Document Type: Research Paper

Authors

Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

Abstract

Here, a new certain class of contractive mappings in the b-metric spaces is introduced. Some fixed point theorems are proved which generalize and modify the recent results in the literature. As an application, some results in the b-metric spaces endowed with a partial ordered are proved.

Keywords

Main Subjects

[1] A. Aghajani, M. Abbas and J.R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered b−metric spaces, Math. Slovaca, 4 (2014), 941-960.

[2] I.A. Bakhtin, The contraction mapping principle in quasimetric spaces, (Russian), Func. An., Gos. Ped. Inst. Unianowsk 30 (1989), 26-37.

[3] V. Berinde, Generalized contractions in quasimetric spaces, Seminar on Fixed Point Theory, Preprint no. 3 (1993), 3-9.

[4] F. Bojor, Fixed point theorems for Reich type contraction on metric spaces with a graph, Nonlinear Anal. 75 (2012), 3895-3901.

[5] M. Boriceanu, Strict fixed point theorems for multivalued operators in b−metric spaces, Int. J. Modern Math., 4(3) (2009), 285-301.

[6] S. Czerwick, Contraction mappings in b-metric spaces, Acta Mathematica et Informatica Universitatis Ostraviensis, 1 (1993), 5-11.

[7] N. Hussain, V. Parvaneh, J.R. Roshan and Z Kadelburg, Fixed points of cyclic weakly (ψ, φ, L, A, B)- contractive mappings in ordered b-metric spaces with applications, Fixed Point Theory Appl. 256 (2013).

[8] J.R. Roshan, V. Parvaneh and I. Altun, Some coincidence point results in ordered b-metric spaces and applications in a system of integral equations, Applied Mathematics and Computation 226 (2014), 725-737.

[9] Z. Mustafa, J.R. Roshan, V. Parvaneh and Z. Kadelburg, Fixed point theorems for weakly T-Chatterjea and weakly T-Kannan contractions in b-metric spaces, Journal of Inequalities and Applications 46 (2014).

[10] N. Hussain, J.R. Roshan, V. Parvaneh and M. Abbas, Common fixed point results for weak contractive mappings in ordered b-dislocated metric spaces with applications, Journal of Inequalities and Applications 486 (2013).

[11] Z. Mustafa, J.R. Roshan, V. Parvaneh and Z. Kadelburg, Some common fixed point results in ordered partial b-metric spaces, Journal of Inequalities and Applications 562 (2013).

[12] N. Hussain, M. A. Kutbi and P. Salimi, Fixed point theory in α-complete metric spaces with applications, Abstract and Applied Analysis 280817 (2014), 11 pages.

[13] N. Hussain, P. Salimi and A. Latif, Fixed point results for single and set-valued α-η-ψ-contractive mappings, Fixed Point Theory and Appl. 212 (2013).

[14] J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 136 (2008), 1359–1373.

[15] R. Johnsonbaugh, Discrete Mathematics, Prentice-Hall, Inc., New Jersey, 1997.

[16] E. Karapınar, P. Kumam and P. Salimi, On α-ψ-Meir-Keeler contractive mappings, Fixed Point Theory and Appl. 94 (2013).

[17] M.S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distancces between the points, Bull. Aust. Math. Soc. 30 (1984), 1-9.

[18] M.A. Khamsi, N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal. 73 (2010), 3123-3129.

[19] P. Kumam, H. Rahimi and G. Soleimani Rad, The existence of fixed and periodic point theorems in cone metric type spaces, J. Nonlinear Sci. Appl. 7 (4) (2014), 255-263.

[20] P. Kumam and A. Roldan, On existence and uniqueness of g-best proximity points under (φ, θ, α, g)- contractivity conditions and consequences, Abstract and Applied Analysis, Article ID 234027 (2014), 14 pages.

[21] J.J. Nieto and R. Rodrıguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005), 223-239.

[22] J.J. Nieto and R. Rodr´ıguez-L´opez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin., (English Ser.), 23 (2007) 2205-2212.

[23] H. Rahimi and G. Soleimani Rad, Fixed point theory in various spaces, Lambert Academic Publishing, Germany, 2013.

[24] H. Rahimi and G. Soleimani Rad, Some fixed point results in metric type space, J. Basic. Appl. Sci. Res. 2 (9) (2012), 9301-9308.

[25] H. Rahimi, P. Vetro, G. Soleimani Rad, Some common fixed point results for weakly compatible mappings in cone metric type space, Miskolc Math. Notes. 14 (1) (2013), 233-243.

[26] A.C.M. Ran and M.C. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), 1435-1443.

[27] J.R. Roshan, V. Parvaneh, S. Sedghi, N. Shobkolaei and W. Shatanawi, Common Fixed Points of Almost Generalized (ψ, φ)s−Contractive Mappings in Ordered b−Metric Spaces, Fixed Point Theory and Appl. 159 (2013).

[28] P. Salimi, A. Latif and N. Hussain, Modified α-ψ-contractive mappings with applications, Fixed Point Theory and Appl. 151 (2013).

[29] P. Salimi and P. Vetro, A result of Suzuki type in partial G-metric spaces, Acta Mathematica Scientia 34B(2) (2014), 1-11.

[30] B. Samet, C. Vetro and P. Vetro, Fixed point theorems for α-ψ-contractive type mappings, Nonlinear Anal. 75 (2012), 2154-2165.