International Journal of Nonlinear Analysis and Applications

Hyers-Ulam stability and well-posedness for fixed point problems on quasi $b$-metric spaces

Document Type : Research Paper

Authors

1 Department of Mathematics, College of Science, University of Al-Qadisiyah, Al-Diwaniya, Iraq

2 Institut Superieur d'Informatique et des Techniques de Communication, Universite de Sousse, Hammam Sousse 4000, Tunisia

3 China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

Abstract

In this paper, we ensure the existence of a unique fixed point in quasi $b$-metric spaces for some contraction mappings requiring the concept of $\Psi ^ {*}$-admissibility. The Ulam-Hyers stability and well-posedness of these fixed point results have been studied and investigated. The obtained results generalize and extend many known results in the literature.

Keywords

[1] Z. Abbasbeygi, A. Bodaghi and A. Gharibkhajeh, On an equation characterizing multi-quartic mappings and its stability, Int. J. Nonlinear Anal. Appl. 13 (2022), no. 1, 991–1002.

[2] H. Aydi, N. Bilgili and E. Karapinar, Common fixed point results from quasi-metric spaces to G-metric spaces, J. Egypt. Math. Soc. 23 (2015), 356–361.

[3] J. Brzdek, J. Chudziak and Z. Pales, A fixed point approach to stability of functional equations, Nonlinear Anal.74 (2011), no. 17, 6728-–6732.

[4] J. Brzdek and K. Ciepliski, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Anal. 74 (2011), no. 18, 6861-–6867.

[5] J. Brzdek and K. Cieplinski, A fixed point theorem and the Hyers-Ulam stability in non-Archimedean spaces, J. Math. Anal. Appl. 400 (2013), no. 1, 68—75.

[6] A. Bodaghi, Th.M. Rassias, and A. Zivari-Kazempour, A fixed point approach to the stability of additive-quadraticquartic functional equations, Int. J. Nonlinear Anal. Appl. 11 (2020), no. 2, 17–28.

[7] M.F. Bota-Borticeanu and A. Petrusel, Ulam-Hyers stability for operatorial equations, Ann. Alexandru Ioan Cuza Univer. Math. 57 (2011), 65–74.

[8] M.F. Bota, E. Karapinar and O. Mlesnite, Ulam -Hyers stability results for fixed point problems via α − ψcontractive mapping in (b)-metric space, Abstr. Appl. Anal. 2013 (2013), Article ID 825293, 2013, 6 pages.

[9] L. Cadariu, L. Gavruta and P. Gavruta, Fixed points and generalized Hyers-Ulam stability, Abstr. Appl. Anal. 2012 (2012), Article ID 712743, 10 pages.

[10] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inf. Univer. Ostraviensis 1 (1993), no. 1, 5–11.

[11] A. Felhi, S. Sahmim and H. Aydi, Ulam-Hyers stability and well-posedness of fixed point problems for α − λcontractions on quasi b-metric spaces, Fixed Point Theory Appl. 2016 (2016), no.1, 20 pages.

[12] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math, Sci. 14 (1991), 431–434.

[13] S.M. Jung, Hyers-Ulam stability of linear partial differential equations of first order, Appl. Math. Lett. 22 (2009), 70–74.

[14] N. Lungu and D. Popa, Hyers-Ulam stability of a first order partial differential equation, J. Math. Anal. Appl. 385 (2012), 86–91.

[15] E. Karapinar, Fixed point theory for cyclic weak-contraction, Appl. Math. Lett. 24 (2011), 822-825.

[16] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Soc. 27 (1941), no.4, 222–224.

[17] D.H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨auser, Boston, 1998.

[18] S.M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York,2011.

[19] M.A. Kutbi and W. Sintunavarat, Ulam-Hyers stability and well-posedness of fixed point problems for λ-contraction mapping in metric spaces, Abstr. Appl. Anal. 2014 (2014), Article ID 268230.

[20] A. Prastaro and T.M. Rassias, Ulam stability in geometry of PDE’s, Nonlinear Funct. Anal. Appl. 8 (2003), 259–278.

[21] I.A. Rus, Remarks on Ulam stability of the operatorial equations, Fixed Point Theory 10 (2009), no. 2, 305–320.

[22] I.A. Rus, Ulam stability of ordinary differential equations, Stud. Univ. Babes Bolyai Math. 54 (2009), 125-–134.

[23] S. Reich and A.J. Zaslawski, Well-posedness of fixed point problems, Far East J. Math. Sci. Special Volume (2001), no. 3, 393–401.

[24] B. Samet, C, Vetro and P. Vetro, Fixed point theorems for α − ψ -contractive type mappings, Nonlinear Anal. 75 (2012), 2154–2165.
25] F.A. Tise and I.C. Tise, Ulam-Hyers-Rassias stability for set integral equations, Fixed Point Theory 13 (2012), no. 2, 659—668.

[26] S.M. Ulam, Problems in Modern Mathematics, Chapter VI, Scince Editors, Wiley, New York, 1960.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 2
February 2023
Pages 101-110
  • Receive Date: 11 January 2022
  • Revise Date: 23 March 2022
  • Accept Date: 01 April 2022