Applications of W. A. Kirk's fixed-point theorem to generalized nonlinear variational-like inequalities in reflexive Ban
- PDF / 568,828 Bytes
- 9 Pages / 468 x 680 pts Page_size
- 88 Downloads / 158 Views
We introduce and study a new class of generalized nonlinear variational-like inequalities, which includes these variational inequalities and variational-like inequalities due to Bose, Cubiotti, Dien, Ding, Ding and Tarafdar, Noor, Parida, Sahoo, and Kumar, and Yao, and others as special cases. By applying Kirk’s fixed-point theorem and Ding-Tan minimax inequality, we establish the existence theorems of solutions for the generalized nonlinear variational-like inequalities in reflexive Banach spaces. 1. Introduction and preliminaries In what follows, let R = (−∞,+∞), let B be a Banach space with norm · , let B ∗ be the topological dual space of B, and let u,v be the pairing between u ∈ B ∗ and v ∈ B. Let D be a nonempty closed convex subset of B and a,b : D × D → R satisfy the following conditions: a is a continuous function which is linear in both arguments; there exist constants α > 0, β > 0 satisfying a(x,x) ≥ αx
(1.1)
2
and a(x, y) ≤ βx y ,
(1.2)
∀x, y ∈ D;
b(x, y) is linear in the first argument and is convex
(1.3)
in the second argument, respectively; there exists a constant γ > 0 satisfying b(x, y) ≤ γx y , b(x, y) − b(x,z) ≤ b(x, y − z),
∀x, y ∈ D;
∀x, y,z ∈ D.
(1.4) (1.5)
Ding and Tarafdar [7] and Ding [4] introduced and studied the following general nonlinear variational inequality problem and general nonlinear variational-like inequality problem, respectively. Find u ∈ D such that
a(u,x − u) + b(u,x) − b(u,u) ≥ Tu,g(x) − g(u) , Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:3 (2005) 251–259 DOI: 10.1155/JIA.2005.251
∀x ∈ D.
(1.6)
252
Applications of W. A. Kirk’s fixed-point theorem
Find u ∈ D such that
Tu − Au,η(x,u) + b(u,x) − b(u,u) ≥ 0,
∀x ∈ D.
(1.7)
They obtained the existence uniqueness theorems of solutions for problems (1.6) and (1.7) in nonempty closed convex subsets of reflexive Banach spaces. Cubiotti [2] established the existence of solution for problem (1.6) in nonempty convex and weakly compact subsets of reflexive Banach spaces. It is well known in the literature that problems (1.6) and (1.7) can characterize a wide class of problems arising in control and optimization, mathematical programming, mechanics, engineering, economics equilibrium, and free boundary-valued problems, and so forth On the other hand, Bose [1], Dien [3], Ding [5], Fang et al. [8], Liu et al. [10], Noor [11], Parida et al. [12], and Yao [13] investigated some special cases of problems (1.6) and (1.7) or a few similar problems in Euclidean spaces, Hilbert spaces, and Banach spaces, respectively. In 1965, Kirk [9] showed the following nice result. Lemma 1.1 (see [9]). Let D be a nonempty bounded closed convex subset of a reflexive Banach space B, and suppose that D has normal structure. If T : D → D is nonexpansive, that is, Tx − T y ≤ x − y ,
∀x ∈ D,
(1.8)
then T has a fixed point in D. Although the result due to Kirk has various applications in different fields, to our knowledge, it never has any applications in variationa
Data Loading...
