On split generalized equilibrium problem with multiple output sets and common fixed points problem. (English) Zbl 1533.47061
Summary: In this article, we introduce and study the notion of split generalized equilibrium problem with multiple output sets (SGEPMOS). We propose a new iterative method that employs viscosity approximation technique for approximating the common solution of the SGEPMOS and common fixed point problem for an infinite family of multivalued demicontractive mappings in real Hilbert spaces. Under mild conditions, we prove a strong convergence theorem for the proposed method. Our method uses self-adaptive step size that does not require prior knowledge of the operator norm. The results presented in this article unify, complement, and extend several existing recent results in the literature.
MSC:
| 47N10 | Applications of operator theory in optimization, convex analysis, mathematical programming, economics |
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H06 | Nonlinear accretive operators, dissipative operators, etc. |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
Keywords:
split generalised equilibrium problem; viscosity approximation method; multivalued demicontractive mapping; fixed point problem; strong convergenceReferences:
| [1] | Y. Censor and T. Elfving, A multi projection algorithms using Bregman projections in a product space, Numer. Algorithms 8 (1994), 221-239. · Zbl 0828.65065 |
| [2] | C. Bryne, Iterative oblique projection onto convex subsets and the split feasibility problems, Inverse Probl. 18 (2002), 441-453. · Zbl 0996.65048 |
| [3] | C. Bryne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl. 20 (2004), 103-120. · Zbl 1051.65067 |
| [4] | E. C. Godwin, C. Izuchukwu, and O. T. Mewomo, An inertial extrapolation method for solving generalized split feasibility problems in real Hilbert spaces, Boll. Unione Mat. Ital. 14 (2021), no. 2, 379-401. · Zbl 1494.47106 |
| [5] | L. O. Jolaoso, M. A. Khamsi, O. T. Mewomo, and C. C. Okeke, On inertial type algorithms with generalized contraction mapping for solving monotone variational inclusion problems, Fixed Point Theory 22 (2021), no. 2, 685-711. · Zbl 1515.47095 |
| [6] | T. O. Alakoya and O. T. Mewomo, S-Iteration inertial subgradient extragradient method for variational inequality and fixed point problems, Optimization (2023), DOI: https://doi.org/10.1080/02331934.2023.2168482. · Zbl 1538.65162 |
| [7] | S. S. Chang, H. W. Lee, and C. K. Chan, A new method for solving equilibrium problem and variational inequality problem with application to optimization, Nonlinear Anal. 70 (2009), 3307-3319. · Zbl 1198.47082 |
| [8] | K. R. Kazmi, R. Ali, and S. Yousuf, Generalized equilibrium and fixed point problems for Bregman relatively nonexpansive mappings in Banach spaces, J. Fixed Point Theory Appl. 20 (2018), no. 151, 21 pp. · Zbl 1401.47006 |
| [9] | E. C. Godwin, C. Izuchukwu, and O. T. Mewomo, Image restoration using a modified relaxed inertial method for generalized split feasibility problems, Math. Methods Appl. Sci. 46 (2023), no. 5, 5521-5544. · Zbl 1530.47072 |
| [10] | P. E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal. 16 (2008), 899-912. · Zbl 1156.90426 |
| [11] | A. Moudafi, A second order differential proximal methods for equilibrium problems, J. Inequal. Pure Appl. Math. 4 (2013), no. 1, 7 pp. · Zbl 1175.90413 |
| [12] | A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl. 150 (2011), 275-283. · Zbl 1231.90358 |
| [13] | T. O. Alakoya, V. A. Uzor, O. T. Mewomo, and J.-C. Yao, On a system of monotone variational inclusion problems with fixed-point constraint, J. Inequ. Appl. 2022 (2022), Paper No. 47, 33 pp. · Zbl 1506.47102 |
| [14] | K. R. Kazmi and S. H. Rizvi, Iterative approximation of a common solution of a split generalized equilibrium problem and a fixed point problem for nonexpansive semigroup, Math. Sci. 7 (2013), no. 1, 10 pp. · Zbl 1277.65051 |
| [15] | F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem, Fixed Point Theory Appl. 2010 (2020), no. 11, Art. ID. 383740. · Zbl 1203.47043 |
| [16] | L.-C. Ceng and J. C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008), 186-201. · Zbl 1143.65049 |
| [17] | S. D. Flam and A. S. Antipin, Equilibrium programming using proximal-like algorithm, Math. Programming Ser. A. 78 (1997), no. 1, 29-41. · Zbl 0890.90150 |
| [18] | O. T. Mewomo and O. K. Oyewole, An iterative approximation of common solutions of split generalized vector mixed equilibrium problem and some certain optimization problems, Demonstr. Math. 54 (2021), no. 1, 335-358. · Zbl 1494.47110 |
| [19] | C. C. Okeke, C. Izuchukwu, and O. T. Mewomo, Strong convergence results for convex minimization and monotone variational inclusion problems in Hilbert space, Rend. Circ. Mat. Palermo (2) 69 (2020), no. 2, 675-693. · Zbl 1442.47051 |
| [20] | O. T. Mewomo and F. U. Ogbuisi, Convergence analysis of iterative method for multiple set split feasibility problems in certain Banach spaces, Quaest. Math. 41 (2018), no. 1, 129-148. · Zbl 1542.47092 |
| [21] | S. Husain and M. Asad, Aninertial extragradient algorithm for split problems in Hilbert spaces. Analysis 43 (2023), no. 2, 89-103. · Zbl 1522.46049 |
| [22] | S. Husain and M. Asad, A modified Picard S-hybrid iterative process for solving split generalized equilibrium problem, Int. J. Appl. Comput. Math. 8 (2022), no. 3, 120. · Zbl 07549860 |
| [23] | F. U. Ogbuisi and O. T. Mewomo, On split generalized mixed equilibrium problems and fixed point problems with no prior knowledge of the operator norm, J. Fixed Point Theory Appl. 19 (2016), no. 3, 2109-2128. · Zbl 1497.47095 |
| [24] | A. Taiwo and O. T. Mewomo, Inertial-viscosity-type algorithms for solving generalized equilibrium and fixed point problems in Hilbert spaces, Vietnam J. Math. 50 (2022), no. 1, 125-149. · Zbl 1500.47100 |
| [25] | A. O.-E. Owolabi, T. O. Alakoya, A. Taiwo, and O. T. Mewomo, A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings, Numer. Algebra Control Optim. 12 (2022), no. 2, 255-278. · Zbl 1487.65077 |
| [26] | E. Blum and W. Oettli, From Optimization and variational inequalities to equilibrium problems, Math. Stud. 63 (1994), 123-145. · Zbl 0888.49007 |
| [27] | S. Suantai, P. Cholamjiak, Y. J. Cho, and W. Cholamjiak, On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces, Fixed Point Theory Appl. 2016 (2016), Paper no. 35, 16pp. · Zbl 1346.47061 |
| [28] | S. Reich and T. M. Tuyen, Iterative methods for solving the generalized split common null point problem in Hilbert spaces, Optimization 69 (2020), 1013-1038. · Zbl 1445.47043 |
| [29] | S. Reich and T. M. Tuyen, Two new self-adaptive algorithms for solving the split common null point problem with multiple output sets in Hilbert spaces, J. Fixed Point Theory Appl. 23 (2021), no. 2, Art. 16. · Zbl 1521.47109 |
| [30] | G. N. Ogwo, C. Izuchukwu, and O. T. Mewomo, Relaxed inertial methods for solving split variational inequality problems without product space formulation, Acta Math. Sci. Ser. B (Engl. Ed.) 42 (2022), no. 5, 1701-1733. · Zbl 1501.47105 |
| [31] | R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33 (1970), 209-216. · Zbl 0199.47101 |
| [32] | Y. Shehu, Iterative methods for split feasibilty problems in certain Banach spaces, J. Nonlinear Convex Anal. 16 (2015), 2351-2364. · Zbl 1334.49037 |
| [33] | Y. Shehu, O. S. Iyiola, and C. D. Enyi, An iterative algorithm for solving split feasibility problems and fixed point problems in Banach spaces, Numer. Algorithm 72 (2016), 835-864. · Zbl 1346.49051 |
| [34] | A. Taiwo, T. O. Alakoya, and O. T. Mewomo, A new efficient algorithm for finding common fixed points of multivalued demicontractive mappings and solutions of split generalized equilibrium problems in Hilbert spaces, Asian-Eur. J. Math. 14 (2021), no. 8, Paper no. 2150137, 31pp. · Zbl 1479.47074 |
| [35] | L. O. Jolaoso, K. O. Oyewole, K. O. Aremu, and O. T. Mewomo, A new efficient algorithm for finding common fixed points of multivalued demicontractive mappings and solutions of split generalized equilibrium problems in Hilbert spaces, Int. J. Comput. Math. 98 (2021), no. 9, 1892-1919. · Zbl 07479099 |
| [36] | W. Phuengrattana and C. Klanarong, Strong convergence of the viscosity approximation method for the split generalized equilibrium problem, Rend. Circ. Mat. Palermo II, 71 (2022), no. 1, 39-64. DOI: https://doi.org/10.1007/s12215-021-00617-7. · Zbl 07501022 |
| [37] | F. O. Isiogugu and M. O. Osilike, Convergence theorems for new classes of multivalued hemicontractive-type mappings, Fixed Point Theory Appl. 2014 (2014), Art. 93, 12pp. · Zbl 1332.47046 |
| [38] | C. Izuchukwu, C. C. Okeke, and O. T. Mewomo, Systems of variational inequalities and multiple-set split equality fixed point problems for countable families of multi-valued type-one demicontractive-type mappings, Ukrainian Math. J. 71 (2019), 1480-1501. |
| [39] | C. Mongkolkeha, Y. J. Cho, and P. Kumam, Convergence theorems for k-demicontractive mappings in Hilbert spaces, Math. Inequal. Appl. 16, no. 4 (2013), 1065-1082. · Zbl 1285.47082 |
| [40] | H. Zegeye and N. Shahzad, Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings, Comp. Math. Appl. 62 (2011), 4007-4014. · Zbl 1471.65043 |
| [41] | P. Cholamjiak and S. Suantai, Viscosity approximation methods for a nonexpansive semigroup in Banach spaces with gauge functions, J Global Optim. 54 (2012), no. 1, 185-197. · Zbl 1252.47072 |
| [42] | C. E. Chidume and J. N. Ezeora, Krasnoselskii-type algorithm for family of multivalued strictly pseudocontractive mappings, Fixed Point Theory Appl. 2014 (2014), Art. 111. · Zbl 1345.47035 |
| [43] | Y. Shehu, P. T. Vuong, and P. Cholamjiak, A self-adaptive projection method with an inertial technique for split feasibility problems in Banach spaces with applications to image restoration problems, J. Fixed Point Theory Appl. 21 (2019), no. 2, 1-24. · Zbl 1415.47010 |
| [44] | S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal. Theory Methods Appl. 75 (2012), no. 2, 742-750. · Zbl 1402.49011 |
| [45] | J. Deepho, W. Kumam, and P. Kumam, A new hybrid projection algorithm for solving the split generalized equilibrium problems and the system of variational inequality problems, J. Math. Model Algorithms 13 (2014), 405-423. · Zbl 1305.65164 |
| [46] | Z. Ma, L. Wang, S. S. Chang, and W. Duan, Convergence theorems for split equality mixed equilibrium problems with applications, Fixed Point Theory Appl. 2015 (2015), no. 31, 18 pp. · Zbl 1310.47094 |
| [47] | S. Suantai, Y. Shehu, P. Cholamjiak, and O. S. Iyiola, Strong convergence of a self-adaptive method for the split feasibility problem in Banach spaces, J. Fixed Point Theory Appl. 20 (2018), no. 2, 1-21. · Zbl 1518.47107 |
| [48] | H. K. Xu, Averaged mappings and the gradient-projection algorithm, J. Optim. Theory Appl. 150 (2011), 360-378. · Zbl 1233.90280 |
| [49] | G. Fichera, Sul problema elastostatico di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei VIII. Ser. Rend. Cl. Sci. Fis. Mat. Nat. 34 (1963), 138-142. · Zbl 0128.18305 |
| [50] | G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413-4416. · Zbl 0124.06401 |
| [51] | A. Gibali, S. Reich, and R. Zalas, Outer approximation methods for solving variational inequalities in Hilbert space, Optimization 66 (2017), 417-437. · Zbl 1367.58006 |
| [52] | G. Kassay, S. Reich, and S. Sabach, Iterative methods for solving systems of variational inequalities in reflexive Banach spaces, SIAM J. Optim. 21 (2011), 1319-1344. · Zbl 1250.47064 |
| [53] | T. O. Alakoya, A. Taiwo, O. T. Mewomo, and Y. J. Cho, An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings, Ann. Univ. Ferrara Sez. VII Sci. Mat. 67 (2021), 1-31. · Zbl 1472.65077 |
| [54] | L.-C. Ceng, A. Petruşel, and J.-C. Yao, On Mann viscosity subgradient extragradient algorithms for fixed point problems of finitely many strict pseudocontractions and variational inequalities, Mathematics 7 (2019), 925. |
| [55] | V. A. Uzor, T. O. Alakoya, and O. T. Mewomo, Strong convergence of a self-adaptive inertial Tseng’s extragradient method for pseudomonotone variational inequalities and fixed point problems, Open Math. 20 (2022), 234-257. · Zbl 1485.65072 |
| [56] | V. A. Uzor, T. O. Alakoya, and O. T. Mewomo, On split monotone variational inclusion problem with multiple output sets with fixed point constraints, Comput. Methods Appl. Math. (2022), DOI: https://doi.org/10.1515/cmam-2022-0199. · Zbl 1506.47102 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
