A Strong Convergence Theorem for an Iterative Method for Finding Zeros of Maximal Monotone Maps with Applications to Convex Minimization and Variational Inequality Problems

Abstract

LetEbe a uniformly convex and uniformly smooth real Banach space, and letE* be its dual. LetA : E→ 2E*be a bounded maximal monotone map. Assume thatA−1(0) ≠ Ø. A new iterative sequence is constructed which convergesstronglyto an element ofA−1(0). The theorem proved complements results obtained on strong convergence ofthe proximal point algorithmfor approximating an element ofA−1(0) (assuming existence) and also resolves an important open question. Furthermore, this result is applied to convex optimization problems and to variational inequality problems. These results are achieved by combining a theorem of Reich on the strong convergence of the resolvent of maximal monotone mappings in a uniformly smooth real Banach space and new geometric properties of uniformly convex and uniformly smooth real Banach spaces introduced by Alber, with a technique of proof which is also of independent interest.

A hybrid inertial algorithm for approximating solution of convex feasibility problems with applications

“…Consequently, approximating solutions of (4.2) is equivalent to approximating J-fixed points of maps T : X → 2 X * defined by T := J -A. This connection is now generating considerable research interest in the study of J-fixed points (see, e.g., Chidume and Idu [11], Chidume and Monday [13], Chidume et al [15,16], and the references contained in them). This notion turns out to be very useful and applicable in approximating solutions of Eq.…”

Iterative Approximation of Solutions of Hammerstein Integral Equations with Maximal Monotone Operators in Banach Spaces

“…Let W : X × X → R be defined by W (x, y) = 1 2 φ(y, x). Then, for all x, y, z ∈ X Lemma 2.8 (Chidume et al, [30]). From Lemma 2.4, setting λn := 1 θn where θn → 0 as n → ∞,…”

New algorithms for approximating zeros of inverse strongly monotone maps and J-fixed points

“…Observe that at equilibrium, u is independent of time so that the equation reduces to Approximation of solutions of Eq. (1.1) has been studied extensively by various authors (see, e.g., Aoyama et al [4], Blum and Oettli [6], Censor, Gibali, Reich and Sabach [12], Censor, Gibali and Reich [9][10][11], Chidume [14], Chidume et al [15,16,18,23,25,26], Gibali, Reich and Zalas [27], Iiduka and Takahashi [29], Iiduka et al [31], Kassay, Reich and Sabach [33], Kinderlehrer and Stampacchia [34], Lions and Stampacchia [36], Liu [37], Liu and Nashed [38], Ofoedu and Malonza [43], Osilike et al [44], Reich and Sabach [48], Reich [46], Rockafellar [49], Su and Xu [51], Zegeye et al [58], Zegeye and Shahzad [57], and the references therein).…”