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Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces. (English) Zbl 0933.47040
Consider a map \(T : K \to K\), where \(K\) is a bounded closed convex subset of a Banach space \(X\). For \(x_0 \in K\) the Ishikawa iteration is defined by \[ x_{n+1} = (1-\alpha_n)x_n + \alpha_n T y_n,\quad y_n = (1-\beta_n)x_n + \beta_n T x_n, \quad n=0,1,2,\ldots, \] where \(\{ \alpha_n \}\), \(\{ \beta_n \}\) are two real sequences in \([0,1]\). Let the set \(F(T)\) of fixed points of \(T\) in \(K\) be nonempty. For a strongly pseudo-contractive \(T\) the authors prove the convergence of the Ishikawa iteration to a fixed point under certain assumptions on \(\alpha_n,\;\beta_n\) and a smoothness condition on \(T\) respectively \(X\). There are also results for strongly accretive \(T\) and for the Mann iteration \[ x_{n+1} = (1-\alpha_n)x_n + \alpha_n T x_n, \quad n=0,1,2,\ldots . \]

MSC:
47J25 Iterative procedures involving nonlinear operators
65J15 Numerical solutions to equations with nonlinear operators
47H06 Nonlinear accretive operators, dissipative operators, etc.
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[1] Asplund, E., Positivity of duality mappings, Bull. amer. math. soc., 73, 200-203, (1967) · Zbl 0149.36202
[2] Browder, F.E., Nonlinear mappings of nonexpansive and accretive type in Banach spaces, Bull. amer. math. soc., 73, 875-882, (1967) · Zbl 0176.45302
[3] Chang, S.S., On Chidume’s open questions and approximate solutions of multivalued strongly accretive mapping equations in Banach spaces, J. math. anal. appl., 216, 94-111, (1997) · Zbl 0909.47049
[4] Chidume, C.E., An iterative process for nonlinear Lipschitzian strongly accretive mappings inL_{p}spaces, J. math. anal. appl., 151, 453-461, (1990) · Zbl 0724.65058
[5] Chidume, C.E., Approximation of fixed points of strongly pseudo-contractive mappings, Proc. amer. math. soc., 120, 545-551, (1994) · Zbl 0802.47058
[6] Chidume, C.E., Iterative solutions of nonlinear equations with strongly accretive operators, J. math. anal. appl., 192, 502-518, (1995) · Zbl 0868.47040
[7] Deimling, K., Zeros of accretive operators, Manuscripta math., 13, 283-288, (1994)
[8] Deng, L., On Chidume’s open questions, J. math. anal. appl., 174, 441-449, (1993) · Zbl 0784.47051
[9] Deng, L., An iterative process for nonlinear Lipschitzian and strongly accretive mappings in uniformly convex and uniformly smooth Banach spaces, Acta appl. math., 32, 183-196, (1993) · Zbl 0801.47040
[10] Deng, L.; Ding, X.P., Iterative approximation of Lipschitz strictly pseudo-contractive mappings in uniformly smooth Banach spaces, Nonlinear anal. TMA, 24, 981-987, (1995) · Zbl 0827.47041
[11] Ishikawa, S., Fixed points by a new iteration method, Proc. amer. math. soc., 149, 147-150, (1974) · Zbl 0286.47036
[12] Ishikawa, S., Fixed point and iteration of a nonexpansive mapping in a Banach spaces, Proc. amer. math. soc., 73, 65-71, (1976) · Zbl 0352.47024
[13] Kato, T., Nonlinear semigroups and evolution equations, J. math. soc. Japan, 18/19, 508-520, (1967) · Zbl 0163.38303
[14] Mann, W.R., Mean value methods in iteration, Proc. amer. math. soc., 4, 506-510, (1953) · Zbl 0050.11603
[15] Morales, C., Pseudo-contractive mappings and leray – schauder boundary condition, Comment. math. univ. carolinea, 20, 745-746, (1979) · Zbl 0429.47021
[16] Reich, S., An iterative procedure for construction zeros of accretive sets in Banach spaces, Nonlinear anal. TMA, 2, 85-92, (1978) · Zbl 0375.47032
[17] Rhoades, B.E., Comments on two fixed point iteration methods, J. math. anal. appl., 56, 741-750, (1976) · Zbl 0353.47029
[18] Tan, K.K.; Xu, H.K., Iterative solution to nonlinear equations and strongly accretive operators in Banach spaces, J. math. anal. appl., 178, 9-21, (1993) · Zbl 0834.47048
[19] Weng, X.L., Fixed point iteration for local strictly pseudo-contractive mapping, Proc. amer. math. soc., 113, 727-731, (1991) · Zbl 0734.47042
[20] Xu, H.K., A note on the Ishikawa iteration scheme, J. math. anal. appl., 167, 582-587, (1992) · Zbl 0776.47042
[21] Xu, Z.B.; Roach, G.F., Characteristic inequalities in uniformly convex and uniformly smooth Banach spaces, J. math. anal. appl., 157, 189-210, (1991) · Zbl 0757.46034
[22] Zhou, H.Y.; Jia, Y.T., Approximation of fixed points of strongly pseudo-contractive maps without Lipschitz assumption, Proc. amer. math. soc., 125, 1705-1709, (1997) · Zbl 0871.47045
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