Abstract
We first introduce a modified proximal point algorithm for maximal monotone operators in a Banach space. Next, we obtain a strong convergence theorem for resolvents of maximal monotone operators in a Banach space which generalizes the previous result by Kamimura and Takahashi in a Hilbert space. Using this result, we deal with the convex minimization problem and the variational inequality problem in a Banach space.
Highlights
Let E be a real Banach space and let T ⊂ E × E∗ be a maximal monotone operator
Such a problem is connected with the convex minimization problem
Using the hybrid method in mathematical programming, Kamimura and Takahashi [9] obtained a strong convergence theorem for maximal monotone operators in a Banach space, which extended the result of Solodov and Svaiter [19] in a Hilbert space
Summary
Let E be a real Banach space and let T ⊂ E × E∗ be a maximal monotone operator. we study the problem of finding a point v ∈ E satisfying. Using the hybrid method in mathematical programming, Kamimura and Takahashi [9] obtained a strong convergence theorem for maximal monotone operators in a Banach space, which extended the result of Solodov and Svaiter [19] in a Hilbert space. In this paper, motivated by Censor and Reich [6], we introduce the following iterative sequence for a maximal monotone operator T ⊂ E × E∗ in a smooth and uniformly convex Banach space: x1 = x ∈ E and xn+1 = J−1 αnJx + 1 − αn JJrn xn We apply Theorem 3.3 to the convex minimization problem and the variational inequality problem
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
