Fixed Point Sets of Multivalued Contractions and Stability Analysis

Year 2018, Volume: 1 Issue: 2, 163 - 171, 24.12.2018

Nikhilesh Metiya , Binayak S. Choudhury , Sunirmal Kundu

https://doi.org/10.33434/cams.454820

Cited By: 3

Abstract

In this paper we derive a fixed point result for a multivalued generalized almost contraction which contains several rational terms through a six variables function and a four variables function. The space is assumed to satisfy some regularity conditions. In another part of the paper we establish stability results for fixed point sets of these contractions. The corresponding singlevalued case is also discussed. The results are obtained without any assumption of continuity. There are two illustrative examples.

Keywords

Hausdorff metric, Generalized almost contraction, Fixed point, Stability

References

• [1] J. P. Aubin, H. Frankowska, Set Valued Analysis, Springer, Revised Version, 2009.
• [2] S. B. Jr. Nadler, Multivalued contraction mapping, Pacific J. Math., 30 (1969), 475–488.
• [3] B. S. Choudhury, N. Metiya, Fixed point theorems for almost contractions in partially ordered metric spaces, Ann. Univ. Ferrara, 58 (2012), 21–36.
• [4] B. S. Choudhury, N. Metiya, C. Bandyopadhyay, Fixed points of multivalued a-admissible mappings and stability of fixed point sets in metric spaces, Rend. Circ. Mat. Palermo, 64 (2015), 43-55.
• [5] B. S. Choudhury, N. Metiya, T. Som, C. Bandyopadhyay, Multivalued fixed point results and stability of fixed point sets in metric spaces, Facta Univ. Ser. Math. Inform., 30(4) (2015), 501–512.
• [6] B. S. Choudhury, N. Metiya, C. Bandyopadhyay, P. Maity, Fixed points of multivalued mappings satisfying hybrid rational Pata-type inequalities, The Journal of Analysis, (2018) https://doi.org/10.1007/s41478-018-0131-4
• [7] M. E. Gordji, H. Baghani, H. Khodaei, M. Ramezani, A generalization of Nadler’s fixed point theorem, J. Nonlinear Sci. Appl., 3 (2010), 148–151.
• [8] A. A. Harandi, End points of setvalued contractions in metric spaces, Nonlinear Anal., 72 (2010), 132–134.
• [9] W. Sintunavarat, P. Kumam, Coincidence and common fixed points for hybrid strict contractions without the weakly commuting condition, Appl. Math. Lett., 22 (2009), 1877–1881.
• [10] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for a $\alpha-\psi$-contractive type mappings, Nonlinear Anal., 75 (2012), 2154–2165.
• [11] S. Chandok, Some fixed point theorems for $(\alpha,\ \beta)$-admissible Geraghty type contractive mappings and related results, Math. Sci., 9 (2015), 127–135.
• [12] B. S. Choudhury, N. Metiya, S. Kundu, Existence and stability results for coincidence points of nonlinear contractions, Facta Univ. Ser. Math. Inform., 32(4) (2017), 469–483.
• [13] A. Felhi, H. Aydi, D. Zhang, Fixed points for $\alpha$- admissible contractive mappings via simulation functions, J. Nonlinear Sci. Appl., 9 (2016), 5544–5560.
• [14] N. Hussain, E. Karapinar, P. Salimi, F. Akbar, $\alpha$-admissible mappings and related fixed point theorems, J. Inequal. Appl., 2013 (2013), page 114.
• [15] E. Karapinar, B. Samet, Generalized $\alpha - \psi $ contractive type mappings and related fixed point theorems with applications, Abstr. Appl. Anal., 2012 (2012), Article ID 793486.
• [16] E. Karapinar, P. Shah, K. Tas, Generalized $\alpha - \psi $ - contractive type mappings of integral type and related fixed point theorems, J. Inequal. Appl., 2014 (2014), page 160.
• [17] V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum, 9(1) (2004), 43–53.
• [18] V. Berinde, General constructive fixed point theorems for $\acute{C}$iri$\acute{c}$-type almost contractions in metric spaces, Carpathian J. Math., 24(2) (2008), 10–19.
• [19] M. Abbas, G. V. R. Babu, G. N. Alemayehu, On common fixed points of weakly compatible mappings satisfying ‘generalized condition (B)’, Filomat, 25(2) (2011), 9–19.
• [20] M. A. Alghamdi, V. Berinde, N. Shahzad, Fixed points of multivalued nonself almost contractions, J. Appl. Math., 2013 (2013), Article ID 621614, 6 pages.
• [21] 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(1) (2008), 08–12.
• [22] V. Berinde, M. Pacurar, Fixed points and continuity of almost contractions, Fixed Point Theory, 9(1) (2008), 23–34.
• [23] R. K. Bose, R. N. Mukherjee, Stability of fixed point sets and common fixed points of families of mappings, Indian J. Pure Appl. Math., 9 (1980), 1130–1138.
• [24] B. S. Choudhury, C. Bandyopadhyay, Stability of fixed point sets of a class of multivalued nonlinear contractions, Journal of Mathematics, 2015 (2015), Article ID 302012, 4 pages.
• [25] T. C. Lim, Fixed point stability for set valued contractive mappings with applications to generalized differential equations, J. Math. Anal. Appl., 110 (1985), 436–441.
• [26] J. T. Markin, A fixed point stability theorem for nonexpansive set valued mappings, J. Math. Anal. Appl., 54 (1976), 441–443.
• [27] B. K. Dass, S. Gupta, An extension of Banach contraction principle through rational expressions, Inidan J. Pure Appl. Math., 6 (1975), 1455–1458.
• [28] M. Abbas, V. C. Rajic, T. Nazir, S. Radenovic, Common fixed point of mappings satisfying rational inequalities in ordered complex valued generalized metric spaces, Afrika Mat., 26 (2015), 17–30.
• [29] I. Cabrera, J. Harjani, K. Sadarangani, A fixed point theorem for contractions of rational type in partially ordered metric spaces , Ann. Univ. Ferrara, 59 (2013), 251–258.
• [30] S. Chandok, J. K. Kim, Fixed point theorem in ordered metric spaces for generalized contractions mappings satisfying rational type expressions, J. Nonlinear Funct. Anal. Appl., 17 (2012), 301–306.
• [31] J. Harjani, B. L´opez, K. Sadarangani, A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space, Abstr. Appl. Anal., (2010), Article ID 190701.
• [32] N. V. Luong, N. X. Thuan, Fixed point theorem for generalized weak contractions satisfying rational expressions in ordered metric spaces, Fixed Point Theory Appl., 46 (2011), 1–10.