Transactions on Modelling and Simulation vol 6, 1993 WIT Press, ISSN X

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1 Auxiliary principle technique for a class of nonlinear variational inequalities M.A. Noor, E.A. Al-Said Mathematics Department, College of Science, King Saud University, PO Box 2455, Riyadh 11451, Saudia Arabia ABSTRACT It is well known that moving, free, obstacle, unilateral, general equilibrium problems arising in elasticity, fluid flow through porous media, economics, transportation, pure and applied sciences can be studied in a unified general framework of variational inequalities. The ideas and techniques of variational inequalities are being applied in a variety of diverse fields and proved to be productive and innovative. One of the most difficult and important problems in this theory is the development of an efficient and implementable numerical method for solving these variational inequalities. In this paper, we use the auxiliary principle technique to prove the existence of a solution of a new class of variational inequalities and to suggest a new novel and general iterative algorithm. We also study the convergence criteria of this algorithm. Several special cases, which can be obtained from our main results, are also discussed. INTRODUCTION The last two centuries have seen increased attention paid by many research workers to the study of the variational principles. During this period, the variational principles have played an important and fundamental part as a unifying influence in pure and applied sciences and as a guide in the

2 420 Free and Moving Boundary Problems mathematical interpretation of many physical phenomena. In recent years, these principles have been enriched by the discovery of variational inequality theory, which is originally due to Stampacchia [l] and Fichera [2]. In 1971, Baiocchi [3] proved that the fluid flow through porous media (free boundary value problems) can be studied effectively in the framework of variational inequalities. Duvaut [4] extended the Baiocchi's technique to characterize the Stefens problems (moving boundary value problems ) by a class of variational inequalities. Since then the variational inequality theory has been developed in several directions and with the aid of many new powerful and varied techniques, a large number of advances were made through cross pollination among many areas of mathematical, social, regional and engineering sciences, see, for example, [4,5,6,7,8] and the references therein for more details. In 1988, Noor [10] introduced and studied a new class of variational inequalities. This new formulation extends various kinds of variational inequality problem formulation that have been introduced and enlarges the class of problems that can be studied by the variational inequality techniques in a unified and general framework. One of the main advantages of this theory is that the location of the free (moving) boundary becomes an intrinsic part of the solution and no special devices are needed to locate it. Inspired and motivated by the recent research work going on in this field, we introduce and consider a new class of variational inequalities. We remark that the projection technique and its variant forms cannot be applied to study the existence of a solution of this new class. In this paper, we use the auxiliary principle technique to study the existence of the solution of these variational inequalities. This technique deals with an auxiliary variational inequality problem and proving that the solution of the auxiliary problems is the solution of the original variational inequality problem. This technique is quite general and is used to suggest an iterative algorithm for computing the approximate solution of variational inequalities and related optimization problems. In section 2, we introduce the variational inequality problem and review some basic results. The main results are proved in Section 3. FORMULATION AND BASIC RESULTS Let H be a real Hilbert space, whose inner product and norm are denoted by <.,. > and. respectively. Let K be a nonempty closed

3 Free and Moving Boundary Problems 421 convex set in H. Given T, g : H > H continuous operators, we consider the problem of finding ueh such that g(u)ek and 0,, for all % #, (1) where <j> : H > R is a convex, lower semi-continuous, proper and nondifferentiable functional. The inequality (1) is known as the mixed variational inequality. SPECIAL CASE I. If g = /, the identity operator, then the problem (l) is equivalent to finding uek such that -4(>(%)>0, ^r all ucjf, (2) a problem originally studied and considered by Duvaut and Lions [4], The existence of its solution has been considered by Glowinski, Lions and Tremolieres [6], Kikuchi and Oden [7] and Noor [10] using the auxiliary variational principle technique. II. If 4>(u) = 0, then problem (l) reduces to the problem of finding ueh such that g(u)ek and < T%,% - #(%) >> o, for all vck, (3) a problem introduced and studied by Oettli [11], Isac [12] and Noor [9] independently in different contexts and applications. III. If (f>(u) =0, K* = {ueh, < u,v >> 0, for all vek} is a polar cone of the convex cone K in H and K C g(k), then problem (l) is equivalent to finding ueh such that #(%) #, r^ejt" and <T%,2(%)»0, (4) which is known as the general nonlinear complementarity problem. The problem (4) is quite general and includes many previously known classes of linear and nonlinear complementarity problems as special cases. For the iterative algorithms, convergence analysis and extensions of the problem (4), see Noor [13].

4 422 Free and Moving Boundary Problems IV. If <t>(u) = 0, and g = /, the identity operator, then problem (l) is equivalent to finding uek such that < T%, v - % >> 0, for all %6#, (5) which is known as the classical variational inequality problem originally introduced and studied by Stampacchia [1 and Fichera [2] in It is clear that problems (2) - (5) are special cases of the problem (l). In brief, the problem (l) is the more general and unifying one, which is one of the main motivations of this paper. DEFINITION. A mapping T : H -> H is said to be: (a) Strongly monotone, if there exists a constant a > 0 such that <Tu-Tv, u - v >> a u - v \ for all u,veh (b) Lipschitz continuous, if there exists a constant /3 > 0 such that \\Tu-Tv \<P \\u-v I!, for all u.veh. In particular, it follows that a < /3. If /3 nonexpansive. 1, then T is said to be MAIN RESULTS In this section, we prove the existence of a solution of the general variational inequality (l) by using the auxiliary principle technique and suggest an iterative algorithm. THEOREM 1. Let the operators T,g : H -» H be both strongly monotone and Lipschitz continuous, then there exists a solution ueh such that g(u)ek satisfying the variational inequality problem (l). PROOF. We use the auxiliary principle technique of Noor 10,13,14] and Glowinski, Lions and Tremolieres [6] to prove the existence of a solution of the problem (l). For a given uth such that g(u)ek, we consider the

5 Free and Moving Boundary Problems 423 problem offindinga unique uch such that g(u)ek satisfying the auxiliary variational inequality <w, v-g(u)> + p(t>(v)-p<f>(g(u))><u,v-g(u)>-p<tu,v-g(u)>, (6j for all vek, where p > 0 is a constant. Let Wi, W2 be two solutions of (6) related to ui^u^eh respectively. It is enough to show that the mapping u > w has a fixed point belonging to H satisfying (6). In other words, it is sufficient to show that for with 0 < 0 < 1, where 0 is independent of %i and u^. Taking v (respectively g(^i)) in (6) related to HI (respectively u^ ), we have and Adding these inequalities, we have < wi - W2,g(wi) - 0(^2) ><< ui - U2 - p(tui - T from which, we obtain 77 uoi - u>2 ^ < HI - U2 - p(tui - Tu-i) H < ( 7/i-722-/)(r^i-^2) wi-w2, (7) where rj > 0 and > 0 are the strongly monotinicity and Lipschitz continuity constants of the operator g. Since T is a strongly monotone Lipschitz continuous operator, so :rw2) ' < ^i-^ ' 2p < Tui Tui, HI HI >

6 424 Free and Moving Boundary Problems Combining (7) and (8), we obtain - w, < k - %2 where a. > /?\/l k* and k < 1. Since 9 < 1, so the mapping u > uj defined by (6) has a fixed point u = ujth, which is the solution of the variational inequality (l). REMARK 3.1. We note that various projection, linear approximation, relaxation and decomposition algorithms that have been proposed and analyzed for solving variational inequalities may be considered as special cases of the auxiliary variational inequality problem (6). To be more specific, we show that the projection technique is a special case of the auxiliary problem (6). For this purpose, we take (j)(v) =0, g = /, the identity operator in (6). In this case, for given uek, the auxiliary problem (6) is equivalent to finding a unique uek such that < w,v w >>< u^v uj > p < Tu,v u >, for all vek, from which it follows that w = P*[%-/r%, (9) where PK is the projection of H into K. It is well known that the map defined by (9) has a fixed point cj = u for 0 < p < f, see Noor [15] for full details. Thus we conclude that u PK[U ptu] is the solution of the variational inequality problem (5) and the converse is also true. This shows that the projection method is a special case of the auxiliary principle technique. We like to point out that the auxiliary principle technique is applicable to study the existence of the solution of some kind of variational inequalities, whereas the projection technique is not.

7 Free and Moving Boundary Problems 425 REMARK 3.2. It is clear that if w = u, the w is the solution of the variational inequality (l). This observation enables to suggest an iterative algorithm for finding the approximate solution of the variational inequality (1) and its various special cases. ALGORITHM 3.1. (a) At n = 0, start with some initial value u^h'. (b) At step n, solve the auxiliary problem (6) with u c<^. Let Wn+i denote the solution of the problem (6). (c) If Un+i UK < e, for given e > 0, stop. Otherwise repeat (b). CONCLUSION In this paper, we have considered and studied a new class of variational inequalities, which includes the known ones as special cases. We have also shown that the auxiliary principle technique can be used not only to study the problem of the existence of solution of variational inequalities, but also to suggest a novel and innovative iterative algorithm. By an appropriate choice of the auxiliary problem, one is able to select a suitable iterative method to solve the variational inequality and related optimization problems. Development and improvement of an implementable algorithm for various classes of variational inequalities deserve further research efforts. REFERENCES 1. Stampacchia, G. 'Formes bilineaires coercities sur les ensembles convexes', C.R. Acad. Sci., Paris, 258 (1964), Fichera, G. Troblemi elastostatici con vincoli unilateral!: il problema di signorini con ambigue condizione al contorno. Atti. Acad. Naz. Lincei. Mem. Cl. Sci. Fiz. Mat. Nat. Sez. la, 7(8), ( ), Baiocchi, C. and Capelo, A. 'Variational and Quasi-variational Inequalities', J. Wiley and Sons, London, Duvaut, D. and Lions, J.L. 'Les Inequations en Mechanique et en

8 426 Free and Moving Boundary Problems 5. Crank, J. 'Free and Moving Boundary Problems'. Clarendon Press, Oxford, Glowinski, R., Lions, J.L. and Tremolieres, R. 'Numerical Analysis of Variational Inequalities ', North-Holland, Amsterdam, Kikuchi, N. and Oden, J.T. 'Contact Problems in Elasticity'. SIAM, Philadelphia, Noor, M.A., Noor, K.I. and Rassias, Th. M. 'Some aspects of variational inequalities', J. Comput. Appl. Math. (1993), in Press. 9. Noor, M.A. 'Quasi variational inequalities'. Appl. Math. Letters, 1(1988), Noor, M.A. 'General nonlinear variational inequalities', J. Math. Anal. Appl. 126(1987), Oettli, W. 'Some remarks on general nonlinear complementarity problems and quasi-variational inequalities', Pre-print, University of Mannheiny Germany, Isac, G. A special variational inequality and the implicit complementarity problem, J. Fac. Sci. Univ. Tokyo, 37(1990), Noor, M.A. 'General algorithm and sensitivity analysis for variational inequalities', J. Appl. Math. Stoch. Anal. 5(1992), Noor, M.A. 'An iterative algorithm for nonlinear variational inequalities', Appl. Math. Letters. 5(4) (1992), Noor, M.A. 'General nonlinear variational inequalities', to appear.