * ) is a
solution of the problem (.)-(.), if α(β) is a strict lower solution (strict upper solution)
for the problem (.)-(.) with α ≤ u (u ≤ β), then α < u (u < β) on (, ).
Definition . Let α be a strict lower solution and β be a strict upper solution for the
problem (.)-(.) satisfying α(t) < β(t) on J. We say that f : J × R → R has property (H)
relative to α and β, if there exists a function ψ ∈ C ([, +∞), (, +∞)) such that
f (t, u, p) < ψ |p| ,
(.)
for all u(t) ∈ (–β(t), –α(t)) ∪ (α(t), β(t)), t ∈ J, and
+∞
s
ds = +∞,
θ s + ψ(s)
where ≤ θ < +∞ with |ρ (t)| ≤ θ , t ∈ J.
3 The key lemmas
Let dom L = C (J * ) ∩ X, and
L : dom L → Z,
N : u → z,
u → ρ(t)u (t) , u(t ), . . . , u(tm ), u (t ), . (...truncated)
