Inertial Extragradient Method for Solving Variational Inequality and Fixed Point Problems of a Bregman Demigeneralized Mapping in a Reflexive Banach Spaces

References

• Fichera, G. (1963). Problemi elasstostatici con vincoli unilaterali: II problema di signorini ambigue condizione al contorno. Atti Accad. Naz. Lincei. Mem. Cl. Sci. Nat. Sez. Ia. 7(8):91–140.
   Google Scholar
• Stampacchia, G. (1964). Formes bilinearieres coercitivities sur les ensembles convexes. C. R. Acad. Sci. Paris 258:4413–4416.
   Google Scholar
• Baiocchi, C., Capello, A. (1984). Variational and Quasi-Variational Inequalities. New York, NY: Wiley.
   Google Scholar
• Bensoussan, A., Lions, J. L. (1982). Applications of Variational Inequalities to Stochastic Control. Amsterdam: North-Holland.
   Google Scholar
• Glowinski, R. (1984). Numerical Methods for Nonlinear Variational Problems. Berlin: Springer.
   Google Scholar
• Grannessi, F., Mangeri, A. (1995). Variational inequalities and network equilibrium problems. New York, NY: Plenum Press.
   Google Scholar
• Pang, J. S. (1985). Asymmetric variational inequalities over product of sets” applications and iterative methods. Math. Program 31(2):206–219. DOI: 10.1007/BF02591749.
   Web of Science ®Google Scholar
• Abass, H. A., Aremu, K. O., Jolaoso, L. O., Mewomo, O. T. (2020). An inertial forward-backward splitting method for approximating solutions of certain optimization problem. J. Nonlinear Funct. Anal.: 2020 (2020):1(1–20). DOI: 10.23952/jnfa.2020.6.
   Web of Science ®Google Scholar
• Abass, H. A., Jolaoso, L. O. (2020). An inertial generalized viscosity approximation method for solving multiple-sets split feasibility problem and common fixed point of strictly pseudo-nonspreading mappings. Axioms. 10(1):1. DOI: 10.3390/axioms10010001.
   Web of Science ®Google Scholar
• Ali, B., Ugwunnadi, G. C., Lawan, M. S., Khan, A. R. (2021). Modified inertial subgradient extragradient method in reflexive Banach space. Bol. Soc. Mat. Mex. 27(1):30. DOI: 10.1007/s40590-021-00332-4.
   Web of Science ®Google Scholar
• Hy Duc, M., Nguyen Thi Thanh, H., Tran Thi Huyen, T., Van Dinh, B. (2020). The Ishikawa subgradient extragradient method for equilibrium problems and fixed point problems in Hilbert spaces. Numer. Funct. Anal. Optim. 41(9):1065–1088. DOI: 10.1080/01630563.2020.1737937.
   Web of Science ®Google Scholar
• Iusem, A. N., Mohebbi, V. (2019). Extragradient methods for vector equilibrium problems in Banach spaces. Numer. Funct. Anal. Optim. 40(9):993–1022. DOI: 10.1080/01630563.2019.1578232.
   Web of Science ®Google Scholar
• Jouymandi, Z., Moradlou, F. (2018). Retraction algorithms for solving variational inequalities, pseudomonotone equilibrium problems and fixed point problems in Banach spaces. Numer. Algorithms 78(4):1153–1182. DOI: 10.1007/s11075-017-0417-7.
   Web of Science ®Google Scholar
• Jolaoso, L. O., Shehu, Y. (2021). Single Bregman projection for solving variational inequalities in reflexive Banach spaces. Appl. Anal. :1–22. DOI: 10.1080/00036811.2020.1869947.
   Web of Science ®Google Scholar
• Jolaoso, L. O., Shehu, Y., Cho, Y., (2021). (2000). Convergence analysis for variational inequalities and fixed point problems in reflexive Banach spaces. J. Inequal. Appl. 2021:44; Kinderlehrer, D., Stampacchia, G. An Introduction to Variational Inequalities and Their Applications. Philadelphia: Society for Industrial and Applied Mathematics.
   Google Scholar
• Mebawondu, A. A., Jolaoso, L. O., Abass, H. A., Oyewole, O. K., Aremu, K. O. (2021). A strong convergence Halpern-type inertial algorithm for solving system of split variational inequalities and fixed point problems. J. Appl. Anal. Comput. 11(6):2762–2791. DOI: 10.11948/20200411.
   Web of Science ®Google Scholar
• Oyewole, O. K., Abass, H. A., Mewomo, O. T. (2021). A strong convergence algorithm for a fixed point constrained split null point problem. Rend. Circ. Mat. Palermo, II. Ser. 70(1):389–408. DOI: 10.1007/s12215-020-00505-6.
   Google Scholar
• Oyewole, O. K., Abass, H. A., Mewomo, O. T. (2024). A totally relaxed self-adaptive subgradient extragradient scheme for equilibrium problem and fixed point problem in a Banach space. Kragujevac J. Math. 48(2):73–95.
   Google Scholar
• Reich, S., Tuyen, T. M., Sunthrayuth, P., Cholamjiak, P. (2021). Two new inertial algorithms for solving variational inequalities in reflexive banach spaces. Numer. Funct. Anal. Optim. 42(16):1954–1984.
   Web of Science ®Google Scholar
• Nadezhkina, N., Takahashi, W. (2006). Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128(1):191–201. DOI: 10.1007/s10957-005-7564-z.
   Web of Science ®Google Scholar
• Fan, K. (1961). A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142(3):305–310. DOI: 10.1007/BF01353421.
   Web of Science ®Google Scholar
• Hadjisavvas, N., Schaible, S. (1996). Quasimonotone variational variational inequalities in Banach spaces. J. Optim. Theory Appl. 90(1):95–111. DOI: 10.1007/BF02192248.
   Web of Science ®Google Scholar
• Korpelevich, G. M. (1976). The extragradient method for finding for finding saddle points and other problems. Ekonomikai Matematicheskie Melody. 12:747–756.
   Google Scholar
• Tseng, P. (2000). A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38(2):431–446. DOI: 10.1137/S0363012998338806.
   Web of Science ®Google Scholar
• Censor, Y., Gibali, A., Reich, S. (2011). The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148(2):318–335.
   PubMed Web of Science ®Google Scholar
• Yang, J., Liu, H. (2019). Strong convergence result for solving monotone variational inequalities in Hilbert space. Numer. Algorithmics 80(3):741–752. DOI: 10.1007/s11075-018-0504-4.
   Web of Science ®Google Scholar
• Shehu, Y., Iyiola, O. S. (2017). Strong convergence result for monotone variational inequalities. Numer Algorithmics 76(1):259–282. DOI: 10.1007/s11075-016-0253-1.
   Web of Science ®Google Scholar
• Chidume, C. E., Adamu, A., Okereke, L. C. (2018). A Krasnoselskii-type algorithm for approximating solutions of variational inequality and convex feasibility problems. J. Nonlinear Var. Anal. 2:203–218.
   Web of Science ®Google Scholar
• Khan, A. R., Ugwunnadi, G. C., Makukula, Z. G., Abbas, M. (2019). Strong convergence of inertial subgradient extragradient method for solving variational inequality in Banach space. Can. J. Math. 35(3):327–338. DOI: 10.37193/CJM.2019.03.07.
   Google Scholar
• Abass, H. A., Izuchukwu, C., Mewomo, O. T., Dong, Q. L. (2020). Strong convergence of an inertial forward-backward splitting method for accretive operators in real Banach spaces. Fixed Point Theory. 21(2):397–412. DOI: 10.24193/fpt-ro.2020.2.28.
   Web of Science ®Google Scholar
• Fan, J., Liu, L., Qin, X. (2020). A subgradient extragradient algorithm with inertial effects for solving strongly pseudomonotne variational inequalities. Optimization. 69(9):2199–2215. DOI: 10.1080/02331934.2019.1625355.
   Web of Science ®Google Scholar
• Wega, G. B., Zegeye, H. (2020). Convergence results of forward-backward method for a zero of the sum of maximally monotone mappings in Banach spaces. Comput. Appl. Math. 39, 223 (2020). DOI: 10.1007/s40314-020-01246-z.
   Web of Science ®Google Scholar
• Polyak, B. T. (1964). Some methods of speeding up the convergence of iterates methods. USSR Comput. Math. Phys. 4(5):1–17. DOI: 10.1016/0041-5553(64)90137-5.
   Google Scholar
• Alvarez, F., Attouch, H. (2001). An Inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9(1–2):3–11. DOI: 10.1023/A:1011253113155.
   Google Scholar
• Cai, G., Dong, Q. L., Peng, Y. (2021). Strong convergence theorems for inertial Tseng’s extragradient method for solving variational inequality and fixed point problems. Optim. Lett. 15(4):1457–1474. DOI: 10.1007/s11590-020-01654-4.
   Web of Science ®Google Scholar
• Bauschke, H. H., Borwein, J. M. (1997). Legendre functions and method of random Bregman functions. J. Convex Anal. 4:27–67.
   Google Scholar
• Bauschke, H. H., Borwein, J. M., Combettes, P. L. (2001). Essentially smoothness, essentially strict convexity and Legendre functions in Banach spaces. Commun. Contemp. Math. 03(04):615–647. DOI: 10.1142/S0219199701000524.
   Web of Science ®Google Scholar
• Butnariu, D., Resmerita, E. (2006). Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal. 2006:1–39. DOI: 10.1155/AAA/2006/84919.
   Web of Science ®Google Scholar
• Zalinescu, C. (2002). Convex Analysis in General Vector Spaces. River Edge NJ: World Scientific Publishing Co. Inc.
   Google Scholar
• Bregman, L. M. (1967). The relaxation method for finding the common point of convex sets and its application to solution of problems in convex programming. U.S.S.R Comput. Math. Phys. 7(3):200–217. DOI: 10.1016/0041-5553(67)90040-7.
   Google Scholar
• Timnak, S., Naraghirad, E., Hussain, N. (2017). Strong convergence of Halpern iteration for products of finitely many resolvents of maximal monotone operators in Banach spaces. Filomat. 31(15):4673–4693. DOI: 10.2298/FIL1715673T.
   Web of Science ®Google Scholar
• Eskandani, G. Z., Raeisi, M., Rassias, T. M. (2018). A hybrid extragradient method for solving pseudomonotone equilibrium problem using Bregman distance. J. Fixed Point Theory Appl. 20(3):132. DOI: 10.1007/s11784-018-0611-9.
   Web of Science ®Google Scholar
• Butnariu, D., Iusem, A. N. (2000). Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization. Dordrecht: Kluwer Academic Publishers.
   Google Scholar
• Reich, S., Sabach, S. (2009). A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces. J. Nonlinear Convex Anal. 10:471–485.
   Web of Science ®Google Scholar
• Mashreghi, J., Nasri, M. (2010). Forcing strong convergence of Korpelevich’s method in Bamach spaces with its applications in game theory. Nonlinear Anal. 72(3–4):2086–2099. DOI: 10.1016/j.na.2009.10.009.
   Web of Science ®Google Scholar
• Gazmeh, H., Naraghirad, E. (2020). Strong converegence theorems for Bregman demigeneralized mappings in Banach spaces with Applications. Numer. Funct. Anal. Optim. 41(6):730–757. DOI: 10.1080/01630563.2019.1673773.
   Web of Science ®Google Scholar
• Hu, X., Wang, J. (2006). Solving pseudomonotone variational inequalities and pseudoconvex optimization problems using the projection neural network. IEEE Trans. Neural Networks 17(6):1487–1499.
   PubMed Web of Science ®Google Scholar
• Bryne, C. (2002). Iterative oblique projection onto convex subsets and the split feasibility problems. Inverse Probl. 18:441–453.
   Web of Science ®Google Scholar
• Burwein, M., Reich, S., Sabach, S. (2011). A characterization of Bregman firmly nonexpansive opertors using a new monotonicity concept. J. Nonlinear Convex Anal. 12:161–184.
   Web of Science ®Google Scholar
• Censor, Y., Elfving, T. (1994). A multiprojection algorithms using Bregman projections in a product space. Numer. Algorithms 8(2):221–239. DOI: 10.1007/BF02142692.
   Google Scholar
• Lions, P. L., Mercier, B. (1979). Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6):964–979. DOI: 10.1137/0716071.
   Web of Science ®Google Scholar
• Shehu, Y., Dong, Q. L., Jiang, D. (2019). Single projection method for pseudomonotone variational inequality in Hilbert spaces. Optimization. 68(1):385–409. DOI: 10.1080/02331934.2018.1522636.
   Web of Science ®Google Scholar
• Reich, S., Sabach, S. (2010). Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer. Funct. Anal. Optim. 31:24–44.
   Web of Science ®Google Scholar
• Schopfer, F., Schuster, T., Louis, A. K. (2008). An iterative regularization method for the solution of the split feasibilty problem in Banach spaces. Inverse Probl. 24(5):055008. DOI: 10.1088/0266-5611/24/5/055008.
   Web of Science ®Google Scholar
• Shehu, Y., Ogbuisi, F. U. (2015). An iterative method for solving split monotone variational inclusion and fixed point problem. RACSAM 110:503–518.
   Google Scholar
• Tie, J. V. (1984). Convex Analysis: An Introductory Text. New York, NY: Wiley.
   Google Scholar