An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces

• Autorzy: Ngwepe Matlhatsi Dorah , Jolaoso Lateef Olakunle , Aphane Maggie , Adiele Ugochukwu Oliver

Identyfikatory

DOI

10.1515/dema-2023-0127

• Języki publikacji: EN

Abstrakty

EN

In this article, we study the split variational inclusion and fixed point problems using Bregman weak relatively nonexpansive mappings in the p-uniformly convex smooth Banach spaces. We introduce an inertial shrinking projection self-adaptive iterative scheme for the problem and prove a strong convergence theorem for the sequences generated by our iterative scheme under some mild conditions in real p-uniformly convex smooth Banach spaces. The algorithm is designed to select its step size self-adaptively and does not require the prior estimate of the norm of the bounded linear operator. Finally, we provide some numerical examples to illustrate the performance of our proposed scheme and compare it with other methods in the literature.

Słowa kluczowe

EN

Bregman distance; split inclusion problem; inertial algorithm; fixed point problem; Banach spaces

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2024
• Tom: Vol. 57, nr 1
• Strony: art. no. 20230127
• Opis fizyczny: Bibliogr. 34 poz., rys., tab.

Twórcy

autor

Ngwepe Matlhatsi Dorah

• Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, P.O. Box 94, Medunsa 0204, Pretoria, South Africa

autor

Jolaoso Lateef Olakunle

• Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, P.O. Box 94, Medunsa 0204, Pretoria, South Africa

autor

Aphane Maggie

• Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, P.O. Box 94, Medunsa 0204, Pretoria, South Africa

autor

Adiele Ugochukwu Oliver

• Department of Mathematics, University of North Texas, Denton, Texas (76203), United States of America

Bibliografia

• [1] Y. Censor and T. Elfving, A multi-projection algorithm using Bregman projection in a product space, Numer. Algorithms 8 (1994), 221–239.
• [2] A. Moudafi and B. S. Thakur, Solving proximal split feasibility problem without prior knowledge of matrix norms, Optim. Lett. 8 (2014), no. 7, 2099–2110.
• [3] C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl. 18 (2002), 441–453.
• [4] P. L. Combettes, The convex feasibility problem in image recovery. In: P. Hawkes, (ed.), Advances in Imaging and Electron Physics, Academic Press, New York, 1996, pp. 155–270.
• [5] S. Riech and S. Sabach, Two strong convergence theorems for a proximal method in Reflexive Banach spaces, Numer. Funct. Anal. Optim. 31 (2010), 24–44.
• [6] Y. Censor, A. Gibali, and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms 59 (2012), 301–323.
• [7] C. S. Chuang, Hybrid inertial proximal algorithm for the split variational inclusion problem in Hilbert spaces with applications, Optimization 66 (2017), 777–792.
• [8] K. R. Kazmi and S. H. Rizvi, An iterative method for split variational inclusion problem and fixed point problem for nonexpansive mapping, Optim. Lett. 8 (2014), 1113–1124.
• [9] D. J. Wen and Y. A. Chen, Iterative methods for split variational inclusion and fixed point problem of nonexpansive semigroup in Hilbert spaces, J. Inequal. Appl. 2015 (2015), 24, DOI: https://doi.org/10.1186/s13660-014-0528-9.
• [10] A. S. Alofi, M. Alsulami, and W. Takahashi, Strongly convergent iterative method for the split common null point problem in Banach spaces, J. Nonlinear Convex. Anal. 2 (2016), 311–324.
• [11] S. Suantai, K. Srisap, N. Naprang, M. Mamat, V. Yundon, and P. Cholamjiak, Convergence theorems for finding the split common null point in Banach spaces, Gen. Topol. Appl. 18 (2017), no. 2, 345–360.
• [12] F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert spaces, SIAM J. Optim. 14 (2004), 773–782.
• [13] 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.
• [14] Y. Tang, New inertial algorithm for solving split common null point problem in Banach spaces, J Inequal Appl. 2019 (2019), 17.
• [15] F. U. Ogbuisi and O. T. Mewomo, Iterative solution of split variational inclusion problem in a real Banach space, Afrik. Math. 28 (2017), 295–309.
• [16] O. K. Oyewole, C. Izuchukwu, C. C. Okeke, and O. T. Mewomo, Inertial approximation method for split variational inclusion problem in Banach spaces, Int. J. Nonlinear Anal. Appl. 11 (2020), 285–304.
• [17] K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984.
• [18] C. E. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, Springer, London, 1965.
• [19] Z. B. Xu and G. F. Roach, Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces, J. Math. Anal. Appl. 157 (1991), no. 1, 189–210.
• [20] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings, and Nonlinear Problems, Kluwer, Dordrecht, 1990, and in its review by S. Reich, Bull. Amer. Math. Soc. 26 (1992), 367–370.
• [21] R. P. Agarwal, D. O. Regan, and D. R. Sahu, Applications of Fixed Point Theorems. In: Fixed Point Theory for Lipschitzian-type Mappings with Applications, Topological Fixed Point Theory and Its Applications, Springer, New York, NY, Vol. 6, 2009.
• [22] Y. Censor, T. Bortfeld, B. Martin, and A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phys. Med. Biol. 51 (2003), 2353–2365.
• [23] A. Taiwo, L. O. Jolaoso, O. T. Mewomo, and A. Gibali, On generalized mixed equilibrium problem with α β μ bifunction and μ τ monotone mapping, J. Nonlinear Convex Anal. 21 (2020), no. 6, 1381–1401.
• [24] F Schopfer, T. Schuster, and A. K. Louis, An iterative regularization method for the solution of the split feasibility problem in Banach spaces, Inverse Probl. 24 (2008), no. 5, 55008.
• [25] D. Reem, S. Reich, and A. De Pierro, Re-examination of Bregman functions and new properties of their divergences, Optimization 68 (2019), 279–348.
• [26] S. Reich and S. Sabach, Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces, Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer, New York, (2010), pp. 299–314.
• [27] F. Kohsaka and W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM J. Optim. 19 (2008), no. 2, 824–835.
• [28] K. Aoyama, F. Kohsaka, and W. Takahashi, Three generalizations of firmly nonexpansive mappings: Their relations and continuity properties, J. Nonlinear Convex Anal. 10 (2009), no. 1, 131–147.
• [29] F. Schopfer, Iterative Regularization Method for the Solution of the Split Feasibility Problem in Banach Spaces, Ph.D. Thesis, Saabrucken, 2007.
• [30] S. Reich, A weak convergence theorem for the alternating method with Bregman distances, In Theory, and Applications of Nonlinear Operators of Accretive and Monotone Type, Marcel Dekker, New York, 1996, pp. 313–318.
• [31] D. Butnariu, S. Reich, and A. J. Zaslavski, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal. 7 (2001), 151–174.
• [32] E. Naraghirad, J. C. Yao, Bregman weak relatively nonexpansive mappings in Banach space, Fixed Point Theory and Applications, 2013 (2013), 141, DOI: https://doi.org/10.1186/1687-1812-2013-141.
• [33] Y. Shehu, Strong convergence theorem for multiple sets split feasibility problems in Banach spaces, Numer. Funct. Anal. Optim. 37 (2016), no. 8, 1021–103.
• [34] Y. Shehu, F. U. Ogbuisi, and O. S. Iyiola, Convergence analysis of an iterative algorithm for fixed point problems and split feasibility problems in specific Banach spaces, Optimization 65 (2016), no. 2, 299–323.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).