Startseite Iterative approximation of common solution of variational inequality and certain optimization problems with multiple output sets in Hadamard space
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Iterative approximation of common solution of variational inequality and certain optimization problems with multiple output sets in Hadamard space

  • Hammed Anuoluwapo Abass ORCID logo , Olawale Kazeem Oyewole ORCID logo EMAIL logo , Olayinka Martins Onifade und Ojen Kumar Narain
Veröffentlicht/Copyright: 3. Januar 2024
Analysis
Aus der Zeitschrift Analysis Band 44 Heft 4

Abstract

In this paper, our main interest is to propose a viscosity iterative method for approximating solutions of variational inequality problems, resolvents of monotone operators and fixed points of ρ-demimetric mappings with multiple output sets in Hadamard spaces. We prove a strong convergence result for approximating the solutions of the aforementioned problems under some mild conditions. Also, we present an application of our main result to a convex minimization problem. Our results improve and generalize many related results in the literature.

References

[1] H. A. Abass, K. O. Aremu, L. O. Jolaoso and O. T. Mewomo, An inertial forward-backward splitting method for approximating solutions of certain optimization problem, J. Nonlinear Funct. Anal. 2020 (2020), Article ID 6. 10.23952/jnfa.2020.6Suche in Google Scholar

[2] H. A. Abass, C. Izuchukwu and K. O. Aremu, A common solution of family of minimization problem and fixed point problem for multivalued type-one demicontractive-type mappings, Adv. Nonlinear Var. Inequal. 21 (2018), no. 2, 94–108. Suche in Google Scholar

[3] H. A. Abass, A. A. Mebawondu, K. O. Aremu and O. K. Oyewole, Generalized viscosity approximation method for minimization and fixed point problems of quasi-pseudocontractive mappings in Hadamard spaces, Asian-Eur. J. Math. 15 (2022), no. 11, Paper No. 2250188. 10.1142/S1793557122501881Suche in Google Scholar

[4] B. Ahmadi Kakavandi and M. Amini, Duality and subdifferential for convex functions on complete CAT ( 0 ) metric spaces, Nonlinear Anal. 73 (2010), no. 10, 3450–3455. 10.1016/j.na.2010.07.033Suche in Google Scholar

[5] S. Alizadeh, H. Dehghan and F. Moradlou, Δ-convergence theorems for inverse-strongly monotone mappings in CAT ( 0 ) spaces, Fixed Point Theory 19 (2018), no. 1, 45–56. 10.24193/fpt-ro.2018.1.05Suche in Google Scholar

[6] K. O. Aremu, H. Abass, C. Izuchukwu and O. T. Mewomo, A viscosity-type algorithm for an infinitely countable family of (f,g)-generalized k-strictly pseudononspreading mappings in CAT(0) spaces, Analysis (Berlin) 40 (2020), no. 1, 19–37. 10.1515/anly-2018-0078Suche in Google Scholar

[7] K. O. Aremu, C. Izuchukwu, H. A. Abass and O. T. Mewomo, On a viscosity iterative method for solving variational inequality problems in Hadamard spaces, Axioms 9 (2020), no. 4, Article ID 143. 10.3390/axioms9040143Suche in Google Scholar

[8] K. O. Aremu, C. Izuchukwu, G. C. Ugwunnadi and O. T. Mewomo, On the proximal point algorithm and demimetric mappings in CAT ( 0 ) spaces, Demonstr. Math. 51 (2018), no. 1, 277–294. 10.1515/dema-2018-0022Suche in Google Scholar

[9] C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, John Wiley & Sons, New York, 1984. Suche in Google Scholar

[10] A. Bensoussan and J.-L. Lions, Applications of Variational Inequalities in Stochastic Control, Stud. Math. Appl. 12, North-Holland, Amsterdam, 1982. Suche in Google Scholar

[11] I. D. Berg and I. G. Nikolaev, Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata 133 (2008), 195–218. 10.1007/s10711-008-9243-3Suche in Google Scholar

[12] H. Dehghan and J. Rooin, A characterization of metric projection in CAT(0) spaces, International Conference on Functional Equation, Geometric Functions and Applications (ICFGA 2012), Payame University, Tabriz (2012), 41–43. Suche in Google Scholar

[13] H. Dehghan and J. Rooin, Metric projection and convergence theorems for nonexpansive mapping in Hadamard spaces, preprint (2014), https://arxiv.org/abs/1410.1137. Suche in Google Scholar

[14] S. Dhompongsa, W. A. Kirk and B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal. 8 (2007), no. 1, 35–45. Suche in Google Scholar

[15] S. Dhompongsa, W. A. Kirk and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal. 65 (2006), no. 4, 762–772. 10.1016/j.na.2005.09.044Suche in Google Scholar

[16] S. Dhompongsa and B. Panyanak, On Δ-convergence theorems in CAT ( 0 ) spaces, Comput. Math. Appl. 56 (2008), no. 10, 2572–2579. 10.1016/j.camwa.2008.05.036Suche in Google Scholar

[17] G. Fichera, Problemi elastostatici con vincoli unilaterali: Il problema di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8) 7 (1963/64), 91–140. Suche in Google Scholar

[18] F. Giannessi and A. Maugeri, Variational Inequalities and Network Equilibrium Problems. Plenum Press, New York, (1995). 10.1007/978-1-4899-1358-6Suche in Google Scholar

[19] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Ser. Comput. Phys., Springer, New York, 1984. 10.1007/978-3-662-12613-4Suche in Google Scholar

[20] C. Izuchukwu, H. A. Abass and O. T. Mewomo, Viscosity approximation method for solving minimization problem and fixed point problem for nonexpansive multivalued mapping in CAT ( 0 ) spaces, Ann. Acad. Rom. Sci. Ser. Math. Appl. 11 (2019), no. 1, 130–157. Suche in Google Scholar

[21] C. Izuchukwu, S. Reich, Y. Shehu and A. Taiwo, Strong convergence of forward-reflected-backward splitting methods for solving monotone inclusions with applications to image restoration and optimal control, J. Sci. Comput. 94 (2023), no. 3, Paper No. 73. 10.1007/s10915-023-02132-6Suche in Google Scholar

[22] T. Kawasaki and W. Takahashi, A strong convergence theorem for countable families of nonlinear nonself mappings in Hilbert spaces and applications, J. Nonlinear Convex Anal. 19 (2018), no. 4, 543–560. Suche in Google Scholar

[23] H. Khatibzadeh and V. Mohebbi, Monotone and pseudo-monotone equilibrium problems in Hadamard spaces, J. Aust. Math. Soc. 110 (2021), no. 2, 220–242. 10.1017/S1446788719000041Suche in Google Scholar

[24] H. Khatibzadeh and S. Ranjbar, Monotone operators and the proximal point algorithm in complete CAT ( 0 ) metric spaces, J. Aust. Math. Soc. 103 (2017), no. 1, 70–90. 10.1017/S1446788716000446Suche in Google Scholar

[25] W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008), no. 12, 3689–3696. 10.1016/j.na.2007.04.011Suche in Google Scholar

[26] P.-E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal. 16 (2008), no. 7–8, 899–912. 10.1007/s11228-008-0102-zSuche in Google Scholar

[27] S. Z. Németh, Variational inequalities on Hadamard manifolds, Nonlinear Anal. 52 (2003), no. 5, 1491–1498. 10.1016/S0362-546X(02)00266-3Suche in Google Scholar

[28] G. N. Ogwo, H. A. Abass, C. Izuchukwu and O. T. Mewomo, Modified proximal point methods involving quasi-pseudocontractive mappings in Hadamard spaces, Acta Math. Vietnam. 47 (2022), no. 4, 847–873. 10.1007/s40306-022-00480-3Suche in Google Scholar

[29] G. N. Ogwo, C. Izuchukwu, K. O. Aremu and O. T. Mewomo, On θ-generalized demimetric mappings and monotone operators in Hadamard spaces, Demonstr. Math. 53 (2020), no. 1, 95–111. 10.1515/dema-2020-0006Suche in Google Scholar

[30] U. A. Osisiogu, F. L. Adum and T. E. Efor, Strong convergence results for variational inequality problem in CAT(0) spaces, Adv. Nonlinear Var. Inequal. 23 (2020), 84–101. Suche in Google Scholar

[31] J.-S. Pang, Asymmetric variational inequality problems over product sets: Applications and iterative methods, Math. Program. 31 (1985), no. 2, 206–219. 10.1007/BF02591749Suche in Google Scholar

[32] S. Ranjbar, Approximating a solution of the inclusion problem for an infinite family of monotone operators in Hadamard spaces and its applications, Numer. Funct. Anal. Optim. 43 (2022), no. 4,412–429. 10.1080/01630563.2022.2042015Suche in Google Scholar

[33] S. Ranjbar and H. Khatibzadeh, Strong and Δ-convergence to a zero of a monotone operator in CAT ( 0 ) spaces, Mediterr. J. Math. 14 (2017), no. 2, Paper No. 56. 10.1007/s00009-017-0885-ySuche in Google Scholar

[34] G. Stampacchia, Formes bilinéaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413–4416. Suche in Google Scholar

[35] W. Takahashi, The split common fixed point problem and the shrinking projection method in Banach spaces, J. Convex Anal. 24 (2017), no. 3, 1015–1028. Suche in Google Scholar

[36] G. C. Ugwunnadi, C. Izuchukwu and O. T. Mewomo, Strong convergence theorem for monotone inclusion problem in CAT ( 0 ) spaces, Afr. Mat. 30 (2019), no. 1–2, 151–169. 10.1007/s13370-018-0633-xSuche in Google Scholar

[37] H.-K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc. (2) 66 (2002), no. 1, 240–256. 10.1112/S0024610702003332Suche in Google Scholar

Received: 2022-06-08
Revised: 2023-06-05
Accepted: 2023-11-21
Published Online: 2024-01-03
Published in Print: 2024-11-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 16.10.2025 von https://www.degruyterbrill.com/document/doi/10.1515/anly-2022-1075/html
Button zum nach oben scrollen