# Fixed point results for generalized contractive mappings involving altering distance functions on complete quasi-metric spaces and applications

Volume 10, Issue 4, pp 1377--1398

Publication Date: 2017-04-20

http://dx.doi.org/10.22436/jnsa.010.04.09

### Authors

Yanbin Sang - School of Science, North University of China, Taiyuan, 030051, China.

### Abstract

In this paper, we introduce $\alpha-\psi-\phi$-Jachymski contractive mappings with generalized altering distance functions in the setting of quasi-metric spaces. Some theorems on the existence and uniqueness of fixed points for such mappings via admissible mappings are established. Utilizing above abstract results, we derive common fixed point theorem for two operators and multidimensional fixed point results for nonlinear mappings satisfying different kinds of contractive conditions on partially ordered metric spaces. Moreover, we present some examples and applications in a Fredholm integral equation and an initial value problem for partial differential equation of parabolic type.

### Keywords

Admissible mapping, altering distance, multidimensional, w-distance, partial order.

### References

[1] C. Alegre, J. Marín, S. Romaguera,/ A fixed point theorem for generalized contractions involving w-distances on complete quasi-metric spaces,/ Fixed Point Theory Appl.,/ 2014 (2014), 8 pages.
[2] S. Al-Homidan, Q. H. Ansari, J.-C. Yao,/ Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory,/ Nonlinear Anal.,/ 69 (2008), 126–139.
[3] H. H. Alsulami, S. Gülyaz I. M. Erhan,/ Fixed points of $\alpha$-admissible Meir-Keeler contraction mappings on quasi-metric spaces,/ J. Inequal. Appl.,/ 2015 (2015), 15 pages.
[4] H. H. Alsulami, S. Gülyaz , E. Karapınar, ˙I. M. Erhan,/ Fixed point theorems for a class of $\alpha$-admissible contractions and applications to boundary value problem,/ Abstr. Appl. Anal.,/ 2014 (2014), 10 pages.
[5] V. Berinde,/ Coupled fixed point theorems for $\phi$-contractive mixed monotone mappings in partially ordered metric spaces,/ Nonlinear Anal.,/ 75 (2012), 3218–3228.
[6] M. Borcut, V. Berinde,/ Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces,/ Appl. Math. Comput.,/ 218 (2012), 5929–5936.
[7] P. Chaipunya, Y. J. Cho, P. Kumam,/ Geraghty-type theorems in modular metric spaces with an application to partial differential equation,/ Adv. Difference Equ.,/ 2012 (2012), 12 pages.
[8] R. H. Haghi, S. Rezapour, N. Shahzad,/ Some fixed point generalizations are not real generalizations,/ Nonlinear Anal.,/ 74 (2011), 1799–1803.
[9] J. Jachymski,/ Equivalent conditions and the Meir-Keeler type theorems,/ J. Math. Anal. Appl.,/ 194 (1995), 293–303.
[10] G. Jungck, B. E. Rhoades,/ Fixed points for set valued functions without continuity,/ Indian J. Pure Appl. Math.,/ 29 (1998), 227–238.
[11] E. Karapınar, I . M. Erhan, A. Öztürk,/ Fixed point theorems on quasi-partial metric spaces,/ Math. Comput. Modelling,/ 57 (2013), 2442–2448.
[12] E. Karapınar, P. Kumam, P. Salimi,/ On $\alpha-\psi$-Meir-Keeler contractive mappings,/ Fixed Point Theory Appl.,/ 2013 (2013), 12 pages.
[13] H. Lakzian, D. Gopal, W. Sintunavarat,/ New fixed point results for mappings of contractive type with an application to nonlinear fractional differential equations,/ J. Fixed Point Theory Appl.,/ 18 (2016), 251–266.
[14] V. La Rosa, P. Vetro,/ Common fixed points for $\alpha-\psi-\phi$-contractions in generalized metric spaces,/ Nonlinear Anal. Model. Control,/ 19 (2014), 43–54.
[15] X.-L. Liu, A. H. Ansari, S. Chandok, C.-K. Park,/ Some new fixed point results in partial ordered metric spaces via admissible mappings and two new functions,/ J. Nonlinear Sci. Appl.,/ 9 (2016), 1564–1580.
[16] N. V. Luong, N. X. Thuan,/ Coupled fixed points in partially ordered metric spaces and application,/ Nonlinear Anal.,/ 74 (2011), 983–992.
[17] J. Marín, S. Romaguera, P. Tirado,/ Generalized contractive set-valued maps on complete preordered quasi-metric spaces,/ J. Funct. Spaces Appl.,/ 2013 (2013), 6 pages.
[18] A. Meir, E. Keeler,/ A theorem on contraction mappings,/ J. Math. Anal. Appl., 28 (1969), 326–329.
[19] S. Park,/ On generalizations of the Ekeland-type variational principles,/ Nonlinear Anal.,/ 39 (2000), 881–889.
[20] P. D. Proinov,/ Fixed point theorems in metric spaces,/ Nonlinear Anal.,/ 64 (2006), 546–557.
[21] A. Roldán, J. Martínez-Moreno, C. Roldán,/ Multidimensional fixed point theorems in partially ordered complete metric spaces,/ J. Math. Anal. Appl.,/ 396 (2012), 536–545.
[22] A. Roldán, J. Martínez-Moreno, C. Roldán, Y. J. Cho,/ Multidimensional fixed point theorems under ( $\psi,\phi$)-contractive conditions in partially ordered complete metric spaces,/ J. Comput. Appl. Math.,/ 273 (2015), 76–87.
[23] A. Roldán, J. Martínez-Moreno, C. Roldán, E. Karapınar,/ Some remarks on multidimensional fixed point theorems,/ Fixed Point Theory,/ 15 (2014), 545–558.
[24] B. Samet, C. Vetro, P. Vetro,/ Fixed point theorems for $\alpha-\psi$-contractive type mappings,/ Nonlinear Anal.,/ 75 (2012), 2154–2165.
[25] Y.-B. Sang, Q. Meng,/ Fixed point theorems with generalized altering distance functions in partially ordered metric spaces via w-distances and applications,/ Fixed Point Theory Appl.,/ 2015 (2015), 25 pages.
[26] T. Suzuki,/ Meir-Keeler contractions of integral type are still Meir-Keeler contractions,/ Int. J. Math. Math. Sci.,/ 2007 (2007), 6 pages.