## Convergence of implicit and explicit schemes for common fixed points for finite families of asymptotically nonexpansive mappings. (Convergence of implicit and explicit schemes for common fixed-points for finite families of asymptotically nonexpansive mappings.)(English)Zbl 1239.49008

Nonlinear Anal., Hybrid Syst. 5, No. 3, 492-501 (2011); corrigendum ibid. 5, No. 3, 603-604 (2011).
Summary: Let $$H$$ be a real Hilbert space and $$T_1,T_2,\dots,T_N$$ be a family of asymptotically nonexpansive self-mappings of $$H$$ with sequences $$\{1+k^{i(n)}_{p(n)}\}$$, such that $$k^{i(n)}_{p(n)}\to 0$$ as $$n\to\infty$$ where $$p(n)=j+1$$ if $$jN<n\leq(j+1)N$$, $$j=0,1,2,\dots$$ and $$n=jN+i(n)$$, $$i(n)\in\{1,2,\dots,N\}$$. Let $$F:=\cap^N_{i=1}\text{Fix}(T_i)\neq\emptyset$$ and let $$f:H\to H$$ be a contraction mapping with coefficient $$\alpha\in(0,1)$$, also let $$A$$ be a strongly positive bounded linear operator with coefficient $$\overline\gamma>0$$, and $$0<\gamma<\frac{\overline\gamma}{\alpha}$$. Let $$\{\alpha_n\}$$, $$\{\beta_n\}$$ be sequences in $$(0,1)$$ satisfying some conditions. Strong convergence of the implicit and explicit schemes are proved for a common fixed-point of the family $$T_1,T_2,\dots,T_n$$, which solves the variational inequality $$\langle(A-\gamma f)\overline x,\overline x-x\rangle\leq 0$$ $$\forall x\in F$$. Our result generalizes and improves several recent results.

### MSC:

 49J40 Variational inequalities 47J25 Iterative procedures involving nonlinear operators 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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### References:

  Goebel, K.; Kirk, W.A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. amer. math. soc., 35, 171-174, (1972) · Zbl 0256.47045  Jianghua, F., 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  Kinderlehrer, D.; Stampacchia, G., An introduction to variational inequalities and their applications, (1980), Academic Press, Inc. · Zbl 0457.35001  Noor, M.A., General variational inequalities and nonexpansive mappings, J. math. anal. appl., 331, 810-822, (2007) · Zbl 1112.49013  Marino, G.; Xu, H.K., A general iterative method for nonexpansive mappings in Hilbert space, J. math. anal. appl., 318, 43-52, (2006) · Zbl 1095.47038  Moudafi, A., Viscosity approximation methods for fixed points problems, J. math. anal. appl., 241, 46-55, (2000) · Zbl 0957.47039  Shahzad, N.; Udomene, A., Fixed point solutions of variational inequalities for asymptotically nonexpansive mappings in Banach spaces, Nonlinear anal., 64, 558-567, (2006) · Zbl 1102.47056  Xu, H.K., Viscosity approximation methods for nonexpansive mappings, J. math. anal. appl., 298, 279-291, (2004) · Zbl 1061.47060  Yamada, I., The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed-point sets of nonexpansive mappings, (), 473-504 · Zbl 1013.49005  Xu, H.K., An iterative approach to quadratic optimization, J. optim. theory appl., 116, 659-678, (2003) · Zbl 1043.90063  Junlouchai, P.; Plubtieng, S., A general iterative method for nonexpansive mappings, NU science journal, 4, SI, 1-12, (2007)  Yao, Y.; Noor, M.A.; Mohyud-Din, S.T., Modified iterative algorithms for nonexpansive mappings, J. appl. math. stoch. anal., 2009, (2009), Article ID 320820 · Zbl 1220.47137  Suzuki, T., Strong convergence of Krasnoselskii and mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. math. anal. appl., 305, 227-239, (2005) · Zbl 1068.47085  Takahashi, W.; Tamura, T.; Toyoda, M., Approximation of common fixed points of a family of finite nonexpansive mappings in Banach spaces, Sci. math. Japan, 56, 475-480, (2002) · Zbl 1026.47042  Xu, H.K., Existence and convergence for fixed points of mappings of asymptotically nonexpansive type, Nonlinear anal., 16, 1139-1146, (1991) · Zbl 0747.47041  Xu, H.K., Iterative algorithms for nonlinear operators, J. lond. math. soc., 66, 2, 240-256, (2002) · Zbl 1013.47032  Chidume, C.E.; Ali, Bashir, Convergence theorems for common fixed points for finite families of nonexpansive mappings in reflexive Banach spaces, Nonlinear anal., 68, 3410-3418, (2008) · Zbl 1223.47074
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