International Journal of Nonlinear Analysis and Applications

Ciric type multi-valued $\alpha _{\ast }$-$\eta _{\ast }$-$\theta $-‎contractions on b-metric spaces with applications

Document Type : Research Paper

Authors

1 Department of Mathematics, Taiz University, Taiz, Yemen

2 University of Dammam, Department of Mathematics. College of Education of Jubail, P.O: 12020, Industrial Jubail 31961. Saudi Arabia

3 ‎Department of Mathematics‎, ‎International Islamic‎ ‎University‎, ‎H-10‎, ‎Islamabad‎ - ‎44000‎, ‎Pakistan

4 Department of Mathematics, Faculty of Natural Sciences, Centre for Mathematical Research, Khawaja Fareed University of Engineering and Information Technology Rahim Yar Khan, Pakistan

5 Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Abstract

‎In this paper‎, ‎we give sufficient conditions for the existence of solutions‎ of a system of Volterra-type integral inclusion equations using new sort of‎ multi-valued contractions‎, ‎named as generalized multi-valued $\alpha _{\ast‎} ‎$-$\eta _{\ast }$-$\theta $-contractions defined on $\alpha $-complete‎ ‎b-metric spaces‎. ‎We give its relevance to fixed point results‎. ‎We set up an‎ ‎example to elucidate our main results‎.

Keywords

[1] A. Felhi, S. Sahmim, H. Aydi, Ulam-Hyers stability and well-posedness of fixed point problems for α-λ-contractions on quasi b-metric spaces, Fixed Point Theory Appl. 2016 (2016).
[2] S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull. 14 (1971) 121–124.
[3] S. Reich, Fixed points of contractive functions, Boll. Un. Mat. Ital. 5 (1972) 26–42.
[4] S. H. Cho, J.S. Bae and E. Karapinar, Fixed point theorems for α-Geraghty contraction type maps in metric spaces, Fixed Point Theory Appl. 2013 (2013) 11 pages.
[5] N. Hussain, A. E. Al-Mazrooei, J. Ahmad, Fixed point results for generalized (α, η)-Θ-contractions with applications, Journal of Nonlinear Sciences and Applications, 10 (2017) 4197–4208.
[6] P. Chuadchawna, A. Kaewcharoen, S. Plubtieng, Fixed point theorems for generalized α-η-ψ-Geraghty contraction type mappings in α-η-complete metric spaces, J. Nonlinear Sci. App., 9 (2016) 471–485.
[7] E. Karapinar, α-ψ-Geraghty contraction type mappings and some related fixed point results, Filomat 28 (2014) 37–48.
[8] M. Jleli, B. Samet, A new generalization of the Banach contraction principle, J. Inequal. Appl. 2014 (2014).
[9] M. Jleli, E. Karapinar, B. Samet, Further generalizations of the Banach contraction principle. J. Inequal. Appl. 2014, 2014:439.
[10] J. Ahmed, A. E. Al-Mazrooei, Y. J. Cho , Y. -O. Yang, Fixed point results for generalized Θ-contractions, Journal of Nonlinear Sciences and Applications, 10 (2017), 2350-2358.
[11] H. Aydi, α-implicit contractive pair of mappings on quasi b-metric spaces and an application to integral equations, Journal of Nonlinear and Convex Analysis, 17 (12) (2016), 2417–2433.
[12] H. Aydi, E. Karapinar, M.F. Bota, S. Mitrovi c, A fixed point theorem for set-valued quasi-contractions in b-metric spaces, Fixed Point Theory Appl. 2012, 2012:88.
[13] H. Aydi, M.F. Bota, E. Karapinar, S.. Moradi, A common fixed point for weak phi-contractions on b-metric spaces, Fixed Point Theory, 13 (2) (2012), 337-346.
[14] H. Aydi, A. Felhi, S. Sahmim, Common fixed points via implicit contractions on b-metric-like spaces, J. Nonlinear Sci. Appl. 10 (4) (2017), 1524–1537.
[15] A.H. Ansari, M.A. Barakat, H. Aydi, New approach for common fixed point theorems via C-class functions in Gp-metric spaces, Journal of Functions Spaces, vol. 2017, Article ID 2624569, 9 pages, 2017.
[16] M. Arshad, E. Ameer, E. Karapinar, Generalized contractions with triangular α-orbital admissible mapping on Branciari metric spaces, Journal of Inequalities and Applications 2016, 2016:63.
[17] E. Ameer, M. Arshad, W. Shatanawi, Common fixed point results for generalized α∗-ψ-contraction multivalued mappings in b-metric spaces, J. Fixed Point Theory Appl. (2017), DOI 10.1007/s11784-017-0477-2.
[18] A. Sˆınt˘am˘arian, Integral inclusions of Fredholm type relative to multivalued φ-contraction, Semin. Fixed Point Theory Cluj-Napoca, 3 (2002), 361–368.
[19] M. Jleli, B. Samet, C. Vetro, F. Vetro, Fixed points for multivalued mappings in b-metric spaces, Abstract and Applied Analysis, Volume 2015, Article ID 718074, 7 pages.
[20] M. Jovanovi´c, Z. Kadelburg, S. Radenovi´c, Common xed point results in metric-type spaces, Fixed Point Theory Appl. Volume 2010, Article ID 978121, 15 pages.
[21] H. Huang, S. Xu, Fixed point theorems of contractive mappings in cone b-metric spaces and applications, Fixed Point Theory Appl. (2013), 2013:112.
[22] N. Hussian, M. H. Shah, KKM mappings in cone b -metric spaces, Comput. Math. Appl. 62 (2011), 1677-168.
[23] E. Karapınar, P. Kumam, P. Salimi, On α-ψ-Meir-Keeler contractive mappings. Fixed Point Theory Appl. 2013, 2013:94.
[24] N. Hussain, M. A. Kutbi, P. Salimi, Fixed point theory in α-complete metric space with applications, Abstr. Appl. Anal. 2014 (2014), 11 pages.
[25] J.R. Roshan, V. Parvaneh, Sh. Sedghi, N. Shobkolaei, W. Shatanawi, Common fixed points of almost generalized(ψ − φ) s-contraction mappings in ordered b-metric spaces, Fixed Point Theory Appl. (2013), 2013:159.
[26] I. A. Bakhtin, The contraction mapping principle in almost metric space, Functional Analysis, vol. 30, pp. 26–37,1989.
[27] L. Shi, S. Xu, Common fixed point theorems for two weakly compatible self-mappings in cone b-metric spaces,
Fixed Point Theory Appl. (2013), 2013:120.
[28] S. Banach, Sur les op´erations dans les ensembles abstraits et leur application aux equations itegrales, Fund. Math. 3 (1922), 133–181.
[29] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Semin. Mat. Fis. Univ. Modena. 46 (2) (1998), 263-276.
[30] S. Czerwik, Contraction mappings in b-metric spaces. Acta Math. Inf. Univ. Ostrav. 1 (1993), 5-11.
[31] P. Salimi, A. Latif, N. Hussain, Modified α − ψ−contractive mappings with applications Fixed Point Theory and Appl. (2013), 2013:151.
[32] M. Berzig, E. Karapınar, On modified α-ψ-contractive mappings with application, Thai Journal of Mathematics. Vol 13, No 1 (2015), 147-152.
[33] B. Mohammadi, Sh. Rezapour, N.Shahzad, Some results of fixed point of α-ψ-quasi-contractive multifunctions, Fixed Point Theory Appl. (2013), 2013:112.
[34] S. B. Nadler, Multivalued contraction mappings, Pac. J. Math. 30 (1969), 475-488.
[35] O. Popescu, Some new fixed point theorems for α-Geraghty contraction type maps in metric spaces. Fixed Point Theory Appl. 2014, 2014:90.
[36] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for α − ψ-contractive type mappings, Nonlinear Anal. 75 (2012), 2154–2165.
[37] W. Shatanawi, Fixed and common fixed point for mappings satisfying nonlinear contractive in b-metric spaces, J. Math. Anal. 7(4) (2016) 1–12.
[38] H. Huang, G. Deng, S. Radenovi´c, Fixed point theorems in b-metric spaces with applications to differential equations, J. Fixed Point Theory Appl. (2018) 20:52, doi.org/10.1007/s11784-018-0491-z.
International Journal of Nonlinear Analysis and Applications
Volume 12, Issue 1
May 2021
Pages 597-614
  • Receive Date: 26 June 2018
  • Revise Date: 06 November 2019
  • Accept Date: 15 February 2020