Abstract
We prove strong and weak convergence theorems for a new resolvent of maximal monotone operators in a Banach space and give an estimate of the convergence rate of the algorithm. Finally, we apply our convergence theorem to the convex minimization problem. The result present in this paper extend and improve the corresponding result of Ibaraki and Takahashi (2007), and Kim and Xu (2005).
Highlights
Let E be a Banach space with norm ·, let E∗ denote the dual of E, and let x, f denote the value of f ∈ E∗ at x ∈ E
We first prove a strong convergence theorem for the algorithm 1.7 which extends the previous result of Ibaraki and Takahashi 7 and we prove a weak convergence theorem for algorithm 1.7 under different conditions on data, respectively
Let {xn} be a sequence generated by x1 x ∈ E and yn αnxn 1 − αn Jrn xn, 3.1 xn 1 βnx 1 − βn yn, for every n 1, 2, . . . , where {αn}, {βn} ⊂ 0, 1, {rn} ⊂ 0, ∞ satisfy limn → ∞αn 0, limn → ∞βn
Summary
Let E be a Banach space with norm · , let E∗ denote the dual of E, and let x, f denote the value of f ∈ E∗ at x ∈ E. They showed that the algorithm 1.2 converges strongly to some element of T −10 and the algorithm 1.3 converges weakly to some element of T −10 provided that the sequences {αn} and {rn} of real numbers are chosen appropriately These results extend the Kamimura and Takahashi 4 results in Hilbert spaces to those in Banach spaces. 1.6 where Jr I rBJ −1, J is the duality mapping of E, and B ⊂ E∗ × E is maximal monotone They proved that Algorithm 1.5 converges strongly to some element of BJ −10 and Algorithm 1.6 converges weakly to some element of BJ −10 provided that the sequences {αn} and {rn} of real numbers are chosen appropriately. By using our main result, we consider the problem of finding minimizes of convex functions defined on Banach spaces
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