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Strong convergence of an iterative algorithm for nonself multimaps in Banach spaces. (English) Zbl 1222.47092

Summary: Let \(E\) be a uniformly convex Banach space having a uniformly Gâteaux differentiable norm, \(D\) a nonempty closed convex subset of \(E\), and \(T:D\rightarrow K(E)\) a nonself multimap such that \(F(T)\neq \varnothing\) and \(P_T\) is nonexpansive, where \(F(T)\) is the fixed point set of \(T\), \(K(E)\) is the family of nonempty compact subsets of \(E\) and \(P_T(x)=\{u_x\in T_X: \|x-u_x\| = d(x,Tx)\}\). Suppose that \(D\) is a nonexpansive retract of \(E\) and that, for each \(v\in D\) and \(t\in (0,1)\), the contraction \(S_t\) defined by \(S_tx=tP_Tx+(1 - t)v\) has a fixed point \(x_t\in D\). Let \(\{\alpha _n\},\{\beta _n\}\) and \(\{\gamma _n\}\) be three real sequences in (0,1) satisfying approximate conditions. Then, for fixed \(u\in D\) and arbitrary \(x_{0}\in D\), the sequence \(\{x_n\}\) generated by
\[ x_n\in \alpha _nu+\beta _nx_{n-1}+\gamma _nP_T(x_n), \quad n\geq 0, \]
converges strongly to a fixed point of \(T\).

MSC:

47J25 Iterative procedures involving nonlinear operators
47H05 Monotone operators and generalizations
47H10 Fixed-point theorems
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[1] Ciorenescu, I., Geometry of Banach spaces duality mapping and nonlinear problems, (1990), Kluwer Academic Publishers
[2] Xu, H.K.; Yin, X.M., Strong convergence theorems for nonexpansive non-self-mappings, Nonlinear anal., 24, 223-228, (1995) · Zbl 0826.47038
[3] Xu, H.K., On weakly nonexpansive and \(\ast\)-nonexpansive multivalued mappings, Math. japonica, 36, 441-445, (1991) · Zbl 0733.54010
[4] Jung, J.S.; Kim, S.S., Strong convergence theorems for nonexpansive nonself-mappings in Banach spaces, Nonlinear anal., 33, 321-329, (1998) · Zbl 0988.47033
[5] Jung, J.S.; Kim, T.H., Strong convergence of approximating fixed points for nonexpansive nonself-mappings in Banach spaces, Kodai math. J., 21, 259-272, (1998) · Zbl 0928.47040
[6] Ceng, L.C.; Cubiotti, P.; Yao, J.C., Strong convergence theorems for finite many nonexpansive mappings and applications, Nonlinear anal., 67, 1464-1473, (2007) · Zbl 1123.47044
[7] Kim, T.H.; Jung, J.S., Approximating fixed points of nonlinear mappings in Banach spaces, in: Proceedings of workshop on fixed point theory (kazimierz dolny, 1997), Ann. univ. mariae Curie-sklodowska sect. A, 51, 149-165, (1997)
[8] Zeng, L.C.; Yao, J.C., Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Nonlinear anal., 64, 2507-2515, (2006) · Zbl 1105.47061
[9] Ceng, L.C.; Xu, H.K., Strong convergence of a hybrid viscosity approximation method with perturbed mappings for nonexpansive and accretive operators, Taiwanese J. math., 11, 661-682, (2007) · Zbl 1219.47102
[10] Shahzad, N.; Zegeye, H., Strong convergence results for nonself multimaps in Banach spaces, Proc. amer. math. soc., 136, 539-548, (2008) · Zbl 1135.47054
[11] Deimling, K., Zeros of accretive operators, Manuscripta math., 13, 365-374, (1974) · Zbl 0288.47047
[12] Takahashi, W.; Jeong, D.H., Fixed point theorem for nonexpansive semigroups on Banach spaces, Proc. amer. math. soc., 122, 1175-1179, (1994) · Zbl 0818.47055
[13] Rafiq, A., On Mann iteration in Hilbert spaces, Nonlinear anal., 66, 2230-2236, (2007) · Zbl 1136.47047
[14] Yao, Y.; Liou, Y.C.; Chen, R., Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces, Nonlinear anal., 67, 3311-3317, (2007) · Zbl 1129.47059
[15] Ceng, L.C.; Xu, H.K.; Yao, J.C., Strong convergence of an iterative method with perturbed mappings for nonexpansive and accretive operators, Numer. funct. anal. optim., 29, 324-345, (2008) · Zbl 1140.47050
[16] Mann, W.R., Mean value methods in iteration, Proc. amer. math. soc., 4, 506-610, (1953) · Zbl 0050.11603
[17] Nadler, S., Multivalued contraction mappings, Pacific J. math., 30, 475-488, (1969) · Zbl 0187.45002
[18] Jung, J.S., Strong convergence theorems for multivalued nonexpansive nonself-mappings in Banach spaces, Nonlinear anal., 66, 2345-2354, (2006) · Zbl 1123.47047
[19] Lim, T.C., A fixed point theorem for weakly inward multivalued contractions, J. math. anal. appl., 247, 323-327, (2000) · Zbl 0957.47040
[20] Sahu, D.R., Strong convergence theorems for nonexpansive type and non-self multi-valued mappings, Nonlinear anal., 37, 401-407, (1999) · Zbl 0938.47039
[21] Deimling, K., Multivalued differential equations, (1992), Walter de Gruyter Berlin · Zbl 0760.34002
[22] Hussain, T.; Latif, A., Fixed points of multivalued nonexpansive maps, Math. japonica, 33, 385-391, (1988) · Zbl 0667.47028
[23] Ceng, L.C.; Ansari, Q.H.; Yao, J.C., Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces, Numer. funct. anal. optim., 29, 987-1033, (2008) · Zbl 1163.49002
[24] Xu, H.K., Multivalued nonexpansive mappings in Banach spaces, Nonlinear anal., 43, 693-706, (2001) · Zbl 0988.47034
[25] Shahzad, N.; Lone, A., Fixed points of multimaps which are not necessarily nonexpansive, Fixed point theory appl, 2005, 2, 169-176, (2005) · Zbl 1112.47050
[26] Barbu, V.; Precupanu, Th., Convexity and optimization in Banach spaces, (1978), Editura Academiei R. S. R. Bucharest · Zbl 0379.49010
[27] Lopez Acedo, G.; Xu, H.K., Remarks on multivalued nonexpansive mappings, Soochow J. math., 21, 107-115, (1995) · Zbl 0826.47037
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