Fixed point results for generalized almost contractions and application to a nonlinear matrix equation

Koti N. V. V. V. Prasad; Vinay Mishra; Zoran D. Mitrović; Ahmad Aloqaily; Nabil Mlaiki; Koti N. V. V. V. Prasad; Vinay Mishra; Zoran D. Mitrović; Ahmad Aloqaily; Nabil Mlaiki

doi:10.3934/math.2024600

AIMS Mathematics

2024, Volume 9, Issue 5: 12287-12304. doi: 10.3934/math.2024600

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Research article

Fixed point results for generalized almost contractions and application to a nonlinear matrix equation

1.
Department of Mathematics, Guru Ghasidas Vishwavidyalaya, A Central University, 495009, Bilaspur (C.G.), India
2.
Faculty of Electrical Engineering, University of Banja Luka, Patre 5, 78000 Banja Luka, Bosnia and Herzegovina
3.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh, 11586 Saudi Arabia
4.
School of computer, Data and Mathematical Sciences, Western Sydney University, Australia-Sydney-2150

Received: 02 December 2023 Revised: 12 March 2024 Accepted: 21 March 2024 Published: 28 March 2024
MSC : 47H10, 54H25

The goal of this paper was to improve some known results of fixed points by using $ w $-distances and properties of locally symmetric $ \mathcal{H} $-transitivity of binary relations. Also, we gave the application of the obtained results for finding the solution of nonlinear matrix equations. Finally, we gave a numerical example to demonstrate the applicability of our results.
Citation: Koti N. V. V. V. Prasad, Vinay Mishra, Zoran D. Mitrović, Ahmad Aloqaily, Nabil Mlaiki. Fixed point results for generalized almost contractions and application to a nonlinear matrix equation[J]. AIMS Mathematics, 2024, 9(5): 12287-12304. doi: 10.3934/math.2024600

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Abstract

The goal of this paper was to improve some known results of fixed points by using $ w $-distances and properties of locally symmetric $ \mathcal{H} $-transitivity of binary relations. Also, we gave the application of the obtained results for finding the solution of nonlinear matrix equations. Finally, we gave a numerical example to demonstrate the applicability of our results.

References

[1]	V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum, 9 (2004), 43–54.
[2]	V. Berinde, General constructive fixed point theorems for Ćirić-type almost contractions in metric spaces, Carpathian J. Math., 24 (2008), 10–19. https://www.jstor.org/stable/43996855
[3]	V. Berinde, M. Pacurar, Fixed points and continuity of almost contractions, Fixed Point Theory, 9 (2008), 23–34.
[4]	G. V. R. Babu, M. L. Sandhya, M. V. R. Kameswari, A note on a fixed point theorem of Berinde on weak contractions, Carpathian J. Math., 24 (2008), 8–12. https://www.jstor.org/stable/43996834
[5]	O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Mathematica Japonicae, 44 (1996), 381–391.
[6]	B. S. Choudhury, P. Chakraborty, Fixed point problem of a multi-valued Kannan-Geraghty type contraction via $w$-distance, J. Anal., 31 (2023), 439–458. https://doi.org/10.1007/s41478-022-00457-3 doi: 10.1007/s41478-022-00457-3
[7]	E. Karapınar, S. Romaguera, P. Tirado, Characterizations of quasi-metric and G-metric completeness involving $w$-distances and fixed points, Demonstr. Math., 55 (2022), 939–951. https://doi.org/10.1515/dema-2022-0177 doi: 10.1515/dema-2022-0177
[8]	S. Barootkoob, E. Karapınar, H. Lakzian, A. Chanda, Extensions of Meir-Keeler contraction via $w$-distances with an application, Kragujev. J. Math., 46 (2022), 533–547. https://doi.org/10.46793/KgJMat2204.533B doi: 10.46793/KgJMat2204.533B
[9]	H. K. Nashine, R. Pant, Feng-Liu type fixed point theorems for $w$-distance spaces and applications, Filomat, 36 (2022), 3899–3917. https://doi.org/10.2298/FIL2211899N doi: 10.2298/FIL2211899N
[10]	R. Jain, H. K. Nashine, Z. Kadelburg, Positive solutions of nonlinear matrix equations via fixed point results in relational metric spaces with $w$-distance, Filomat, 36 (2022), 4811–4829. https://doi.org/10.2298/FIL2214811J doi: 10.2298/FIL2214811J
[11]	H. Lakzian, B. E. Rhoades, Some fixed point theorems using weaker Meir-Keeler function in metric spaces with $w$-distance, Appl. Math. Comput., 342 (2019), 18–25. https://doi.org/10.1016/j.amc.2018.06.048 doi: 10.1016/j.amc.2018.06.048
[12]	H. Lakzian, D. Kocev, V. Rakočević, Ćirić-generalized contraction via $wt$-distance, Appl. Gen. Topol., 24 (2023), 267–280. http://dx.doi.org/10.4995/agt.2023.19268 doi: 10.4995/agt.2023.19268
[13]	S. Reich, A. J. Zaslavski, Genericity in nonlinear analysis, Springer Science & Business Media, 2013.
[14]	A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2004), 1435–1443. https://doi.org/10.1090/S0002-9939-03-07220-4 doi: 10.1090/S0002-9939-03-07220-4
[15]	A. Alam, M. Imdad, Relation-theoretic contraction principle, J. Fixed Point Theory Appl., 17 (2015), 693–702. https://doi.org/10.1007/s11784-015-0247-y doi: 10.1007/s11784-015-0247-y
[16]	A. Alam, M. Imdad, Nonlinear contractions in metric spaces under locally $t$-transitive binary relations, Fixed Point Theory, 19 (2018), 13–24. https://doi.org/10.24193/fpt-ro.2018.1.02 doi: 10.24193/fpt-ro.2018.1.02
[17]	G. Prasad, Fixed point theorems via $w$-distance in relational metric spaces with an application, Filomat, 34 (2020), 1889–1898. https://doi.org/10.2298/FIL2006889P doi: 10.2298/FIL2006889P
[18]	G. Prasad, R. C. Dimri, Fixed point theorems for weakly contractive mappings in relational metric spaces with an application, J. Anal., 26 (2018), 151–162. https://doi.org/10.1007/s41478-018-0076-7 doi: 10.1007/s41478-018-0076-7
[19]	W. Shatanawi, T. A. M. Shatnawi, New fixed point results in controlled metric type spaces based on new contractive conditions, AIMS Math., 8 (2023), 9314–9330. https://doi.org/10.3934/math.2023468 doi: 10.3934/math.2023468
[20]	A. Z. Rezazgui, A. A. Tallafha, W. Shatanawi, Common fixed point results via $A\nu$-$\alpha$-contractions with a pair and two pairs of self-mappings in the frame of an extended quasi b-metric space, AIMS Math., 8 (2023), 7225–7241. https://doi.org/10.3934/math.2023363 doi: 10.3934/math.2023363
[21]	N. H. Altaweel, N. H. E. Eljaneid, H. I. A. Mohammed, I. M. Alanazi, F. A. Khan, Coincidence theorems under generalized nonlinear relational contractions, Symmetry, 15 (2023), 1–17. https://doi.org/10.3390/sym15020434 doi: 10.3390/sym15020434
[22]	S. Shil, H. K. Nashine, Unique positive definite solution of non-linear matrix equation on relational metric spaces, Fixed Point Theory, 24, (2023), 367–382. https://doi.org/10.24193/fpt-ro.2023.1.20 doi: 10.24193/fpt-ro.2023.1.20
[23]	S. Antal, D. Khantwal, S. Negi, U. C. Gairola, Fixed points theorems for $(\phi, \psi, p)$-weakly contractive mappings via $w$-distance in relational metric spaces with applications, Filomat, 37 (2023), 7319–7328.
[24]	R. D. Maddux, Relation Algebras, Amsterdam: Elsevier Science Limited, 2006.
[25]	A. Alama, M. Imdad, Relation-theoretic metrical coincidence theorems, Filomat, 31 (2017), 4421–4439. https://doi.org/10.2298/FIL1714421A doi: 10.2298/FIL1714421A
[26]	M. S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points, B. Aust. Math. Soc., 30 (1984), 1–9. https://doi.org/10.1017/S0004972700001659 doi: 10.1017/S0004972700001659
[27]	T. Senapati, L. K. Dey, Relation-theoretic metrical fixed-point results via $w$-distance with applications, J. Fixed Point Theory Appl., 19 (2017), 2945–2961. https://doi.org/10.1007/s11784-017-0462-9 doi: 10.1007/s11784-017-0462-9
[28]	T. Wongyat, W. Sintunavarat, The existence and uniqueness of the solution for nonlinear Fredholm and Volterra integral equations together with nonlinear fractional differential equations via $w$-distances, Adv. Differ. Equ., 2017 (2017), 1–15. https://doi.org/10.1186/s13662-017-1267-2 doi: 10.1186/s13662-017-1267-2
[29]	G. V. R. Babu, P. D. Sailaja, A fixed point theorem of generalized weakly contractive maps in orbitally complete metric spaces, Thai J. Math., 9 (2012), 1–10.
[30]	M. R. Turinici, Contractive operators in relational metric spaces, Handbook of Functional Equations: Functional Inequalities, 95 (2014), 419–458. https://doi.org/10.1007/978-1-4939-1246-9 doi: 10.1007/978-1-4939-1246-9
[31]	B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. Theor., 47 (2001), 2683–2693.
[32]	P. N. Dutta, B. S. Choudhury, A generalisation of contraction principle in metric spaces, Fixed Point Theory A., 2008 (2008), 1–8.
[33]	Q. Zhang, Y. Song, Fixed point theory for generalized $\psi$-weak contractions, Appl. Math. Lett., 22 (2009), 75–78. https://doi.org/10.1016/j.aml.2008.02.007 doi: 10.1016/j.aml.2008.02.007
[34]	S. Radenović, Z. Kadelburg, Generalized weak contractions in partially ordered metric spaces, Comput. Math. Appl., 60 (2010), 1776–1783. https://doi.org/10.1016/j.camwa.2010.07.008 doi: 10.1016/j.camwa.2010.07.008
[35]	A. C. M. Ran, M. C. B. Reurings, On the nonlinear matrix equation $X+ A^* F (X) A = Q:$ Solutions and perturbation theory, Linear Algebra Appl., 346 (2002), 15–26. https://doi.org/10.1016/S0024-3795(01)00508-0 doi: 10.1016/S0024-3795(01)00508-0
[36]	S. Shil, H. K. Nashine, Positive definite solution of non-linear matrix equations through fixed point technique, AIMS Math., 7 (2022), 6259–6281. https://doi.org/10.3934/math.2022348 doi: 10.3934/math.2022348

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