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Solvability of Multivalued General Mixed Variational Inequalities
Journal of Mathematical Analysis and Applications 261, 390᎐402 Ž2001. doi:10.1006rjmaa.2001.7533, available online at http:rrwww.idealibrary.com on
Solvability of Multivalued General Mixed Variational Inequalities Muhammad Aslam Noor Mathematics, Etisalat College of Engineering, P.O. Box 980, Sharjah, United Arab Emirates E-mail: [email protected] Submitted by William F. Ames Received January 26, 2001
In this paper, we use the auxiliary principle technique to suggest a new class of predictor-corrector algorithms for solving multivalued general mixed variational inequalities. The convergence of the proposed method only requires the partially relaxed strong monotonicity of the operator, which is weaker than co-coercivity. As special cases, we obtain a number of known and new results for solving various classes of variational inequalities. 䊚 2001 Academic Press Key Words: variational inequalities; auxiliary principle; iterative methods; convergence.
1. INTRODUCTION Variational inequalities theory has emerged an interesting and fascinating branch of applicable mathematics with a wide range of applications in industry, physical, regional, social, pure, and applied sciences. This field is dynamic and is experiencing an explosive growth in both theory and applications; as a consequence, research techniques and problems are drawn from various fields. Variational inequalities have been generalized and extended in different directions using novel and innovative techniques. An important and useful generalization of variational inequalities is called the multivalued general mixed variational inequality. For applications and numerical methods, see w12, 14᎐19x and the references therein. There are several numerical methods for solving variational inequalities and related optimization problems. Among the most efficient numerical techniques are projection and its variant forms, Wiener᎐Hopf equations, auxiliary principle, and the penalty function methods. It is well known that the conver390 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
MIXED VARIATIONAL INEQUALITIES
391
gence analysis of the projection method requires that the underlying operator must be strongly monotone and Lipschitz continuous. These strict conditions rule out many applications of the projection methods. These facts motivated us to modify the projection methods using the updating technique of the solution and the Wiener᎐Hopf equations methods; see, for example, w17, 19, 21, 22, 29x and the references therein for recent state-of-the-art techniques. It is well known that all the projection type methods cannot be extended and generalized to suggest and analyze iterative methods for solving the mixed variational inequalities involving the nonlinear terms. Second the evaluation of the projection of the operator is very expansive. To overcome these drawbacks, we use the resolvent operator methods. In fact, if the nonlinear term involving the mixed variational inequalities is proper, convex, and lower-semicontinuous, then the mixed variational inequalities are equivalent to the fixed-point problem and the resolvent equations. In this technique, the given operator is decomposed into the sum of Žmaximal. monotone operators, whose resolvents are easier to evaluate than the resolvent of the given original operator. In the context of the mixed variational inequalities, Noor w17᎐19x has used the resolvent operator and resolvent equations techniques to develop various splitting and predictor-corrector type methods for solving mixed variational inequalities and related optimization problems using the updating technique of the solution. A useful feature of the forward᎐backward splitting methods is that the resolvent operator involves the subdifferential of the proper, convex, and lower-semicontinuous function and the other part facilitates the problem decomposition only. If the nonlinear term involving the mixed variational inequalities is an indicator function of a closed convex set in a space, then the resolvent operator is exactly the projection operator from the space into the convex set. Consequently, the resolvent equations are equivalent to the Wiener᎐Hopf equations, which were introduced by Shi w28x and Robinson w27x. For the recent applications of the Wiener᎐Hopf equations, see w13x. In passing, we remark that the resolvent equations play the same role in the mixed variational inequalities as the Wiener᎐Hopf equations in variational inequalities. To implement these methods, one has to evaluate the resolvent of the operator, which is itself a difficult problems. If the nonlinear term is nondifferentiable, then one cannot use the resolvent type methods for solving the mixed type variational inequalities. Furthermore, the updating technique of the solution cannot be extended to suggest two-step, three-step splitting, and predictor-corrector type methods for multivalued Žmixed. variational inequalities. These facts motivated us to consider and develop other methods. One of these techniques is called the auxiliary principle, the origin of which can be traced back to Lions and Stampacchia w9x. This technique deals with finding the auxiliary variational inequality and prov-
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MUHAMMAD ASLAM NOOR
ing that the solution of the auxiliary problem is the solution of the original variational inequality by using the fixed-point technique. It turned out that this technique can be used to find the equivalent differentiable optimization problem, which enables us to construct a gap Žmerit. function. These gap Žmerit. functions have played an important part in developing some efficient iterative methods for solving variational inequalities; see w5, 22, 24, 30x. Glowinski et al. w6x used this technique to study the existence of a solution of the mixed variational inequalities. Noor w11, 12, 14᎐16x has used the auxiliary principle technique to develop some iterative methods for solving various classes of variational inequalities and optimization problems. It has been shown that a substantial number of numerical methods can be obtained as special cases from this technique; see w10, 11᎐16, 30x and references therein. In this paper, we use the auxiliary principle technique to suggest a class of three-step predictor-corrector iterative methods for multivalued general mixed variational inequalities. In particular, we show that one can obtain various forward᎐backward splitting, modified resolvent, and other methods as special cases from these methods. We also prove that the convergence of the suggested methods requires only the partially relaxed strong monotonicity, which is a weaker condition than the co-coercivity. Consequently, our results represent an improvement and refinement of the previously known results. Our results can be considered as an extension of the results of Noor w11, 12, 14, 15x for solving general mixed variational inequalities and complementarity problems.
2. PRELIMINARIES Let H be a real Hilbert space whose inner product and norm are denoted by ² ⭈ ,⭈ : and 5 ⭈ 5, respectively. Let C Ž H . be a family of all nonempty compact subset of H. Let T : H ª C Ž H . be a multivalued operator and g: H ª H be a single-valued operator. Let K be a nonempty closed convex set in H. Let : H ª R j q⬁4 be a function. For a given single-valued operator N Ž⭈,⭈ .: H = H ª H, we consider the problem of finding u g H, g T Ž u. such that ² N Ž , . , g Ž ¨ . y g Ž u . : q Ž g Ž ¨ . . y Ž g Ž u . . G 0, ᭙ g Ž ¨ . g H.
Ž 2.1. The inequality of type Ž2.1. is called the multi¨ alued general mixed ¨ ariational inequality. It can be shown that a wide class of multivalued odd order and nonsymmetric free, obstacle, moving, equilibrium, and optimization problems arising in pure and applied sciences can be studied via the multivalued variational inequalities Ž2.1..
393
MIXED VARIATIONAL INEQUALITIES
If : H ª R j q⬁4 is a proper, convex, and lower semicontinuous function, then problem Ž2.1. is equivalent to finding u g H, g T Ž u. such that 0 g N Ž , . q ⭸ Ž g Ž u . . ,
Ž 2.2.
where ⭸ is the subdifferential of and is a maximal monotone operator. Problem Ž2.2. is also known as finding a zero of the sum of Žmaximal. monotone mappings. Problems of type Ž2.2. has been extensively studied recently; see w5, 17᎐19x. We note that if T : H ª H is a single-valued operator, then problem Ž2.1. is equivalent to finding u g H such that ² N Ž u, u . , g Ž ¨ . y g Ž u . : q Ž g Ž ¨ . . y Ž g Ž u . . G 0,
᭙g Ž ¨ . g H, Ž 2.3.
which is known as the general mixed variational inequality. If N Ž u, u. ' Tu, T : H ª H and is the indicator function of a closed convex set K in H, then problem Ž2.3. is equivalent to finding u g H, g Ž u. g K such that ²Tu, g Ž ¨ . y g Ž u . : G 0,
᭙g Ž ¨ . g K ,
Ž 2.4.
which is called the general ¨ ariational inequality introduced and studied by Noor w21x in 1988. It can be shown that a class of quasi variational inequalities and nonconvex programming problems can be studied by the general variational inequality approach; see Noor w15, 16, 25, 26x. We remark that if g ' I, the identity operator, then problem Ž2.1. is equivalent to finding u g H, g T Ž u. such that ² N Ž , . , ¨ y u: q Ž ¨ . y Ž u . G 0,
᭙¨ g H,
Ž 2.5.
which are called the generalized mixed variational inequalities. If is the indicator function of a closed convex set K in H, then problem Ž2.1. is equivalent to finding u g H, g Ž u. g K, g T Ž u. such that ² N Ž , . , g Ž ¨ . y g Ž u . : G 0,
᭙g Ž ¨ . g K ,
Ž 2.6.
which is known as the multivalued variational inequality, introduced and studied by Noor w25x recently. In particular, for g ' I, the identity operator, the problem is called the generalized variational inequality problem introduced and is studied by Fang and Peterson w4x. If K * s u g H: ² u, ¨ : G 0, ᭙ ¨ g K 4 is a polar cone of a convex cone K in H, then problem Ž2.6. is equivalent to finding u g H such that g Ž u. g K ,
N Ž , . g K *,
and
² N Ž , . , g Ž u . : s 0, Ž 2.7.
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which is known as the multivalued complementarity problem. We note that if g Ž u. s u y mŽ u., where m is a point-to-point mapping, then problem Ž2.7. is called the multivalued quasiŽimplicit. complementarity problem. For N Ž u, u. ' Tu, the problem is called the general nonlinear complementarity problem; see the references for the formulation and numerical methods. It is clear that problems Ž2.2. ᎐ Ž2.7. are special cases of the multivalued variational inequality Ž2.1.. In brief, for a suitable and appropriate choice of the operators T, g, N Ž⭈,⭈ . and the space H, one can obtain a wide class of variational inequalities and complementarity problems. This clearly shows that problem Ž2.1. is quite general and a unifying one. Furthermore, problem Ž2.1. has many important applications in various branches of pure and applied sciences; see w2᎐30x. We also need the following well known result and concepts. LEMMA 2.1. ᭙ u, ¨ g H, we ha¨ e 2² u, ¨ : s 5 u q ¨ 5 2 y 5 u 5 2 y 5 ¨ 5 2 .
Ž 2.8.
DEFINITION 2.1. ᭙ u1 , u 2 , z g H, w 1 g T Ž u1 ., w 2 g T Ž u 2 ., the operator N Ž⭈,⭈ .: H = H ª C Ž H . is said to be: Ži. g-partially relaxed strongly monotone, if there exists a constant ␣ ) 0 such that ² N Ž w 1 , w 1 . y N Ž w 2 , w 2 . , g Ž z . y g Ž u 2 . : G y␣ 5 g Ž u1 . y g Ž z . 5 2 , Žii.
g-co-coercive, if there exists a constant ) 0 such that ² N Ž w 1 , w 1 . y N Ž w 2 , w 2 . , g Ž u1 . y g Ž u 2 . : G 5 N Ž w1 , w1 . y N Ž w 2 , w 2 . 5 2 ,
Žiii.
M-Lipschitz continuous, if there exists a constant ␦ ) 0 such
that M Ž T Ž u1 . , T Ž u 2 . . F ␦ 5 u1 y u 2 5 , where M Ž⭈,⭈ . is the Hausdorff metric on C Ž H .. We remark that if z s u1 , then g-partially relaxed strong monotonicity is exactly g-monotonicity of the operator N Ž⭈,⭈ .. For g ' I, the indentity operator and N Ž u, u. s Tu, T : H ª H is an operator, Definition 2.1 reduces to the definition of partially relaxed strong monotonicity and co-coercivity of the operator. Using the technique of Noor w11x, it can be shown that g-co-coercivity implies g-partially relaxed strong monotonicity, but not conversely. Consequently, it follows that the concept of g-partially relaxed strong monotonicity is weaker than co-coercivity.
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MIXED VARIATIONAL INEQUALITIES
3. MAIN RESULTS In this section, we suggest and analyze a new iterative method for solving the problem Ž2.1. by using the auxiliary principle technique. For a given u g H, g T Ž u. consider the problem of finding a unique w g H, satisfying the auxiliary variational inequality ² N Ž , . q g Ž w . y g Ž u. , g Ž ¨ . y g Ž w . : q Ž g Ž ¨ . . y Ž g Ž u . . G 0,
᭙g Ž ¨ . g H,
Ž 3.1.
where ) 0 is a constant. We note that if w s u, then clearly w is a solution of the multivalued variational inequality Ž2.1.. This observation enables us to suggest the following predictor-corrector method for solving the multivalued mixed variational inequalities Ž2.1.. ALGORITHM 3.1. For a given u 0 g H, compute the approximate solution u nq 1 by the iterative schemes ² N Ž n , n . q g Ž u nq1 . y g Ž wn . , g Ž ¨ . y g Ž u nq1 . : q Ž g Ž ¨ . . y Ž g Ž u nq 1 . . G 0,
᭙g Ž ¨ . g H
n g T Ž wn . : 5nq1 y n 5 F M Ž T Ž wnq1 . , T Ž wn . .
Ž 3.2. Ž 3.3.
²  N Ž n , n . q g Ž wn . y g Ž yn . , g Ž ¨ . y g Ž wn . : q  Ž g Ž ¨ . . y  Ž g Ž wn . . G 0,
᭙g Ž ¨ . g H
n g T Ž yn . : 5 nq1 y n 5 F M Ž T Ž ynq1 . , T Ž yn . .
Ž 3.4. Ž 3.5.
and ² N Ž n , n . q g Ž yn . y g Ž u n . , g Ž ¨ . y g Ž yn . : q Ž g Ž ¨ . . y Ž g Ž yn . . G 0,
n g T Ž u n . : 5 nq1 y n 5 F M Ž T Ž u nq1 . , T Ž u n . . ,
᭙ g Ž ¨ . g H.
Ž 3.6.
n s 0, 1, 2, . . . ,
Ž 3.7. where ) 0, ) 0, and  ) 0 are constants. Note that if g ' I, the identity operator, then Algorithm 3.1 reduces to the following predictor-corrector method for solving the mixed variational inequalities Ž2.3..
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MUHAMMAD ASLAM NOOR
ALGORITHM 3.2. For a given u 0 g H, compute u nq1 by the iterative schemes ² N Ž n , n . q u nq1 y wn , ¨ y u nq1 : q Ž ¨ . y Ž u nq1 . G 0, ᭙¨ g H,
n g T Ž wn . : 5nq1 y n 5 F M Ž T Ž wnq1 . , T Ž wn . . ²  N Ž n , n . q wn y yn , ¨ y wn : q  Ž ¨ . y  Ž wn . G 0,
᭙¨ g H
n g T Ž yn . : 5 nq1 y n 5 F M Ž T Ž ynq1 . , T Ž yn . . ² N Ž n , n . q yn y u n , ¨ y yn : q Ž ¨ . y Ž yn . G 0,
n g T Ž u n . : 5 nq1 y n 5 F M Ž T Ž u nq1 . , T Ž u n . . ,
᭙ ¨ g H.
n s 0, 1, 2 . . . .
If is a proper, convex, and lower-semicontinuous function, Algorithm 3.1 can be written as ALGORITHM 3.3. For a given u 0 g H, compute u nq1 such that n g T Ž wn ., n g T Ž yn ., n g T Ž u n . by the iterative schemes g Ž u nq 1 . s J g Ž wn . y N Ž n , n . , g Ž wn . s J g Ž yn . y  N Ž n , n . , g Ž yn . s J g Ž u n . y N Ž n , n . ,
n s 0, 1, 2, . . . ,
where J is the resolvent operator associated with the subdifferential ⭸ , which is a maximal monotone operator; see w17, 18x. Algorithm 3.3 is a three-step forward᎐backward splitting method for solving multivalued mixed variational inequalities Ž2.1., which appears to be a new one. If T is a single-valued operator, then Algorithm 3.1 collapses to the following predictor-corrector method for solving general mixed variational inequalities Ž2.2. and appears to be a new one. ALGORITHM 3.4. For a given u 0 g H, compute u nq1 by the iterative schemes ² N Ž wn , wn . q g Ž u nq1 . y g Ž wn . , g Ž ¨ . y g Ž u nq1 . : q Ž g Ž ¨ . . y Ž g Ž u nq 1 . . G 0,
᭙g Ž ¨ . g H
²  N Ž yn , yn . q g Ž wn . y g Ž yn . , g Ž ¨ . y g Ž wn . : q  Ž g Ž ¨ . . y  Ž g Ž wn . . G 0,
᭙g Ž ¨ . g H
² N Ž u n , u n . q g Ž yn . y g Ž u n . , g Ž ¨ . y g Ž yn . : q Ž g Ž ¨ . . y Ž g Ž yn . . G 0,
᭙ g Ž ¨ . g H.
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MIXED VARIATIONAL INEQUALITIES
We remark that Algorithm 3.4 can be written in the equivalent form as ALGORITHM 3.5. For a given u 0 g H, compute u nq1 by the iterative schemes g Ž yn . s J g Ž u n . y N Ž u n , u n . g Ž wn . s J g Ž yn . y  N Ž yn , yn . g Ž u nq 1 . s J g Ž wn . y N Ž wn , wn . ,
n s 0, 1, 2 . . . .
If N Ž u, u. ' Tu, T : H ª H is a single-valued operator, then Algorithm 3.1 becomes: ALGORITHM 3.6. For a given u 0 g H, compute the approximate solution u nq 1 by the iterative schemes ² T Ž wn . q g Ž u nq1 . y g Ž wn . , g Ž ¨ . y g Ž u nq1 . : q Ž g Ž ¨ . . y Ž g Ž u nq 1 . . G 0,
᭙g Ž ¨ . g H
²  T Ž yn . q g Ž wn . y g Ž yn . , g Ž ¨ . y g Ž wn . : q  Ž g Ž ¨ . . y  Ž g Ž yn . . G 0,
᭙g Ž ¨ . g H
² Tu n q g Ž yn . y g Ž u n . , g Ž ¨ . y g Ž yn . : q Ž g Ž ¨ . . y Ž g Ž yn . . , ᭙ g Ž ¨ . g H. If is the indicator function of a closed convex set K in H, then Algorithm 3.6 reduces to the following algorithm of Noor w12x for solving the general variational inequalities Ž2.4.. ALGORITHM 3.7. For a given u 0 g H, g Ž u 0 . g K, compute the approximate solution u nq 1 by the iterative schemes ² T Ž wn . q g Ž u nq1 . y g Ž wn . , g Ž ¨ . y g Ž u nq1 . : G 0,
᭙g Ž ¨ . g K
²  T Ž yn . q g Ž wn . y g Ž yn . , g Ž ¨ . y g Ž wn . : G 0,
᭙g Ž ¨ . g K
² Tu n q g Ž yn . y g Ž u n . , g Ž ¨ . y g Ž yn . : G 0,
᭙g Ž ¨ . g K .
For the convergence analysis of Algorithm 3.7, see Noor w12x. For a suitable choice of the operators N Ž⭈,⭈ ., T and the space H, one can obtain various new and known methods for solving variational inequalities and complementarity problems. For the convergence analysis of Algorithm 3.1, we need the following result, which is proved by using the technique of Noor w11x.
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MUHAMMAD ASLAM NOOR
LEMMA 3.1. Let u g H be the exact solution of Ž2.1. and u nq 1 be the approximate solution obtained from Algorithm 3.1. If the operator N Ž⭈,⭈ .: H = H ª C Ž H . is a g-partially relaxed strongly monotone operator with constant ␣ ) 0, then 5 g Ž u nq 1 . y g Ž u . 5 2 F 5 g Ž u n . y g Ž u . 5 2 y Ž 1 y 2 ␣ . 5 g Ž u nq1 . y g Ž u n . 5 2 . Ž 3.8. Proof. Let u g H, g T Ž u. be a solution of Ž2.1.. Then ² N Ž , . , g Ž ¨ . y g Ž u . : q Ž g Ž ¨ . . y Ž g Ž u . . G 0, ᭙g Ž ¨ . g H
Ž 3.9.
²  N Ž , . g Ž ¨ . y g Ž u . : q  Ž g Ž ¨ . . y  Ž g Ž u . . G 0, ᭙g Ž ¨ . g H
Ž 3.10.
² N Ž , . , g Ž ¨ . y g Ž u . : q Ž g Ž ¨ . . y Ž g Ž ¨ . . G 0, ᭙g Ž ¨ . g H,
Ž 3.11.
where ) 0,  ) 0, and ) 0 are constants. Now taking ¨ s u nq 1 in Ž3.9. and ¨ s u in Ž3.2., we have ² N Ž , . , g Ž u nq 1 . y g Ž u . : q Ž g Ž u nq1 . . y Ž g Ž u . . G 0 Ž 3.12. and ² N Ž n , n . q g Ž u nq1 . y g Ž wn . , g Ž u . y g Ž u nq1 . : q Ž g Ž u . . y Ž g Ž u nq 1 . . G 0.
Ž 3.13.
Adding Ž3.12. and Ž3.13., we have ² g Ž u nq 1 . y g Ž wn . , g Ž u . y g Ž u nq1 . : G ² N Ž n , n . y N Ž , . , g Ž u nq1 . y g Ž u . : G y␣ 5 g Ž u nq 1 . y g Ž wn . 5 2 ,
Ž 3.14.
where we have used the fact that N Ž⭈,⭈ . is g-partially relaxed strongly monotone with constant ␣ ) 0. Setting u s g Ž u. y g Ž u nq 1 . and ¨ s g Ž u nq1 . y g Ž wn . in Ž2.6., we obtain ² g Ž u nq 1 . y g Ž wn . , g Ž u . y g Ž u nq1 . : s
1 2
5 g Ž u . y g Ž wn . 5 2 y 5 g Ž u . y g Ž u nq1 . 5 2 y5 g Ž u nq 1 . y g Ž wn . 5 2 4 .
Ž 3.15.
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MIXED VARIATIONAL INEQUALITIES
Combining Ž3.14. and Ž3.15., we have 5 g Ž u nq 1 . y g Ž u . 5 2 F 5 g Ž wn . y g Ž u . 5 2 y Ž 1 y 2 ␣ . 5 g Ž u nq1 . y g Ž wn . 5 2 . Ž 3.16. Taking ¨ s u in Ž3.4. and ¨ s wn in Ž3.10., we have ²  N Ž , . , g Ž wn . y g Ž u . : q  Ž g Ž wn . . y  Ž g Ž u . . G 0 Ž 3.17. and ²  N Ž n , n . q g Ž wn . y g Ž yn . , g Ž u . y g Ž wn . : q  Ž g Ž u . . y  Ž g Ž wn . . G 0.
Ž 3.18.
Adding Ž3.18. and Ž3.17. and rearranging the terms, we have ² g Ž wn . y g Ž yn . , g Ž u . y g Ž wn . : G  ² N Ž n , n . y N Ž , . , g Ž wn . y g Ž u . : G y␣ 5 g Ž yn . y g Ž wn . 5 2 ,
Ž 3.19.
since N Ž⭈,⭈ . is a g-partially relaxed strongly monotone operator with constant ␣ ) 0. Now taking ¨ s g Ž wn . y g Ž yn . and u s g Ž u. y g Ž wn . in Ž2.6., Ž3.19. can be written as 5 g Ž u . y g Ž wn . 5 2 F 5 g Ž u . y g Ž yn . 5 2 y Ž 1 y 2 ␣ . 5 g Ž yn . y g Ž wn . 5 2 F 5 g Ž u . y g Ž yn . 5 2 ,
for 0 -  - 1r2 ␣ .
Ž 3.20.
Similarly, by taking ¨ s u in Ž3.5. and ¨ s u nq 1 in Ž3.11. and using the g-partially relaxed strong monotonicity of the operator N Ž⭈,⭈ ., we have ² g Ž yn . y g Ž u n . , g Ž u . y g Ž yn . : G y␣ 5 g Ž yn . y g Ž u n . 5 2 . Ž 3.21. Letting ¨ s yn y u n , and u s u y yn in Ž2.6., and combining the resultant with Ž3.21., we have 5 g Ž yn . y g Ž u . 5 2 F 5 g Ž u . y g Ž u n . 5 2 y Ž 1 y 2 ␣ . 5 g Ž yn . y g Ž u n . 5 2 F 5 g Ž u. y g Ž un . 5 2 ,
for 0 - -
1 2␣
.
Ž 3.22.
Now 5 g Ž u nq 1 . y g Ž wn . 5 2 s 5 g Ž u nq1 . y g Ž u n . q g Ž u n . y g Ž wn . 5 2 s 5 g Ž u nq 1 . y g Ž u n . 5 2 q 5 g Ž u n . y g Ž wn . 5 2 q 2² g Ž u nq 1 . y g Ž u n . , g Ž u n . y g Ž wn . : . Ž 3.23.
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MUHAMMAD ASLAM NOOR
Combining Ž3.16., Ž3.20., Ž3.22., and Ž3.23., we obtain 5 g Ž u nq 1 . y g Ž u . 5 2 F 5 g Ž u n . y g Ž u . 5 2 y Ž 1 y 2 ␣ . 5 g Ž u nq1 . y g Ž u n . 5 2 , the required result Ž3.8.. THEOREM 3.1. Let H be a finite dimensional space. Let g: H ª H be in¨ ertible and 0 - - 21␣ . Let T : H ª C Ž H . be an M-Lipschitz continuous operator. If u nq 1 is the approximate solution obtained from Algorithm 3.1 and u g H is the exact solution of Ž2.1., then lim nª⬁ u n s u. Proof. Let u g H be a solution of Ž1.. Since 0 - - 21␣ , from Ž3.8., it follows that the sequence 5 g Ž u. y g Ž u n .54 is nonincreasing and consequently u n4 is bounded. Furthermore, we have ⬁
Ý Ž 1 y 2 ␣ . 5 g Ž u nq 1 . y g Ž u n . 5 2 F 5 g Ž u 0 . y g Ž u . 5 2 , ns0
which implies that lim 5 g Ž u nq 1 . y g Ž u n . 5 s 0.
nª⬁
Ž 3.24.
Let u ˆ be the cluster point of u n4 and let the subsequence u n j 4 of the sequence u n4 converge to u ˆ g H. Replacing wn and yn by u n j in Ž3.2., Ž3.4., and Ž3.6., taking the limit n j ª ⬁ and using Ž3.24., we have ² N Ž ˆ , ˆ . , g Ž ¨ . y g Ž u ˆ. : q Ž g Ž ¨ . . y Ž g Ž uˆ. . G 0,
᭙g Ž ¨ . g H,
which implies that u ˆ solves the multivalued mixed variational inequality Ž2.1. and 5 g Ž u nq 1 . y g Ž u . 5 2 F 5 g Ž u n . y g Ž u . 5 2 . Thus it follows from the above inequality that the sequence u n4 has exactly one cluster point u ˆ and lim g Ž u n . s g Ž u ˆ. .
nª⬁
Since g is invertible, thus lim Ž u n . s u. ˆ
nª⬁
It remains to show that g T Ž u.. From Ž3.7. and using the M-Lipschitz continuity of T, we have 5 n y 5 F M Ž T Ž u n . , T Ž u . . F ␦ 5 u n y u 5 ,
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MIXED VARIATIONAL INEQUALITIES
which implies that n ª as n ª ⬁. Now consider d Ž , T Ž u . . F 5 y n 5 q d Ž , T Ž u . . F 5 y n 5 q M Ž T Ž u n . , T Ž u . . F 5 y n 5 q ␦ 5 u n y u 5 ª 0
as n ª ⬁,
where d Ž , T Ž u.. s inf5 y z 5: z g T Ž u.4 , and ␦ ) 0 is the M-Lipschitz continuity constant. From the above inequality, it follows that d Ž , T Ž u.. s 0. This implies that g T Ž u., since T Ž u. g C Ž H .. This completes the proof.
REFERENCES 1. W. F. Ames, ‘‘Numerical Methods for Partial Differential Equations,’’ 3rd ed., Academic Press, New York, 1992. 2. E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 Ž1994., 325᎐333. 3. G. Cohen, Auxiliary problem principle extended to variational inequalities, J. Optim. Theory Appl. 59 Ž1988., 325᎐333. 4. S. C. Fang and E. L. Peterson, Generalized variational inequalities, J. Optim. Theory Appl. 38 Ž1982., 363᎐383. 5. F. Giannessi and A. Maugeri, ‘‘Variational Inequalities and Network Equilibrium Problems,’’ Plenum, New York, 1995. 6. R. Glowinski, J. L. Lions, and R. Tremolieres, ‘‘Numerical Analysis of Variational Inequalities,’’ North-Holland, Amsterdam, 1981. 7. R. Glowinski and P. Le Tallec, ‘‘Augmented Lagrangians and Operator Splitting Methods in Nonlinear Mechanics,’’ SIAM, Philadelphia, 1989. 8. D. Kinderleher and G. Stampacchia, ‘‘An Introduction to Variational Inequalities and Their Applications,’’ Academic Press, New York, 1980. 9. J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 Ž1967., 493᎐512. 10. P. Marcotte and J. H. Wu, On the convergence of projection methods: Applications to decomposition of affine variational inequalities, J. Optim. Theory Appl. 85 Ž1995., 347᎐362. 11. M. Aslam Noor, Some predictor-corrector algorithms for multivalued variational inequalities, J. Optim. Theory Appl. 108 Ž2001., 659᎐670. 12. M. Aslam Noor, A predictor-corrector method for general variational inequalities, Appl. Math. Lett. 14 Ž2001., 53᎐87. 13. M. Aslam Noor, Wiener᎐Hopf equations technique for variational inequalities, Korean J. Comput. Appl. Math. 7 Ž2000., 581᎐598. 14. M. Aslam Noor, Some new iterative methods for general mixed variational inequalities, Southeast Asian Bull. Math., in press. 15. M. Aslam Noor, A class of new iterative methods for general variational inequalities, Math. Comput. Modelling 31 Ž2000., 11᎐19. 16. M. Aslam Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 Ž2000., 217᎐229. 17. M. Aslam Noor, Splitting methods for pseudomonotone general mixed variational inequalities, J. Global Optim. 18 Ž2000., 75᎐89.
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18. M. Aslam Noor, Some algorithms for general monotone mixed variational inequalities, Math. Comput. Modelling 29 Ž1999., 1᎐9. 19. M. Aslam Noor, Algorithms for general monotone mixed variational inequalities, J. Math. Anal. Appl. 229 Ž1999., 330᎐343. 20. M. Aslam Noor, Set-valued mixed quasi variational inequalities and implicit resolvent equations, Math. Comput. Modelling 29 Ž1999., 1᎐11. 21. M. Aslam Noor, General variational inequalities, Appl. Math. Lett. 1 Ž1988., 119᎐121. 22. M. Aslam Noor, Some recent advances in variational inequalities. Part I. Basic concepts, New Zealand J. Math. 26 Ž1997., 53᎐80. 23. M. Aslam Noor, Some recent advances in variational inequalities. Part II. Other concepts, New Zealand J. Math. 26 Ž1997., 229᎐255. 24. M. Aslam Noor, Equivalence of differentiable optimization problems for variational inequalities, J. Natural Geom. 8 Ž1995., 117᎐128. 25. M. Aslam Noor, Generalized set-valued variational inequalities, Matematiche 52 Ž1997., 3᎐24. 26. M. Aslam Noor, K. Inayat Noor, and Th. M. Rassias, Some aspects of variational inequalities, J. Comput. Appl. Math. 47 Ž1993., 285᎐312. 27. M. S. Robinson, Normal maps induced by linear transformations, Math. Oper. Res. 17 Ž1992., 691᎐714. 28. P. Shi, Equivalence of variational inequalities with Wiener᎐Hopf equations, Proc. Amer. Math. Soc. 111 Ž1991., 339᎐346. 29. P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control. Optim. 38 Ž2000., 431᎐446. 30. D. L. Zhu and P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities, SIAM J. Optim. 6 Ž1996., 714᎐726.
Solvability of Multivalued General Mixed Variational Inequalities Muhammad Aslam Noor Mathematics, Etisalat College of Engineering, P.O. Box 980, Sharjah, United Arab Emirates E-mail: [email protected] Submitted by William F. Ames Received January 26, 2001
In this paper, we use the auxiliary principle technique to suggest a new class of predictor-corrector algorithms for solving multivalued general mixed variational inequalities. The convergence of the proposed method only requires the partially relaxed strong monotonicity of the operator, which is weaker than co-coercivity. As special cases, we obtain a number of known and new results for solving various classes of variational inequalities. 䊚 2001 Academic Press Key Words: variational inequalities; auxiliary principle; iterative methods; convergence.
1. INTRODUCTION Variational inequalities theory has emerged an interesting and fascinating branch of applicable mathematics with a wide range of applications in industry, physical, regional, social, pure, and applied sciences. This field is dynamic and is experiencing an explosive growth in both theory and applications; as a consequence, research techniques and problems are drawn from various fields. Variational inequalities have been generalized and extended in different directions using novel and innovative techniques. An important and useful generalization of variational inequalities is called the multivalued general mixed variational inequality. For applications and numerical methods, see w12, 14᎐19x and the references therein. There are several numerical methods for solving variational inequalities and related optimization problems. Among the most efficient numerical techniques are projection and its variant forms, Wiener᎐Hopf equations, auxiliary principle, and the penalty function methods. It is well known that the conver390 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
MIXED VARIATIONAL INEQUALITIES
391
gence analysis of the projection method requires that the underlying operator must be strongly monotone and Lipschitz continuous. These strict conditions rule out many applications of the projection methods. These facts motivated us to modify the projection methods using the updating technique of the solution and the Wiener᎐Hopf equations methods; see, for example, w17, 19, 21, 22, 29x and the references therein for recent state-of-the-art techniques. It is well known that all the projection type methods cannot be extended and generalized to suggest and analyze iterative methods for solving the mixed variational inequalities involving the nonlinear terms. Second the evaluation of the projection of the operator is very expansive. To overcome these drawbacks, we use the resolvent operator methods. In fact, if the nonlinear term involving the mixed variational inequalities is proper, convex, and lower-semicontinuous, then the mixed variational inequalities are equivalent to the fixed-point problem and the resolvent equations. In this technique, the given operator is decomposed into the sum of Žmaximal. monotone operators, whose resolvents are easier to evaluate than the resolvent of the given original operator. In the context of the mixed variational inequalities, Noor w17᎐19x has used the resolvent operator and resolvent equations techniques to develop various splitting and predictor-corrector type methods for solving mixed variational inequalities and related optimization problems using the updating technique of the solution. A useful feature of the forward᎐backward splitting methods is that the resolvent operator involves the subdifferential of the proper, convex, and lower-semicontinuous function and the other part facilitates the problem decomposition only. If the nonlinear term involving the mixed variational inequalities is an indicator function of a closed convex set in a space, then the resolvent operator is exactly the projection operator from the space into the convex set. Consequently, the resolvent equations are equivalent to the Wiener᎐Hopf equations, which were introduced by Shi w28x and Robinson w27x. For the recent applications of the Wiener᎐Hopf equations, see w13x. In passing, we remark that the resolvent equations play the same role in the mixed variational inequalities as the Wiener᎐Hopf equations in variational inequalities. To implement these methods, one has to evaluate the resolvent of the operator, which is itself a difficult problems. If the nonlinear term is nondifferentiable, then one cannot use the resolvent type methods for solving the mixed type variational inequalities. Furthermore, the updating technique of the solution cannot be extended to suggest two-step, three-step splitting, and predictor-corrector type methods for multivalued Žmixed. variational inequalities. These facts motivated us to consider and develop other methods. One of these techniques is called the auxiliary principle, the origin of which can be traced back to Lions and Stampacchia w9x. This technique deals with finding the auxiliary variational inequality and prov-
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ing that the solution of the auxiliary problem is the solution of the original variational inequality by using the fixed-point technique. It turned out that this technique can be used to find the equivalent differentiable optimization problem, which enables us to construct a gap Žmerit. function. These gap Žmerit. functions have played an important part in developing some efficient iterative methods for solving variational inequalities; see w5, 22, 24, 30x. Glowinski et al. w6x used this technique to study the existence of a solution of the mixed variational inequalities. Noor w11, 12, 14᎐16x has used the auxiliary principle technique to develop some iterative methods for solving various classes of variational inequalities and optimization problems. It has been shown that a substantial number of numerical methods can be obtained as special cases from this technique; see w10, 11᎐16, 30x and references therein. In this paper, we use the auxiliary principle technique to suggest a class of three-step predictor-corrector iterative methods for multivalued general mixed variational inequalities. In particular, we show that one can obtain various forward᎐backward splitting, modified resolvent, and other methods as special cases from these methods. We also prove that the convergence of the suggested methods requires only the partially relaxed strong monotonicity, which is a weaker condition than the co-coercivity. Consequently, our results represent an improvement and refinement of the previously known results. Our results can be considered as an extension of the results of Noor w11, 12, 14, 15x for solving general mixed variational inequalities and complementarity problems.
2. PRELIMINARIES Let H be a real Hilbert space whose inner product and norm are denoted by ² ⭈ ,⭈ : and 5 ⭈ 5, respectively. Let C Ž H . be a family of all nonempty compact subset of H. Let T : H ª C Ž H . be a multivalued operator and g: H ª H be a single-valued operator. Let K be a nonempty closed convex set in H. Let : H ª R j q⬁4 be a function. For a given single-valued operator N Ž⭈,⭈ .: H = H ª H, we consider the problem of finding u g H, g T Ž u. such that ² N Ž , . , g Ž ¨ . y g Ž u . : q Ž g Ž ¨ . . y Ž g Ž u . . G 0, ᭙ g Ž ¨ . g H.
Ž 2.1. The inequality of type Ž2.1. is called the multi¨ alued general mixed ¨ ariational inequality. It can be shown that a wide class of multivalued odd order and nonsymmetric free, obstacle, moving, equilibrium, and optimization problems arising in pure and applied sciences can be studied via the multivalued variational inequalities Ž2.1..
393
MIXED VARIATIONAL INEQUALITIES
If : H ª R j q⬁4 is a proper, convex, and lower semicontinuous function, then problem Ž2.1. is equivalent to finding u g H, g T Ž u. such that 0 g N Ž , . q ⭸ Ž g Ž u . . ,
Ž 2.2.
where ⭸ is the subdifferential of and is a maximal monotone operator. Problem Ž2.2. is also known as finding a zero of the sum of Žmaximal. monotone mappings. Problems of type Ž2.2. has been extensively studied recently; see w5, 17᎐19x. We note that if T : H ª H is a single-valued operator, then problem Ž2.1. is equivalent to finding u g H such that ² N Ž u, u . , g Ž ¨ . y g Ž u . : q Ž g Ž ¨ . . y Ž g Ž u . . G 0,
᭙g Ž ¨ . g H, Ž 2.3.
which is known as the general mixed variational inequality. If N Ž u, u. ' Tu, T : H ª H and is the indicator function of a closed convex set K in H, then problem Ž2.3. is equivalent to finding u g H, g Ž u. g K such that ²Tu, g Ž ¨ . y g Ž u . : G 0,
᭙g Ž ¨ . g K ,
Ž 2.4.
which is called the general ¨ ariational inequality introduced and studied by Noor w21x in 1988. It can be shown that a class of quasi variational inequalities and nonconvex programming problems can be studied by the general variational inequality approach; see Noor w15, 16, 25, 26x. We remark that if g ' I, the identity operator, then problem Ž2.1. is equivalent to finding u g H, g T Ž u. such that ² N Ž , . , ¨ y u: q Ž ¨ . y Ž u . G 0,
᭙¨ g H,
Ž 2.5.
which are called the generalized mixed variational inequalities. If is the indicator function of a closed convex set K in H, then problem Ž2.1. is equivalent to finding u g H, g Ž u. g K, g T Ž u. such that ² N Ž , . , g Ž ¨ . y g Ž u . : G 0,
᭙g Ž ¨ . g K ,
Ž 2.6.
which is known as the multivalued variational inequality, introduced and studied by Noor w25x recently. In particular, for g ' I, the identity operator, the problem is called the generalized variational inequality problem introduced and is studied by Fang and Peterson w4x. If K * s u g H: ² u, ¨ : G 0, ᭙ ¨ g K 4 is a polar cone of a convex cone K in H, then problem Ž2.6. is equivalent to finding u g H such that g Ž u. g K ,
N Ž , . g K *,
and
² N Ž , . , g Ž u . : s 0, Ž 2.7.
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which is known as the multivalued complementarity problem. We note that if g Ž u. s u y mŽ u., where m is a point-to-point mapping, then problem Ž2.7. is called the multivalued quasiŽimplicit. complementarity problem. For N Ž u, u. ' Tu, the problem is called the general nonlinear complementarity problem; see the references for the formulation and numerical methods. It is clear that problems Ž2.2. ᎐ Ž2.7. are special cases of the multivalued variational inequality Ž2.1.. In brief, for a suitable and appropriate choice of the operators T, g, N Ž⭈,⭈ . and the space H, one can obtain a wide class of variational inequalities and complementarity problems. This clearly shows that problem Ž2.1. is quite general and a unifying one. Furthermore, problem Ž2.1. has many important applications in various branches of pure and applied sciences; see w2᎐30x. We also need the following well known result and concepts. LEMMA 2.1. ᭙ u, ¨ g H, we ha¨ e 2² u, ¨ : s 5 u q ¨ 5 2 y 5 u 5 2 y 5 ¨ 5 2 .
Ž 2.8.
DEFINITION 2.1. ᭙ u1 , u 2 , z g H, w 1 g T Ž u1 ., w 2 g T Ž u 2 ., the operator N Ž⭈,⭈ .: H = H ª C Ž H . is said to be: Ži. g-partially relaxed strongly monotone, if there exists a constant ␣ ) 0 such that ² N Ž w 1 , w 1 . y N Ž w 2 , w 2 . , g Ž z . y g Ž u 2 . : G y␣ 5 g Ž u1 . y g Ž z . 5 2 , Žii.
g-co-coercive, if there exists a constant ) 0 such that ² N Ž w 1 , w 1 . y N Ž w 2 , w 2 . , g Ž u1 . y g Ž u 2 . : G 5 N Ž w1 , w1 . y N Ž w 2 , w 2 . 5 2 ,
Žiii.
M-Lipschitz continuous, if there exists a constant ␦ ) 0 such
that M Ž T Ž u1 . , T Ž u 2 . . F ␦ 5 u1 y u 2 5 , where M Ž⭈,⭈ . is the Hausdorff metric on C Ž H .. We remark that if z s u1 , then g-partially relaxed strong monotonicity is exactly g-monotonicity of the operator N Ž⭈,⭈ .. For g ' I, the indentity operator and N Ž u, u. s Tu, T : H ª H is an operator, Definition 2.1 reduces to the definition of partially relaxed strong monotonicity and co-coercivity of the operator. Using the technique of Noor w11x, it can be shown that g-co-coercivity implies g-partially relaxed strong monotonicity, but not conversely. Consequently, it follows that the concept of g-partially relaxed strong monotonicity is weaker than co-coercivity.
395
MIXED VARIATIONAL INEQUALITIES
3. MAIN RESULTS In this section, we suggest and analyze a new iterative method for solving the problem Ž2.1. by using the auxiliary principle technique. For a given u g H, g T Ž u. consider the problem of finding a unique w g H, satisfying the auxiliary variational inequality ² N Ž , . q g Ž w . y g Ž u. , g Ž ¨ . y g Ž w . : q Ž g Ž ¨ . . y Ž g Ž u . . G 0,
᭙g Ž ¨ . g H,
Ž 3.1.
where ) 0 is a constant. We note that if w s u, then clearly w is a solution of the multivalued variational inequality Ž2.1.. This observation enables us to suggest the following predictor-corrector method for solving the multivalued mixed variational inequalities Ž2.1.. ALGORITHM 3.1. For a given u 0 g H, compute the approximate solution u nq 1 by the iterative schemes ² N Ž n , n . q g Ž u nq1 . y g Ž wn . , g Ž ¨ . y g Ž u nq1 . : q Ž g Ž ¨ . . y Ž g Ž u nq 1 . . G 0,
᭙g Ž ¨ . g H
n g T Ž wn . : 5nq1 y n 5 F M Ž T Ž wnq1 . , T Ž wn . .
Ž 3.2. Ž 3.3.
²  N Ž n , n . q g Ž wn . y g Ž yn . , g Ž ¨ . y g Ž wn . : q  Ž g Ž ¨ . . y  Ž g Ž wn . . G 0,
᭙g Ž ¨ . g H
n g T Ž yn . : 5 nq1 y n 5 F M Ž T Ž ynq1 . , T Ž yn . .
Ž 3.4. Ž 3.5.
and ² N Ž n , n . q g Ž yn . y g Ž u n . , g Ž ¨ . y g Ž yn . : q Ž g Ž ¨ . . y Ž g Ž yn . . G 0,
n g T Ž u n . : 5 nq1 y n 5 F M Ž T Ž u nq1 . , T Ž u n . . ,
᭙ g Ž ¨ . g H.
Ž 3.6.
n s 0, 1, 2, . . . ,
Ž 3.7. where ) 0, ) 0, and  ) 0 are constants. Note that if g ' I, the identity operator, then Algorithm 3.1 reduces to the following predictor-corrector method for solving the mixed variational inequalities Ž2.3..
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ALGORITHM 3.2. For a given u 0 g H, compute u nq1 by the iterative schemes ² N Ž n , n . q u nq1 y wn , ¨ y u nq1 : q Ž ¨ . y Ž u nq1 . G 0, ᭙¨ g H,
n g T Ž wn . : 5nq1 y n 5 F M Ž T Ž wnq1 . , T Ž wn . . ²  N Ž n , n . q wn y yn , ¨ y wn : q  Ž ¨ . y  Ž wn . G 0,
᭙¨ g H
n g T Ž yn . : 5 nq1 y n 5 F M Ž T Ž ynq1 . , T Ž yn . . ² N Ž n , n . q yn y u n , ¨ y yn : q Ž ¨ . y Ž yn . G 0,
n g T Ž u n . : 5 nq1 y n 5 F M Ž T Ž u nq1 . , T Ž u n . . ,
᭙ ¨ g H.
n s 0, 1, 2 . . . .
If is a proper, convex, and lower-semicontinuous function, Algorithm 3.1 can be written as ALGORITHM 3.3. For a given u 0 g H, compute u nq1 such that n g T Ž wn ., n g T Ž yn ., n g T Ž u n . by the iterative schemes g Ž u nq 1 . s J g Ž wn . y N Ž n , n . , g Ž wn . s J g Ž yn . y  N Ž n , n . , g Ž yn . s J g Ž u n . y N Ž n , n . ,
n s 0, 1, 2, . . . ,
where J is the resolvent operator associated with the subdifferential ⭸ , which is a maximal monotone operator; see w17, 18x. Algorithm 3.3 is a three-step forward᎐backward splitting method for solving multivalued mixed variational inequalities Ž2.1., which appears to be a new one. If T is a single-valued operator, then Algorithm 3.1 collapses to the following predictor-corrector method for solving general mixed variational inequalities Ž2.2. and appears to be a new one. ALGORITHM 3.4. For a given u 0 g H, compute u nq1 by the iterative schemes ² N Ž wn , wn . q g Ž u nq1 . y g Ž wn . , g Ž ¨ . y g Ž u nq1 . : q Ž g Ž ¨ . . y Ž g Ž u nq 1 . . G 0,
᭙g Ž ¨ . g H
²  N Ž yn , yn . q g Ž wn . y g Ž yn . , g Ž ¨ . y g Ž wn . : q  Ž g Ž ¨ . . y  Ž g Ž wn . . G 0,
᭙g Ž ¨ . g H
² N Ž u n , u n . q g Ž yn . y g Ž u n . , g Ž ¨ . y g Ž yn . : q Ž g Ž ¨ . . y Ž g Ž yn . . G 0,
᭙ g Ž ¨ . g H.
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MIXED VARIATIONAL INEQUALITIES
We remark that Algorithm 3.4 can be written in the equivalent form as ALGORITHM 3.5. For a given u 0 g H, compute u nq1 by the iterative schemes g Ž yn . s J g Ž u n . y N Ž u n , u n . g Ž wn . s J g Ž yn . y  N Ž yn , yn . g Ž u nq 1 . s J g Ž wn . y N Ž wn , wn . ,
n s 0, 1, 2 . . . .
If N Ž u, u. ' Tu, T : H ª H is a single-valued operator, then Algorithm 3.1 becomes: ALGORITHM 3.6. For a given u 0 g H, compute the approximate solution u nq 1 by the iterative schemes ² T Ž wn . q g Ž u nq1 . y g Ž wn . , g Ž ¨ . y g Ž u nq1 . : q Ž g Ž ¨ . . y Ž g Ž u nq 1 . . G 0,
᭙g Ž ¨ . g H
²  T Ž yn . q g Ž wn . y g Ž yn . , g Ž ¨ . y g Ž wn . : q  Ž g Ž ¨ . . y  Ž g Ž yn . . G 0,
᭙g Ž ¨ . g H
² Tu n q g Ž yn . y g Ž u n . , g Ž ¨ . y g Ž yn . : q Ž g Ž ¨ . . y Ž g Ž yn . . , ᭙ g Ž ¨ . g H. If is the indicator function of a closed convex set K in H, then Algorithm 3.6 reduces to the following algorithm of Noor w12x for solving the general variational inequalities Ž2.4.. ALGORITHM 3.7. For a given u 0 g H, g Ž u 0 . g K, compute the approximate solution u nq 1 by the iterative schemes ² T Ž wn . q g Ž u nq1 . y g Ž wn . , g Ž ¨ . y g Ž u nq1 . : G 0,
᭙g Ž ¨ . g K
²  T Ž yn . q g Ž wn . y g Ž yn . , g Ž ¨ . y g Ž wn . : G 0,
᭙g Ž ¨ . g K
² Tu n q g Ž yn . y g Ž u n . , g Ž ¨ . y g Ž yn . : G 0,
᭙g Ž ¨ . g K .
For the convergence analysis of Algorithm 3.7, see Noor w12x. For a suitable choice of the operators N Ž⭈,⭈ ., T and the space H, one can obtain various new and known methods for solving variational inequalities and complementarity problems. For the convergence analysis of Algorithm 3.1, we need the following result, which is proved by using the technique of Noor w11x.
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LEMMA 3.1. Let u g H be the exact solution of Ž2.1. and u nq 1 be the approximate solution obtained from Algorithm 3.1. If the operator N Ž⭈,⭈ .: H = H ª C Ž H . is a g-partially relaxed strongly monotone operator with constant ␣ ) 0, then 5 g Ž u nq 1 . y g Ž u . 5 2 F 5 g Ž u n . y g Ž u . 5 2 y Ž 1 y 2 ␣ . 5 g Ž u nq1 . y g Ž u n . 5 2 . Ž 3.8. Proof. Let u g H, g T Ž u. be a solution of Ž2.1.. Then ² N Ž , . , g Ž ¨ . y g Ž u . : q Ž g Ž ¨ . . y Ž g Ž u . . G 0, ᭙g Ž ¨ . g H
Ž 3.9.
²  N Ž , . g Ž ¨ . y g Ž u . : q  Ž g Ž ¨ . . y  Ž g Ž u . . G 0, ᭙g Ž ¨ . g H
Ž 3.10.
² N Ž , . , g Ž ¨ . y g Ž u . : q Ž g Ž ¨ . . y Ž g Ž ¨ . . G 0, ᭙g Ž ¨ . g H,
Ž 3.11.
where ) 0,  ) 0, and ) 0 are constants. Now taking ¨ s u nq 1 in Ž3.9. and ¨ s u in Ž3.2., we have ² N Ž , . , g Ž u nq 1 . y g Ž u . : q Ž g Ž u nq1 . . y Ž g Ž u . . G 0 Ž 3.12. and ² N Ž n , n . q g Ž u nq1 . y g Ž wn . , g Ž u . y g Ž u nq1 . : q Ž g Ž u . . y Ž g Ž u nq 1 . . G 0.
Ž 3.13.
Adding Ž3.12. and Ž3.13., we have ² g Ž u nq 1 . y g Ž wn . , g Ž u . y g Ž u nq1 . : G ² N Ž n , n . y N Ž , . , g Ž u nq1 . y g Ž u . : G y␣ 5 g Ž u nq 1 . y g Ž wn . 5 2 ,
Ž 3.14.
where we have used the fact that N Ž⭈,⭈ . is g-partially relaxed strongly monotone with constant ␣ ) 0. Setting u s g Ž u. y g Ž u nq 1 . and ¨ s g Ž u nq1 . y g Ž wn . in Ž2.6., we obtain ² g Ž u nq 1 . y g Ž wn . , g Ž u . y g Ž u nq1 . : s
1 2
5 g Ž u . y g Ž wn . 5 2 y 5 g Ž u . y g Ž u nq1 . 5 2 y5 g Ž u nq 1 . y g Ž wn . 5 2 4 .
Ž 3.15.
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MIXED VARIATIONAL INEQUALITIES
Combining Ž3.14. and Ž3.15., we have 5 g Ž u nq 1 . y g Ž u . 5 2 F 5 g Ž wn . y g Ž u . 5 2 y Ž 1 y 2 ␣ . 5 g Ž u nq1 . y g Ž wn . 5 2 . Ž 3.16. Taking ¨ s u in Ž3.4. and ¨ s wn in Ž3.10., we have ²  N Ž , . , g Ž wn . y g Ž u . : q  Ž g Ž wn . . y  Ž g Ž u . . G 0 Ž 3.17. and ²  N Ž n , n . q g Ž wn . y g Ž yn . , g Ž u . y g Ž wn . : q  Ž g Ž u . . y  Ž g Ž wn . . G 0.
Ž 3.18.
Adding Ž3.18. and Ž3.17. and rearranging the terms, we have ² g Ž wn . y g Ž yn . , g Ž u . y g Ž wn . : G  ² N Ž n , n . y N Ž , . , g Ž wn . y g Ž u . : G y␣ 5 g Ž yn . y g Ž wn . 5 2 ,
Ž 3.19.
since N Ž⭈,⭈ . is a g-partially relaxed strongly monotone operator with constant ␣ ) 0. Now taking ¨ s g Ž wn . y g Ž yn . and u s g Ž u. y g Ž wn . in Ž2.6., Ž3.19. can be written as 5 g Ž u . y g Ž wn . 5 2 F 5 g Ž u . y g Ž yn . 5 2 y Ž 1 y 2 ␣ . 5 g Ž yn . y g Ž wn . 5 2 F 5 g Ž u . y g Ž yn . 5 2 ,
for 0 -  - 1r2 ␣ .
Ž 3.20.
Similarly, by taking ¨ s u in Ž3.5. and ¨ s u nq 1 in Ž3.11. and using the g-partially relaxed strong monotonicity of the operator N Ž⭈,⭈ ., we have ² g Ž yn . y g Ž u n . , g Ž u . y g Ž yn . : G y␣ 5 g Ž yn . y g Ž u n . 5 2 . Ž 3.21. Letting ¨ s yn y u n , and u s u y yn in Ž2.6., and combining the resultant with Ž3.21., we have 5 g Ž yn . y g Ž u . 5 2 F 5 g Ž u . y g Ž u n . 5 2 y Ž 1 y 2 ␣ . 5 g Ž yn . y g Ž u n . 5 2 F 5 g Ž u. y g Ž un . 5 2 ,
for 0 - -
1 2␣
.
Ž 3.22.
Now 5 g Ž u nq 1 . y g Ž wn . 5 2 s 5 g Ž u nq1 . y g Ž u n . q g Ž u n . y g Ž wn . 5 2 s 5 g Ž u nq 1 . y g Ž u n . 5 2 q 5 g Ž u n . y g Ž wn . 5 2 q 2² g Ž u nq 1 . y g Ž u n . , g Ž u n . y g Ž wn . : . Ž 3.23.
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Combining Ž3.16., Ž3.20., Ž3.22., and Ž3.23., we obtain 5 g Ž u nq 1 . y g Ž u . 5 2 F 5 g Ž u n . y g Ž u . 5 2 y Ž 1 y 2 ␣ . 5 g Ž u nq1 . y g Ž u n . 5 2 , the required result Ž3.8.. THEOREM 3.1. Let H be a finite dimensional space. Let g: H ª H be in¨ ertible and 0 - - 21␣ . Let T : H ª C Ž H . be an M-Lipschitz continuous operator. If u nq 1 is the approximate solution obtained from Algorithm 3.1 and u g H is the exact solution of Ž2.1., then lim nª⬁ u n s u. Proof. Let u g H be a solution of Ž1.. Since 0 - - 21␣ , from Ž3.8., it follows that the sequence 5 g Ž u. y g Ž u n .54 is nonincreasing and consequently u n4 is bounded. Furthermore, we have ⬁
Ý Ž 1 y 2 ␣ . 5 g Ž u nq 1 . y g Ž u n . 5 2 F 5 g Ž u 0 . y g Ž u . 5 2 , ns0
which implies that lim 5 g Ž u nq 1 . y g Ž u n . 5 s 0.
nª⬁
Ž 3.24.
Let u ˆ be the cluster point of u n4 and let the subsequence u n j 4 of the sequence u n4 converge to u ˆ g H. Replacing wn and yn by u n j in Ž3.2., Ž3.4., and Ž3.6., taking the limit n j ª ⬁ and using Ž3.24., we have ² N Ž ˆ , ˆ . , g Ž ¨ . y g Ž u ˆ. : q Ž g Ž ¨ . . y Ž g Ž uˆ. . G 0,
᭙g Ž ¨ . g H,
which implies that u ˆ solves the multivalued mixed variational inequality Ž2.1. and 5 g Ž u nq 1 . y g Ž u . 5 2 F 5 g Ž u n . y g Ž u . 5 2 . Thus it follows from the above inequality that the sequence u n4 has exactly one cluster point u ˆ and lim g Ž u n . s g Ž u ˆ. .
nª⬁
Since g is invertible, thus lim Ž u n . s u. ˆ
nª⬁
It remains to show that g T Ž u.. From Ž3.7. and using the M-Lipschitz continuity of T, we have 5 n y 5 F M Ž T Ž u n . , T Ž u . . F ␦ 5 u n y u 5 ,
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MIXED VARIATIONAL INEQUALITIES
which implies that n ª as n ª ⬁. Now consider d Ž , T Ž u . . F 5 y n 5 q d Ž , T Ž u . . F 5 y n 5 q M Ž T Ž u n . , T Ž u . . F 5 y n 5 q ␦ 5 u n y u 5 ª 0
as n ª ⬁,
where d Ž , T Ž u.. s inf5 y z 5: z g T Ž u.4 , and ␦ ) 0 is the M-Lipschitz continuity constant. From the above inequality, it follows that d Ž , T Ž u.. s 0. This implies that g T Ž u., since T Ž u. g C Ž H .. This completes the proof.
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