Relation theoretic metrical fixed point results for Suzuki type $\mathcal{Z_\mathcal{R}}$-contraction with an application

Md Hasanuzzaman; Mohammad Imdad; Md Hasanuzzaman; Mohammad Imdad

doi:10.3934/math.2020137

AIMS Mathematics

2020, Volume 5, Issue 3: 2071-2087. doi: 10.3934/math.2020137

Previous Article Next Article

Research article

Relation theoretic metrical fixed point results for Suzuki type $\mathcal{Z_\mathcal{R}}$-contraction with an application

Md Hasanuzzaman ^,,
Mohammad Imdad

Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India

Received: 09 December 2019 Accepted: 17 February 2020 Published: 24 February 2020
MSC : 47H10, 54H25

In this paper, we introduce the concept of Suzuki type $\mathcal{Z_\mathcal{R}}$-contraction by unifying the definitions of Suzuki type $\mathcal{Z}$-contraction and $\mathcal{Z_\mathcal{R}}$-contraction and also provide examples to highlight the genuineness of our newly introduced contraction over earlier mentioned ones. Chiefly, we prove an existence and corresponding uniqueness fixed point results for Suzuki type $\mathcal{Z_\mathcal{R}}$-contraction employing an amorphous binary relation on metric spaces without completeness and also furnish an illustrative example to demonstrate the utility of our main results. Finally, we utilize our main results to discuss the existence and uniqueness of solutions of a family of nonlinear matrix equations.
- fixed points,
- Suzuki type $\mathcal{Z}_{\mathcal{R}}$-contraction,
- simulation functions,
- binary relations,
- matrix equations
Citation: Md Hasanuzzaman, Mohammad Imdad. Relation theoretic metrical fixed point results for Suzuki type $\mathcal{Z_\mathcal{R}}$-contraction with an application[J]. AIMS Mathematics, 2020, 5(3): 2071-2087. doi: 10.3934/math.2020137

Related Papers:

Abstract

In this paper, we introduce the concept of Suzuki type $\mathcal{Z_\mathcal{R}}$-contraction by unifying the definitions of Suzuki type $\mathcal{Z}$-contraction and $\mathcal{Z_\mathcal{R}}$-contraction and also provide examples to highlight the genuineness of our newly introduced contraction over earlier mentioned ones. Chiefly, we prove an existence and corresponding uniqueness fixed point results for Suzuki type $\mathcal{Z_\mathcal{R}}$-contraction employing an amorphous binary relation on metric spaces without completeness and also furnish an illustrative example to demonstrate the utility of our main results. Finally, we utilize our main results to discuss the existence and uniqueness of solutions of a family of nonlinear matrix equations.

References

[1]	S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1992), 133-181.
[2]	M. Edelstein, On fixed and periodic points under contractive mappings, J. Lond. Math. Soc., 1 (1962), 74-79.
[3]	J. Matkowski, Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc., 62 (1977), 344-348. doi: 10.1090/S0002-9939-1977-0436113-5
[4]	D. W. Boyd, J. S. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., 20 (1969), 458-464. doi: 10.1090/S0002-9939-1969-0239559-9
[5]	J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc., 136 (2008), 1359-1373.
[6]	B. E. Rhoades, A comparison of various definitions of contractive mappings, T. Am. Math. Soc., 226 (1977), 257-290. doi: 10.1090/S0002-9947-1977-0433430-4
[7]	D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory A., 2012 (2012), 94.
[8]	A. Alam, M. Imdad, Relation-theoretic contraction principle, J. Fixed Point Theory A., 17 (2015), 693-702. doi: 10.1007/s11784-015-0247-y
[9]	L. Ćirić, Some recent results in fixed point theory, University of Belgrade, Beograd, Serbia, 2003.
[10]	A. Razani, Results in fixed point theory, Andisheh Zarrin, Ghazvin, Iran, 2010.
[11]	A. Razani, R. Moradi, Fixed point theory in modular space, Saieh Ghostar publisher, Qazvin, 2006.
[12]	A. Razani, A fixed point theorem in the menger probabilistic metric space, New Zealand J. Math., 35 (2006), 109-114.
[13]	A. Razani, Existence of fixed point for the nonexpansive mapping of intuitionistic fuzzy metric spaces, Chaos Soliton. Fract., 30 (2006), 367-373. doi: 10.1016/j.chaos.2005.10.010
[14]	A. Razani, An existence theorem for ordinary differential equation in menger probabilistic metric space, Miskolc Math. Notes, 15 (2014), 711-716. doi: 10.18514/MMN.2014.640
[15]	A. Razani, A contraction theorem in fuzzy metric spaces, Fixed Point Theory Appl., 2005 (2005), 257-265.
[16]	F. Khojasteh, A. Razani, S. Moradi, A fixed point of generalized TF-contraction mappings in cone metric spaces, Fixed Point Theory A., 2011 (2011), 14.
[17]	M. Turinici, Fixed points for monotone iteratively local contractions, Demonstr. Math., 19 (1986), 171-180.
[18]	A. C. Ran, M. C. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 2004 (2004), 1435-1443.
[19]	B. Samet, M. Turinici, Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications, Commun. Math. Anal., 13 (2012), 82-97.
[20]	A. Alam, M. Imdad, Relation-theoretic metrical coincidence theorems, Filomat, 31 (2017), 4421-4439. doi: 10.2298/FIL1714421A
[21]	F. Khojasteh, S. Shukla, S. Radenović, A new approach to the study of fixed point theory for simulation functions, Filomat, 29 (2015), 1189-1194. doi: 10.2298/FIL1506189K
[22]	K. Sawangsup, W. Sintunavarat, On modified Z-contractions and an iterative scheme for solving nonlinear matrix equations, J. Fixed Point Theory A., 20 (2018), 80.
[23]	T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (2009), 5313-5317. doi: 10.1016/j.na.2009.04.017
[24]	P. Kumam, D. Gopal, L. Budhiyi, A new fixed point theorem under Suzuki type Z-contraction mappings, J. Math. Anal., 8 (2017), 113-119.
[25]	E. Karapınara, Fixed points results via simulation functions, Filomat, 30 (2016), 2343-2350. doi: 10.2298/FIL1608343K
[26]	H. H. Alsulami, E. Karapınar, F. Khojasteh, et al. A proposal to the study of contractions in quasimetric spaces, Discrete Dyn. Nat. Soc., 2014 (2014), 269286.
[27]	A. F. Roldán López-de Hierro, E. Karapınar, C. Roldán López-de Hierro, et al. Coincidence point theorems on metric spaces via simulation functions, J. Comput. Appl. Math., 275 (2015), 345-355. doi: 10.1016/j.cam.2014.07.011
[28]	A. Hussain, T. Kanwal, Z. Mitrović, et al. Optimal solutions and applications to nonlinear matrix and integral equations via simulation function, Filomat, 32 (2018), 6087-6106. doi: 10.2298/FIL1817087H
[29]	A. Chanda, L. K. Dey, S. Radenović, Simulation functions: A survey of recent results, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A. Mat., 113 (2019), 2923-2957.
[30]	S. Radenovic, F. Vetro, J. Vujaković, An alternative and easy approach to fixed point results via simulation functions, Demonstr. Math., 50 (2017), 223-230. doi: 10.1515/dema-2017-0022
[31]	X. L. Liu, A. H. Ansari, S. Chandok, et al. On some results in metric spaces using auxiliary simulation functions via new functions, J. Comput. Anal. Appl., 24 (2018), 1103-1114.
[32]	S. Radenovic, S. Chandok, Simulation type functions and coincidence points, Filomat, 32 (2018), 141-147. doi: 10.2298/FIL1801141R
[33]	E. Karapınar, F. Khojasteh, An approach to best proximity points results via simulation functions, J. Fixed Point Theory A., 19 (2017), 1983-1995. doi: 10.1007/s11784-016-0380-2
[34]	A. Alam, M. Imdad, Nonlinear contractions in metric spaces under locally T-transitive binary relations, Fixed Point Theor-Ro, 19 (2018), 13-24. doi: 10.24193/fpt-ro.2018.1.02
[35]	B. Kolman, R. C. Busby, S. Ross, Discrete mathematical structures, 3 Eds., PHI Pvt. Ltd., New Delhi, 2000.
[36]	A. Razani, Weak and strong detonation profiles for a qualitative model, J. Math. Anal. Appl., 276 (2002), 868-881. doi: 10.1016/S0022-247X(02)00459-6
[37]	A. Razani, Subsonic detonation waves in porous media, Phys. Scripta, 94 (2019), 085209.
[38]	A. Razani, Fixed points for total asymptotically nonexpansive mappings in a new version of bead space, Int. J. Ind. Math., 6 (2014), 329-332.
[39]	M. Berzig, B. Samet, Solving systems of nonlinear matrix equations involving Lipshitzian mappings, Fixed Point Theory A., 2011 (2011), 89.
[40]	M. Berzig, Solving a class of matrix equations via the Bhaskar-Lakshmikantham coupled fixed point theorem, Appl. Math. Lett., 25 (2012), 1638-1643. doi: 10.1016/j.aml.2012.01.028
[41]	J. Long, X. Hu, L. Zhang, On the hermitian positive definite solution of the nonlinear matrix equation X + A^X^-1 A+ B^X^-1 B=I, Bull. Braz. Math. Soc., 39 (2008), 371-386. doi: 10.1007/s00574-008-0011-7

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)