# Contractive mapping theorems in Partially ordered metric spaces

### Abstract

The purpose of this paper is to establish some coincidence, common fixed point theorems for monotone \(f\)-non decreasing self mappings satisfying certain rational type contraction in the context of a metric spaces endowed with partial order. Also, the results involving an integral type of such classes of mappings are discussed in application point of view. These results generalize and extend well known existing results in the literature.

### References

S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133–181.

B. K. Dass and S. Gupta, An extension of Banach contraction principle through rational expression, Indian J. Pure Appl. Math. 6 (1975), no. 12, 1455–1458.

S.K. Chetterjee, Fixed point theorems, C.R. Acad. Bulgare Sci. 25 (1972), 727–730.

M. Edelstein, On fixed points and periodic points under contraction mappings, J. Lond. Math. Soc. 37 (1962), 74–79.

G.C. Hardy, T.Rogers, A generalization of fixed point theorem of S. Reich, Can. Math. Bull. 16 (1973), 201–206.

D.S. Jaggi, Some unique fixed point theorems, Indian J. Pure Appl. Math. 8, (1977) 223–230.

R. Kannan, Some results on fixed points. II, Amer. Math. Monthly 76 (1969), 405–408.

S. Reich, Some remarks concerning contraction mappings, Canadian Math. Bull. 14 (1971), 121–124.

M.R. Singh, A.K. Chatterjee, Fixed point theorems, Commun. Fac. Sci. Univ. Ank. Series A1 37 (1988), 1–4.

D. R. Smart, Fixed point theorems, Cambridge University Press, London, 1974.

C. S. Wong, Common fixed points of two mappings, Pacific J. Math. 48 (1973), 299–312.

R. P. Agarwal, M. A. El-Gebeily and D. O’Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal. 87 (2008), no. 1, 109–116.

I. Altun, B. Damjanovi ́c and D. Djori ́c, Fixed point and common fixed point theorems on ordered cone metric spaces, Appl. Math. Lett. 23 (2010), no. 3, 310–316.

A. Amini-Harandi and H. Emami, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal. 72 (2010), no. 5, 2238–2242.

A. Chanda, B. Damjanovi ́c and L. K. Dey, Fixed point results on θ-metric spaces via simulation functions, Filomat 31 (2017), no. 11, 3365–3375.

M. Arshad, A. Azam and P. Vetro, Some common fixed point results in cone metric spaces, Fixed Point Theory Appl. 2009, Art. ID 493965, 11 pp.

M. Arshad, J. Ahmad and E. Karapınar, Some common fixed point results in rectangular metric spaces, Int. J. Anal. 2013, Art. ID 307234, 7 pp.

T.G. Bhaskar, V. Lakshmikantham, Fixed point theory in partially ordered metric spaces and applications, Nonlinear Anal. Theory Methods Appl. 65 (2006), 1379–1393.

J. Harjani, B. Lo ́pez and K. Sadarangani, A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space, Abstr. Appl. Anal. 2010, Art. ID 190701, 8 pp.

S. Hong, Fixed points of multivalued operators in ordered metric spaces with applications, Nonlinear Anal. 72 (2010), no. 11, 3929–3942.

I.X. Liu, M. Zhou, B. Damjanovic, Nonlinear Operators in Fixed point theory with Applications to Fractional Differential and Integral Equations, Journal of Function spaces 2018, Article ID 9863267, 11 pages (2018), Doi:10,1155/2018/9063267.

M. Zhou, X. Liu, B. Damjanovic, Arslan Hojat Ansari, Fixed point theorems for several types of Meir-Keeler contraction mappings in Ms-metric spaces, J. Comput. Anal. Appl. 25 (2018), no. 7, 1337–1353.

J.J. Nieto, R.R. Lopéz, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005), 223–239.

J.J. Nieto, R.R. Lopéz: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equation, Acta Math. Sin. Engl. Ser. 23(12), 2205–2212 (2007).

Öztürk, Mahpeyker; Başarır, Metin, On some common fixed point theorems with rational expressions on cone metric spaces over a Banach algebra, Hacet. J. Math. Stat. 41 (2012), no. 2, 211–222.

Ran, André C. M.; Reurings, Martine C. B. A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1435-1443.

S. Chandok, Some common fixed point results for generalized weak contractive mappings in partially ordered matrix spaces, Journal of Nonlinear Anal. Opt. 4 (2013), 45–52 .

S. Chandok, Some common fixed point results for rational type contraction mappings in partially ordered metric spaces, Math. Bohem. 138 (2013), no. 4, 407–413.

E. S. Wolk, Continuous convergence in partially ordered sets, General Topology and Appl. 5, (1975), no. 3, 221–234.

X. Zhang, Fixed point theorems of multivalued monotone mappings in ordered metric spaces, Appl. Math. Lett. 23 (2010), no. 3, 235–240.

*CUBO, A Mathematical Journal*,

*22*(2), 203–214. https://doi.org/10.4067/S0719-06462020000200203

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