AUXILIARY PRINCIPLE FOR GENERALIZED NONLINEAR VARIATIONAL-LIKE INEQUALITIES

We introduce and study a new class of generalized nonlinear variational-like inequalities and prove an existence theorem of solutions for this kind of generalized nonlinear variational-like inequalities. By using the auxiliary principle technique, we construct a new iterative scheme for solving the class of the generalized nonlinear variational-like inequalities. The convergence of the sequence generated by the iterative algorithm is also discussed. Our results extend and unify the corresponding results due to Ding, Liu, Ume, Kang, Yao, and others.


Introduction
Variational inequality theory has become a very effective and powerful tool for studying a wide range of problems arising in many diverse fields of pure and applied sciences.It is well known that one of the most important problems in variational inequality theory is the development of efficient and implementable iterative algorithms for solving various classes of variational inequalities and variational inclusions.In  there are a lot of iterative algorithms for finding the approximate solutions of various variational inequalities.Glowinski et al. [8] had developed the auxiliary principle technique.By using the auxiliary principle technique, Ding [3,4], Ding and Tan [5], and Ding and Yao [6], Liu et al. [21,28], Zeng et al. [37], Zeng et al. [38], and others suggested several iterative algorithms to compute approximate solutions for some classes of general nonlinear mixed variational inequalities and variational-like inequalities in reflexive Banach spaces.
Motivated and inspired by the research work in , in this paper, we introduce and study a new class of generalized nonlinear variational-like inequalities and prove an existence theorem of solutions for this kind of generalized nonlinear variational-like inequalities.By applying the result due to Chang [1,2] and the auxiliary principle technique, we suggest a new iterative scheme for solving the class of generalized nonlinear variational-like inequalities.The convergence of the sequence generated by the iterative 2 Generalized nonlinear variational-like inequalities algorithm is also discussed.Our results extend and unify the corresponding results due to Ding [3], Liu et al. [21], Yao [36], and others.

Preliminaries
Throughout this paper, we assume that H is a real Hilbert space with dual space H * and that u,v is the dual pairing between u ∈ H and v ∈ H * .Let K be a nonempty closed convex subset of H, and let A,B,C : ) be nondifferentiable and satisfy the following conditions: for all u,v,w ∈ K. Now we consider the following generalized nonlinear variational-like inequality.
For given which was introduced and studied by Ding [3].
(5) N is said to be η-hemicontinuous with respect to A and B in the first and second arguments if for any x, y,z ∈ K, the mapping g (2.9) (7) η is said to be strongly monotone with constant t if there exists a constant t > 0 such that Let X be a nonempty closed convex subset of a Hausdorff linear topological space E, and let φ,ψ : X × X → R be mappings satisfying the following conditions: (a) ψ(x, y) ≤ φ(x, y) for all x, y ∈ X, and ψ(x,x) ≥ 0 for all x ∈ X; (d) there exists a nonempty compact set K ⊂ X and x 0 ∈ K such that ψ(x 0 , y) < 0 for all y ∈ X \ K. Then there exists y ∈ K such that φ(x, y) ≥ 0 for all x ∈ X.

Auxiliary problem and algorithm
Now we consider the following auxiliary problem with respect to the generalized nonlinear variational-like inequality (2.2).For any given u ∈ K, find w ∈ K such that where ρ > 0 is a constant.
Theorem 3.1.Let K be a nonempty closed convex subset of H and f ∈ H. Suppose that a : Proof.Let u be in K. Define the functionals φ and ψ : for all v,w ∈ K.
We check that the functionals φ and ψ satisfy all the conditions of Lemma 2.2 in the weak topology.It is easy to see for all v,w ∈ K, which implies that φ and ψ satisfy the condition (1) of Lemma 2.2.Since a is a coercive continuous bilinear form, it follows that a(v,v − w) is weakly upper semicontinuous with respect to w.Note that b is convex and lower semicontinuous in the second argument and for given x, y,z ∈ H, v ∈ K, the mapping N(x, y,z),η(v,•) is concave and upper semicontinuous.Therefore φ(v,•) is weakly upper semicontinuous in the second argument and the set {v ∈ K : ψ(v,w) < 0} is convex for each w ∈ K.That is, the conditions (2) and (3) of Lemma 2.2 hold.Let v ∈ K. Put (3.4) Zeqing Liu et al. 5 Clearly, M is a weakly compact subset of K and for any w which means that the condition (4) of Lemma 2.2 holds.Thus Lemma 2.2 ensures that there exists Let t be in (0,1] and let v be in K. Replacing v by v t = tv + (1 − t) w in (3.6), we see that Notice that b is convex in the second argument and N(x, y,z),η(v,•) is concave and upper semicontinuous.From (C6) and (3.7) we infer that which implies that 6 Generalized nonlinear variational-like inequalities Letting t → 0 + in the above inequality, we conclude that That is, w is a solution of (3.1).Now we prove the uniqueness.For any two solutions w 1 ,w 2 ∈ K of (3.1) with respect to u, we know that for all v ∈ K. Taking v = w 2 in (3.11) and v = w 1 in (3.12), we get that (3.13) Adding these inequalities, we deduce that which yields w 1 = w 2 .That is, w is the unique solution of (3.1).This completes the proof.
By Theorem 3.1, we suggest the following algorithms for solving the generalized nonlinear variational-like inequality (2.2). and A,B,C : K → H, N : H × H × H → H and η : K × K → H * are mappings.For given f ∈ H and u 0 ∈ K, compute the sequence {u n } n≥0 ⊂ K by the following iterative scheme: for all v ∈ K and n ≥ 0, where {e n } n≥0 ⊂ H and ρ > 0 is a constant.

Existence and convergence
In this section, we prove the existence of solution for the generalized nonlinear variational-like inequality (2.2) and discuss the convergence of the sequence generated by Algorithm 3.2.
Theorem 4.1.Let a, b, A, B, N, η be as in Theorem 3.1.Let C : K → H be Lipschitz continuous with constant ξ.Assume that N is Lipschitz continuous with constant σ in the third argument and strongly monotone with constant β with respect to C in the third argument and If there exist a constant ρ satisfying and one of the following conditions: then the iterative sequence {u n } n≥0 generated by Algorithm 3.2 converges strongly to some u ∈ K and u is a solution of the generalized nonlinear variational-like inequality (2.2).
Proof.It follows from the proof of Theorem 3.1 that there exists a mapping G : where w is the unique solution of (3.1) for each u ∈ K. Next we show that G is a contraction mapping.Let u 1 and u 2 be arbitrary elements in K. Using (3.1), we see that 8 Generalized nonlinear variational-like inequalities for all v ∈ K. Letting v = Gu 2 in (4.4) and v = Gu 1 in (4.5), and adding these inequalities, we arrive at that is, where by ( 4.2) and one of (4.3).Therefore, G : K → K is a contraction mapping and has a unique fixed point u ∈ K.It follows from (3.1) that which implies that that is, u is a solution of the generalized nonlinear variational-like inequality (2.2).
Zeqing Liu et al. 9 Next, we consider the convergence of the iterative sequence generated by Algorithm 3.2.Taking v = u n+1 in (4.9) and v = u in (3.15), and adding these inequalities, we have where θ is defined by (4.8).It follows from (4.1) and (4.12) that the iterative sequence {u n } n≥0 generated by Algorithm 3.2 converges strongly to u.This completes the proof.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

First
Round of ReviewsMay 1, 2009 and η(u,v) = gu − gv for all u,v ∈ K, where g : K → H * is a mapping, then the generalized nonlinear variational-like inequality (2.2) is equivalent to finding u ∈ K such that