Multivalued -Lienard systems
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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 Man´asevich 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
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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 = Σˆ = the universal σfield)
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