Positive periodic solution of p-Laplacian Liénard type differential equation with singularity and deviating argument
Xin and Cheng Advances in Difference Equations
Positive periodic solution of p-Laplacian
Yun Xin 0 1 3
Zhibo Cheng 0 1 2
0 University , Jiaozuo, 454000 , China
1 Technology , Henan Polytechnic
2 School of Mathematics and Information Science, Henan Polytechnic University , Jiaozuo, 454000 , China
3 College of Computer Science and Technology, Henan Polytechnic University , Jiaozuo, 454000 , China
In this paper, we consider the following p-Laplacian Liénard type differential equation with singularity and deviating argument: By applications of coincidence degree theory and some analysis techniques, sufficient conditions for the existence of positive periodic solutions are established.
positive solution; p-Laplacian; Liénard equation; singularity; deviating
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available at the end of the article
argument
1 Introduction
with singularity and deviating argument:
ϕp x (t)
+ f x(t) x (t) + g t, x(t – σ ) = e(t),
(.)
solutions of the Liénard equation with a singularity and a deviating argument,
x (t) + f x(t) x (t) + g t, x(t – σ ) = ,
where σ is a constant. When g has a strong singularity at x = and satisfies a new small
force condition at x = ∞, the author proved that the given equation has at least one positive
T -periodic solution.
However, the Liénard type differential equation (.), in which there is a p-Laplacian
Liénard type differential equation, has not attracted much attention in the literature. There
are not so many existence results for (.) even as regards the p-Laplacian Liénard type
differential equation with singularity and deviating argument. In this paper, we try to fill
this gap and establish the existence of a positive periodic solution of (.) using coincidence
degree theory. Our new results generalize in several aspects some recent results contained
in [, ].
2 Preparation
Let X and Y be real Banach spaces and L : D(L) ⊂ X → Y be a Fredholm operator with
index zero, here D(L) denotes the domain of L. This means that Im L is closed in Y and
dim Ker L = dim(Y / Im L) < +∞. Consider supplementary subspaces X, Y of X, Y ,
respectively, such that X = Ker L ⊕ X, Y = Im L ⊕ Y. Let P : X → Ker L and Q : Y → Y denote the
natural projections. Clearly, Ker L ∩ (D(L) ∩ X) = {} and so the restriction LP := L|D(L)∩X
is invertible. Let K denote the inverse of LP.
Let be an open bounded subset of X with D(L) ∩ = ∅. A map N : → Y is said to
be L-compact in if QN ( ) is bounded and the operator K (I – Q)N : → X is compact.
Lemma . (Gaines and Mawhin []) Suppose that X and Y are two Banach spaces, and
L : D(L) ⊂ X → Y is a Fredholm operator with index zero. Let ⊂ X be an open bounded
set and N : → Y be L-compact on . Assume that the following conditions hold:
() Lx = λNx, ∀x ∈ ∂ ∩ D(L), λ ∈ (, );
() Nx ∈/ Im L, ∀x ∈ ∂ ∩ Ker L;
() deg{JQN , ∩ Ker L, } = , where J : Im Q → Ker L is an isomorphism.
Then the equation Lx = Nx has a solution in ∩ D(L).
For the sake of convenience, throughout this paper we will adopt the following notation:
|u|p =
T
|u|∞ = max u(t) ,
t∈[,T]
|u| = min u(t) ,
t∈[,T]
T
|u|p dt
p
T
h(t) dt.
Lemma . ([]) If ω ∈ C(R, R) and ω() = ω(T ) = , then
ω(t) p dt ≤
T
πp
p
T
ω (t) p dt,
where ≤ p < ∞, πp = (p–)/p (– psd–ps )/p = pπs(ipn–(π)/p/)p .
T
T
x(t) p dt
x(t) p dt
p
p
T
≤ πp
T
T
=
≤
T
≤ πp
= πTp
T
T
T
x (t) p dt
+ dT p .
p
p
T
p
p
ω(t) + x(t) p dt
ω(t) p dt
p
+
x(t) p dt
p
ω (t) p dt
x (t) p dt
+ dT p
+ dT p .
Lemma . If x ∈ C(R, R) with x(t + T ) = x(t), and t ∈ [, T ] such that |x(t)| < d, then
In order to apply the topological degree theorem to study the existence of a positive
periodic solution for (.), we rewrite (.) in the form
⎧⎨ x(t) = ϕq(x(t)),
⎩ x(t) = –f (x(t))x(t) – g(t, x(t – σ )) + e(t),
where p + q = . Clearly, if x(t) = (x(t), x(t)) is an T -periodic solution to (.), then
x(t) must be an T -periodic solution to (.). Thus, the problem of finding an T -periodic
solution for (.) reduces to finding one for (.).
Now, set X = Y = {x = (x(t), x(t)) ∈ C(R, R) : x(t + T ) ≡ x(t)} with the norm x =
max{|x|∞, |x|∞}. Clearly, X and Y are both Banach spaces. Meanwhile, define
L : D(L) = x ∈ C R, R : x(t + T ) = x(t), t ∈ R ⊂ X → Y
by
(Lx)(t) =
x(t)
x(t)
and N : X → Y by
(Nx)(t) =
T
T
Ker L ∼= R,
Im L = y ∈ Y :
y(s)
y(s)
ds =
ψ (t) = lim sup g(t, x)
x→+∞ xp–
g(t, x) ≤ ψ (t) + ε x + gε(t),
exists uniformly a.e. t ∈ [, T ], i.e., for any ε > there is gε ∈ L(, T ) such that
for all x > and a.e. t ∈ [, T ]. Moreover, ψ ∈ C(R, R) and ψ (t + T ) = ψ (t).
For the sake of convenience, we list the following assumptions which will be used
repeatedly in the sequel:
(H) (Balance condition) There exist constants < D < D such that if x is a positive
continuous T -periodic function satisfying
g t, x(t) dt = ,
then
T
D ≤ x(τ ) ≤ D,
for some τ ∈ [, T ].
(.)
(.)
(.)
(H) (Degree condition) g¯(x) < for all (...truncated)