On monotone mappings in modular function spaces
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2025
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Authors
Buthinah A. Bin Dehaish
- Department of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah 21593, Saudi Arabia.
Mohamed A. Khamsi
- Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia.
- Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, TX 79968, U. S. A..
Abstract
We prove the existence of fixed points of monotone \(\rho\)-nonexpansive mappings in \(\rho\)-uniformly convex
modular function spaces. This is the modular version of Browder and Göhde fixed point theorems for
monotone mappings. We also discuss the validity of this result in modular function spaces where the
modular is uniformly convex in every direction. This property has never been considered in the context of
modular spaces.
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ISRP Style
Buthinah A. Bin Dehaish, Mohamed A. Khamsi, On monotone mappings in modular function spaces, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 8, 5219--5228
AMA Style
Dehaish Buthinah A. Bin, Khamsi Mohamed A., On monotone mappings in modular function spaces. J. Nonlinear Sci. Appl. (2016); 9(8):5219--5228
Chicago/Turabian Style
Dehaish, Buthinah A. Bin, Khamsi, Mohamed A.. "On monotone mappings in modular function spaces." Journal of Nonlinear Sciences and Applications, 9, no. 8 (2016): 5219--5228
Keywords
- Fixed point
- Krasnoselskii iteration
- modular function spaces
- monotone mapping
- nonexpansive mapping
- partially ordered
- uniformly convex
- uniformly convex in every direction.
MSC
References
-
[1]
V. A. Akimovič , The uniform convexity and uniform smoothness of Orlicz spaces, (Russian) Teor. Funkciĭ Funkcional. Anal. i Priložen., 15 (1972), 114--121
-
[2]
M. Bachar, M. A. Khamsi , Fixed points of monotone mappings and application to integral equations, Fixed Point Theory Appl., 2015 (2015), 7 pages
-
[3]
S. Banach , Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales, Fund. Math., 3 (1922), 133--181
-
[4]
B. A. Bin Dehaish, W. M. Kozlowski, Fixed point iteration processes for asymptotic pointwise nonexpansive mapping in modular function spaces, Fixed Point Theory Appl., 2012 (2012), 23 pages
-
[5]
F. E. Browder , Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A., 54 (1965), 1041--1044
-
[6]
S. Chen , Geometry of Orlicz spaces , With a preface by Julian Musielak, Dissertationes Math. (Rozprawy Mat.), 356 (1996), 204 pages
-
[7]
S. M. El-Sayed, A. C. M. Ran, On an iteration method for solving a class of nonlinear matrix equations, SIAM J. Matrix Anal. Appl., 23 (2002), 632--645
-
[8]
A. L. Garkavi, On the optimal net and best cross-section of a set in a normed space, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 26 (1962), 87--106
-
[9]
K. Goebel, W. A. Kirk, Topics in metric fixed point theory , Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1990)
-
[10]
D. Göhde, Zum Prinzip der kontraktiven Abbildung, (German) Math. Nachr., 30 (1965), 251--258
-
[11]
H. Hudzik, A. Kamińska, M. Masty lo , Geometric properties of some Caldern-Lozanovski spaces and Orlicz-Lorentz spaces , Houston J. Math., 22 (1996), 639--663
-
[12]
S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc., 59 (1976), 65--71
-
[13]
A. Kamińska, On uniform convexity of Orlicz spaces , Nederl. Akad. Wetensch. Indag. Math., 44 (1982), 27--36
-
[14]
M. A. Khamsi, W. A. Kirk, An introduction to metric spaces and fixed point theory , Pure and Applied Mathematics, Wiley-Interscience, New York (2001)
-
[15]
M. A. Khamsi, W. M. Kozlowski , On asymptotic pointwise contractions in modular function spaces, Nonlinear Anal., 73 (2010), 2957--2967
-
[16]
M. A. Khamsi, W. M. Kozlowski, Fixed point theory in modular function spaces, With a foreword by W. A. Kirk. Birkhäuser/Springer, Cham (2015)
-
[17]
M. A. Khamsi, W. M. Kozlowski, S. T. Chen, Some geometrical properties and fixed point theorems in Orlicz spaces, J. Math. Anal. Appl., 155 (1991), 393--412
-
[18]
M. A. Khamsi, W. M. Kozlowski, S. Reich, Fixed point theory in modular function spaces, Nonlinear Anal., 14 (1990), 935--953
-
[19]
W. M. Kozlowski, Modular function spaces, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York (1988)
-
[20]
M. A. Krasnoselskiĭ , Two remarks on the method of successive approximations, (Russian) Uspehi Mat. Nauk (N.S.), 10 (1955), 123--127
-
[21]
M. A. Krasnoselskiĭ, J. B. Rutickiĭ, Convex functions and Orlicz spaces, Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen (1961)
-
[22]
W. A. J. Luxemburg , Banach function spaces, Thesis, Technische Hogeschool te Delft, 1955 (1955), 70 pages
-
[23]
H. W. Milnes, Convexity of Orlicz spaces, Pacific J. Math., 7 (1957), 1451--1486
-
[24]
J. J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223--239
-
[25]
A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2004), 1435--1443
-
[26]
J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43 (1991), 153--159
-
[27]
V. Zizler , On some rotundity and smoothness properties of Banach spaces, Dissertationes Math. Rozprawy Mat., 87 (1971), 33 pages