New Iteration Algorithms for Finite Family of Two Quasi-nonexpansive Mappings Satisfying Jointly Demiclosedness Principle in Banach Spaces

Document Type : Original Article

Author

Department of Mathematics, Micheal Okpara University of Agriculture, Umudike, Umuahia, Abia State, Nigeria

Abstract

In this paper, we propose and study two iteration schemes (modified Halpern's type and HS-iteration schemes). Furthermore, it is proved that if two infinite families of quasi-nonexpansive mappings satisfy jointly demiclosedness principle, then under appropriate conditions on the iteration parameters, the schemes so introduced strongly converge to the common fixed points of the mappings. Our main results improve and generalize the results in literture and many other existing results currently in literature.

Keywords


[1] E. Naraghirad, Approximation of common fixed points of nonlinear mappings satisfying jointly demiclosedness
principle in Banach spaces, Mediterr. J. Math. 14(2017),162. 
[2] R. E. Bruck, Properties of fixed point sets of nonexpansive mappings in Banach spaces, J. Amer. Math. Soc.,
179(1973), 251-262.
[3] K. Goebel, S. Reich, Uniform convexity, hyperbolic geometry and nonexpansive mappings, Monograph and Text-
books in Pure and Applied Mathematics, Vol. 83, Marcel Dekker, New York, 1984. 
[4] A. Aleyner, S. Reich, An explicit construction of sunny nonexpansive retractions in Banach spaces, Fixed Point
Theory and Appl. 3(2005), 295-365. 
[5] S. H. Khan, W. Takahashi, Approximating common fixed points of two asymptotically nonexpansive mappings, Sci., Math. Japon, 53(1)(2001), 143-148.
[6] Z. Opial, Weak convergence of the sequence of successive approximation for nonexpansive mappings , Bull Amer:
Math Soc., 73(1967), 591-597. 
[7] B. E. Rhoades, Fixed point iteration for certain nonlinear mappings, J. Math. Anal. Appl., 183(1994), 118-120.
[8] W. Takahashi, G. E. Kim, Approximating fixed points of nonexpansive mappings in Banach spaces, Math.
Japonoica, 48(1)(1998), 1-9.
[9] K. K. Tan, H. K. Xu, Approximating fixed point of nonexpansive mappings by the lshikawa iteration process, J.
Math. Anal. Appl., 178(1993),301-308. 
[10] D. I. Igbokwe, S. J. Uko, Weak and strong convergence theorems for approximating fixed points of nonexpansive
mappings using composite hybrid iteration method, J Nig Math Soc., 33(2014), 129-144. 
[11] M. Eslamian, Weak and strong convergence theorems of iterative process for two nite families of mappings, Sci.
Bull. Politeh. Univ. Buchar. ; Ser. A. Appl. Math. Phys., 75(4)(2013), 81-90.
Imo Kalu Agwu, Commun. Nonlinear Anal. 2 (2020), 28-43
[12] J. Garcia-Falset, E. Llorens-Fuster, T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings,
J. Math. Anal. Appl.,375(2011), 185-195.
[13] F. E. Browder, Fixed point theorems for noncompact mappings in Hilbert space, Proc. Nut. Sci., 43 (1965),
1272-1276.
[14] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nut. Sci. , 43 (1965), 1041-1044. 
[15] D. Gohde, Zum prinzip der kontraktiven abbildung, Math. Nach., 30 (1965), 251-258. 
[16] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72
(1965), 1004-1006. 
[17] J. B. . Baillon and R. Schoneberg, Asymptotic normal structure and fixed points of nonexpansive mappings, Proc,
Arner. Math. Soc., 81 (1981), 257-264.
[18] B. Halpern, Fixed points of nonexpansive mappings, Bull. Amer. Math. Soc., 73(1967), 957-961.
[19] F. Kohasaka, W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone
operator in Banach spaces , Arch. Math., 91(2008), 166-177. 
[20] F. Kohasaka, W. Takahashi, Existence and approximationn of fixed points of firmly nonexpansive-type mappings
in Banach spaces, SIAM. J. Optim., 19(2008), 824-835. 
[21] W. P. Cholamjiak, S. Suantai, Y. J. Cho, Fixed points for nonspreading-type multivalued mappings: Existence
and convergence results, Ann. Acad. Rom. Sci. Ser. Math. Apll., 10(2)(2018), 838-844.
[22] W. R. Mann, Mean value method in iteration, Proc. Ame. Math. Soc., 4(1953), 506-510.
[23] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl.,
67(2)(1979), 274-276. 
[24] A. Genel, J. Lindenstrauss, An example concerning fixed points, lsr. J. Math.,22(1975),81-86.
[25] C. E. Chidume, C. O. Chidume, lterative approximation of fixed points of nonexpansive mappings, J. Math. Anal.
Appl., 318(1)(2006), 288-295. 
[26] P. L. Lions, Approximation de points fixes de contraction, C. R. Acad. Sci., Ser. A-B Paris, 284(1977), 1357-1359.
[27] S. lshikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44(1974), 147-150. 
[28] R. P. Agarwal, D. O'Regan, D. R. Sahu, lterative construction of fixed points of nearly asymptotically nonexpansive
mappings, Int. J. Nonlinear Convex Anal., 8(1)(2007), 61-69.