## Approximation of fixed points of strongly pseudocontractive mappings.(English)Zbl 0802.47058

Summary: Let $$E$$ be a real Banach space with a uniformly convex dual, and let $$K$$ be a nonempty closed convex and bounded subset of $$E$$. Let $$T: K\to K$$ be a continuous strongly pseudocontractive mapping of $$K$$ into itself. Let $$\{c_ n\}^ \infty_{n=1}$$ be a real sequence satisfying:
(i) $$0<c_ n <1$$ for all $$n\geq 1$$;
(ii) $$\sum_{n=1}^ \infty c_ n= \infty$$; and
(iii) $$\sum_{n=1}^ \infty c_ n b(c_ n) <\infty$$, where $$b: [0,\infty)\to [0,\infty)$$ is some continuous nondecreasing function satisfying $$b(0)=0$$, $$b(ct)\leq cb(t)$$ for all $$c\geq 1$$.
Then the sequence $$\{x_ n\}^ \infty_{n=1}$$ generated by $$x_ 1\in K$$, $x_{n+1}= (1- c_ n) x_ n+ c_ n Tx_ n, \qquad n\geq 1,$ converges strongly to the unique fixed point of $$T$$. A related result deals with the Ishikawa iteration scheme when $$T$$ is Lipschitzian and strongly pseudocontractive.

### MSC:

 47J25 Iterative procedures involving nonlinear operators
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### References:

 [1] Nadim A. Assad and W. A. Kirk, Fixed point theorems for set-valued mappings of contractive type, Pacific J. Math. 43 (1972), 553 – 562. · Zbl 0239.54032 [2] J. Bogin, On strict pseudo-contractions and a fixed point theorem, Technion Preprint Series No. MT-219, Haifa, Israel, 1974. [3] Felix E. Browder, The solvability of non-linear functional equations, Duke Math. J. 30 (1963), 557 – 566. · Zbl 0119.32503 [4] Felix E. Browder, Nonlinear monotone and accretive operators in Banach spaces, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 388 – 393. · Zbl 0167.15205 [5] Felix E. Browder, Nonlinear mappings of nonexpansive and accretive type in Banach spaces., Bull. Amer. Math. Soc. 73 (1967), 875 – 882. · Zbl 0176.45302 [6] F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197 – 228. · Zbl 0153.45701 [7] C. E. Chidume, Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings, Proc. Amer. Math. Soc. 99 (1987), no. 2, 283 – 288. · Zbl 0646.47037 [8] C. E. Chidume, Iterative solution of nonlinear equations of the monotone type in Banach spaces, Bull. Austral. Math. Soc. 42 (1990), no. 1, 21 – 31. · Zbl 0703.47047 [9] Klaus Deimling, Zeros of accretive operators, Manuscripta Math. 13 (1974), 365 – 374. · Zbl 0288.47047 [10] Juan A. Gatica and W. A. Kirk, Fixed point theorems for Lipschitzian pseudo-contractive mappings, Proc. Amer. Math. Soc. 36 (1972), 111 – 115. · Zbl 0254.47076 [11] J. Gwinner, On the convergence of some iteration processes in uniformly convex Banach spaces, Proc. Amer. Math. Soc. 71 (1978), no. 1, 29 – 35. · Zbl 0393.47040 [12] Troy L. Hicks and John D. Kubicek, On the Mann iteration process in a Hilbert space, J. Math. Anal. Appl. 59 (1977), no. 3, 498 – 504. · Zbl 0361.65057 [13] Shiro Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147 – 150. · Zbl 0286.47036 [14] Shiro Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976), no. 1, 65 – 71. · Zbl 0352.47024 [15] Tosio Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508 – 520. · Zbl 0163.38303 [16] W. A. Kirk, A fixed point theorem for local pseudocontractions in uniformly convex spaces, Manuscripta Math. 30 (1979/80), no. 1, 89 – 102. · Zbl 0422.47032 [17] W. A. Kirk, Remarks on pseudo-contractive mappings, Proc. Amer. Math. Soc. 25 (1970), 820 – 823. · Zbl 0203.14603 [18] W. A. Kirk and Claudio Morales, On the approximation of fixed points of locally nonexpansive mappings, Canad. Math. Bull. 24 (1981), no. 4, 441 – 445. · Zbl 0475.47041 [19] W. Robert Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506 – 510. · Zbl 0050.11603 [20] R. H. Martin Jr., A global existence theorem for autonomous differential equations in a Banach space, Proc. Amer. Math. Soc. 26 (1970), 307 – 314. · Zbl 0202.10103 [21] Claudio H. Morales, Surjectivity theorems for multivalued mappings of accretive type, Comment. Math. Univ. Carolin. 26 (1985), no. 2, 397 – 413. · Zbl 0595.47041 [22] R. N. Mukerjee, Construction of fixed points of strictly pseudocontractive mappings in generalized Hilbert spaces and related applications, Indian J. Pure Appl. Math. 15 (1966), 276-284. [23] Olavi Nevanlinna and Simeon Reich, Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces, Israel J. Math. 32 (1979), no. 1, 44 – 58. · Zbl 0427.47049 [24] Simeon Reich, An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear Anal. 2 (1978), no. 1, 85 – 92. · Zbl 0375.47032 [25] Simeon Reich, Constructing zeros of accretive operators, Applicable Anal. 8 (1978/79), no. 4, 349 – 352. , https://doi.org/10.1080/00036817908839243 Simeon Reich, Constructing zeros of accretive operators. II, Applicable Anal. 9 (1979), no. 3, 159 – 163. · Zbl 0424.47034 [26] Simeon Reich, Constructive techniques for accretive and monotone operators, Applied nonlinear analysis (Proc. Third Internat. Conf., Univ. Texas, Arlington, Tex., 1978) Academic Press, New York-London, 1979, pp. 335 – 345. [27] Simeon Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), no. 1, 287 – 292. · Zbl 0437.47047 [28] B. E. Rhoades, Comments on two fixed point iteration methods, J. Math. Anal. Appl. 56 (1976), no. 3, 741 – 750. · Zbl 0353.47029
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