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A new method for solving a system of generalized nonlinear variational inequalities in Banach spaces. (English) Zbl 1215.65119
The essential aim of the present paper is to consider a new iterative scheme to study the approximate solvability problem for a system of generalized nonlinear variational inequalities in the framework of uniformly smooth and strictly convex Banach spaces by using the generalized projection approach. Here the authors consider some special cases of the problem (the so-called system of generalized nonlinear variational inequalities problem, SGNIVP): find $x^*, y^*, z^* \in K$ (a nonempty closed convex subset of real strictly convex Banach space) such that for all $x \in K$ ($S,T,U: K^3 \rightarrow X^*$ are the nonlinear mappings) (1)$\langle S(y^*, z^*, x^*), x-x^*\rangle \geq 0$, $\langle T(z^*, x^*, y^*), x-y^*\rangle \geq 0$, $\langle U(x^*, y^*, z^*), x-z^*\rangle \geq 0$. (I) If $U=0$ and $(S,T)$ are bifurcations from $K^2$ to $X^*$ (the dual space) with $S(y,x) = \varrho T_1(y,x) + Jx - Jy$ ($J: E \rightarrow 2 ^{E^*}$ is the normalized duality mapping) and $T(x,y) = \eta T_2 (x,y) + Jy - Jx$, where $T_i: K^2 \rightarrow X^*$ is a mapping, $i = 1,2$, then the problem (1) reduces to find $x^*, y^* \in K$ such that for all $x \in X$: (2) $\langle \varrho T_1 (y^*, x^*)+Jx^* - Jy^*, x-x^* \rangle \geq 0$, $\langle \eta T_2 (x^*, y^*)+Jy^* - Jx^*, x-y^* \rangle \geq 0$, $\forall x \in K$; where $\varrho$ and $\eta$ are two positive constants. (II) If $S$ and $T$ both are univariate mappings then the problem (2) reduces to the (SGNVIP). Find $x^*, y^* \in K$ such that for all $x \in X$; (3) $\langle \varrho T_1 (y^*) + Jx^* - Jy^*, x-x^*\rangle \geq 0$, $\langle \eta T_2 (x^*) + Jy^* - Jx^*, x-y^*\rangle \geq 0$. Main result: If $X$ is a real smooth and strictly convex Banach space with Kadec-Klee property, $K$ is a nonempty closed and convex subset of $X$ with $\theta \in K$ and $S,T,U,: K^3 \rightarrow X^*$ be continuous mappings satisfying the condition: There exist a compact subset $C \subset X^*$ and constants $\varrho > 0$, $\eta > 0$, $\xi > 0$ such that $(J- \varrho S) (K^3) \cup (J- \eta T) (K^3) \cup (J- \xi U)(K^3) \cup C$, where $J(x,y,z) = Jz$, $\forall (x,y,z) \in K^3$ and $\langle S(x,y,z), J^{-1}(Jz- \varrho S (x,y,z)\rangle \geq 0$, $\langle T(x,y,z), J^{-1}(Jz- \eta T (x,y,z)\rangle \geq 0$ $\langle U(x,y,z)$, $J^{-1}(Jz- \eta U (x,y,z)\rangle \geq 0$, then problem (1) has a solution $ (x^*, y^*, z^*) \in K^3$ and $ x_n \rightarrow x^*$, $y_n \rightarrow y^*$, $z_n \rightarrow z^*$, and the sequences converge strongly to a unique solution $[x^*, y^*, z^*] \in K^3$. The problem (1) is a more general system of generalized nonlinear variational inequality problem, which includes many kinds of well-known systems of variational inequalities as its special case.

65K15Numerical methods for variational inequalities and related problems
49J40Variational methods including variational inequalities
49M25Discrete approximations in calculus of variations
Full Text: DOI
[1] He, X. F.; Chen, J. M.; He, Z.: Generalized projection method for a variational inequality system with different mappings in Banach spaces, Comput. math. Appl. 58, 1391-1396 (2009) · Zbl 1189.49008 · doi:10.1016/j.camwa.2009.07.021
[2] Qin, X. L.; Cho, S. Y.; Kang, S. M.: On hybrid projection methods for asymptotically quasi-$\phi $-nonexpansive mappings, Appl. math. Comput. 215, No. 11, 3874-3883 (2010) · Zbl 1225.47105 · doi:10.1016/j.amc.2009.11.031
[3] Qin, X. L.; Cho, S. Y.; Kang, S. M.: Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications, J. comput. Appl. math. 233, No. 2, 231-240 (2009) · Zbl 1204.65081 · doi:10.1016/j.cam.2009.07.018
[4] Gu, F.: Some convergence theorems of non-implicit iteration process with errors for a finite families of II-asymptotically nonexpansive mappings, Appl. math. Comput. 216, No. 1, 161-172 (2010) · Zbl 1225.47095 · doi:10.1016/j.amc.2010.01.025
[5] He, Z. H.; Gu, F.: Generalized system for relaxed cocoercive mixed variational inequalities in Hilbert spaces, Appl. math. Comput. 214, No. 1, 26-30 (2009) · Zbl 1166.49011 · doi:10.1016/j.amc.2009.03.056
[6] Cho, Y. J.; Qin, X.: System of generalized nonlinear variational inequalities and its projection methods, Nonlinear anal.: theory meth. Appl. 69, No. 12, 4443-4451 (2008) · Zbl 1153.49009 · doi:10.1016/j.na.2007.11.001
[7] Chang, S. S.; Joseph, H. W.; Chan, C. K.: Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces, Appl. math. Lett. 20, No. 3, 329-334 (2007) · Zbl 1114.49008 · doi:10.1016/j.aml.2006.04.017
[8] Fan, J. H.: A Mann type iterative scheme for variational inequalities in noncompact subsets of Banach spaces, J. math. Anal. appl. 337, 1041-1047 (2008) · Zbl 1140.49011 · doi:10.1016/j.jmaa.2007.04.025
[9] Li, J.: On the existence of solutions of variational inequalities in Banach spaces, J. math. Anal. appl. 295, 115-126 (2004) · Zbl 1045.49008 · doi:10.1016/j.jmaa.2004.03.010
[10] Cioranescu, I.: Geometry of Banach spaces, Duality mappings and nonlinear problems (1990) · Zbl 0712.47043
[11] Alber, Y. I.: Metric and generalized projection operators in Banach spaces: properties and applications, Theory and applications of nonlinear operators of accretive and monotone type, 15-50 (1996) · Zbl 0883.47083
[12] Zhang, S. S.: On the generalized mixed equilibrium problem in Banach spaces, Appl. math. Mech. 30, No. 9, 1105-1112 (2009) · Zbl 1178.47051 · doi:10.1007/s10483-009-0904-6
[13] Xu, H. K.: Inequalities in Banach spaces with applications, Nonlinear anal. 16, 1127-1138 (1991) · Zbl 0757.46033 · doi:10.1016/0362-546X(91)90200-K
[14] Chang, Shih-Sen: On chidume’s open questions and approximate solutions of multivalued strong accretive mappings in Banach spaces, J. math. Anal. appl. 216, 94-111 (1997) · Zbl 0909.47049 · doi:10.1006/jmaa.1997.5661
[15] Verma, R. U.: General convergence analysis for two-step projection methods and application to variational problems, Appl. math. Lett. 18, No. 11, 1286-1292 (2005) · Zbl 1099.47054 · doi:10.1016/j.aml.2005.02.026
[16] Agarwal, Ravi P.; Verma, Ram U.: General implicit variational inclusion problems based on AA-maximal (m)$(m)$-relaxed monotonicity (AMRM) frameworks, Appl. math. Comput. 215, No. 1, 367-379 (2009) · Zbl 1178.65073 · doi:10.1016/j.amc.2009.04.078