Home Iterative approximation of common solution of variational inequality and certain optimization problems with multiple output sets in Hadamard space
Article
Licensed
Unlicensed Requires Authentication

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 and Ojen Kumar Narain
Published/Copyright: January 3, 2024

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.2020.6Search 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. Search 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/S1793557122501881Search 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.2010.07.033Search 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.2018.1.05Search 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-2018-0078Search 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/axioms9040143Search 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-2018-0022Search in Google Scholar

[9] C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, John Wiley & Sons, New York, 1984. Search 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. Search 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/s10711-008-9243-3Search 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. Search 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. Search 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. Search 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.2005.09.044Search 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.2008.05.036Search 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. Search in Google Scholar

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

[19] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Ser. Comput. Phys., Springer, New York, 1984. 10.1007/978-3-662-12613-4Search 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. Search 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/s10915-023-02132-6Search 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. Search 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/S1446788719000041Search 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/S1446788716000446Search 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.2007.04.011Search 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-zSearch 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)00266-3Search 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/s40306-022-00480-3Search 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-2020-0006Search 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. Search 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/BF02591749Search 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. 10.1080/01630563.2022.2042015Search 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-ySearch in Google Scholar

[34] G. Stampacchia, Formes bilinéaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413–4416. Search 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. Search 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-xSearch 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/S0024610702003332Search 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

Downloaded on 16.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/anly-2022-1075/html?lang=en
Scroll to top button