**Volume 10, Issue 4, pp 1424--1432**

**Publication Date**: 2017-04-20

**Cüneyt Çevik**
- Department of Mathematics, Faculty of Science, Gazi University, 06500 Teknikokullar Ankara, Turkey.

**Ishak Altun**
- Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia.

**Hakan Şahin**
- Department of Mathematics, Faculty of Science, Gazi University, 06500 Teknikokullar Ankara, Turkey.

**Çetin Cemal Özeken**
- Department of Mathematics, Faculty of Science, Gazi University, 06500 Teknikokullar Ankara, Turkey.

In this paper, considering an order relation on a vector metric space which is introduced by Çevik and Altun in 2009, we present some fundamental fixed point results. Then, we provide some nontrivial examples show that the investigation of this work is significant.

Fixed point, Riesz space, vector metric space.

[1] C. D. Aliprantis, K. C. Border,/ Infinite dimensional analysis: A hitchhiker’s guide,/ Springer-Verlag,/ Berlin (1999).

[2] C. Çevik,/ On continuity of functions between vector metric spaces,/ J. Funct. Space,/ 2014 (2014), 6 pages.

[3] C. Çevik, I. Altun,/ Vector metric spaces and some properties,/ Topol. Methods Nonlinear Anal.,/ 34 (2009), 375–382.

[4] N. Hussain, C. Vetro, F. Vetro,/ Fixed point results for \(\alpha\)-implicit contractions with application to integral equations,/ Nonlinear Anal. Model. Control,/ 21 (2016), 362–378.

[5] P. Kumam, C. Vetro, F. Vetro,/ Fixed points for weak \(\alpha-\psi\)-contractions in partial metric spaces,/ Abstr. Appl. Anal.,/ 2013 (2013), 9 pages.

[6] W. A. J. Luxemburg, A. C. Zaanen,/ Riesz space: Vol. I,/ North-Holland Publishing Co., Amsterdam-London,/ (1971).

[7] J. J. Nieto, R. Rodríguez-López,/ Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,/ Order,/ 22 (2005), 223–239.

[8] J. J. Nieto, R. Rodríguez-López,/ Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations,/ Acta Mathematica Sinica (English Series),/ 23 (2007), 2205–2212.

[9] 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.