Multivalued -Lienard systems
MULTIVALUED p-LIENARD SYSTEMS
MICHAEL E. FILIPPAKIS AND NIKOLAOS S. PAPAGEORGIOU
Received 7 October 2003 and in revised form 9 March 2004
We examine p-Lienard systems driven by the vector p-Laplacian differential operator and
having a multivalued nonlinearity. We consider Dirichlet systems. Using a fixed point
principle for set-valued maps and a nonuniform nonresonance condition, we establish
the existence of solutions.
1. Introduction
In this paper, we use fixed point theory to study the following multivalued p-Lienard
system:
p−2 d
x (t) x (t) + ∇G x(t) + F t,x(t),x (t) 0
dt
a.e. on T = [0,b],
(1.1)
x(0) = x(b) = 0, 1 < p < ∞.
In the last decade, there have been many papers dealing with second-order multivalued boundary value problems. We mention the works of Erbe and Krawcewicz [5, 6],
Frigon [7, 8], Halidias and Papageorgiou [9], Kandilakis and Papageorgiou [11], Kyritsi
et al. [12], Palmucci and Papalini [17], and Pruszko [19]. In all the above works, with
the exception of Kyritsi et al. [12], p = 2 (linear differential operator), G = 0, and g = 0.
Moreover, in Frigon [7, 8] and Palmucci and Papalini [17], the inclusions are scalar (i.e.,
N = 1). Finally we should mention that recently single-valued p-Lienard systems were
studied by Mawhin [14] and Manásevich and Mawhin [13].
In this work, for problem (1.1), we prove an existence theorem under conditions of
nonuniform nonresonance with respect to the first weighted eigenvalue of the negative
vector ordinary p-Laplacian with Dirichlet boundary conditions [15, 20]. Our approach
is based on the multivalued version of the Leray-Schauder alternative principle due to
Bader [1] (see Section 2).
Copyright © 2004 Hindawi Publishing Corporation
Fixed Point Theory and Applications 2004:2 (2004) 71–80
2000 Mathematics Subject Classification: 34B15, 34C25
URL: http://dx.doi.org/10.1155/S1687182004310016
72
Multivalued p-Lienard systems
2. Mathematical background
In this section, we recall some basic definitions and facts from multivalued analysis, the
spectral properties of the negative vector p-Laplacian, and the multivalued fixed point
principles mentioned in the introduction. For details, we refer to Denkowski et al. [3] and
Hu and Papageorgiou [10] (for multivalued analysis), to Denkowski et al. [2] and Zhang
[20] (for the spectral properties of the p-Laplacian), and to Bader [1] (for the multivalued
fixed point principle; similar results can also be found in O’Regan and Precup [16] and
Precup [18]).
Let (Ω,Σ) be a measurable space and X a separable Banach space. We introduce the
following notations:
P f (c) (X) = A ⊆ X : nonempty, closed (and convex) ,
P(w)k(c) (X) = A ⊆ X : nonempty, (weakly) compact (and convex) .
(2.1)
A multifunction F : Ω → P f (X) is said to be measurable if, for all x ∈ X, ω → d(x,
F(ω)) = inf [x − y : y ∈ F(ω)] is measurable. A multifunction F : Ω → 2X \{∅} is said
to be “graph measurable” if GrF = {(ω,x) ∈ Ω × X : x ∈ F(ω)} ∈ Σ × B(X), with B(X)
being the Borel σ-field of X. For P f (X)-valued multifunctions, measurability implies
graph measurability and the converse is true if Σ is complete (i.e., Σ = Σ̂ = the universal σfield). Let µ be a finite measure on (Ω,Σ), 1 ≤ p ≤ ∞, and F : Ω → 2X \{∅}. We introduce
p
the set SF = { f ∈ L p (Ω,X) : f (ω) ∈ F(ω) µ-a.e.}. This set may be empty. For a graphmeasurable multifunction, it is nonempty if and only if inf [ y : y ∈ F(ω)] ≤ ϕ(ω) µ-a.e.
on Ω, with ϕ ∈ L p (Ω)+ .
Let Y , Z be Hausdorff topological spaces. A multifunction G : Y → 2Z \{∅} is said
to be “upper semicontinuous” (usc for short) if, for all C ⊆ Z closed, G− (C) = { y ∈ Y :
G(y) ∩ C = ∅} is closed or equivalently for all U ⊆ Z open, G+ { y ∈ Y : G(y) ⊆ U } is
open. If Z is a regular space, then a P f (Z)-valued multifunction which is usc has a closed
graph. The converse is true if the multifunction G is locally compact (i.e., for every y ∈ Y ,
there exists a neighborhood U of y such that G(U) is compact in Z). A Pk (Z)-valued
multifunction which is usc maps compact sets to compact sets.
Consider the following weighted nonlinear eigenvalue problem in RN :
p−2
p−2
− x (t) x (t) = λθ(t)x(t) x(t) a.e. on T = [0,b],
x(0) = x(b) = 0, 1 < p < ∞, θ ∈ L∞ (T), {θ > 0} > 0, λ ∈ R.
(2.2)
1
Here by | · |1 we denote the 1-dimensional Lebesgue measure. The real parameters
λ, for which problem (2.3) has a nontrivial solution, are called eigenvalues of the neg1,p
ative vector p-Laplacian with Dirichlet boundary conditions denoted by (− p ,W0 (T,
RN )), with weight θ ∈ L∞ (T). The corresponding nontrivial solutions are known as
eigenfunctions. We know that the eigenvalues of problem (2.3) are the same as those of
the corresponding scalar problem [13]. Then from Denkowski et al. [2] and Zhang [20],
we know that there exist two sequences {λn (θ)}n≥1 and {λ−n (θ)}n≥1 such that λn (θ) > 0,
λn (θ) → +∞ and λ−n (θ) < 0, λ−n (θ) → −∞ as n → ∞. Moreover, if θ(t) ≥ 0 a.e. on T with
strict inequality on a set of positive Lebesgue measure, then we have only the positive
M. E. Filippakis and N. S. Papageorgiou 73
sequence {λn (θ)}n≥1 . Also, for λ1 (θ) > 0, we have the following variational characterization:
p
λ1 (θ) = inf b
0
x p
1,p
N
p : x ∈ W0 T, R , x = 0 .
θ(t)x(t) dt
(2.3)
The infimum is attained at the normalized principal eigenfunction u1 (λ1 (θ) > 0 is
simple) and u1 (t) = 0 a.e. on T. Also, λ1 (θ) is strictly monotone with respect to θ, namely,
if θ1 (t) ≤ θ2 (t) a.e. on T with strict inequality on a set of positive measure, then λ1 (θ2 ) <
λ1 (θ1 ) (see (3.2)).
Finally we state the multivalued fixed point principle that we will use in the study of
problem (1.1). So let Y , Z be two Banach spaces and C ⊆ Y , D ⊆ Z two nonempty closed
and convex sets. We consider multifunctions G : C → 2C \{∅} which have a decomposition G = K ◦ N, satisfying the following: K : D → C is completely continuous, namely, if
w
zn −
→ z in D, then K(zn ) → K(z) in C and N : C → Pwkc (D) is usc from C, furnished with
the strong topology into D, furnished with the weak topology.
Theorem 2.1. If C, D, and G = K ◦ N are as above, 0 ∈ C, and G is compact (namely, G
maps bounded subsets of C into relatively compact subsets of D), then one of the following
alternatives holds:
(a) S = { y ∈ C : y ∈ µG(y) for some µ ∈ (0,1)} is unbounded or
(b) G has a fixed point, that is, there exists y ∈ C such that y ∈ G(y).
Remark 2.2. Evidently this is a multivalued version of the classical Leray-Schauder alternative principle [2, page 206]. In contrast to previous multivalued extensions of the
Leray-Schauder alternative principal [4, page 61], Theorem 2.1 does not require G to
have convex values, which is important when dealing with nonlinear problems such as
(1.1).
3. Nonuniform nonresonance
In this section, we deal with problem (1.1) using a condition of nonuniform nonresonance with respect to the first eigenvalue λ1 (θ) > 0. Our hypotheses on t (...truncated)
