İbrahim KARAHAN, Lateef Olakunle JOLAOSO

A THREE STEPS ITERATIVE PROCESS FOR APPROXIMATING THE FIXED POINTS OF MULTIVALUED GENERALIZED α-NONEXPANSIVE MAPPINGS IN UNIFORMLY CONVEX HYPERBOLIC SPACES

In this paper, we prove some fixed point properties and demiclosedness principle for multivalued generalized α-nonexpansive mappings in uniformly convex hyperbolic spaces. We also proposed a three steps iterative scheme for approximating the common fixed points of generalized α-nonexpansive mapping and prove some strong and Δ-convergence theorems for such operator in the setting of uniformly convex hyperbolic space. We provide a numerical example to show that the three steps scheme proposed in this paper performs better than the modified SP-iterative scheme. The results obtained in this paper extend and generalized the corresponding results in uniformly convex Banach spaces, CAT(0) space and many other results in this direction.

Keywords:

Generalized nonexpansive, three steps iteration, multivalued mappings hyperbolic spaces, fixed point theorem,

PDF

___

[1] Abbas M., Khan S. H., Khan A. R., Agarwal R. P., (2011) Common fixed points of two multivalued nonexpansive mappings by one-step iterative scheme, Appl. Math. Lett., 24, 97-102.
[2] Aoyama K. and Kohsaka F., (2011) Fixed point theorem for α-nonexpansive mappings in Banach spaces, Nonlinear Anal., 74, 4387-4391. [3]Bauschke H. H. and Combettes P. L., (2011)Convex analysis and monotone operator theory in Hilbert spaces, ser. CMS Books in Mathematics, Berlin, Germany. [4]Browder F. E., (1965) Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Natl. Acad. Sci. 53, 1272-1276. [5]Browder F. E., (1965) Nonexpansive nonlinear operators in Banach spaces, Proc. Natl. Acad. Sci., 54, 1041-1044. [6]Chang S. S., Wang G., Wang L., Tang Y. K. and Ma Z. L., (2014) Δ-convergence theorems for multi-valued nonexpansive mappings in hyperbolic spaces, Appl. Math. Comp., 249, 535-540. [7]Glowinski R. and Le Tallec P., (1989) Augmented Lagrangian and operator-splitting methods in non linear mechanics, 9, SIAM. [8]Goebel K. and Kirk W. A., (1983) Iteration processes for nonexpansive mappings, In Topological Methods in Nonlinear Functional Analysis, S. P. Singh, S. Thomeier, and B.Watson, Eds., vol. 21 of Contemporary Mathematics, 115-123, American Mathematical Society, Providence, RI, USA. [9]Goebel K. and Reich S., (1984) Uniform convexity, Hyperbolic Geometry and Nonexpansive mappings, Marcel Dekket, New York. [10]Goebel K. and Kirk W. A., (1990) Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, England.
[11]Göhde D., (1965) Zum Prinzip def Kontraktiven Abbilding, Math. Nachr., 30, 251-258.
[12]Gunduz B. and Karahan I., (2018) Convergence of SP iterative scheme for three multivalued mappings in hyperbolic space, J. Comput. Analy. Appl., 24, 815-827.
[13][aubruge S., Nguyen V. H. and Strodiot J., (1998) Convergence analysis and applications of the glowinski-le tallec splitting method for finding a zero of the sum of two maximal monotone operators, J. Optim. Theory Appl., 97 (3), 645-673.
[14]Ishikawa S., (1974) Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44, 147- 150.
[15][han S H., Abbas M., Rhoades B. E., (2010) A new one-step iterative scheme for approximating common fixed points of two multivalued nonexpansive mappings, Rend del Circ Mat, 59, 149-157.
[16]Khan A. R., Fukhar-ud-din H. and Khan M. A. A., (2012) An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl. 2012, 54.
[17]Khan A. R., Khamsi M. A. and Fukhar-ud-din H., (2011) Strong convergence of a general iteration scheme in CAT(0) spaces, Nonlinear Anal, 74, 783-791.
[18]Kirk W. A., (1965) A fixed point theorem for mappings which do not increase distances, Am. Math. Monthly, 72, 1004-1006.
[19]Kirk W. A. and Panyanak B., (2008) A concept of convergence in geodesic spaces, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 68, 12, 3689-3696.
[20]Kohlenbach U., (2005) Some logical metathorems with applications in functional analysis, Trans. Amer. Math. Soc., 357 (1), 89-128.
[21]Kuczumow T., (1978) An almost convergence and its applications, Ann. Univ. Mariae Curie-SkÅ ,odowska, Sect. A, 32, 79-88.
[22]Leustean L., (2010) Nonexpansive iteration in uniformly convex W-hyperbolic spaces, In A. Leizarowitz, B.S. Mordukhovich, I. Shafrir, A. Zaslavski, Nonlinear Analysis and Optimization I. Nonlinear analysis Contemporary Mathematics. Providence. RI Ramat Gan American Mathematical Soc. Bar Ilan University, 513, 193-210.
[23]Mann W.R., (1953) Mean value methods in iteration, Proc. Amer. Math. Soc., 4 , 506-510.
[24]Markin J. T., (1973) Continuous dependence of fixed point sets, Proc. Amer. Math. Soc., 38, 545-547.
[25]Mebawondu A. A., Jolaoso L. O. and Abass H. A., (2017) On Some Fixed Points Properties and Convergence Theorems for a Banach Operator in Hyperbolic Spaces, Inter J Nonlinear Analy Appl, 8(2), 293-306.
[26]Nadler S. B. , Jr., (1969) Multivalued contraction mappings, Pacific J. Math., 30, 475-488.
[27]Noor M. A., (2000) New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251, 217-229.
[28]Pant R. and Shukla R., (2017) Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optim., 38(2), 248-266.
[29]Panyanak B., (2007) Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces, Comp Math Appl., 54, 872-877.
[30]Phuengrattana W. and Suantai S., (2011) On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math., 235, 3006-3014.
[31]Reich S. and Shafrir I., (1990) Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15, 537-558.
[32]Sastry K. P. R. and Babu G. V. R., (2005) Convergence of Ishikawa iterates for a multivalued mapping with a fixed point, Czechoslovak Math J., 55, 817-826.
[33]Senter H.F. and Dotson W.G., (1974) Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44(2), 375-380.
[34]Shahzad N. and Zegeye H., (2009) On Mann and Ishikawa iteration schemes for multivalued maps in Banach space, Nonlinear Anal. 71, 838-844.
[35]Shimizu T. and Takahashi W., (1996) Fixed points of multivalued mappings in certain convex metric spaces, Topol. Methods Nonlinear Anal 8,197-203.
[36]Song Y. and Cho Y. J., (2011) Some notes on Ishikawa iteration for multivalued mappings, Bull. Korean Math Soc., 48, 575-584.
[37]Song Y., Wang H., (2008) Erratum to Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces, Comp Math Appl 55, , 2999-3002.
[38]Song Y. and Wang H., (2009) Convergence of iterative algorithms for multivalued mappings in Banach spaces, Nonlinear Anal., 70, 1547-1556.
[39]Suanoom C. and Klin-eam C., (2016) Remark on fundamentally nonexpansive mappings in hyperbolic spaces, Bull. Austral J. Nonlinear Sci. Appl., 9, 1952-1956.
[40]Suzuki T., (2008) Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl., 340, 1088-1095.
[41]Takahashi W. A., (1970) A convexity in metric space and nonexpansive mappings, I. Kodai Math. Sem. Rep., 22, 142-149.

Sigma Mühendislik ve Fen Bilimleri Dergisi-Cover

ISSN: 1304-7191
Yayın Aralığı: Yılda 4 Sayı
Yayıncı: Yıldız Teknik Üniversitesi

Arşiv