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Resolvent dynamical systems for set-valued quasi variational inclusions in Banach spaces.

[section] 1. Introductions

In recent years, variational inequalities have been extended and generalized in different directions by using novel and innovative techniques and ideas both for their own sake and for their applications. An important and useful generalization is called the variational inclusion, see [1-8] and references therein. But almost all discussions about variational inclusions are limited in Hilbert spaces. In this paper, we introduce a new class of set-valued quasi-variational inclusions in Banach spaces. In recent years, much attention has been given to consider and analyze the projected dynamics systems associated with variational inequalities and nonlinear programming problems, in which the righthand side of the ordinary differential equation is a projection operator. Such types of projected dynamical systems were introduced and studied by Dupuis and Nagurney [9]. Projected dynamical systems are characterized by a discontinuous right-hand side. The discontinuity arises from the constraint governing the question. The innovative and novel feature of a projected dynamical system is that its set of stationary points corresponds to the set of solutions of the corresponding variational inequality problems. It has been shown in [9-15] that the dynamical systems are useful in developing efficient and powerful numerical technique for solving variational inequalities and related optimization problems. Xia and Wang [13], Zhang and Nagurner [15] and Nagurner and Zhang [9] have studied the globally asymptotic stability of these projected dynamics systems. Noor [16,17] has also suggested and analyzed similar resolvent dynamical systems for mixed variational inequalities and quasi variational inclusions by extending and modifying there techniques. It is worth mentioning that there is no such type of dynamical systems for quasi variational inclusions in Banach spaces. In this paper, we suggest and analyze dynamical systems for quasi variational inclusions in Banach spaces. Using the resolvent operator method, we establishes the equivalence between the quasi variational inclusions in Banach spaces, resolvent equations and fixed-point problems. We use this alternative formation to suggest a class of resolvent dynamical systems associated with quasi variational inclusions in Banach spaces. We show that the trajectory of the solutions of these dynamical systems converges globally exponentially to the unique solution of the related quasi variational inclusions in Banach spaces. The analysis is in the spirit of Xia and Wang [14] and Noor [16,17]. Since the quasi variational inequalities and nonlinear programming problems as special cases, the results obtained in this paper continue to hold for these problems.

[section] 2. Preliminary

Let E be a real Banach space, E* is the topological dual space of E, CB(E) is the family of all nonempty closed and bounded subsets of E, < ., . > is the dual pair between E and E*, D(T) denotes the domain of T, and J : E [right arrow] [2.sup.E*] is the normalized duality mapping defined by

J(x) = {[x.sup.*] [member of] [E.sup.*] :< x, [x.sup.*] > = [[parallel] x [parallel].sup.2] = [[parallel] [x.sup.*] [parallel].sup.2]}, x [member of] E.

We now give the following well-known concepts and notions.

Definition 2.1. Let A : D(A) [subset] E x E [right arrow] [2.sup.E] be a set-value mapping. P : E x E [right arrow] E is a projection mapping, that is P(x, y) = x, for any (x, y) [member of] E x E.

(i) A is said to be accretive (k-strongly accretive) with respect to the first argument, if for any x, y [member of] D(A), u [member of] Ax, v [member of] Ay, there exist j(P(x - y)) [member of] J(P(x - y)), such that

< u - v, j(P(x - y)) > [greater than or equal to] 0 ([greater than or equal to] k [[parallel]u - v [parallel].sup.2]);

(ii) A is said to be an m-accretive mapping with respect to the first argument, if A is accretive with respect to the first argument and (I + [rho]A(u))(P(D(A)) - E, for every u [member of] E and [rho] > 0 (equivalently, if A is accretive with respect to the first argument and (I + A(u))(P(D(A))) - E, for all u [member of] E ), where A(., u) [equivalent of] A (u).

Remark 2.1. If A(u, v) = A(u), definition 2.1 is the very definition2.1 proposed by S. S. Chang [2]. Furthermore, if E = [E.sup.*] = H is a Hilbert space, the definition of an accretive mapping with respect to the first argument is in fact the definition of a monotone mapping with respect to the first argument proposed by Noor [1].

Proposition 2.1. If E = H is a Hilbert space, an m-accretive mapping A with respect to the first argument is a maximal monotone mapping with respect to the first argument.

Proof. If we use the technique given in S. S. Chang [2], we can prove this proposition immediately.

Definition 2.2. Let A : D(A) [subset] E x E [right arrow] [2.sup.E] be an m-accretive mapping with respect to the first argument, for any [rho] > 0 , the mapping defined by:

[R.sub.A(u)](u) = [(I + [rho]A(u)).sup.-1](u),

for any u [member of] E, which is called the resolvent operator, where A(., u) = A(u).

Problem 2.1. Let E be a real Banach space, T, V : E [right arrow] CB(E) set-valued mappings, g : E [right arrow] E a single-valued mapping, A(., .) : E x E [right arrow] [2.sup.E] be an m-accretive mapping with respect to the first argument, and N(., .) : E x E [right arrow] E be a nonlinear mapping, now let us to consider the following problem of finding u [member of] E, w [member of] T(u), y [member of] V (u) such that

0 [member of] N(w, y) + A(g(u), u). (2.1)

Remark 2.2. For a suitable choice of the mappings N, T, V, A, g, and the space E, we can obtain a number of known and new classes of variational inequalities, variational inclusions and the corresponding optimization problems. Furthermore, these variational inclusions provide us with a general and unified framework for studying a wide class of problems arising in mathematics, physics and engineering science [1-4].

Definition 2.3. [3] Let A : E [right arrow] CB(E) be a set-valued mapping and H(., .) be a hausdorff metric on CB(E), T is said to be [xi]-Lipschitz continuous if, for any x, y [member of] E,

H(Tx, Ty) [less than or equal to] [xi] [parallel]x - y[parallel],

Where [xi] > 0 is a constant.

Definition 2.4. [17] The dynamical system is said to converge to the solution set [K.sup.*] of (2.1), if, irrespective of the initial point, the trajectory of the dynamical system satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 2.5. [17] The dynamical system is said to be globally exponentially stable with degree [eta] at [u.sup.*], if , irrespective of the initial point, the trajectory of the system satisfies

[parallel] u(t) - [u.sup.*] [parallel] [less than or equal to] [[mu].sub.1] [parallel] u([t.sub.0]) - [u.sup.*] [parallel] exp(-[eta](t - [t.sub.0])), [for all]t [greater than or equal to] [t.sub.0],

where [[mu].sub.1] and [eta] are positive constants independent of the initial point. It is clear that globally exponentially stability is necessarily globally asymptotically stable and the dynamical system converges arbitrarily fast.

Lemma 2.1. [2,7] Let E is a real Banach space and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a normalized duality mapping, then for any x, y [member of] E,

[[parallel] x + y [parallel].sup.2] [less than or equal to] [[parallel] x [parallel].sup.2] + 2 < y, j(x + y) >, [for all]j(x + y) [member of] J(x + y).

Lemma 2.2. [17] Let [??] and [??] be real-valued nonnegative continuous functions with domain {t : t [less than or equal to] [t.sub.0]} and let [alpha](t) = [[alpha].sub.0]([absolute value of t - [t.sub.0]]), where [[alpha].sub.0] is a monotone increasing function. If, for t [greater than or equal to] [t.sub.0],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Assumption 2.1. For all u, v, w [member of] E, the resolvent operator [R.sub.A(u)] satisfying

[parallel][R.sub.A(u)] w - [R.sub.A(v)] w[parallel] [less than or equal to] [upsilon] [parallel] u - v [parallel],

where [upsilon] > 0 is a constant.

[section] 3. Main result

In this section, by using the technique of resolvent operator we establish the equivalence between the variational inclusion (2.1) and the fixed point problem. This equivalence is used to suggest a class of resolvent dynamical systems for the quasi variational inclusions (2.1). For this purpose, we need the following well-known result.

Lemma 3.1. The following statements are equivalent:

(1) (u, w, y), where u [member of] E, w [member of] T (u), y [member of] V (u) , is the solution of generalized set-valued quasi-variational inclusion (2.1);

(2) (u, w, y) is the solution of resolvent equation

g(u) - [R.sub.A(u)] [g(u) - [rho]N(w, y)], (3.1)

where [rho] > 0 is a constant, and [R.sub.A(u)] is a resolvent operator;

(3) (z, u, w, y) is the solution of implicit resolvent equation

z = g(u) - [rho]N(w, y), g(u) - [R.sub.A(u)] (z). (3.2)

Proof. If we use the technique given in Noor [1], we can prove this lemma immediately.

From Lemma 3.1, we conclude that the set-valued quasi variational inclusion (2.1) is equivalent to the fixed point problem (3.1). We use this equivalence to suggest a class of resolvent dynamical system associated with quasi variational inclusion (2.1) as

du / dt = [lambda]{[R.sub.A](u)[g(u) - [rho]N(w, y)] - g(u)}, u([t.sub.0]) - [u.sub.0] [member of] E, (3.3)

where [lambda] is a parameter. The system of type (3.3) is called the resolvent dynamical system associated with quasi variational inclusion (2.1). Here the right-hand side is related to the resolvent operator and is discontinuous on the boundary. It is clear from the definition that the solution to (3.3) always stays in the constraint set. This implies that the qualitative results such that the existence, uniqueness and continuous dependence of the solution to (3.3) can be studied.

We now show that the trajectory of the solution of the resolvent dynamical system (3.3) converges to the unique solution of quasi variational inclusion (2.1) by using the technique of Xia and Wang [13,14] as extended by Noor [16,17].

Theorem 3.1. Let E be a real Banach space, T, V : E [right arrow] [2.sup.E] be set-valued mappings, g : E [right arrow] E be a single valued mapping, A(., .) : E x E [right arrow] [2.sup.E] be an m-accretive with respect to the first argument, N(., .) : E x E [right arrow] E be nonlinear mappings satisfying the following conditions

(i) g is Lipschitz continuous with constants [delta] and k-strongly accretive, where k is a constant;

(ii) A(., .) : E x E [right arrow] [2.sup.E] is m-accretive with respect to the first argument;

(iii) T, V : E [right arrow] CB(E) are Lipschitz continuous with respect to constants [mu], [xi];

(iv) for a given y [member of] E, the mapping x [right arrow] N(x, y) is [beta]-Lipschitz continuous with respect to the set-valued mapping T;

(v) for a given x [member of] E, the mapping y [right arrow] N(x, y) is [[gamma].sub.1] -Lipschitz continuous with respect to the set-valued mapping B;

if the Assumption 2.1 holds, then, for each [u.sub.0] [member of] E, [w.sub.0] [member of] T [u.sub.0], [y.sub.0] [member of] V [u.sub.0], there exists a unique continuous solution u(t) of dynamical system (3.3) with u([t.sub.0]) = [u.sub.0] over [[t.sub.0], [infinity]).

Proof. Let

G(u) = [lambda]{[R.sub.A(u)] [g(u) - [rho]N(w, y)] - g(u)},

where [lambda] > 0 is a constant, w [member of] T u, y [member of] V u. For all u, v [member of] E, w [member of] T u, y [member of] V u, w' [member of] T v, y' [member of] V v, and using conditions (i),(iii),(iv),(v) and Assumption 2.1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies that operator G(u) is Lipschitz continuous in E. So, for each [u.sub.0] [member of] E, there exists a unique and continuous solution u(t) of the dynamical system (3.3), defined in an interval to [less than or equal to] t [less than or equal to] [T.sub.1] with the initial condition u([t.sub.0]) - [u.sub.0]. Let [[t.sub.0], [T.sub.1]] be its maximal interval of existence. Then we have to show that [T.sub.1] - [infinity]. Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any u [member of] E, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [k.sub.2] = [lambda](2[delta] + [rho]([mu][beta] + [gamma][xi] + [upsilon]). Hence by invoking Lemma 2.2, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This show that the solution is bounded on [[t.sub.0], [T.sub.1]). So [T.sub.1] = [infinity].

Theorem 3.2. Let the operators N, T, V, A, g be as Theorem 3.1, if Assumption 2.1 holds, then the resolvent dynamical system (3.3) converges globally exponentially to the unique solution of the quasi variational inclusion (2.1).

Proof. Since the operator N, T, V, A, g be as Theorem 3.1, it follows from Theorem 3.1 that the resolvent dynamical system (3.3) has a unique solution u(t) over [[t.sub.0], [T.sub.1]) for any fixed [u.sub.0] [member of] E. Let u(t) = u(t, [t.sub.0] : [u.sub.0]) be the initial value problem (3.3). For a given [u.sup.*] [member of] E, [w.sup.*] [member of] T [u.sup.*], [y.sup.*] [member of] V [u.sup.*], satisfying (2.1), consider the following Lyapunov function:

L(u) = [lambda] [parallel] u - [u.sup.*] [parallel], u [member of] E. (3.4)

From (3.3) and (3.4), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

where [u.sup.*], [w.sup.*] [member of] T [u.sup.*], [y.sup.*] [member of] V [u.sup.*] is the solution of the quasi variational inclusion (2.1), that is ,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now using Assumption 2.1 and conditions (i), (iii), (iv)and (v), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

From (3.5) and (3.6), we have d / dt [parallel] u(t) - [u.sup.*] [parallel] [less than of equal to] 2 [alpha] [lambda] [parallel] u(t) - [u.sup.*] [parallel], where [alpha] = [upsilon] + [delta] + [rho]([mu][beta] + [gamma][xi]) -k. Thus, for [lambda] = - [[lambda].sub.1], where [[lambda].sub.1] is a positive constant, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which shows that the trajectory of the solution of resolvent dynamical system (3.3) converges globally exponentially to the unique solution of the quasi variational inclusion (2.1).

References

[1] M. A. Noor, Three-step iterative algorithms for multivalued quasi variational inclusions, J. Math. Anal. Appl., 255(2001), 589-604.

[2] S. S. Chang, Y. J Cho, B. S. Lee and I. H. Jung, Generalized set-valued variational inclusions in Banach spaces, J. Math. Anal. Appl., 246(2000), 409-422.

[3] M. A. Noor, K. I. Noor and T. M. Rassias, Set-valued Resolvent Equations and Mixed Variational Inequalities, J. Math. Anal. Appl., 220(1998), 741-759.

[4] Z. Q. Liu, debnath Lokenath, Kang Shin Min and Jeong Sheok Ume. Completely generalized multivalued nonlinear quasi-variational inclusions, IJMMS, 30(2002), 593-604.

[5] S. B. Naddler, Multivalued contraction Mappings, Palific J. Math., 30(1969), 475-488.

[6] N. J. Huang, A general class of nonlinear variational inclusions for fuzzy mappings, Indian J. Pure Appl. Math., 29(1998), 957-964.

[7] S. S. Chang, Some problems and results in the study of nonlinear analysis, Nonlinear Anal., 30(1997), 4197-4208.

[8] X. P. Ding, Perturbed proximal point algorithms for generalized quasi variational inclusions, J. Math. Anal. Appl., 210(1997), 88-101.

[9] A. Nagurney D. Zhang, Projected dynamical systems and variational inequalities with applications, Kluwer Academic, Dordrecht, 1995.

[10] P. Dupuis, A. Nagurney, dynamical systems and variational inequalities with applications, Ann. Oper. Res., 44 (1993), 19-42.

[11] J. Dong, D. Zhang, A. Nagurney, A projected dynamical systems model of general financial equilibrium with stability analysis, Math. Comput. Appl., 24 (1996), 35-44.

[12] T. L. Riesz, D. H. Bernstein, R. Stough, Dynamical systems, variational inequalities and control theoretic medels for predicting time-varying urban network flows, Trans. Sci., 30(1996), 14-31.

[13] Y. S. Xia, J. Wang, A recurrent neural network for solving linear projection equations, Neural Network, 13(2000), 337-350.

[14] Y. S. Xia, J. Wang, On the stability of globally projected dynamical systems, J. Optim. Theory Appl., 106 (2000), 129-150.

[15] D. Zhang, A. Nagurney On the stability of globally projected dynamical systems, J. Optim. Theory Appl., 85 (1995), 97-124.

[16] M. A. Noor, Resolvent dynamical systems for mixed variational inequalities, Korean J. Math. Appl., 9(2002), 15-26.

[17] M. A. Noor, Implicit resolvent dynamical systems for quasi variational inclusions, J. Math. Anal. Appl., 269 (2002), 216-226.

(1) This research is supported by the natural foundation of Shaanxi province of China(Grant. No.: 2006A14) and the natural foundation of Shaanxi educational department of China(Grant. No.:07JK421).

Baoan Chen and Chaofeng Shi

Department of Mathematics, Xianyang Normal University,

Xianyang, Shaanxi, 712000, P. R. China

E-mail: shichf@163.com
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