## Nihonkai Mathematical Journal

### Fixed point theorems for asymptotic mappings of a generalized contractive type in complete metric spaces

#### Abstract

We consider an asymptotic version of $\alpha$-$\psi$ contractive mappings. We show the existence and uniqueness of fixed points. Caccioppoli's fixed point theorem is deduced from main results in this paper. Moreover, we discuss an asymptotic version of mappings related with $(c)$-comaprison functions.

#### Note

The authors would like to thank the referees for valuable suggestions and comments.

#### Article information

Source
Nihonkai Math. J., Volume 29, Number 1 (2018), 21-28.

Dates
Revised: 11 June 2018
First available in Project Euclid: 6 February 2019

https://projecteuclid.org/euclid.nihmj/1549422081

Mathematical Reviews number (MathSciNet)
MR3908816

Zentralblatt MATH identifier
07063838

#### Citation

Toyoda, Masashi; Watanabe, Toshikazu. Fixed point theorems for asymptotic mappings of a generalized contractive type in complete metric spaces. Nihonkai Math. J. 29 (2018), no. 1, 21--28. https://projecteuclid.org/euclid.nihmj/1549422081

#### References

• H. Ayde and E. Karapinar, Fixed point results for generalized $\alpha$-$\psi$-contractions in metric-like spaces and applications, Electronic J. Diff. Equ. 2015 (2015), 1–15.
• S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133–181.
• V. Berinde, Iterative approximation of fixed points, Springer-Verlag Berlin Heidelberg, 2007.
• D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458–464.
• S. Chandok, M. S. Khan and T. D. Narang, Fixed point theorem in partially ordered metric spaces for generalized contraction mappings, Azerb. J. Math. 5 (2015), 89–96.
• G. Durmaz, G. Minak and I. Altun, Fixed point results for $\alpha$-$\psi$-contractive mappings including almost contractions and applications, Abstr. Appl. Anal. 2014, Art. ID 869123, 10 pp.
• A. P. Farajzadeh, A. Kaewcharoen and S. Plubtieng, An application of fixed point theory to a nonlinear differential equation, Abstr. Appl. Anal. 2014, Art. ID 605405, 7 pp.
• S. Gülyaz, Fixed points of $\alpha$-$\psi$ contraction mappings on quasi-$b$-metric-like spaces, J. Nonlinear Convex Anal. 17 (2016), 1439–1447.
• N. Hussain, J. Ahmad and A. Azam, Generalized fixed point theorems formulti-valued $\alpha$-$\psi$-contractive mappings, J. Inequal. Appl. 2014, 2014:348, 15 pp.
• N. Hussain, M. A. Kutbi, S. Khaleghizadeh and P. Salimi, Discussions on recent results for $\alpha$-$\psi$-contractive mappings, Abstr. Appl. Anal. 2014, Art. ID 456482, 13 pp.
• J. Jachymski and I. Jóźwik, On Kirk's asymptotic contractions, J. Math. Anal. Appl. 300 (2004), 147–159.
• E. Karapinar, Discussion on $\alpha$-$\psi$ contractions on generalized metric spaces, Abstr. Appl. Anal. 2014, Art. ID 962784, 7 pp.
• E. Karapinar and B. Samet, Generalized $\alpha$-$\psi$ contractive tyep mappings and related fixed point theorems with applications, Abstr. Appl. Anal. 2012, Art. ID 793486, 17 pp.
• W. A. Kirk, Contraction mappings and extensions, Handbook of metric fixed point theory, 1–34, Kluwer Acad. Publ., Dordrecht, 2001.
• W. A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003), 645–650.
• M. A. Kutbi and W. Sintunavarat, On the weakly $(\alpha,\psi,\xi)$-contractive condition for multi-valued operators in metric spaces and related fixed point results, Open Math. 2016; 14: 167–180.
• A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969), 326–329.
• J J. Nieto and R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005), 223–239.
• A. C. M. Ran and 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.
• B. Samet, C. Vetro and P. Vetro, Fixed point theorems for $\alpha$-$\psi$ contractive type mappings, Nonlinear Anal. 75 (2012), 2154–2165.
• T. Suzuki, Fixed-point theorem for asymptotic contractions of Meir-Keeler type in complete metric spaces, Nonlinear Anal. 64 (2006) 971–978.
• M. Toyoda and T. Watanabe, Caccioppoli's fixed point theorem in the setting of metric spaces with a partial order, to appear in the proceedings of the fifth Asian conference on Nonlinear Analysis and Optimization.
• J. Weissinger, Zur Theorie und Anwendung des Iterationsverfahrens, Math. Nachr. 8 (1952), 193–212.