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Approximation of fixed point of multivalued \(\rho\)-quasi-contractive mappings in modular function spaces. (English) Zbl 1483.47115

Summary: The purpose of this paper is to extend the recent results of G. A. Okeke et al. [J. Funct. Spaces 2018, Article ID 1785702, 9 p. (2018; Zbl 1442.47062)] to the class of multivalued \(\rho\)-quasi-contractive mappings in modular function spaces. We approximate fixed points of this class of nonlinear multivalued mappings in modular function spaces. Moreover, we extend the concepts of \(T\)-stability, almost \(T\)-stability and summably almost \(T\)-stability to modular function spaces and give some results.

MSC:

47J26 Fixed-point iterations
47H04 Set-valued operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Citations:

Zbl 1442.47062
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References:

[1] R.P. Agarwal, D. O’Regan, D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8 (1) (2007) 61-79. · Zbl 1134.47047
[2] T.D. Benavides, M.A. Khamsi, S. Samadi, Asymptotically non-expansive mappings in modular function spaces, J. Math. Anal. Appl. 265 (2002) 249-263. · Zbl 1014.47031
[3] V. Berinde, Iterative Approximation of Fixed Points, Editura Efemeride, Baia Mare, 2002. · Zbl 1036.47037
[4] V. Berinde, Summable almost stability of fixed point iteration procedures, Carpathian J. Math. 19 (2) (2003) 81-88. · Zbl 1086.47510
[5] V. Berinde, A convergence theorem for some mean value fixed point iteration procedures, Demonstr. Math. 38 (1) (2005) 177-184. · Zbl 1074.47030
[6] B.A.B. Dehaish, W.M. Kozlowski, Fixed point iteration for asymptotic pointwise nonexpansive mappings in modular function spaces, Fixed Point Theory Appl. 2012 (2012) 118. · Zbl 1333.47049
[7] A.M. Harder, T.L. Hicks, Stability results for fixed point iteration procedures, Math. Japon. 33 (5) (1988) 693-706. · Zbl 0655.47045
[8] A.M. Harder, T.L. Hicks, A stable iteration procedure for nonexpansive mappings, Math. Japon. 33 (5) (1988) 687-692. · Zbl 0655.47046
[9] N. Hussain, A. Rafiq, B. Damjanović, R. Lazović, On rate of convergence of various iterative schemes, Fixed Point Theory Appl. 2011 (45) (2011) 6. · Zbl 1315.47065
[10] M.A. Khamsi, W.M. Kozlowski, Fixed Point Theory in Modular Function Spaces, Springer International Publishing, Switzerland, 2015. · Zbl 1318.47002
[11] S.H. Khan, Approximating fixed points of \(( \lambda , \rho )\)-firmly nonexpansive mappings in modular function spaces, Arab. J. Math. (2018) 7, http://dx.doi.org/10.1007/s40065-018-0204-x. · Zbl 06989344
[12] S.H. Khan, M. Abbas, Approximating fixed points of multivalued \(\rho \)-nonexpansive mappings in modular function spaces, Fixed Point Theory Appl. 2014 (2014) 34. · Zbl 1332.47049
[13] S.H. Khan, M. Abbas, S. Ali, Fixed point approximation of multivalued \(\rho \)-quasi-nonexpansive mappings in modular function spaces, J. Nonlinear Sci. Appl. 10 (2017) 3168-3179. · Zbl 1412.47063
[14] S.J. Kilmer, W.M. Kozlowski, G. Lewicki, Sigma order continuity and best approximation in \(L_\rho \)-spaces, Comment. Math. Univ. Carolin. 3 (1991) 2241-2250.
[15] M.A. Kutbi, A. Latif, Fixed points of multivalued mappings in modular function spaces, Fixed Point Theory Appl. (2009) 786357, 12 pages. · Zbl 1183.47051
[16] G.A. Okeke, M. Abbas, A solution of delay differential equations via Picard-Krasnoselskii hybrid iterative process, Arab. J. Math. 6 (2017) 21-29. · Zbl 1445.47048
[17] G.A. Okeke, S.A. Bishop, S.H. Khan, Iterative approximation of fixed point of multivalued \(\rho \)-quasi-nonexpansive mappings in modular function spaces with applications, J. Funct. Spaces 2018 (2018) Article ID 1785702, 9 pages. · Zbl 1442.47062
[18] M. Öztürk, M. Abbas, E. Girgin, Fixed points of mappings satisfying contractive condition of integral type in modular spaces endowed with a graph, Fixed Point Theory Appl. 2014 (2014) 220. · Zbl 1462.54088
[19] T. Zamfirescu, Fixed point theorems in metric spaces, Arch. Math. 23 (1992) 292-298. · Zbl 0239.54030
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