A Strong Convergence Theorem For Zeros Of Bounded Maximal Monotone Mappings In Banach Spaces With Applications

ABSTRACT

Let E be a uniformly convex and uniformly smooth real Banach space and E ∗ be its dual. Let A : E → 2 E∗ be a bounded maximal monotone map. Assume that A−1 (0) 6= ∅. A new iterative sequence is constructed which converges strongly to an element of A−1 (0). The theorem proved, complements results obtained on strong convergence of the proximal point algorithm for approximating an element of A−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 Riech on the strong convergence of the resolvent of maximal monotone mappings in a uniformly smooth real Banach space; 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.

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APA

OLUNNA., U (2021). A Strong Convergence Theorem For Zeros Of Bounded Maximal Monotone Mappings In Banach Spaces With Applications. Afribary. Retrieved from https://afribary.com/works/a-strong-convergence-theorem-for-zeros-of-bounded-maximal-monotone-mappings-in-banach-spaces-with-applications

MLA 8th

OLUNNA., UBA "A Strong Convergence Theorem For Zeros Of Bounded Maximal Monotone Mappings In Banach Spaces With Applications" Afribary. Afribary, 13 Apr. 2021, https://afribary.com/works/a-strong-convergence-theorem-for-zeros-of-bounded-maximal-monotone-mappings-in-banach-spaces-with-applications. Accessed 14 Jun. 2024.

MLA7

OLUNNA., UBA . "A Strong Convergence Theorem For Zeros Of Bounded Maximal Monotone Mappings In Banach Spaces With Applications". Afribary, Afribary, 13 Apr. 2021. Web. 14 Jun. 2024. < https://afribary.com/works/a-strong-convergence-theorem-for-zeros-of-bounded-maximal-monotone-mappings-in-banach-spaces-with-applications >.

Chicago

OLUNNA., UBA . "A Strong Convergence Theorem For Zeros Of Bounded Maximal Monotone Mappings In Banach Spaces With Applications" Afribary (2021). Accessed June 14, 2024. https://afribary.com/works/a-strong-convergence-theorem-for-zeros-of-bounded-maximal-monotone-mappings-in-banach-spaces-with-applications