Existence and Uniqueness of Fixed Point for Cyclic Mappings in Quasi-αb-Metric Spaces

Ainun Sukmawati Al Idrus, Resmawan Resmawan, Muhammad Rezky Friesta Payu, Salmun K. Nasib, Asriadi Asriadi

Abstract


The fixed point theory remains the most important and preferred topic studied in mathematical analysis. This study discusses sufficient conditions to prove a unique fixed point in quasi-αb-metric spaces with cyclic mapping. The analysis starts by showing fulfillment of the cyclic Banach contraction and proving the Cauchy sequence as a condition for proving a unique fixed point in quasi-αb-metric spaces with cyclic mapping. Furthermore, it's shown that the cyclic mappings, T have a unique fixed point in quasi-αb-metric spaces. Finally, an example is given to strengthen the proof of the theorems that have been done.

Keywords: fixed point theory; Quasi -Metric spaces; Cyclic Banach Contraction; Cauchy sequence.

 

Abstrak

Teori titik tetap termasuk salah satu topik penting dan menarik untuk diteliti pada bidang analisis. Pada penelitian ini, dibahas tentang syarat cukup dalam membuktikan bahwa terdapat titik tetap tunggal dalam ruang quasi- b-metrik pada pemetaan siklik. Analisis diawali dengan menunjukkan pemenuhan kondisi kontraksi Banach siklik dan pembuktian barisan Cauchy sebagai syarat pembuktian bahwa terdapat titik tetap tunggal pada pemetaan siklik dalam ruang quasi- b-metrik. Selanjutnya ditunjukkan bahwa pemetaan siklik  memiliki titik tetap tunggal dalam ruang quasi b-metrik. Terakhir, diberikan contoh untuk memperkuat pembuktian teorema yang telah dilakukan.

Kata Kunci: teori titik tetapruang Quasi -Metrik; Kontraksi Banach Siklik; barisan Cauchy.


Keywords


Fixed Point Theory; Quasi αb-Metric Spaces; Cyclic Banach Contraction; Cauchy Sequence

References


S. Cho, “Fixed point theorems for generalized weakly contractive mappings in metric spaces with applications,” Fixed Point Theory Appl., vol. 2018, no. 1, p. 3, Dec. 2018, doi: 10.1186/s13663-018-0628-1.

S. Banach, “Sur les Integrals dans les ensembles abstraits et leur application aux équations Integrals”, Fund. Math., vol. 3, pp. 133-181, 1922.

L. N. Mishra, V. N. Mishra, P. Gautam, and K. Negi, “Fixed point theorems for Cyclic-Ćirić-Reich-Rus contraction mapping in quasi-partial b-metric spaces,” Sci. Publ. State Univ. Novi Pazar Ser. A Appl. Math. Informatics Mech., vol. 12, no. 1, pp. 47–56, 2020, doi: 10.5937/SPSUNP2001047M.

N. Hussain, M. A. Kutbi, and P. Salimi, “Fixed Point Theory in α-Complete Metric Spaces with Applications,” Abstr. Appl. Anal., vol. 2014, pp. 1–11, 2014, doi: 10.1155/2014/280817.

A. A. Khan and B. Ali, “Completeness of b−Metric Spaces and Best Proximity Points of Nonself Quasi-Contractions,” Symmetry (Basel)., vol. 13, no. 11, p. 2206, Nov. 2021, doi: 10.3390/sym13112206.

M. Malahayati, “Ketunggalan Titik Tetap di Ruang Dislocated Quasi b-Metrik pada Pemetaan Siklik”, Jurnal Matematika MANTIK, vol. 3, no. 1, pp. 39-43, 2017.

K. X. Zoto and E. Hoxha, “Fixed Point Theorems in Dislocated and Dislocated Quasi-metric Spaces,” J. Adv. Stud. Topol., vol. 3, no. 4, pp. 119–124, Aug. 2012, doi: 10.20454/jast.2012.430.

K. Zoto, E. Hoxha, and A. Isufati, “Some New Result on Fixed Point in Dislocated and Dislocated Quasi-Metric Spaces”, Applied Mathematical Sciences, vol. 6, no. 71, pp. 3519-3526, 2012.

H. Piri, S. Rahrovi, H. Marasi, and P. Kumam, “A fixed point theorem for F-Khan-contractions on complete metric spaces and application to integral equations,” J. Nonlinear Sci. Appl., vol. 10, no. 09, pp. 4564–4573, Sep. 2017, doi: 10.22436/jnsa.010.09.02.

P. L. Powar, A. K. Pathak, L. N. Mishra, R. Tiwari, and R. Kushwaha, “Fixed Point Theorems Concerning Hausdorff F-PGA Contraction in Complete Metric Space,” J. Phys. Conf. Ser., vol. 1724, no. 1, p. 012030, Jan. 2021, doi: 10.1088/1742-6596/1724/1/012030.

H. Aydi, M.-F. Bota, E. Karapınar, and S. Mitrović, “A fixed point theorem for set-valued quasi-contractions in b-metric spaces,” Fixed Point Theory Appl., vol. 2012, no. 1, p. 88, Dec. 2012, doi: 10.1186/1687-1812-2012-88.

B. S. Chaudhury, P. N. Dutta, and P. Maiti, “Weak Contraction Principle in b-Metric Spaces”, Journal of Mathematics and Informatics, vol. 6, pp. 15-19, 2016.

B. Nurwahyu, “Fixed Point Theorems for Cyclic Weakly Contraction Mappings in Dislocated Quasi Extended -Metric Space,” J. Funct. Spaces, vol. 2019, pp. 1–10, Aug. 2019, doi: 10.1155/2019/1367879.

B. Nurwahyu, M. S. Khan, N. Fabiano, and S. N. Radenovic, “Common Fixed Point on Generalized Weak Contraction Mappings in Extended Rectangular b-Metric Spaces”, Filomat, vol. 35, no. 11, pp. 3621-3634, 2021.

B. Nurwahyu, “Fixed Point Theorems fot Generalized Contraction Mappings in Quasi αb-Metric Space”, Far East Journal of Mathematical Sciences (FJMS), vol. 101, no. 8, Allahabad: India. Pushpa Publishing House, 2016.

B. Nurwahyu, “Fixed Point Theorems for The Multivalued Contraction Mapping in The Quasi αb-Metric Space”, Far East Journal of Mathematical Sciences (FJMS), vol. 102, no. 9, pp. 2105-2119, Allahabad: India. Pushpa Publishing House, 2017.

B. Nurwahyu, A. Nasrun, and N. Aris, “Some Properties of Fixed Point for Contraction Mappings in Quasi αb-Metric Spaces”, IOP Conf. Series: Journal of Physics, doi:10.1088/1742-6596/979/012068, 2018.

[B. Nurwahyu and N. Aris, “Fixed point theorems on some weak contraction mappings in quasi αb-metric space,” J. Phys. Conf. Ser., vol. 1013, p. 012151, May 2018, doi: 10.1088/1742-6596/1013/1/012151.

M. Kir and H. Kiziltunc, “On Some Well Known Fixed Point Theorems in b-Metric Spaces,” Turkish J. Anal. Number Theory, vol. 1, no. 1, pp. 13–16, Jan. 2016, doi: 10.12691/tjant-1-1-4.

B. Nurwahyu, “Common Fixed Point Theorems on Generalized Ratio Contraction Mapping in Extended Rectangular b -Metric Spaces,” Int. J. Math. Math. Sci., vol. 2019, pp. 1–14, Dec. 2019, doi: 10.1155/2019/2756870.

C. Klin-eam and C. Suanoom, “Dislocated quasi-b-metric spaces and fixed point theorems for cyclic contractions,” Fixed Point Theory Appl., vol. 2015, no. 1, p. 74, Dec. 2015, doi: 10.1186/s13663-015-0325-2.

S. Weng and Q. Zhu, “Some Fixed-Point Theorems on Generalized Cyclic Mappings in B-Metric-Like Spaces,” Complexity, vol. 2021, pp. 1–7, Aug. 2021, doi: 10.1155/2021/9042402.

D. R. Sherbert, Introduction to Real Analysis, 4th ed. New Jersey: John Wiley and Sons, Inc., 2011.

Z. Wu, “Two Convergence Theorems and an Extension of the Ratio Test for a Series,” Math. Mag., vol. 92, no. 3, pp. 222–227, May 2019, doi: 10.1080/0025570X.2019.1560841.


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DOI: 10.15408/inprime.v4i1.24462

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