On Reich type λ−α-nonexpansive mapping in Banach spaces with applications to L1([0,1])

[EN] In this manuscript we introduce a new class of monotone generalized nonexpansive mappings and establish some weak and strong convergence theorems for Krasnoselskii iteration in the setting of a Banach space with partial order. We consider also an application to the space L1([0,1]). Our results generalize and unify the several related results in the literature.The authors thanks to anonymous referees for their remarkable comments, suggestion and ideas that helps to improve this paper.Belbaki, R.; Karapinar, E.; Ould-Hammouda, A. (2018). On Reich type λ−α-nonexpansive mapping in Banach spaces with applications to L1([0,1]). Applied General Topology. 19(2):291-305. doi:10.4995/agt.2018.10213291305192K. Aoyama and F. Kohsaka, Fixed point theorem for α-nonexpansive mappings in Banach spaces, Nonlinear Anal. 74 (2011), 4387-4391. https://doi.org/10.1016/j.na.2011.03.057J.-B. Baillon, Quelques aspects de la théorie des points fixes dans les espaces de Banach. I, II. In : Séminaire d'analyse fonctionnelle (1978-1979), pp. 7-8. Ecole Polytech., Palaiseau (1979).H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. Math. Soc. 88, no. 3 (1983), 486-490. https://doi.org/10.2307/2044999F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci. USA 54 (1965) 1041-1044. https://doi.org/10.1073/pnas.54.4.1041F. E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Natl. Acad. Sci. USA 53 (1965), 1272-1276. https://doi.org/10.1073/pnas.53.6.1272J. B. Diaz and F. T. Metcalf, On the structure of the set of subsequential limit points of successive approximations, Bull. Am. Math. Soc.73 (1967), 516-519. https://doi.org/10.1090/S0002-9904-1967-11725-7J. G. Falset, E. L. Fuster and T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, J. Math. Anal. Appl. 375 (2011), 185-195. https://doi.org/10.1016/j.jmaa.2010.08.069K. Goebel and W. A. Kirk, Iteration processes for nonexpansive mappings, Contemp. Math. 21 (1983), 115-123. https://doi.org/10.1090/conm/021/729507K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28, p.244. Cambridge University Press (1990). https://doi.org/10.1017/CBO9780511526152D. Gohde, Zum prinzip der dertraktiven abbildung, Math. Nachr. 30 (1965), 251-258. https://doi.org/10.1002/mana.19650300312J. P. Gossez and E. Lami Dozo, Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math. 40 (1972), 565-573. https://doi.org/10.2140/pjm.1972.40.565E. Karapinar, Remarks on Suzuki (C)-condition, dynamical systems and methods, Springer-Verlag New York, 2012, Part 2, 227-243.E. Karapinar and K. Tas, Generalized (C)-conditions and related fixed point theorems, Comput. Math. Appl. 61, no. 11 (2011), 3370-3380. https://doi.org/10.1016/j.camwa.2011.04.035M. A. Khamsi, and A. R. Khan, On monotone nonexpansive mappings in L1[0,1]. Fixed point theory Appl. 2015, Article ID 94 (2015). https://doi.org/10.1186/s13663-015-0346-xW. A. Kirk, A fixed point theorem for mappings which do not increase distances, Am. Math. Mon. 72 (1965), 1004-1006. https://doi.org/10.2307/2313345W. A. Kirk, Krasnoselskii's iteration process in hyperbolic space, Numer. Func. Anal. Opt. 4, no. 4 (1982), 371-381. https://doi.org/10.1080/01630568208816123Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597. https://doi.org/10.1090/S0002-9904-1967-11761-0R. Shukla, R. Pant and M. De la Sen, Generalized $alpha$-nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications (2017) 2017:4. https://doi.org/10.1186/s13663-017-0597-9Y. Song, K. Promluang, P. Kuman and Y. Je Cho, Some convergence theorems of the Mann iteration for monotone α-nonexpansive mappings, Appl. Math. Comput. 287/288 (2016), 74-82. https://doi.org/10.1016/j.amc.2016.04.011T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340, no. 2 (2008), 1088-1095. https://doi.org/10.1016/j.jmaa.2007.09.023D. van Dulst, Equivalent norms and the fixed point property for nonexpansive mappings, J. London Math. Soc. 25 (1982), 139-144.https://doi.org/10.1112/jlms/s2-25.1.139P. Veeramani, On some fixed point theorems on uniformly convex Banach spaces, J. Math. Anal. Appl. 167 (1992), 160-166. https://doi.org/10.1016/0022-247X(92)90243-