Convergence of a hybrid iterative process for asymptotically quasi -nonexpansive mappings and generalized mixed variational $\Phi$-like inequality problems in Banach spaces
Main Article Content
Abstract
In this paper, we introduce a hybrid iterative process for approximating common elements of fixed points of asymptotically quasi $\phi$-nonexpansive mappings and solutions of generalized mixed variational-like inequality problems. After that, we prove a strong convergence result for the proposed iteration in Banach spaces. In addition, we give a numerical example to illustrate the convergence result of the proposed iteration.
Keywords
asymptotically unexpanded quasi $\phi$-mapping, generalized mixed variational-like inequality problem, hybrid iterative process
Article Details
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References
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Anh, P. K., & Hieu, D. V. (2015). Parallel and sequential hybrid methods for a finite family of asymptotically quasi-φ-nonexpansive mappings. J. Appl. Math. Comput, 48(1), 241-263. J. Appl. Math. Comput, 48(1), 241-263.
https://doi.org/10.1007/s12190-014-0801-6
Blum, E., & W. Oettli, W. (1994). From optimization and variational inequalities to equilibrium problems. Math. Stud.,63, 123-145.
Farid, M., Cholamjiak, W., Ali, R., & Kazmi, K. R. (2021). A new shrinking projection algorithm for a generalized mixed variational-like inequality problem and asymptotically quasi- -nonexpansive mapping in a Banach space. R. Acad. Cienc. Exactas Fi's. Nat. Ser. A Mat., 115(3), 114. https://doi.org/10.1016/j.na.2009.04.023
Hao, Y. (2013). Some results on a modified Mann iterative scheme in a reflexive Banach spaces. J. Fixed Point Theory Appl., 2013(227), 1-14.
https://doi.org/10.1186/1687-1812-2013-227
Kazmi, K. R., & Ali, R. (2019). Hybrid projection method for a system of unrelated generalized mixed variational-like inequality problems, Georgian Math. J., 26(1), 63-78.
https://doi.org/10.1515/gmj-2017-0027
Kamimura, S., & Takahashi, W. (2002) Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938-945
https://doi.org/10.1137/S105262340139611X
Muu, L.D., & Oettli, W. (1992). Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal., 18, 1159-1166
https://doi.org/10.1016/0362-546X(92)90159-C
Noor, M. A., & Oettli, W. (1994). On general nonlinear complementarity problems and quasi-equilibria. Le Matematiche (Catania), 49, 313-331.
Preda, V., Beldiman, M. & Bătătoresou, A. (2007). On Variational-like inequalities with generalized monotone mappings, in: Generalized Convexity and Related Topics, Lecture Notes in Econom. Math. Springer, Berlin, 415–431.
https://doi.org/10.1007/978-3-540-37007-9_25
Qin, X., Cho, Y.J., Kang, S.M., & Zhou, H. (2009). Convergence of a modified Halpern-type iteration algorithm for quasi-φ- nonexpansive mappings. Appl. Math. Lett, 22, 1051-1055.
https://doi.org/10.1016/j.aml.2009.01.015
Qin, X., Cho, S.Y. , & Kang, S.M. (2010). On hybrid projection methods for asymptotically quasi-φ-nonexpansive mappings. Appl. Math. Comput, 215, 3874-3883. https://doi.org/10.1016/j.amc.2009.11.031
Tada, A. & Takahashi, W. (2007). Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl., 133, 359-370.
https://doi.org/10.1007/s10957-007-9187-z
Takahashi, W. & Zembayashi, K. (2009). Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal., 70, 45-57.
https://doi.org/10.1016/j.na.2007.11.031
Thianwan, T., & Yambangwai, D. (2019). Convergence analysis for a new two-step iteration process for G-nonexpansive mappings with directed graphs. J. Fixed Point Theory Appl. 21(44),1-16. https://doi.org/10.1007/s11784-019-0681-3
Yang, F., Zhao, L., & Kim., J. K. (2012). Hybrid projection method for generalized mixed equilibrium problem and fixed point problem of infinite family of asymptotically quasi-φ-nonexpansive mappings in Banach spaces. Appl. Math. Comput. 218(10), 6072-6082. https://doi.org/10.1016/j.amc.2011.11.091
Anh, P. K., & Hieu, D. V. (2015). Parallel and sequential hybrid methods for a finite family of asymptotically quasi-φ-nonexpansive mappings. J. Appl. Math. Comput, 48(1), 241-263. J. Appl. Math. Comput, 48(1), 241-263.
https://doi.org/10.1007/s12190-014-0801-6
Blum, E., & W. Oettli, W. (1994). From optimization and variational inequalities to equilibrium problems. Math. Stud.,63, 123-145.
Farid, M., Cholamjiak, W., Ali, R., & Kazmi, K. R. (2021). A new shrinking projection algorithm for a generalized mixed variational-like inequality problem and asymptotically quasi- -nonexpansive mapping in a Banach space. R. Acad. Cienc. Exactas Fi's. Nat. Ser. A Mat., 115(3), 114. https://doi.org/10.1016/j.na.2009.04.023
Hao, Y. (2013). Some results on a modified Mann iterative scheme in a reflexive Banach spaces. J. Fixed Point Theory Appl., 2013(227), 1-14.
https://doi.org/10.1186/1687-1812-2013-227
Kazmi, K. R., & Ali, R. (2019). Hybrid projection method for a system of unrelated generalized mixed variational-like inequality problems, Georgian Math. J., 26(1), 63-78.
https://doi.org/10.1515/gmj-2017-0027
Kamimura, S., & Takahashi, W. (2002) Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938-945
https://doi.org/10.1137/S105262340139611X
Muu, L.D., & Oettli, W. (1992). Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal., 18, 1159-1166
https://doi.org/10.1016/0362-546X(92)90159-C
Noor, M. A., & Oettli, W. (1994). On general nonlinear complementarity problems and quasi-equilibria. Le Matematiche (Catania), 49, 313-331.
Preda, V., Beldiman, M. & Bătătoresou, A. (2007). On Variational-like inequalities with generalized monotone mappings, in: Generalized Convexity and Related Topics, Lecture Notes in Econom. Math. Springer, Berlin, 415–431.
https://doi.org/10.1007/978-3-540-37007-9_25
Qin, X., Cho, Y.J., Kang, S.M., & Zhou, H. (2009). Convergence of a modified Halpern-type iteration algorithm for quasi-φ- nonexpansive mappings. Appl. Math. Lett, 22, 1051-1055.
https://doi.org/10.1016/j.aml.2009.01.015
Qin, X., Cho, S.Y. , & Kang, S.M. (2010). On hybrid projection methods for asymptotically quasi-φ-nonexpansive mappings. Appl. Math. Comput, 215, 3874-3883. https://doi.org/10.1016/j.amc.2009.11.031
Tada, A. & Takahashi, W. (2007). Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl., 133, 359-370.
https://doi.org/10.1007/s10957-007-9187-z
Takahashi, W. & Zembayashi, K. (2009). Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal., 70, 45-57.
https://doi.org/10.1016/j.na.2007.11.031
Thianwan, T., & Yambangwai, D. (2019). Convergence analysis for a new two-step iteration process for G-nonexpansive mappings with directed graphs. J. Fixed Point Theory Appl. 21(44),1-16. https://doi.org/10.1007/s11784-019-0681-3
Yang, F., Zhao, L., & Kim., J. K. (2012). Hybrid projection method for generalized mixed equilibrium problem and fixed point problem of infinite family of asymptotically quasi-φ-nonexpansive mappings in Banach spaces. Appl. Math. Comput. 218(10), 6072-6082. https://doi.org/10.1016/j.amc.2011.11.091
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