Strong Convergence Theorems for Zeros of Bounded Maximal Monotone Nonlinear Operators

Abstract

An iteration process studied by Chidume and Zegeye 2002 is proved to convergestronglyto a solution of the equationAu=0whereAis a boundedm-accretive operator on certain real Banach spacesEthat includeLpspaces2≤p<∞.The iteration process does not involve the computation of the resolvent at any step of the process and does not involve the projection of an initial vector onto the intersection of two convex subsets ofE, setbacks associated with the classicalproximal point algorithmof Martinet 1970, Rockafellar 1976 and its modifications by various authors for approximating of a solution of this equation. The ideas of the iteration process are applied to approximate fixed points of uniformly continuous pseudocontractive maps.

Iterative algorithm for computing fixed points of demicontractive and zeros points of multivalued accretive operators in certain Banach spaces with application

“…Remark 3.5. Many already studied problems in the literature can be considered as special cases of this paper; see, for example, [20,26,8,21,25,28,14] and the references therein.…”

“…x n+1 = J rn (x n + e n ), ∀ n ≥ 0, where J r = (I+rA) −1 for all r > 0, is the resolvent of A and {e n } is an error sequence in a Hilbert space. In the recent years, inclusion problem (1) in real Hilbert spaces, Banach spaces and complete CAT(0) (Hadamard) spaces have been intensively studied by many authors; see, for example, [8,11,16,20,29,30,3,23,15,2] and the references therein.…”

A new iterative algorithm for solving some nonlinear problems in Hilbert spaces

“…In response to Q2, several works have been done (see, e.g., Güler [8], Kamimura and Takahashi [10], Chidume and Djitte [5] and the references therein). In the recent years, the problem of finding a common element of the set of solutions of various convex minimization problems and the set of fixed points for nonlinear mapping in the framework of Hilbert spaces and Banach spaces have been intensively studied by many authors.…”

New semi-implicit midpoint rule for zero of monotone mappings in Banach spaces

“…However, we have to acknowledge that the technique of converting the inclusion 0 ∈ Au into a fixed point problem for (I − A) : E → E is not applicable in Banach spaces since, in this case when A is monotone, A maps E into E * and the identity map does not make sense (see, e.g., Chidume [20], Xu [21], Sina [6] and other authors). Consequently, algorithms for approximating solutions of equations 0 ∈ Au when A : E → E * is of monotone type in Banach spaces has been focused and explored by increasing authors (see, e.g., Zegeye [22], Ibaraki [23]).…”