Abstract
AbstractIn this article, we introduce composite iterative schemes for finding a zero point of a finite family of maximal monotone operators in a reflexive Banach space. Then, we prove strong convergence theorems by using a shrinking projection method. Moreover, we also apply our results to a system of convex minimization problems in reflexive Banach spaces.AMS Subject Classification: 47H09, 47H10
Highlights
Let E be a real Banach space and C a nonempty subset of E
A set-valued mapping A is said to be monotone if 〈x* - y*, x - y〉 ≥ 0 whenever (x, x*); (y, y*) Î G(A)
It is said to be maximal monotone if its graph is not contained in the graph of any other monotone operator on E
Summary
Let E be a real Banach space and C a nonempty subset of E. In 2005, Kohsaka and Takahashi [10] studied the above iteration process in a more general setting, reflexive Banach spaces Those authors proposed the following algorithm: xn+1 = ∇f ∗ αn∇f (xn) + (1 − αn)∇f (Jλnxn) , ∀n ≥ 1,. In 2010, Reich and Sabach [11] proposed an algorithm for finding a zero point of maximal monotone operators Ai : E ® 2E* (i = 1, 2,..., N) in a general reflexive Banach space E as follows:. The function f is said to be totally convex on bounded sets if νf (B, t) is positive for any nonempty bounded subset B of E and t >0, where the modulus of total convexity of the function f on the set B is the function νf : int dom f × [0, +∞) ® [0, +∞] defined by νf (B, t) := inf νf (x, t) : x ∈ B ∩ dom f. Df (p, ResfλA(x)) + Df (ResfλA(x), x) ≤ Df (p, x) for all l > 0, p Î A -1(0*) and x Î E
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