General strongly nonlinear variational inequalities

JOURNAL

OF MATHEMATICAL

ANALYSIS

AND

APPLlCATlONS

166,

386-392 (1992)

General Strongly Nonlinear Variational Inequalities A. H. SIDDIQI* AND Q. H. ANSARI* International

Centre .for Theoretical Physics, Trieste, Italy Submitted by E. Stanley Lee Received August 23, 1990

In this paper we develop iterative algorithms for linding approximate solutions for new classes of variational and quasivariational inequalities which include, as a special case, some known results in this field. It is shown that the solutions of the iterative schemes converge to the exact solutions. c 1992 Academic Press, Inc.

1. INTRODUCTION Variational inequalities are a very powerful tool of the current mathematical technology, introduced by Stampacchia in the early sixties. These have been extended and generalized to study a wide class of problems arising in mechanics, optimization and control problems, operations research and engineering sciences, etc. Quasivariational inequalities are the extended form of variational inequalities in which the convex set does depend upon the solution. These were introduced and studied by Bensoussan, Goursat, and Lions [3]. For further details we refer to Bensoussan and Lions [4], Baiocchi and Capelo [l], Mosco [9], and Bensoussan [2]. In 1982, Noor [lo] introduced a more general class of variational inequalities, called strongly nonlinear variational inequalities, where the operators are nonlinear. Siddiqi and Ansari [14] have considered the same inequalities in which the convex set depends upon the solution. For the applications of this type of problems we refer the reader to [S, 8, 9, 151. Noor [ 123 and Isac [7] separately introduced and studied another class of variational inequalities, called general variational inequalities, which include the variational inequality introduced by Hartman and Stampacchia [6] as a special case. * Permanent 202002, India.

address: Mathematics

Department,

386 0022-247X/92 85.00 CopyrIght All rghts

#$~’1992 by Academic Press. Inc of reproducwn m any form reserved

Aligarh

Muslim

University,

Aligarh

387

NONLINEAR VARIATIONAL INEQUALITIES

In this paper, we consider and study a new class of variational and quasivariational inequalities. Using the projection technique, we suggest the iterative algorithms for finding the approximate solutions and prove this approximate solution converges to the exact solution. Our results are the extension and improvements of the results of Noor [ 10, 131 and Siddiqi and Ansari [ 141. 2. PRELIMINARIES AND FORMULATION

Let H be a Hilbert space with norm and inner product denoted by II.11 and ( ., .), respectively. Let K be a nonempty closed convex subset of H and T and A be nonlinear operators from H into itself. Let g be a continuous mapping from H into itself; then we consider a problem of finding u E H such that g(u) E K, and (T(u), u-g(u))

3 (A(u), u-g(u)),

for all

v E K,

which we shall call the general strongly nonlinear variational

(2.1) inequality

problem (GSNVIP).

If the convex set K depends upon the solution u, then the inequality (2.1) is called the general strongly nonlinear quasivariational inequality (GSNQVI). More precisely, given a point-to-set mapping K from H into itself, the general strongly nonlinear quasivariational inequality problem (GSNQVIP) is to find u E H such that g(u) E K(u), and (T(u), u - g(u) > 3 ,

for all

v E K(u).,

(2.2)

which is more general than problem (2.1). In many important applications K(u) has the form K(u)=m(u)+K,

(2.3)

where m is a point-to-point mapping from H into itself. Special cases. (i) Note tha t for g(u) = UE K(u), the problem (2.2) is equivalent to finding u E K(u) such that



2 ,

for all

v E K(u),

(2.4)

which is known as strongly nonlinear quasivariational inequality problem (SNQVIP) introduced and studied by Siddiqi and Ansari [14]. If K is independent of the solution u, that is K(u) = K, then (2.4) is called strongly nonlinear variational inequality, considered by Noor [lo].

388

SIDDIQI AND ANSARI

(ii) If A(u)=O, then (a) Equation (2.1) is equivalent to finding u E H such that g(u) E K, and
for all

30,

L’E K,

(2.5)

which is called a general variational inequality problem, introduced and studied by Noor [12] and Isac [7]. (b) Equation (2.2) takes the form for all


v E K(u),

(2.6)

which is known as a general quasivariational inequality, introduced by Noor [13]. (iii) If A(u) = 0 and g is the identity mapping, then (2.1) takes the form for all u E K. (T(u), u-u) 30, (2.7) Equation (2.7) is called a variational Hartman and Stampacchia [6].

inequality,

which was studied by

It is clear from these special casesthat our GSNQVI is the most general unifying one, which is one of the main motivations of this paper. We use the following definitions: DEFINITION

(i)

2.1. An operator T from H into itself is called

Lipschitz continuous, if there exists a constant t( > 0 such that

/IT(u) - T(v)11G LYllu - 41, (ii)

for all

U, v E H,

strongly monotone, if there exists a constant /?> 0 such that

(T(u)-T(v),v-u)>fl

llu-vl12, 3.

MAIN

for all

U, v E M.

RESULTS

We need the following lemma in order to suggest an iterative algorithm for finding the approximate solution of GSNQVIP. LEMMA 3.1. Zf K(u) is of the form (2.3), then UE H is a solution of (2.2) if and only if u E H satisfies the relation

u = F(u),

NONLINEAR VARIATIONAL INEQUALITIES

389

where

F(u)=u-g(u)+m(u)+P,[g(u)-i;(T(u)-A(u))-m(u)Il for some positive Proof

constant

5, where P, is the projection

(3.1)

of H into K.

It is similar to the proof of Lemma 3.1 [ 141.

Now, in the light of this lemma we can suggest an iterative algorithm such as

ALGORITHM

3.1. Given u,, E H, compute u, + , by the iterative scheme

uH+]=u,,- g(u,)+m(u,)+P,Cg(u,)--(T(u,) - A(4)) -m(u,Jl,

n = 0, 1, 2...,

(3.2)

where 4 > 0 is a constant. Now, we shall prove that the approximate solution obtained from the iterative scheme (3.2) converges strongly to the exact solution of (2.2).

THEOREM 3.1. Let T and g he strongly monotone and Lipschitz continuous and A and m be Lipschitz continuous. If u,+ , and u are solutions of (3.2) and (2.2), respectively, then u, + , strongly converges to u in H, ,for

5&D+Pv-l) 2-g

<~(B+p(k-1))2-((a2-~2)k(2-k) (x2-$ p(l -k)
a>~(l-k)+J(a2-~2)k(2-k), k = 2(v + ,/m)

and

< 1,

where fi and 6 are strongly monotonicity constants of T and g, and a, u, a, and v are Lipschitz constants of T, A, g, and m, respectively. Proof By Lemma 3.1, we see that u E K(u) satisfying (3.2) is also a solution of (3.1) and conversely. Thus from (3.1) and (3.2) we obtain

390

SIDDIQI

AND

ANSARI

(since P, is nonexpansive)

By using the method of Noor [ 131 and the Lipschitz continuity and strong monotonicity of T and g, we get

ll~,-~-(g(u,)-g(~))l126(1

-2d+a2)

lb,--ul12

(3.4)

and

IIu,-u-5(~(~,)-T(u))lI~d(1-2B5+~~~~)

ll~,--u11*.

(3.5)

Therefore, from (3.3), (3.4) (3.5) and by using the Lipschitz continuity of A, we have

IIUII+1-uu(I < {2v+2J~ +Ju -wt+~252)+rP) = {k+t(t)+bj

Ilk--ulI

II&-43

where k = 2v + 2 Jm and t(5) = (1 -2/I< + a2t2). Hence IIUn+l-~nll =8 llun-ull, where O=k+t(t)+tp. Now we have to show that 9 < 1. For this, we assume that the minimum value of t(t) at [=8/a* with r(g) =Jm.

391

NONLINEAR VARIATIONAL INEQUALITIES

For 5 = 5, k + t(g) + r/~ < 1 implies that k < 1 and b>u(l-k)+&*-p*)k(2-k).

Thus, it follows that 8 = k + t(g) + 5~ < 1 for all 5 with r-P+Ak-l) ci2- p*


/I>/41 -k)+J(a2-y*)k(2-k),

~(1 -k)
and k< 1. Since 8 < 1, we have that U, + I converges to u strongly in H. As a particular case of Algorithm 3.1 now we suggest an algorithm for finding the approximate solution of GSNVIP. ALGORITHM u,+

3.2. For any given q, E H, compute 1=

42

-

dun)

+

PKCd%J

-

t(n%7)

-

4%J)l,

(3.6)

where < > 0 is a constant. We can see from the next corollary that the approximate solution of GSNVIP obtained by the iterative scheme(3.6) converges to the exact solution. COROLLARY

3.1. Let T and g be strongly monotone and Lipschitz con-

tinuous and A be a Lipschitz continuous. If u, + , and u are solutions of (3.6) and (2.1), respectively, then u, + , strongly converges to u in H, for

t-P+p(k-1) ff*-/I*

,J(P+p(k-1))2-(22-~2)k(2-k) 2*-p*

/?>/~(l -k)+J(a*-u2)k(2-k),

p(l -k)
and

k=2,,‘-4, where /I and 6 are strongly monotone constants of T and g, and ~1,u, and o are Lipschitz constants of T, A, and g, respectively.

ACKNOWLEDGMENTS The authors thank Professor Abdus Salam, the International Atomic Energy Agency, and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. They also thank Professor A. Verjovsky of the Mathematics Section, ICTP, for his valuable help.

392

SIDDIQI AND ANSARI REFERENCES

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