Fixed points of nonlinear and asymptotic contractions in the modular space. (English) Zbl 1152.47043

The main result of the paper is the following fixed point theorem for nonlinear contractions in modular spaces. Theorem 2.1.Let \(X_{\rho}\) be a \(\rho\)-complete modular space, where \(\rho\) satisfies the following condition:
\[ \rho(2x_n)\to 0 \text{ as } n\to +\infty \text{ whenever } \rho(x_n)\to 0 \text{ as } n\to +\infty. \]
Assume that \(\psi:\mathbb{R}_+\to \mathbb{R}_+\) is an increasing, upper semicontinuous function such that \(\psi(t)<t\) for each \(t>0\). Let \(B\) be a \(\rho\)-closed subset of \(X_{\rho}\) and \(T:B\to B\) be an operator such that there exist \(c,l\in \mathbb{R}_+\) with \(c>l\) so that
\[ \rho(c(Tx-Ty))\leq \psi(\rho(l(x-y))) \text{ for all } x,y\in B. \]
Then \(T\) has a fixed point.
An asymptotic version of this result is also given.


47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
54H25 Fixed-point and coincidence theorems (topological aspects)
Full Text: DOI EuDML


[1] H. Nakano, Modular Semi-Ordered Spaces, Tokyo, Japan, 1959. · Zbl 0086.09104
[2] J. Musielak and W. Orlicz, “On modular spaces,” Studia Mathematica, vol. 18, pp. 49-65, 1959. · Zbl 0086.08901
[3] T. Dominguez Benavides, M. A. Khamsi, and S. Samadi, “Uniformly Lipschitzian mappings in modular function spaces,” Nonlinear Analysis, vol. 46, no. 2, Ser. A: Theory Methods, pp. 267-278, 2001. · Zbl 1001.47039 · doi:10.1016/S0362-546X(00)00117-6
[4] A. Hajji and E. Hanebaly, “Fixed point theorem and its application to perturbed integral equations in modular function spaces,” Electronic Journal of Differential Equations, vol. 2005, no. 105, pp. 1-11, 2005. · Zbl 1110.47045
[5] E. Hanebaly, “Fixed point theorems in modular space,” November 2005, http://arxiv.org/abs/math.FA/0511319v1.
[6] M. A. Khamsi, “Nonlinear semigroups in modular function spaces,” Mathematica Japonica, vol. 37, no. 2, pp. 291-299, 1992. · Zbl 0766.47031
[7] D. W. Boyd and J. S. W. Wong, “On nonlinear contractions,” Proceedings of the American Mathematical Society, vol. 20, no. 2, pp. 458-464, 1969. · Zbl 0175.44903 · doi:10.2307/2035677
[8] I. D. Arandelović, “On a fixed point theorem of Kirk,” Journal of Mathematical Analysis and Applications, vol. 301, no. 2, pp. 384-385, 2005. · Zbl 1075.47031 · doi:10.1016/j.jmaa.2004.07.031
[9] M. Edelstein, “On fixed and periodic points under contractive mappings,” Journal of the London Mathematical Society, vol. 37, no. 1, pp. 74-79, 1962. · Zbl 0113.16503 · doi:10.1112/jlms/s1-37.1.74
[10] L. B. Ćirić, “A generalization of Banach’s contraction principle,” Proceedings of the American Mathematical Society, vol. 45, no. 2, pp. 267-273, 1974. · Zbl 0291.54056 · doi:10.2307/2040075
[11] E. Rakotch, “A note on contractive mappings,” Proceedings of the American Mathematical Society, vol. 13, no. 3, pp. 459-465, 1962. · Zbl 0105.35202 · doi:10.2307/2034961
[12] S. Reich, “Fixed points of contractive functions,” Bollettino dell’Unione Matematica Italiana (4), vol. 5, pp. 26-42, 1972. · Zbl 0249.54026
[13] W. A. Kirk, “Contraction mappings and extensions,” in Handbook of Metric Fixed Point Theory, W. A. Kirk and B. Sims, Eds., pp. 1-34, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. · Zbl 1019.54001
[14] A. Ait Taleb and E. Hanebaly, “A fixed point theorem and its application to integral equations in modular function spaces,” Proceedings of the American Mathematical Society, vol. 128, no. 2, pp. 419-426, 2000. · Zbl 0928.46005 · doi:10.1090/S0002-9939-99-05546-X
[15] R. Caccioppoli, “Una teorem general sull’esistenza di elementi uniti in una transformazione funzionale,” Rendiconti dell’Accademia Nazionale dei Lincei, vol. 11, pp. 794-799, 1930.
[16] F. E. Browder, “Nonlinear operators and nonlinear equations of evolution in Banach spaces,” in Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968), pp. 1-308, American Mathematical Society, Providence, RI, USA, 1976. · Zbl 0327.47022
[17] Y.-Z. Chen, “Asymptotic fixed points for nonlinear contractions,” Fixed Point Theory and Applications, vol. 2005, no. 2, pp. 213-217, 2005. · Zbl 1097.54039 · doi:10.1155/FPTA.2005.213
[18] P. Gerhardy, “A quantitative version of Kirk’s fixed point theorem for asymptotic contractions,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 339-345, 2006. · Zbl 1094.47050 · doi:10.1016/j.jmaa.2005.04.039
[19] J. Jachymski and I. Jóźwik, “On Kirk’s asymptotic contractions,” Journal of Mathematical Analysis and Applications, vol. 300, no. 1, pp. 147-159, 2004. · Zbl 1064.47052 · doi:10.1016/j.jmaa.2004.06.037
[20] W. A. Kirk, “Fixed points of asymptotic contractions,” Journal of Mathematical Analysis and Applications, vol. 277, no. 2, pp. 645-650, 2003. · Zbl 1022.47036 · doi:10.1016/S0022-247X(02)00612-1
[21] T. Suzuki, “Fixed-point theorem for asymptotic contractions of Meir-Keeler type in complete metric spaces,” Nonlinear Analysis, vol. 64, no. 5, pp. 971-978, 2006. · Zbl 1101.54047 · doi:10.1016/j.na.2005.04.054
[22] H.-K. Xu, “Asymptotic and weakly asymptotic contractions,” Indian Journal of Pure and Applied Mathematics, vol. 36, no. 3, pp. 145-150, 2005. · Zbl 1090.47047
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