Submit your papersSubmit Now
For Enquiries: [email protected]
IIARD LogoIIARD

Common Attractive Point and Viscosity Approximation Method of Two Widely More Generalized Hybrid Mappings

Malik Ahmad, Abbas Yusuf Balarabe, and Hamza Sani

Abstract

In this research work, common attractive point problem involving widely more generalized hybrid mappings is studied. Using viscosity approximation method, we establish strong convergence theorem for common attractive points of two widely more generalized hybrid mappings and solutions of some variational inequality problems in a real Hilbert space. The result presented in this paper improves and extends some recent results in the literature.

Keywords

Viscosity approximation methodCommon Attractive PointWidely more generalized hybrid mappingHilbert Space.

References

[1] Takahashi, W. Wong, N. C. and Yao J. C., Attractive point and Halpharn-type strong convergence theorem in Hilbert space, J. Fixed Point Theory Appl. 17(2015), 301-311. [2] Halpharn, B., Fixed point of nonspreading maps, Bull. Amer. Math. Soc. 73(1967), 957- 961. [3] Zheng Y., Attractive points and convergence theorem of generalized hybrid mapping, J.Non.Sci.Appl.8(2015), 354-362. [4] Takahashi, W. Takeuchi, Y., Nonlinear ergodic theorem without convexity for generalized hybrid mappings in Hilbert space. j.nonlinear convex Anal.(2011), 12, 399-406. [5] Baillon, j. B., Un theorem detype ergodique pour les contractions nonlinear dans un espaces de Hilbert. C.R. Acad. Sci. paris, ser. A-B 280, 1511-1541 (1975). [6] Takahashi, W. Wong, N. C. Yao, J. C., Attractive point and weak convergence theorems for new generalized hybrid mappings in Hilbert space. J. Nonlinear Convex Anal.13(4), 745- 757,(2012). [7] Khan, S. H., Iterative approximation of common attractive points of further generalized hybrid mappings. j. fixed point theory Appl.(2018), 8, 1-10. [8] Kocourk, P. Takahashi, W. Yao, J. C., Fixed point theorem for weak convergence Theorems for generalized hybrid mappings in Hilbert spaces. J. Taiwan. Math 2010, 8, 2497-2511 (2010). [9] Hsu, M. H. Takahashi, W., Generalized hybrid mappings in Hilbert space and Banach space. J.Taiwan.Math.2012, 16 129-149(2012). [10] Phuengrattana, W. Suantai, S., Existance and convergence theorems for generalized hybrid mappings in uniformly convex metric space. J.Taiwan, Indian, Pure and Apply Math.45, 121-136, (2014). [11] Lemoto, S. and Takahashi, W., Approximation common fixed point of nonexpansive mappings and nonspreading mapping in Hilbert space. Nonlinear Anal. 71(2009), 2082- 2089. [12] Kasawaki T. and Takahashi W., Existance and approximation of fixed point of generalized hybrid mapping in Hilbert space. J. Nonlinear Convex Anal.14(2013), 71-87. [13] Ming. Guu Sy., Takahashi W., Existence and approximation of attractive points of widely more generalized hybrid mappings in Hilbert space. Abstract. Appl. Anal. 152-185 (2013). [14] Zheng, Y., Attractive points of convergence theorem of generalized hybrid mapping. J. Nonlinear. Sci. Appl.(2015), 8, 354-362. [15] Ishikawa, S., Fixed point by a new iterative method. Proc. Amer. Math. Soc (1974), 44, 45- 58. [16] Das, G. Debata, J. P., Fixed points of quasi-nonexpansive mapping. Indian. J. Pure. Appl. Math. (1968), 17, 1263-1269. [17] Takahashi, W. Tamura, T., Convergence theorem for a pair of non-expansive mappings. J. Convex. Anal.(1998), 5, 45-58. [18] Takahashi, W., Iterative method for approximation of fixed points and their application. J. Oper. Res. Soc. Jpn, (2000), 43, 87-108. [19] Thonpaen, P. Inthakon, W., Common attractive points of generalized hybrid mappings in Hilbert space. Thai. J. Math, (2020), 18, 861-869. [20] Chen, L. Yang, N. Zhou, J., Common attractive points of generalized multi-valued mappings and applications. mathematics, (2020), 8, 130. E- ISSN 2489-009X , [21] Chen, L. Zou, J. Zhao, Y. Zhang, M., Iterative approximation of common attractive points of (α,β) generalized hybrid set-valued mappings. J. fixed point theory .Appl. (2019), 21, 1- 17. [22] Khan, S. H., Convergence of one step iterative scheme for asymtotically non-expansive mappings. World. Acad. Sci. Eng. Technol. (2012), 63, 504-506. [23] Yao, Y. Chen, R., Weak and strong convergence of a modified mann iteration for asymtotically non-expansive mappings. Nonlinear. Funct. Anal. Appl, (2007), 12, 307-315. [24] Ali, J. Ali, F., Approximation of common fixed points and solution of image recovery problems. Result. Math. (2019), 74, 130. [25] Panadda, T. Attapol, K. Narawadee, P. Sulhep, S. Inthakon, W., Weak and Strong convergence theorem for common attractive points of widely more generalized hybrid mappings in Hilbert space. Mathematics (2021), 9, 2491. [26] Moudafi, A., Viscosity approximation methods for fixed points problems. J. Math. Anal. Appl. 241,46-55,(2000). [27] Mainge, P. E., The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces. Comput. Math. Appl. 59, 74-79, (2010). [28] Tian, M. Jin, X., A general iterative method for quasi-nonexpansive mappings in Hilbert space. J. Inequal. Appl. 2012, 38(2012). [29] Marino, G. Scardamaglia, B. Zaccono, R., A general viscosity explicit midpoint rule for quasi-nonexpansive mappings. J. Nonlinear convex. Anal. 1, 137-148 ,(2017). [30] Lim, T. C. Xu, H. K., Fixed point theorem asymtotically non-expansive mappings. Nonlinear. Anal., 22, (1994), 1345-1355. [31] Xu, H. K., Iterative algorithms for nonlinear operators. J. London Math. Sci, 66 (2002), 240-256. [32] Kawasaki, T. Kobayashi, T., Existence and mean approximation of fixed points of generalized hybrid non-self mappings in Hilbert space. Scientiae. Mathematicae Japonicae. Online e2014(2014), 29-42. [33] Farid M. Irfan S. S. Khan M. F., Attractive point and the viscosity iterative algorithm for the implicit midpoint rule. vol 9,2. (2018), 134139. [34] Yao Y. Maruster S., Strong convergence of an iterative algorithm for variational inequalities in Banach spaces. Math. Compt. Modell. (2011), 54, 325-329. [35] Khan, S. H., A Picard-Mann hybrid iterative process. Fixed point Theory Appl. 2013, 69(2013). [36] Kohsaka, F. and Takahashi W., Fixed point theorem for class of nonlinear mappings related to maximal monotone operator in Banach space. Arch. Math. ,91 (2008), 166-177. [37] Takahashi, W., Fixed point theorems for new nonlinear mappings in a Hilbert space. J.Nonlinear Convex Anal., 11(2010),79-88. [38] Chang, S. S. and Jinfang, T., Viscosity approximatio methods for two nonexpansive semigroups in CAT(0) spaces, Journal of Inequalities and applications, (2014) : 283.3. [39] Chidume, E. C., Foundation of mathematical analysis. Ibadan university press Publishin House, University of ibadan (2013), 978-978-8456-32-2. [40] Naraghirad, E. and Lin, L. J., Strong convergence Theorems for generalized nonexpansive mappings on star-shaped set with applications, Fixed point Theory and Applications, (2014), 2014:72.