Forward–backward splitting algorithm for fixed point problems and zeros of the sum of monotone operators
In this paper, we construct a forward–backward splitting algorithm for approximating a zero of the sum of an α -inverse strongly monotone operator and a maximal monotone operator. The strong convergence theorem is then proved under mild conditions. Then, we add a nonexpansive mapping in the algorith...
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| Published in: | Arabian Journal of Mathematics Vol. 9; no. 1; pp. 89 - 99 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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01.04.2020
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| ISSN: | 2193-5343, 2193-5351, 2193-5351 |
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| Abstract | In this paper, we construct a forward–backward splitting algorithm for approximating a zero of the sum of an
α
-inverse strongly monotone operator and a maximal monotone operator. The strong convergence theorem is then proved under mild conditions. Then, we add a nonexpansive mapping in the algorithm and prove that the generated sequence converges strongly to a common element of a fixed points set of a nonexpansive mapping and zero points set of the sum of monotone operators. We apply our main result both to equilibrium problems and convex programming. |
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| AbstractList | In this paper, we construct a forward-backward splitting algorithm for approximating a zero of the sum of an [alpha]-inverse strongly monotone operator and a maximal monotone operator. The strong convergence theorem is then proved under mild conditions. Then, we add a nonexpansive mapping in the algorithm and prove that the generated sequence converges strongly to a common element of a fixed points set of a nonexpansive mapping and zero points set of the sum of monotone operators. We apply our main result both to equilibrium problems and convex programming. In this paper, we construct a forward-backward splitting algorithm for approximating a zero of the sum of an [alpha]-inverse strongly monotone operator and a maximal monotone operator. The strong convergence theorem is then proved under mild conditions. Then, we add a nonexpansive mapping in the algorithm and prove that the generated sequence converges strongly to a common element of a fixed points set of a nonexpansive mapping and zero points set of the sum of monotone operators. We apply our main result both to equilibrium problems and convex programming. Mathematics Subject Classification 47H05 * 47H09 In this paper, we construct a forward–backward splitting algorithm for approximating a zero of the sum of an α -inverse strongly monotone operator and a maximal monotone operator. The strong convergence theorem is then proved under mild conditions. Then, we add a nonexpansive mapping in the algorithm and prove that the generated sequence converges strongly to a common element of a fixed points set of a nonexpansive mapping and zero points set of the sum of monotone operators. We apply our main result both to equilibrium problems and convex programming. In this paper, we construct a forward–backward splitting algorithm for approximating a zero of the sum of an α-inverse strongly monotone operator and a maximal monotone operator. The strong convergence theorem is then proved under mild conditions. Then, we add a nonexpansive mapping in the algorithm and prove that the generated sequence converges strongly to a common element of a fixed points set of a nonexpansive mapping and zero points set of the sum of monotone operators. We apply our main result both to equilibrium problems and convex programming. |
| Audience | Academic |
| Author | Postolache, Mihai Dadashi, Vahid |
| Author_xml | – sequence: 1 givenname: Vahid surname: Dadashi fullname: Dadashi, Vahid organization: Department of Mathematics, Sari Branch, Islamic Azad University – sequence: 2 givenname: Mihai surname: Postolache fullname: Postolache, Mihai email: mihai@mathem.pub.ro organization: China Medical University, Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Department of Mathematics and Computer Science, University Politehnica of Bucharest |
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| Cites_doi | 10.1186/1687-1812-2013-93 10.1155/2016/2371857 10.1007/978-94-009-2121-4 10.1007/s10898-011-9772-4 10.1007/s11590-010-0176-z 10.1017/S0004972700028884 10.1080/02331930801951116 10.1137/S0363012998339745 10.1016/j.cam.2008.06.011 10.1017/S000497271700017X 10.1016/S0362-546X(99)00136-4 10.1007/s10957-005-7564-z 10.23952/jnfa.2018.26 10.1007/s10898-012-0017-y 10.1023/A:1026592623460 10.1023/A:1022643914538 10.1007/s10957-011-9973-5 10.1007/s10957-017-1117-0 10.1016/S0377-0427(02)00906-8 10.1137/0314056 10.1080/02331934.2013.798320 10.1137/S0363012998338806 10.1006/jmaa.1995.1289 10.1007/s11590-011-0286-2 10.1007/s10957-007-9329-3 10.1155/FPTA.2005.103 10.1007/s10957-008-9370-x 10.1007/s10957-010-9713-2 10.1016/0362-546X(79)90083-X 10.1090/S0002-9939-2011-10883-9 10.1016/j.cam.2007.06.013 10.1017/CBO9780511526152 10.1016/0022-247X(79)90234-8 10.1007/s10483-008-0502-y 10.2140/pjm.1970.33.209 10.1016/j.na.2008.04.035 10.1016/j.jmaa.2006.06.067 10.1186/1687-1812-2014-75 10.1016/j.na.2008.04.019 10.1080/02331934.2017.1337764 |
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| Snippet | In this paper, we construct a forward–backward splitting algorithm for approximating a zero of the sum of an
α
-inverse strongly monotone operator and a... In this paper, we construct a forward-backward splitting algorithm for approximating a zero of the sum of an [alpha]-inverse strongly monotone operator and a... In this paper, we construct a forward–backward splitting algorithm for approximating a zero of the sum of an α-inverse strongly monotone operator and a maximal... |
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| SubjectTerms | Algorithms Computational geometry Convergence Convexity Fixed points (mathematics) Mapping Mathematical programming Mathematics Mathematics and Statistics Operators (mathematics) Splitting |
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| Title | Forward–backward splitting algorithm for fixed point problems and zeros of the sum of monotone operators |
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