# Some new cyclic admissibility type with uni-dimensional and multidimensional fixed point theorems and its applications

Volume 11, Issue 9, pp 1056--1069 Publication Date: June 19, 2018       Article History
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### Authors

Chirasak Mongkolkeha - Department of Mathematics, Statistics and Computer Sciences, Faculty of Liberal Arts and Science, Kasetsart University, Kamphaeng-Saen Campus, Nakhonpathom 73140, Thailand. Wutiphol Sintunavarat - Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani 12121, Thailand.

### Abstract

In this paper, we introduce the concept of a cyclic $(\alpha,\beta)$-admissible mapping type $S$ and the notion of an $(\alpha,\beta)-(\psi,\varphi)$-contraction type $S$. We also establish fixed point results for such contractions along with the cyclic $(\alpha,\beta)$-admissibility type $S$ in complete $b$-metric spaces and provide some examples for supporting our result. Applying our new results, we obtain fixed point results for cyclic mappings and multidimensional fixed point results. As application, the existence of a solution of the nonlinear integral equation is discussed.

### Keywords

• $\alpha$-admissible mappings
• cyclic $(\alpha,\beta)$-admissible mappings
• generalized weak contraction mappings
• multidimensional fixed points
• nonlinear integral equations

•  47H09
•  47H10
•  54H25

### References

• [1] M. Abbas, D. Dorić, Common fixed point theorem for four mappings satisfying generalized weak contractive condition, Filomat, 24 (2010), 1–10.

• [2] A. Aghajani, M. Abbas, J. R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca, 64 (2014), 941–960.

• [3] Y. I. Alber, S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert spaces, in: New Results in Operator Theory and its Applications, Theory Adv. Appl., 98 (1997), 7–22.

• [4] S. Alizadeh., F. Moradloub, P. Salimic, Some Fixed Point Results for ($\alpha,\beta)-(\psi,\phi$)-Contractive Mappings, Filomat, 28 (2014), 635–647.

• [5] A. D. Arvanitakis, A proof of the generalized Banach contraction conjecture, Proc. Amer. Math. Soc., 131 (2003), 3647– 3656.

• [6] S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales, Fund. Math., 3 (1922), 133–181.

• [7] V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum, 9 (2004), 43–53.

• [8] M. Boriceanu, M. Bota, A. Petrusel, Multivalued fractals in b-metric spaces, Cent. Eur. J. Math., 8 (2010), 367–377.

• [9] L. B. Ćirić, A generalization of Banach principle, Proc. Amer. Math. Soc., 45 (1947), 267–273.

• [10] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1 (1993), 5–11.

• [11] D. D. Dorić , Common fixed point for generalized ( $\psi,\phi$)-weak contractions, Appl. Math. Lett., 22 (2009), 1896–1900.

• [12] P. N. Dutta, B. S. Choudhury, A generalisation of contraction principle in metric spaces, Fixed Point Theory Appl., 2008 (2008), 8 pages.

• [13] M. A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), 604–608.

• [14] M. S. Khan, M. Swaleh, S. Sessa, Fixed points theorems by altering distances between the points, Bull. Aust. Math. Soc., 30 (1984), 1–9.

• [15] B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 226 (1977), 257–90.

• [16] B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal., 47 (2001), 2683–2693

• [17] 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.

• [18] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for $\alpha-\psi$-contractive type mappings, Nonlinear Anal., 4 (2012), 2154–2165.

• [19] W. Sintunavarat, Nonlinear integral equations with new admissibility types in b-metric spaces, J. Fixed Point Theory Appl., 18 (2016), 397–416.