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In this paper, we introduce a new type of a generalized split monotone variational inclusion (GSMVI) problem in the framework of real Hilbert spaces. By incorporating an inertial extrapolation method and an Halpern iterative technique, we establish a strong convergence result for approximating a solution of GSMVI and fixed point problems of certain nonlinear mappings in the framework of real Hilbert spaces. Many existing results are derived as corollaries to our main result. Furthermore, we present a numerical example to support our main result and propose an open problem for interested researchers in this area. The result obtained in this paper improves and generalizes many existing results in the literature.
Wydawca
Czasopismo
Rocznik
Tom
Strony
359--376
Opis fizyczny
Bibliogr. 38 poz., rys.
Twórcy
autor
- Department of Computer Science and Mathematics, Mountain Top University, Prayer City, Ogun State, Nigeria
- School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
- DST-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Lower Groud Floor, TW Kambule Mathematical Science Building, West Campus Private Bag 3, Wit 2050, Gauteng, South Africa
autor
- School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
- DST-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Lower Groud Floor, TW Kambule Mathematical Science Building, West Campus Private Bag 3, Wit 2050, Gauteng, South Africa
autor
- School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
- DST-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Lower Groud Floor, TW Kambule Mathematical Science Building, West Campus Private Bag 3, Wit 2050, Gauteng, South Africa
autor
- Department of Mathematics, Usmanu Danfodiyo University Sokoto, P.M.B. 2346, Sokoto State, Nigeria
- Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria, P.O. Box 60, 0204, South Africa
autor
- School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
Bibliografia
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- [2] C. Byrne, Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse Prob. 18 (2002), 441–453.
- [3] Y. Censor, X. A. Motova, and A. Segal, Perturbed projections and subgradient projections for the multiple-set split feasibility problem, J. Math. Anal. Appl. 327 (2007), 1224–1256.
- [4] Y. Censor, T. Elfving, N. Kopt, and T. Bortfeld, The multiple-sets split feasibility problem and its applications, Inverse Prob. 21 (2005), 2071–2084.
- [5] P. Cholamjiak and Y. Shehu, Inertial forward-backward splitting method in Banach spaces with application to compressed sensing, Appl. Math. 64 (2019), 409–435.
- [6] H. A. Abass, C. Izuchukwu, F. U. Ogbuisi, and O. T. Mewomo, An iterative algorithm for finite family of split minimization problem and fixed point problem, Novi Sad J. Math. 49 (2019), no. 1, 117–136.
- [7] S. S. Chang, L. Wang, and L. J. Qin, Split equality fixed point problem for quasi-pseudo-contractive mappings with applications, Fixed Point Theory Appl. 2015 (2015), 208.
- [8] S. Suantai, N. Pholosa, and P. Cholamjiak, The modified inertial relaxed CQ algorithm for solving the split feasibility problems, J. Indust. Manag. Optim. 14 (2018), no. 4, 1595–1615.
- [9] S. Suantai, Y. Shehu, and P. Cholamjiak, Nonlinear iterative methods for solving the split common null point problems in Banach spaces, Optim. Meth. Softw. 34 (2019), 853–874.
- [10] S. Suantai, N. Pholasa, and P. Cholamjiak, Relaxed CQ algorithms involving the inertial technique for multiple-sets split feasibility problems, RACSAM 113 (2019), 1081–1099.
- [11] F. Wang and H. K. Xu, Approximating curve and strong convergence of the CQ algorithm for split feasibility problem, J. Inequal. Appl. 2010 (2010), 102085.
- [12] Y. Censor and A. Segal, The split common fixed point for directed operators, J. Convex Anal. 16 (2009), 587–600.
- [13] M. Abbas, M. Alshahrani, Q. H. Ansari, O. S. Iyiola, and Y. Shehu, Iterative methods for solving proximal split minimization problem, Numer. Algor. 78 (2018), 193–215.
- [14] H. A. Abass, K. O. Aremu, and C. Izuchukwu, A common solution of family of minimization problem and fixed point problem for multivalued type one demicontractive type mapping, Adv. Nonlinear Var. Inequal. 21 (2018), no. 2, 94–108.
- [15] S. Y. Cho, X. Qin, and L. Wang, Strong convergence of a splitting algorithm for treating monotone operators, Fixed Point Theory Appl. 2014 (2014), 94.
- [16] B. Martinet, Réegularisation ďinequalities variationnelles par approximation successives, Rev. Franaise Informat. Recherche Opérationnelle 4 (1970), 154–158.
- [17] C. Byrne, Y. Censor, and A. Gibali, Weak and strong convergence of algorithms for the split common null point problem, J. Nonlinear Convex Anal. 13 (2012), 759–775.
- [18] C. S. Chuang, Algorithms with new parameter conditions for split variational inclusion problems in Hilbert spaces with application to split feasibility problem, Optimization 65 (2016), 859–876.
- [19] V. Dadashi, Shrinking projection algorithms for the split common null point problem, Bull. Aust. Math. Soc. 96 (2017), 299–306.
- [20] S. Takahashi and W. Takahashi, The split common null point problem and the shrinking projection method in Banach spaces, Optimization 65 (2016), 281–287.
- [21] W. Takahashi, The split common null point problem in Banach spaces, Arch Math. 104 (2015), 357–365.
- [22] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics/Ouvrages de Mathmatiques de la SMC, Springer, New York, 2011.
- [23] P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal. 16 (1979), 964–979.
- [24] D. A. Lorenz and T. Pock, An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vis. 51 (2015), 311–325.
- [25] A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl. 150 (2011), 275–283.
- [26] H. Attouch, X. Goudon, and P. Redont, The heavy ball with friction method, I. The continuous dynamical system: Global exploration of the local minima of a real-valued function by asymptotic analysis of a dissipative dynamical system, Commun. Contemp. Math. 2 (2000), no. 1, 1–34.
- [27] H. Attouch and M. O. Czarnecki, Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria, J. Diff. Equ. 179 (2002), 278–310.
- [28] F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal. 9 (2001), 3–11.
- [29] P. E. Mainge, Regularized and inertial algorithms for common fixed points of nonlinear operators, J. Math. Anal. Appl. 34 (2008), 876–887.
- [30] A. Moudafi and M. Oliny, Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math. 155 (2013), 447–454.
- [31] P. L. Combettes, M. Defrise, and C. De Mol, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul. 4 (2005), 1168–1200.
- [32] I. Daubechies, M. Defrise, and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pure Appl. Math. 57 (2004), 1413–1457.
- [33] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003), 417–428.
- [34] W. Takahashi, Nonlinear Functional Analysis-Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, 2000.
- [35] G. Lopez, M. V. Marquez, F. Wang, and H. K. Xu, Forward-backward splitting method for accretive operators in Banach space, Abstr. Appl. Anal. 2012 (2012), 109236.
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- [38] D. Van Hieu, L. Van Vy, and P. K. Quy, Three-operators splitting algorithm for a class of variational inclusion problems, Bull. Iran. Math. Soc. 46 (2020), 1055–1071.
Uwagi
PL
Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-470ec09d-baff-4bc6-b313-35ed70f110aa