A Variational Inequality Theory with Applications to P $P$ -Laplacian Elliptic Inequalities
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A Variational Inequality Theory with Applications to P -Laplacian Elliptic Inequalities Yi-rong Jiang1 · Nan-jing Huang2 · Donal O’Regan3
Received: 29 March 2016 / Accepted: 1 July 2017 © Springer Science+Business Media B.V. 2017
Abstract The main purpose of this paper is to establish variational inequality theory in connection with demicontinuous ψp -dissipative maps in reflexive smooth Banach spaces by considering the convergence of approximants. As an application of this variational inequality theory, existence, uniqueness and convergence of approximants of positive weak solution for p-Laplacian elliptic inequalities are obtained under suitable conditions. Keywords Variational inequality theory · Demicontinuous ψp -dissipative map · p-Laplacian elliptic inequality · Positive weak solution Mathematics Subject Classification (2000) 49J40 · 35R45 · 47J20
1 Introduction Let X be a Banach space with its dual X ∗ and K a closed convex subset of X. Let A : D ⊂ ∗ K → X ∗ be a map and J : X → 2X be a duality map defined by J (x) = x ∗ ∈ X ∗ x ∗ , x = x ∗ · x, x ∗ = xp−1 , ∀x ∈ X, 2 ≤ p < ∞. If p = 2, then J is the classical normalized duality map. In this paper, we focus on existence, uniqueness and convergence of approximants of solutions for the following variational inequality in a smooth Banach space: find u ∈ K such that (J u − Au, u − v) ≤ 0,
∀v ∈ K.
(1.1)
This work was supported by the National Natural Science Foundation of China (11471230, 11671282).
B N.-j. Huang
[email protected]
1
College of Science, Guilin University of Technology, Guilin, Guangxi 541004, P.R. China
2
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P.R. China
3
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
Y.-r. Jiang et al.
It is well known that a normalized duality map J is an identity when X is a Hilbert space (Proposition 1.4.8 in [5]). Thus, if X is a Hilbert space, then (1.1) reduces to the following variational inequality: find u ∈ K such that (u − Au, u − v) ≤ 0,
∀v ∈ K.
(1.2)
Lan in [12] discussed the existence of nonzero solutions for (1.2) involving demicontinuous S-contractive maps A. Lan in [13] investigated existence, uniqueness and convergence of approximants of solutions of (1.2) with demicontinuous pseudo-contractive maps A. Lan and Lin [15] obtained the existence of nonzero solutions for (1.2) by establishing a theory of variational inequality index concerning γ -condensing maps A. Jiang [11] studied the existence of nonzero solutions of (1.1) with demicontinuous S-contractive maps A in reflexive smooth Banach spaces and recently, Lan [14] generalized the main results in connection with existence of nonzero solutions of (1.2) for demicontinuous S-contractive maps of [12] from Hilbert spaces to reflexive smooth Banach spaces. In order to discuss the solvability of (1.1), Lan [14] developed a theory of variational inequality for demicontinuous S-contractive maps on bounded closed convex subsets o
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