Extended Newton-type Method for Nonsmooth Generalized Equation under n,α-point-based Approximation
Let X and Y be Banach spaces and Ω⊆X. Let f:Ω⟶Y be a single valued function which is nonsmooth. Suppose that F:X⇉2Y is a set-valued mapping which has closed graph. In the present paper, we study the extended Newton-type method for solving the nonsmooth generalized equation 0∈fx+Fx and analyze its se...
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Wiley
2022-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2022/7108996 |
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| author | M. Z. Khaton M. H. Rashid |
| author_facet | M. Z. Khaton M. H. Rashid |
| author_sort | M. Z. Khaton |
| collection | DOAJ |
| description | Let X and Y be Banach spaces and Ω⊆X. Let f:Ω⟶Y be a single valued function which is nonsmooth. Suppose that F:X⇉2Y is a set-valued mapping which has closed graph. In the present paper, we study the extended Newton-type method for solving the nonsmooth generalized equation 0∈fx+Fx and analyze its semilocal and local convergence under the conditions that f+F−1 is Lipschitz-like and f admits a certain type of approximation which generalizes the concept of point-based approximation so-called n,α-point-based approximation. Applications of n,α-point-based approximation are provided for smooth functions in the cases n=1 and n=2 as well as for normal maps. In particular, when 0<α<1 and the derivative of f, denoted ∇f, is ℓ,α-Hölder continuous, we have shown that f admits 1,α-point-based approximation for n=1 while f admits 2,α-point-based approximation for n=2, when 0<α<1 and the second derivative of f, denoted ∇2f, is K,α-Hölder. Moreover, we have constructed an n,α-point-based approximation for the normal maps fC+F when f has an n,α-point-based approximation. Finally, a numerical experiment is provided to validate the theoretical result of this study. |
| format | Article |
| id | doaj-art-de97a18a380f4db78700680d801139c1 |
| institution | Kabale University |
| issn | 1687-0425 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-de97a18a380f4db78700680d801139c12025-02-03T06:12:27ZengWileyInternational Journal of Mathematics and Mathematical Sciences1687-04252022-01-01202210.1155/2022/7108996Extended Newton-type Method for Nonsmooth Generalized Equation under n,α-point-based ApproximationM. Z. Khaton0M. H. Rashid1Department of MathematicsDepartment of MathematicsLet X and Y be Banach spaces and Ω⊆X. Let f:Ω⟶Y be a single valued function which is nonsmooth. Suppose that F:X⇉2Y is a set-valued mapping which has closed graph. In the present paper, we study the extended Newton-type method for solving the nonsmooth generalized equation 0∈fx+Fx and analyze its semilocal and local convergence under the conditions that f+F−1 is Lipschitz-like and f admits a certain type of approximation which generalizes the concept of point-based approximation so-called n,α-point-based approximation. Applications of n,α-point-based approximation are provided for smooth functions in the cases n=1 and n=2 as well as for normal maps. In particular, when 0<α<1 and the derivative of f, denoted ∇f, is ℓ,α-Hölder continuous, we have shown that f admits 1,α-point-based approximation for n=1 while f admits 2,α-point-based approximation for n=2, when 0<α<1 and the second derivative of f, denoted ∇2f, is K,α-Hölder. Moreover, we have constructed an n,α-point-based approximation for the normal maps fC+F when f has an n,α-point-based approximation. Finally, a numerical experiment is provided to validate the theoretical result of this study.http://dx.doi.org/10.1155/2022/7108996 |
| spellingShingle | M. Z. Khaton M. H. Rashid Extended Newton-type Method for Nonsmooth Generalized Equation under n,α-point-based Approximation International Journal of Mathematics and Mathematical Sciences |
| title | Extended Newton-type Method for Nonsmooth Generalized Equation under n,α-point-based Approximation |
| title_full | Extended Newton-type Method for Nonsmooth Generalized Equation under n,α-point-based Approximation |
| title_fullStr | Extended Newton-type Method for Nonsmooth Generalized Equation under n,α-point-based Approximation |
| title_full_unstemmed | Extended Newton-type Method for Nonsmooth Generalized Equation under n,α-point-based Approximation |
| title_short | Extended Newton-type Method for Nonsmooth Generalized Equation under n,α-point-based Approximation |
| title_sort | extended newton type method for nonsmooth generalized equation under n α point based approximation |
| url | http://dx.doi.org/10.1155/2022/7108996 |
| work_keys_str_mv | AT mzkhaton extendednewtontypemethodfornonsmoothgeneralizedequationundernapointbasedapproximation AT mhrashid extendednewtontypemethodfornonsmoothgeneralizedequationundernapointbasedapproximation |