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Some aspects of variational inequalities. (English) Zbl 0788.65074

The paper written in an expository style provides a brief review of some modern trends and achievements in the variational inequality theory. In particular the following aspects of variational inequalities are considered.
1) Iterative methods for solving variational inequalities of the form \[ \langle T(u), v-u\rangle\geq \langle f, v-u\rangle\quad \forall v\in K\;(u\in K),\tag{1} \] where \(T: H\to H\) is a nonlinear strongly monotone operator, \(K\) is a closed convex subset of a real Hilbert space \(H\) and \(f\in H\). The presented methods are based on the equivalence between (1), the fixed point problem \(u= P_ k(u-\rho(T(u)-f))\), \(\rho>0\) and the Wiener-Hopf equation \(T(P_ k(v))+\rho^{-1} Q_ k(v)= f\), where \(P_ k\) is the projection of \(H\) onto \(K\), \(Q_ k= I-P_ k\).
2) The sensitivity analysis of quasivariational inequalities \[ \langle T(u;\lambda), v-u\rangle\geq 0\quad \forall v\in K_ \lambda(u)\quad (u\in K_ \lambda(u))\tag{2} \] with a parameter \(\lambda\in H\). The main result of this section establishes those conditions under which (2) has a locally unique solution \(u(\lambda)\) and the function \(\lambda\to u(\lambda)\) is continuous or Lipschitz continuous.
3) The constructing of iterative methods for solving generalized variational inequalities \[ \langle T(u),g(v)-g(u)\rangle\geq \langle f,g(v)- g(u)\rangle\quad\forall g(v)\in K\;(u\in H, g(u)\in K)\tag{3} \] by transforming (3) to the fixed point problem or the general Wiener-Hopf equation. Here \(g: H\to H\) is a continuous operator.
4) Variational inequalities for fuzzy mappings and iterative methods for solving such inequalities.
5) Finite element approximation and error estimation for (1).

MSC:

65K10 Numerical optimization and variational techniques
47J20 Variational and other types of inequalities involving nonlinear operators (general)
49M05 Numerical methods based on necessary conditions
49J40 Variational inequalities
49J27 Existence theories for problems in abstract spaces
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[1] Ahn, H. B., Iterative methods for linear complementarity problems with upper bounds on primary variables, Math. Programming, 26, 295-315 (1983) · Zbl 0506.90081
[2] Baiocchi, C.; Capelo, A., Variational and Quasi-Variational Inequalities (1984), Wiley: Wiley New York · Zbl 1308.49002
[3] Bensoussan, A.; Lions, J. L., Applications des Inéquations Variationnelles en Control et Stochastiques (1978), Dunod: Dunod Paris · Zbl 0411.49002
[4] Bertsekas, D. P.; Gafni, E. M., Projection methods for variational inequalities with applications to the traffic assignment problem, Math. Prog. Study, 17, 139-159 (1982) · Zbl 0478.90071
[5] Butt, R., Optimal shape design for differential inequalities, Ph.D. Thesis (1988), School of Math., Univ. Leeds · Zbl 0800.93421
[6] Chan, D.; Pang, J. S., The generalized quasi variational inequalities problems, Math. Oper. Res., 7, 211-222 (1982) · Zbl 0502.90080
[7] Chang, S.; Zhu, Y., On variational inequalities for fuzzy mappings, Fuzzy Sets and Systems, 32, 359-367 (1989) · Zbl 0677.47037
[8] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, 4 (1978), North-Holland: North-Holland Amsterdam, Stud. Math. Appl. · Zbl 0445.73043
[9] Cohen, G., Auxiliary problem principle extended to variational inequalities, J. Optim. Theory Appl., 59, 325-333 (1988) · Zbl 0628.90066
[10] Cottle, R. W.; Dantzig, G. B., Complementarity pivot theory of mathematical programming, Linear Algebra Appl., 1, 103-105 (1968) · Zbl 0155.28403
[11] Cottle, R. W.; Giannessi, F.; Lions, J.-L., Variational Inequalities and Complementarity Problems: Theory and Applications (1980), Wiley: Wiley New York · Zbl 0476.00017
[12] Crank, J., Free and Moving Boundary Problems (1984), Clarendon Press: Clarendon Press Oxford · Zbl 0547.35001
[13] Dafermos, S., Traffic equilibria and variational inequalities, Transportation Sci., 14, 42-54 (1980)
[14] Dafermos, S., Sensitivity analysis in variational inequalities, Math. Oper. Res., 13, 421-434 (1988) · Zbl 0674.49007
[15] Dafermos, S., Exchange price equilibria and variational inequalities, Math. Programming, 46, 391-402 (1990) · Zbl 0709.90013
[16] De Luca, M.; Maugeri, A., Quasi-variational inequalities and applications to the traffic equilibrium problem; discussion of a paradox, J. Comput. Appl. Math., 28, 163-171 (1989) · Zbl 0683.90093
[17] Dolcetta, I., Sistemi di complementarita a disequaglianze variationale, Ph.D. Thesis (1972), Univ. Rome
[18] Falk, R. S., Error estimates for the approximation of a class of variational inequalities, Math. Comp., 28, 963-972 (1974) · Zbl 0297.65061
[19] Fichera, G., Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia, 7, 8, 91-140 (1963-1964) · Zbl 0146.21204
[20] Glowinski, R.; Lions, J.-L.; Trémolières, R., Numerical Analysis of Variational Inequalities, 8 (1981), North-Holland: North-Holland Amsterdam, Stud. Math. Appl. · Zbl 0508.65029
[21] Harker, P. T., Predicting Intercity Freight Flows (1987), VNU Science Press: VNU Science Press Utrecht
[22] P.T. Harker, Generalized Nash games and quasi variational inequalities, European J. Oper. Res., to appear.; P.T. Harker, Generalized Nash games and quasi variational inequalities, European J. Oper. Res., to appear. · Zbl 0754.90070
[23] Harker, P. T.; Pang, J. S., Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Programming, 48, 161-220 (1990) · Zbl 0734.90098
[24] Haslinger, J.; Neittaanmäki, P., Finite Element Approximation for Optimal Shape Design, Theory and Applications (1988), Wiley: Wiley New York · Zbl 0713.73062
[25] Hlavaček, I.; Haslinger, J.; Necas, J.; Lovisek, J., Solution of Variational Inequalities in Mechanics (1988), Springer: Springer Berlin · Zbl 0654.73019
[26] G. Isac and D. Goeleven, Existence theorems for the implicit complementarity problem, to appear.; G. Isac and D. Goeleven, Existence theorems for the implicit complementarity problem, to appear. · Zbl 0770.90077
[27] G. Isac and M. Kostreva, The generalized order complementarity problem, J. Optim. Theory Appl., to appear.; G. Isac and M. Kostreva, The generalized order complementarity problem, J. Optim. Theory Appl., to appear. · Zbl 0795.90073
[28] Isac, G.; Thera, M., Complementarity problem and the existence of the post-critical equilibrium state of the thin elastic plate, J. Optim. Theory Appl., 58, 241-257 (1988) · Zbl 0631.49005
[29] Karamardian, S., Generalized complementarity problem, J. Optim. Theory Appl., 8, 161-168 (1971) · Zbl 0218.90052
[30] Khalifa, A. K.; Noor, M. A., Quintic splines solutions of a class of contact problems, Math. Comput. Modelling, 13, 51-58 (1990) · Zbl 0705.65045
[31] Kikuchi, N.; Oden, J. T., Contact Problems in Elasticity (1988), SIAM: SIAM Philadelphia, PA · Zbl 0685.73002
[32] Kinderlehrer, D.; Stampacchia, G., An Introduction to Variational Inequalities and their Applications (1980), Academic Press: Academic Press New York · Zbl 0457.35001
[33] Kyparisis, J., Sensitivity analysis framework for variational inequalities, Math. Programming, 38, 203-213 (1987)
[34] Kyparisis, J., Perturbed solutions of variational inequality problems over polyhedral sets, J. Optim. Theory Appl., 57, 295-305 (1988) · Zbl 0621.49004
[35] Lemke, C. E., Bimatrix equilibrium points and mathematical programming, Management Sci., 11, 681-689 (1965) · Zbl 0139.13103
[36] Lewy, H.; Stampacchia, G., On the regularity of the solutions of the variational inequalities, Comm. Pure Appl. Math., 22, 153-188 (1969) · Zbl 0167.11501
[37] Lions, J.-L., Optimal Control of Systems Governed by Partial Differential Equations (1971), Springer: Springer Berlin · Zbl 0203.09001
[38] Mancino, O.; Stampacchia, G., Convex progrmaming and variational inequalities, J. Optim. Theory Appl., 9, 3-23 (1972) · Zbl 0223.90031
[39] Mosco, U., Implicit variational problems and quasi variational inequalities, (Nonlinear Operators and the Calculus of Variations, 543 (1976), Springer: Springer New York), 83-126, Lecture Notes in Math. · Zbl 0346.49003
[40] Mosco, U.; Strang, G., One-sided approximation and variational inequalities, Bull. Amer. Math. Soc., 80, 308-312 (1974) · Zbl 0278.35026
[41] Noor, M. A., On variational inequalities, Ph.D. Thesis (1975), Brunel Univ · Zbl 0859.49009
[42] Noor, M. A., An iterative scheme for a class of quasi variational inequalities, J. Math. Anal. Appl., 110, 463-468 (1985) · Zbl 0581.65051
[43] Noor, M. A., Variational inequalities related with a Signorini problem, C.R. Math. Rep. Acad. Sci. Canada, 7, 267-272 (1985) · Zbl 0592.49006
[44] Noor, M. A., General nonlinear variational inequalities, J. Math. Anal. Appl., 126, 78-84 (1987) · Zbl 0645.49004
[45] Noor, M. A., On a class of variational inequalities, J. Math. Anal. Apl., 128, 138-155 (1987) · Zbl 0631.49004
[46] Noor, M. A., General variational inequalities, Appl. Math. Lett., 1, 119-122 (1988) · Zbl 0655.49005
[47] Noor, M. A., Quasi variational inequalities, Appl. Math. Lett., 1, 367-370 (1988) · Zbl 0708.49015
[48] Noor, M. A., Nonlinear variational inequalities in elastostatics, Internat. J. Engrg. Sci., 26, 1043-1051 (1988) · Zbl 0671.73017
[49] Noor, M. A., Convergence analysis of the iterative methods for quasi complementarity problems, Internat. J. Math. Math. Sci., 11, 319-344 (1988) · Zbl 0651.49003
[50] Noor, M. A., Iterative methods for a class of complementarity problems, J. Math. Anal. Appl., 133, 366-382 (1988) · Zbl 0649.65036
[51] Noor, M. A., Nonlinear quasi complementarity problems, Appl. Math. Lett., 2, 251-254 (1989) · Zbl 0705.65044
[52] Noor, M. A., An iterative algorithm for variational inequalities, J. Math. Anal. Appl., 158, 448-455 (1991) · Zbl 0733.65047
[53] Noor, M. A., Some classes of variational inequalities, (Rassias, Th. M., Constantin Caratheodory: An International Tribute (1991), World Scientific: World Scientific Singapore), 996-1019 · Zbl 0747.49011
[54] Noor, M. A., Iterative methods for quasi-variational inequalities, Panamer. Math. J., 2, 2, 17-26 (1992) · Zbl 0842.49012
[55] M.A. Noor, Finite element estimates for a class of nonlinear variational inequalities, Internat. J. Math. Math. Sci., to appear.; M.A. Noor, Finite element estimates for a class of nonlinear variational inequalities, Internat. J. Math. Math. Sci., to appear. · Zbl 0809.65070
[56] Noor, M. A., General algorithm and sensitivity analysis for variational inequalities, J. Appl. Math. Stochastic Anal., 5, 1, 29-42 (1992)
[57] Noor, M. A., General algorithm for variational inequalities (I), Math. Japon., 38, 47-53 (1993) · Zbl 0769.49008
[58] M.A. Noor, General nonlinear complementarity problems, in: Th.M. Rassias and H.M. Srivastava, Eds., Analysis, Geometry and Groups: A Riemann Legacy Volume (Hardonic, to appear).; M.A. Noor, General nonlinear complementarity problems, in: Th.M. Rassias and H.M. Srivastava, Eds., Analysis, Geometry and Groups: A Riemann Legacy Volume (Hardonic, to appear).
[59] M.A. Noor, General auxiliary principle and variational inequalities, to appear.; M.A. Noor, General auxiliary principle and variational inequalities, to appear. · Zbl 0845.49005
[60] Noor, M. A., Iterative methods and sensitivity analysis for general quasi variational inequalities, J. Nat. Geometry, 3, 1, 39-58 (1993) · Zbl 0769.49007
[61] M.A. Noor, Wiener—Hopf equations and variational inequalities, J. Optim. Theory Appl., to appear.; M.A. Noor, Wiener—Hopf equations and variational inequalities, J. Optim. Theory Appl., to appear. · Zbl 0799.49010
[62] Noor, M. A., Variational inequalities for fuzzy mappings (I), Fuzzy Sets and Systems, 55, 3, 309-312 (1993) · Zbl 0785.49006
[63] Noor, M. A.; Khalifa, A. K., Cubic splines collocation methods for unilateral problems, Internat. J. Engrg. Sci., 25, 1527-1530 (1987) · Zbl 0624.73120
[64] Noor, M. A.; Noor, K. I., Iterative methods for variational inequalities and nonlinear programming, Oper. Res. Verfahren, 31, 455-463 (1979) · Zbl 0403.35026
[65] M.A. Noor, K.I. Noor and Th.M. Rassias, Invitation to variational inequallities, in: Th.M. Rassias and H.M. Srivastava, Eds., Analysis, Geometry and Groups: A Riemann Legacy Volume (Hardonic, to appear).; M.A. Noor, K.I. Noor and Th.M. Rassias, Invitation to variational inequallities, in: Th.M. Rassias and H.M. Srivastava, Eds., Analysis, Geometry and Groups: A Riemann Legacy Volume (Hardonic, to appear).
[66] Noor, M. A.; Tirmizi, S. I.A., Finite difference techniques for solving obstacle problems, Appl. Math. Lett., 1, 267-271 (1988) · Zbl 0659.49006
[67] Oden, J. T.; Kikuchi, N., Theory of variational inequalities with applications to flow through porous media, Internat. J. Engrg. Sci., 18, 1173-1284 (1980) · Zbl 0444.76069
[68] Oettli, W., Some remarks on general complementarity problems and quasi-variational inequalities, Preprint (1987), Univ. Mannheim
[69] Panagiotopoulos, P. D., Inequality Problems in Mechanics and Applications (1985), Birkhäuser: Birkhäuser Boston · Zbl 0579.73014
[70] Pang, J. S., On the convergence of a basic iterative method for the implicit complementarity problem, J. Optim. Theory Appl., 37, 149-162 (1982) · Zbl 0482.90084
[71] Pires, E. B.; Oden, J. T., Error estimates for the approximations of a class of variational inequalities arising in unilateral problems with friction, Numer. Funct. Anal. Optim., 4, 397-412 (1981/1982) · Zbl 0506.65024
[72] Pitonyak, A.; Shi, P.; Shillor, M., On an iterative method for variational inequalities, Numer. Math., 58, 231-242 (1990) · Zbl 0689.65043
[73] Qiu, Y.; Magnanti, T. L., Sensitivity analysis for variational inequalities defined on polyhedral sets, Math. Oper. Res., 14, 410-432 (1989) · Zbl 0698.90069
[74] (Rassias, G. M.; Rassias, Th. M., Differential Geometry, Calculus of Variations, and their Applications, 100 (1985), Marcel Dekker: Marcel Dekker New York), Lecture Notes in Pure and Appl. Math.
[75] Rassias, Th. M., Morse theory and Plateau’s problem, (Rassias, Th. M.; Rassias, G. M., Selected Studies: Physics-Astrophysics, Mathematics, History of Science (1982), North-Holland: North-Holland Amsterdam), 261-292
[76] (Rassias, Th. M., Global Analysis — Analysis on Manifolds, 57 (1983), Teubner: Teubner Leipzig), Teubner-Texte Math.
[77] Rassias, Th. M., Morse theory in global variational analysis, (Rassias, Th. M., Global Analysis — Analysis on Manifolds, 57 (1983), Teubner: Teubner Leipzig), 7-16, Teubner-Texte Math.
[78] Rassias, Th. M., Foundation of Global Nonlinear Analysis (1986), Teubner: Teubner Leipzig
[79] (Rassias, Th. M., Nonlinear Analysis (1987), World Scientific: World Scientific Singapore) · Zbl 0668.00008
[80] (Rassias, Th. M., Topics in Mathematical Analysis (1989), World Scientific: World Scientific Singapore)
[81] Rassias, Th. M., Eigenvalues of the Laplacian, (Francaviglia, M., Mechanics, Analysis and Geometry: 200 Years after Lagrange (1991), North-Holland: North-Holland Amsterdam), 315-332
[82] Rodrigues, J.-F., Obstacle Problems in Mathematical Physics, 134 (1987), North-Holland: North-Holland Amsterdam, North-Holland Math. Stud. · Zbl 0606.73017
[83] Shi, P., An iterative method for obstacle problems via Green’s functions, Nonlinear Anal. Theory Methods Appl., 15, 339-344 (1990) · Zbl 0725.65068
[84] Shi, P., Equivalence of variational inequalities with Wiener—Hopf equations, Proc. Amer. Math. Soc., 111, 339-346 (1991) · Zbl 0881.35049
[85] Sibony, M., Sur l’approximation d’équations et inéquations aux derivées partielles nonlineaires de type monotone, J. Math. Anal. Appl., 34, 502-504 (1971) · Zbl 0216.42201
[86] Smith, M. J., The existence, uniqueness and stability of traffic equilibria, Transportation Research, 133, 295-304 (1979)
[87] Speck, F. O., General Wiener—Hopf Factorization Methods, 119 (1985), Longman: Longman Harlow, Pitman Res. Notes Math. Ser. · Zbl 0588.35090
[88] Stampacchia, G., Formes bilinéaires coercitives sur les ensembles convexes, C.R. Acad. Sci. Paris Sér I. Math., 258, 4413-4416 (1964) · Zbl 0124.06401
[89] Strang, G.; Fix, G. J., An Analysis of the Finite Element Method (1973), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0278.65116
[90] Tobin, R. L., Sensitivity analysis for variational inequalities, J. Optim. Theory Appl., 48, 191-204 (1986) · Zbl 0557.49004
[91] Tonti, E., Variational formulation for every nonlinear problem, Internat. J. Engrg. Sci., 22, 1343-1371 (1984) · Zbl 0558.49022
[92] Zimmermann, H. I., Fuzzy Set Theory and its Applications (1988), Kluwer: Kluwer Dordrecht
[93] Cottle, R. W.; Pang, J. S.; Stone, R. E., The Linear Complementarity Problem (1992), Academic Press: Academic Press London · Zbl 0757.90078
[94] Isac, G., Complementarity Problems, 1528 (1993), Springer: Springer Berlin, Lecture Notes in Math. · Zbl 0784.47047
[95] Noor, M. A., General quasi complementarity problems, Math. Japon., 36, 113-119 (1991) · Zbl 0729.90084
[96] Noor, M. A., Generalized Wiener—Hopf equations and nonlinear quasi variational inequalities, Panamer. Math. J., 2, 4, 51-70 (1992) · Zbl 0842.49011
[97] Noor, M. A., Wiener—Hopf equations and complementarity problems, Panamer. Math. J., 3, 1, 59-68 (1993) · Zbl 0847.35006
[98] M.A. Noor, Some recent advances in variational inequalities, in: R.U. Verma, Ed., New Developments in Approximation — Solvability of Nonlinear Equations — Abstract, Differential and Integral Equations: Theory and Applications (Marcel Dekker, New York, to appear).; M.A. Noor, Some recent advances in variational inequalities, in: R.U. Verma, Ed., New Developments in Approximation — Solvability of Nonlinear Equations — Abstract, Differential and Integral Equations: Theory and Applications (Marcel Dekker, New York, to appear). · Zbl 0889.49006
[99] Noor, M. A.; El-Shemas, A. H., Iterative methods for general quasi complementarity problems, Honam Math. J., 14, 107-121 (1992) · Zbl 0997.90541
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