Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Physical Sciences, Ahmadu Bello University, Zaria, Nigeria.

Abstract

In this paper, a new general class of contraction, namely admissible hybrid $(G$-$\alpha$-$\phi)$-contraction is introduced and some fixed point theorems that cannot be deduced from their corresponding ones in metric spaces are proved. The distinction of this family of contractions is that its contractive inequality can be specialized in several ways, depending on multiple parameters. Consequently, several corollaries, including some recently announced results in the literature are highlighted and analyzed. Nontrivial comparative examples are constructed to validate the assumptions of our obtained theorems. We further examine Ulam-type stability and well-posedness for the new contraction proposed herein. In addition, one of our obtained corollaries is applied to set up novel existence conditions for the solution of a class of integral equations. There is an open problem concerning the discretized population balance model, whose solution may be analyzed using the methods established here.

Keywords

[1] M. Alghamdi and E. Karapınar, $G-\beta-\psi$-contractive type mappings in $G$-metric spaces, Fixed Point Theory Appl., 1(123) (2013), pp. 1-17.
[2] E. Altiparmak and I. Karahan, Fixed point theorems for Geraghty type contraction mappings in complete partial $b_v$-metric spaces, Sahand Commun. Math. Anal., 18(2) (2021), pp. 45-62.
[3] A.H. Ansari, S. Chandok, N. Hussain, Z. Mustafa and M.M.M. Jaradat, Some common fixed point theorems for weakly $\alpha$-admissible pairs in $G$-metric spaces with auxiliary functions, J. Math. Anal., 8(3) (2017), pp. 80-107.
[4] J. Chen, C. Zhu and L. Zhu, A note on some fixed point theorems on $G$-metric spaces, J. Appl. Anal. Comput., 11(1) (2021), pp. 101-112.
[5] M. Jleli and B. Samet, Remarks on $G$-metric spaces and fixed point theorems, Fixed Point Theory Appl., 1(210) (2012), pp. 1-7.
[6] E. Karapınar and R.P. Agarwal, Further fixed point results on $G$-metric space, Fixed Point Theory Appl., 1(154) (2013), pp. 1-14.
[7] E. Karapınar and A. Fulga, An admissible hybrid contraction with an Ulam type stability, Demonstr. Math., 52 (2019), pp. 428-436.
[8] E. Karapınar, M. De La Sen and A. Fulga, A note on the Gornicki-Proinov type contraction, J. Funct. Spaces, 2021 (2021), pp. 1-8.
[9] S.S. Mohammed, I. Fulatan and Y. Sirajo, Fixed points of $p$-hybrid-fuzzy contractions, Sahand Commun. Math. Anal., 18(3) (2021), pp. 1-25.
[10] Z. Mustafa, A New Structure for Generalized Metric Spaces - with Applications to Fixed Point Theory, PhD Thesis, University of Newcastle, Australia, 2005.
[11] Z. Mustafa and B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal., 7(2) (2006), pp. 289-297.
[12] Z. Mustafa, H. Obiedat and F. Awawdeh, Some fixed point theorem for mapping on complete $G$-metric spaces, Fixed Point Theory Appl., 2008 (2008), pp. 1-12.
[13] M. Noorwali and S.S. Yeşilkaya, On Jaggi-Suzuki-type hybrid contraction mappings, J. Funct. Spaces, 2021 (2021) pp. 1-7.
[14] O. Popescu, Some new fixed point theorems for $\alpha$-Geraghty contraction type maps in metric spaces, Fixed Point Theory Appl., 1(190) (2014), pp. 1-12.
[15] B. Samet, C. Vetro and F. Vetro, Remarks on $G$-metric spaces, Int. J. Anal., 2013 (2013), pp. 1-6.
[16] W. Shatanawi, Fixed point theory for contractive mappings satisfying $\Phi$-maps in $G$-metric spaces, Fixed Point Theory Appl., 2010 (2010), pp. 1-9.
[17] I. Yildirim, Fixed point results for $\mathcal{F}$-Hardy-Rogers contractions via Mann's iteration process in complete convex $b$-metric spaces, Sahand Commun. Math. Anal., 19(2) (2022), pp. 15-32.