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Algorithm for solving a generalized mixed equilibrium problem with perturbation in a Banach space. (English) Zbl 1213.49033

Summary: Let \(B\) be a real Banach space with the dual space \(B^*\). Let \(\varphi :B\rightarrow\mathbb R\cup+\infty\) be a proper functional and let \(\Theta :B\times B\rightarrow\mathbb R\) be a bifunction. In this paper, a new concept of \(\eta \)-proximal mapping of \(\varphi \) with respect to \(\Theta \) is introduced. The existence and Lipschitz continuity of the \(\eta \)-proximal mapping of \(\varphi \) with respect to \(\Theta \) are proved. By using properties of the \(\eta \)-proximal mapping of \(\varphi \) with respect to \(\Theta \), a Generalized Mixed Equilibrium Problem with Perturbation (for short, GMEPP) is introduced and studied in Banach space \(B\). An existence theorem of solutions of the GMEPP is established and a new iterative algorithm for computing approximate solutions of the GMEPP is suggested. Strong convergence criteria of the iterative sequence generated by the new algorithm are established in a uniformly smooth Banach space \(B\), and the weak convergence criteria of the iterative sequence generated by this new algorithm are also derived if \(B\) is \(H\) a Hilbert space.

MSC:

49K40 Sensitivity, stability, well-posedness
49J27 Existence theories for problems in abstract spaces
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References:

[1] Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244. · Zbl 0708.47031 · doi:10.1017/CBO9780511526152
[2] Baiocchi C, Capelo A: Variational and Quasivariational Inequalities, Applications to Free Boundary Problems. John Wiley & Sons, New York, NY, USA; 1984:ix+452. · Zbl 0551.49007
[3] Facchinei F, Pang JS: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York, NY, USA; 2003. · Zbl 1062.90002
[4] Glowinski R, Lions J-L, Trémolières R: Numerical Analysis of Variational Inequalities, Studies in Mathematics and Its Applications. Volume 8. North-Holland, Amsterdam, The Netherlands; 1981:xxix+776.
[5] Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics. Volume 88. Academic Press, New York, NY, USA; 1980:xiv+313. · Zbl 0988.49003
[6] Xiu N, Zhang J: Some recent advances in projection-type methods for variational inequalities.Journal of Computational and Applied Mathematics 2003,152(1-2):559-585. 10.1016/S0377-0427(02)00730-6 · Zbl 1018.65083 · doi:10.1016/S0377-0427(02)00730-6
[7] Cho YJ, Kim JK, Verma RU: A class of nonlinear variational inequalities involving partially relaxed monotone mappings and general auxiliary problem principle.Dynamic Systems and Applications 2002,11(3):333-337. · Zbl 1014.49007
[8] Cohen G: Auxiliary problem principle extended to variational inequalities.Journal of Optimization Theory and Applications 1988,59(2):325-333. · Zbl 0628.90066
[9] Schaible S, Yao JC, Zeng L-C: Iterative method for set-valued mixed quasivariational inequalities in a Banach space.Journal of Optimization Theory and Applications 2006,129(3):425-436. 10.1007/s10957-006-9077-9 · Zbl 1123.49006 · doi:10.1007/s10957-006-9077-9
[10] Zeng L-C, Guu S-M, Yao J-C: An iterative method for generalized nonlinear set-valued mixed quasi-variational inequalities with -monotone mappings.Computers & Mathematics with Applications 2007,54(4):476-483. 10.1016/j.camwa.2007.01.025 · Zbl 1131.49012 · doi:10.1016/j.camwa.2007.01.025
[11] Zeng L-C: Iterative algorithms for finding approximate solutions for general strongly nonlinear variational inequalities.Journal of Mathematical Analysis and Applications 1994,187(2):352-360. 10.1006/jmaa.1994.1361 · Zbl 0820.49005 · doi:10.1006/jmaa.1994.1361
[12] Zeng L-C: Iterative algorithm for finding approximate solutions to completely generalized strongly nonlinear quasivariational inequalities.Journal of Mathematical Analysis and Applications 1996,201(1):180-194. 10.1006/jmaa.1996.0249 · Zbl 0853.65073 · doi:10.1006/jmaa.1996.0249
[13] Huang N-J, Deng C-X: Auxiliary principle and iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities.Journal of Mathematical Analysis and Applications 2001,256(2):345-359. 10.1006/jmaa.2000.6988 · Zbl 0972.49008 · doi:10.1006/jmaa.2000.6988
[14] Xia F-Q, Huang N-J: Algorithm for solving a new class of general mixed variational inequalities in Banach spaces.Journal of Computational and Applied Mathematics 2008,220(1-2):632-642. 10.1016/j.cam.2007.09.011 · Zbl 1157.65043 · doi:10.1016/j.cam.2007.09.011
[15] Zeng L-C, Schaible S, Yao JC: Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities.Journal of Optimization Theory and Applications 2005,124(3):725-738. 10.1007/s10957-004-1182-z · Zbl 1067.49007 · doi:10.1007/s10957-004-1182-z
[16] Ding XP: General algorithm of solutions for nonlinear variational inequalities in Banach space.Computers & Mathematics with Applications 1997,34(9):131-137. 10.1016/S0898-1221(97)00194-6 · Zbl 0888.65080 · doi:10.1016/S0898-1221(97)00194-6
[17] Ding XP: General algorithm for nonlinear variational-like inequalities in reflexive Banach spaces.Indian Journal of Pure and Applied Mathematics 1998,29(2):109-120. · Zbl 0908.49009
[18] Ding XP, Yao J-C, Zeng L-C: Existence and algorithm of solutions for generalized strongly nonlinear mixed variational-like inequalities in Banach spaces.Computers & Mathematics with Applications 2008,55(4):669-679. 10.1016/j.camwa.2007.06.004 · Zbl 1291.49004 · doi:10.1016/j.camwa.2007.06.004
[19] Chang SS: Set-valued variational inclusions in Banach spaces.Journal of Mathematical Analysis and Applications 2000,248(2):438-454. 10.1006/jmaa.2000.6919 · Zbl 1031.49018 · doi:10.1006/jmaa.2000.6919
[20] Huang N-J, Fang Y-P: Iterative processes with errors for nonlinear set-valued variational inclusions involving accretive type mappings.Computers & Mathematics with Applications 2004,47(4-5):727-738. 10.1016/S0898-1221(04)90060-0 · Zbl 1081.47064 · doi:10.1016/S0898-1221(04)90060-0
[21] Huang N-J, Yuan GX-Z: Approximating solution of nonlinear variational inclusions by Ishikawa iterative process with errors in Banach spaces.Journal of Inequalities and Applications 2001,6(5):547-561. 10.1155/S1025583401000339 · Zbl 1010.49004 · doi:10.1155/S1025583401000339
[22] Zeng LC, Hu HY, Wong MM: Strong and weak convergence theorems for generalized mixed equilibrium problem with perturbation and fixed point problem of infinitely many nonexpansive mappings. to appear in Taiwanese Journal of Mathematics · Zbl 0628.90066
[23] Peng J-W, Yao J-C: A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems.Taiwanese Journal of Mathematics 2008,12(6):1401-1432. · Zbl 1185.47079
[24] Ding XP, Tan K-K: A minimax inequality with applications to existence of equilibrium point and fixed point theorems.Colloquium Mathematicum 1992,63(2):233-247. · Zbl 0833.49009
[25] Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings.Bulletin of the American Mathematical Society 1967, 73: 591-597. 10.1090/S0002-9904-1967-11761-0 · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0
[26] van Dulst D: Equivalent norms and the fixed point property for nonexpansive mappings.The Journal of the London Mathematical Society 1982,25(1):139-144. 10.1112/jlms/s2-25.1.139 · Zbl 0453.46017 · doi:10.1112/jlms/s2-25.1.139
[27] Petryshyn WV: A characterization of strict convexity of Banach spaces and other uses of duality mappings.Journal of Functional Analysis 1970, 6: 282-291. 10.1016/0022-1236(70)90061-3 · Zbl 0199.44004 · doi:10.1016/0022-1236(70)90061-3
[28] Xu HK, Kim TH: Convergence of hybrid steepest-descent methods for variational inequalities.Journal of Optimization Theory and Applications 2003,119(1):185-201. · Zbl 1045.49018 · doi:10.1023/B:JOTA.0000005048.79379.b6
[29] Tan K-K, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.Journal of Mathematical Analysis and Applications 1993,178(2):301-308. 10.1006/jmaa.1993.1309 · Zbl 0895.47048 · doi:10.1006/jmaa.1993.1309
[30] Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244. · Zbl 0708.47031 · doi:10.1017/CBO9780511526152
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