Weak convergence of an iterative sequence for accretive operators in Banach spaces.

*(English)*Zbl 1128.47056The authors study the generalized variational inequality problem of finding \(u \in C\) such that
\[
\langle Au, J(v-u) \rangle \geq 0\;\forall v \in C,
\]
where \(C\) is a nonempty closed convex subset of a smooth Banach space \(E\), \(A\) is an accretive operator of \(C\) into \(E\), \(J\) is the duality mapping of \(E\) into \(E^*\), and \(\langle \cdot, \cdot \rangle\) is the duality paring between \(E\) and \(E^*\). To solve this problem, the authors propose the following iterative scheme: \(x_1 =x \in C\) and
\[
x_{n+1}=\alpha_{n}x_n +(1-\alpha_{n})Q_{C}(x_n -\lambda_{n}Ax_n)
\]
for \(n=1, 2, 3,\dots\), where \(Q_C\) is a sunny nonexpansive retraction from \(E\) onto \(C\), \(\{\alpha_n\}\) is a sequence in \([0,1]\), and \(\{\lambda_n\}\) is a sequence of real numbers.

For this iterative scheme, the authors establish a weak convergence result (Theorem 3.1) in a uniformly convex and \(2\)-uniformly smooth Banach space for an \(\alpha\)-inverse strongly accretive operator. Applications to finding a zero point of an inverse strongly accretive operator and to finding a fixed point of a strictly pseudocontractive mapping are given.

For this iterative scheme, the authors establish a weak convergence result (Theorem 3.1) in a uniformly convex and \(2\)-uniformly smooth Banach space for an \(\alpha\)-inverse strongly accretive operator. Applications to finding a zero point of an inverse strongly accretive operator and to finding a fixed point of a strictly pseudocontractive mapping are given.

Reviewer: Jen-Chih Yao (Kaohsiung)

##### MSC:

47J25 | Iterative procedures involving nonlinear operators |

47H10 | Fixed-point theorems |

47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |

##### References:

[1] | Ball, K; Carlen, EA; Lieb, EH, Sharp uniform convexity and smoothness inequalities for trace norms, Inventiones Mathematicae, 115, 463-482, (1994) · Zbl 0803.47037 |

[2] | Beauzamy B: Introduction to Banach Spaces and Their Geometry, North-Holland Mathematics Studies. Volume 68. 2nd edition. North-Holland, Amsterdam; 1985:xv+338. |

[3] | Brezis H: Analyse Fonctionnelle. Théorie et Applications, Collection of Applied Mathematics for the Master’s Degree. Masson, Paris; 1983:xiv+234. |

[4] | Browder, FE, Nonlinear operators and nonlinear equations of evolution in Banach spaces, 1-308, (1976), Rhode Island |

[5] | Browder, FE; Petryshyn, WV, Construction of fixed points of nonlinear mappings in Hilbert space, Journal of Mathematical Analysis and Applications, 20, 197-228, (1967) · Zbl 0153.45701 |

[6] | Bruck, RE, Nonexpansive retracts of Banach spaces, Bulletin of the American Mathematical Society, 76, 384-386, (1970) · Zbl 0224.47034 |

[7] | Bruck, RE, A simple proof of the Mean ergodic theorem for nonlinear contractions in Banach spaces, Israel Journal of Mathematics, 32, 107-116, (1979) · Zbl 0423.47024 |

[8] | Gol’shteĭn, EG; Tret’yakov, NV, Modified Lagrangians in convex programming and their generalizations, 86-97, (1979) · Zbl 0404.90069 |

[9] | Iiduka H, Takahashi W: Strong convergence of a projection algorithm by hybrid type for monotone variational inequalities in a Banach space. in preparation · Zbl 1220.47095 |

[10] | Iiduka H, Takahashi W: Weak convergence of a projection algorithm for variational inequalities in a Banach space. in preparation · Zbl 1129.49012 |

[11] | Iiduka, H; Takahashi, W; Toyoda, M, Approximation of solutions of variational inequalities for monotone mappings, Panamerican Mathematical Journal, 14, 49-61, (2004) · Zbl 1060.49006 |

[12] | Kamimura, S; Takahashi, W, Weak and strong convergence of solutions to accretive operator inclusions and applications, Set-Valued Analysis, 8, 361-374, (2000) · Zbl 0981.47036 |

[13] | Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics. Volume 88. Academic Press, New York; 1980:xiv+313. · Zbl 0988.49003 |

[14] | Kirk, WA, A fixed point theorem for mappings which do not increase distances, The American Mathematical Monthly, 72, 1004-1006, (1965) · Zbl 0141.32402 |

[15] | Kitahara, S; Takahashi, W, Image recovery by convex combinations of sunny nonexpansive retractions, Topological Methods in Nonlinear Analysis, 2, 333-342, (1993) · Zbl 0815.47068 |

[16] | Lau, AT; Takahashi, W, Weak convergence and nonlinear ergodic theorems for reversible semigroups of nonexpansive mappings, Pacific Journal of Mathematics, 126, 277-294, (1987) · Zbl 0587.47058 |

[17] | Lions, J-L; Stampacchia, G, Variational inequalities, Communications on Pure and Applied Mathematics, 20, 493-519, (1967) · Zbl 0152.34601 |

[18] | Liu, F; Nashed, MZ, Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Analysis, 6, 313-344, (1998) · Zbl 0924.49009 |

[19] | Osilike, MO; Udomene, A, Demiclosedness principle and convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type, Journal of Mathematical Analysis and Applications, 256, 431-445, (2001) · Zbl 1009.47067 |

[20] | Reich, S, Asymptotic behavior of contractions in Banach spaces, Journal of Mathematical Analysis and Applications, 44, 57-70, (1973) · Zbl 0275.47034 |

[21] | Reich, S, Weak convergence theorems for nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications, 67, 274-276, (1979) · Zbl 0423.47026 |

[22] | Takahashi W: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama; 2000:iv+276. · Zbl 0997.47002 |

[23] | Takahashi, Y; Hashimoto, K; Kato, M, On sharp uniform convexity, smoothness, and strong type, cotype inequalities, Journal of Nonlinear and Convex Analysis, 3, 267-281, (2002) · Zbl 1030.46012 |

[24] | Takahashi, W; Kim, G-E, Approximating fixed points of nonexpansive mappings in Banach spaces, Mathematica Japonica, 48, 1-9, (1998) · Zbl 0913.47056 |

[25] | Xu, HK, Inequalities in Banach spaces with applications, Nonlinear Analysis, 16, 1127-1138, (1991) · Zbl 0757.46033 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.