Existence and stability results for fixed points of multivalued F contractions and application to Volterra type non homogeneous integral equation of second kind

Binayak S. Choudhury, Nikhilesh Metiya, T. Som, Sunirmal Kundu

Abstract


In this paper we introduce multivalued modified F-contraction on a metric space. This is a multivalued mapping obtained by incorporating the idea of the recently introduced F-contraction which has attracted much attention in contemporary research. We explore the fixed point problem associated with the above contractive mapping. We also investigate the data dependence and stability properties of the fixed point sets associated with these multivalued contractions. We discuss an illustration of the main result and present an application of the single valued version of the main theorem to a problem of an integral equation of Volterra type. The domain of the study is fixed point theory and set valued analysis.

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R. P. Agarwal, N. Hussain, and M.-A. Taoudi, Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations, Abstr. Appl. Anal. (2012) 2012, Article ID 245872. DOI: 10.1155/2012/245872

M. U. Ali and T. Kamran, Multivalued F- contractions and related fixed point theorems with application, Filomat 30 (2016), no. 14, 3779-3793. DOI: 10.2298/FIL1614779A

I. Altun, G. Durmaz, G. Minak, and S. Romaguera, Multivalued almost F- contractions on complete metric spaces, Filomat 30 (2016), no. 2, 441-448. DOI: 10.2298/FIL1602441A

I. Altun, G. Minak, and M. Olgun, Fixed points of multivalued nonlinear $F$- contraction on complete metric space, Nonlinear Anal. Model. Control 21 (2016), no. 2, 201-210. DOI: 10.15388/NA.2016.2.4

E. Ameer and M. Arshad, Two new generalization for F- cvontraction on closed ball and fixed point theorems wioth application, Jouurnal of mathematical Extension 11 (2017), no. 3, 43-67. URL: http://www.ijmex.com

G. V. R. Babu and M. V. R. Kameswari, Coupled fixed points for generalized contractive maps with rational expressions in partially ordered metric spaces, Journal of Advanced Research in Pure Mathematics 6 (2014), 43--57. DOI: 10.5373/jarpm.1686.020513

S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales, Fund Math. 3(1922), 133--181.

I. Beg, A. R. Butt, and S. Radojević, The contraction principle for set valued mappings on a metric space with a graph, Comput. Math. Appl. 60 (2010), 1214–-1219. 10.1016/j.camwa.2010.06.003

R. K. Bose and 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.

I. Cabrera, J. Harjani, and K. Sadarangani, A fixed point theorem for contractions of rational type in partially ordered metric spaces, Ann. Univ. Ferrara 59 (2013), 251--258. DOI: 10.1007/s11565-013-0176-x

S. Chandok and J. K. Kim, Fixed point theorems in ordered metric spaces for generalized contractions mappings satisfying rational type expressions, J. Nonlinear Functional Anal. Appl. 17 (2012), 301--306.

C. Chifu and G. Petruşel, Coupled fixed point results for $(varphi,G)$-contractions of type (b) in b-metric spaces endowed with a graph, J. Nonlinear Sci. Appl. 10 (2017), 671--683. DOI: 10.22436/jnsa.010.02.29

A. Chiş-Novac, R. Precup, and I. A. Rus, Data dependence of fixed points for non-self generalized contractions, Fixed Point Theory 10 (2009), no. 1, 73--87.

B. S. Choudhury, N. Metiya, and C. Bandyopadhyay, Fixed points of multivalued α-admissible mappings and stability of fixed point sets in metric spaces, Rend. Circ. Mat. Palermo 64 (2015), 43--55. DOI: 10.1007/s12215-014-0177-3

B. S. Choudhury, N. Metiya, T. Som, and C. Bandyopadhyay, Multivalued fixed point results and stability of fixed point sets in metric spaces, Facta Universitatis (NIŠ) Ser. Math. Inform. 30 (2015), 501--512.

B. S. Choudhury, N. Metiya, and S. Kundu, Existence and stability results for coincidence points of nonlinear contractions, Facta Universitatis (NIŠ) Ser. Math. Inform. 32 (2017), no. 4, 469 - 483. DOI: 10.22190/FUMI1704469C

B. S. Choudhury, N. Metiya, and S. Kundu, Fixed point sets of multivalued contractions and stability analysis, Commun. Math. Sci. 2 (2018), 163-171. DOI: 10.33434/cams.454820

B. K. Dass and S. Gupta, An extension of Banach contraction principle through rational expressions, Indian J. Pure Appl. Math. 6(1975), 1455--1458.

G. Durmaz and I. Altun, Fixed point results for α- admissible multivalued $F$- contraction, Miskolc Mathematical Notes 17 (2016), no. 1, 187-199. DOI: 10.18514/MMN.2016.1478

A. Felhi, H. Aydi, and D. Zhang, Fixed points for α- admissible contractive mappings via simulation functions, J. Nonlinear Sci. Appl. 9 (2016), 5544--5560. DOI: 10.22436/jnsa.009.10.05

J. Harjani, B. López, and K. Sadarangani, Fixed point theorems for mixed monotone operators and applications to integral equations, Nonlinear Anal. 74 (2011), 1749--1760. 10.1016/j.na.2010.10.047

N. Hussain and M. A. Taoudi, Krasnosel’skii-type fixed point theorems with applications to Volterra integral equations, Fixed Point Theory Appl. 2013 (2013) article 196. DOI: 10.1186/1687-1812-2013-196

A. Hussain, H.F. Ahmad, M. Arshad, and M. Nazam, New type of multivalued $F$- contraction involving fixed points on closed ball, J. Math. Computer Sci. 10 (2017), 246-254. DOI: 10.22436/jmcs.017.02.06

A. Hussain, New approach of F- contraction involving fixed points on a closed ball, Turkish journal of Analysis and number Theory 4 (2016), no. 6, 159-163. DOI: 10.12691/tjant-4-6-2

I. Iqbal and N. Hussain, Fixed point theorems for generalized multivalued nonlinear F- contractions, J. Nonlinear Sci. Appl. 9 (2016), 5870-5893. DOI: 10.22436/jnsa.009.11.15

E. Karapinar and B. Samet, Generalized α - ᴪ contractive type mappings and related fixed point theorems with applications, Abstr. Appl. Anal. 2012 (2012), Article ID 793486. DOI: 10.1155/2012/793486

M.A. Kutbi and W. Sintunavarat, On new fixed point results for (α, ᴪ, ξ)-contractive multivalued mappings on α- complete metric spaces and their consequences, Fixed Point Theory Appl. 2015 (2015), 2. DOI: 10.1186/1687-1812-2015-2

D. S. Jaggi and B. K. Das, An extension of Banach's fixed point theorem through rational expression, Bull. Cal. Math. Soc. 72 (1980), 261 - 264.

T. C. Lim, On fixed point stability for set valued contractive mappings with applications to generalized differential equations, J. Math. Anal. Appl. 110 (1985), 436--441. DOI: 10.1016/0022-247X(85)90306-3

J. T. Markin, A fixed point stability theorem for nonexpansive set valued mappings, J. Math. Anal. Appl. 54 (1976), 441--443. DOI: 10.1016/0022-247X(76)90212-2

S. B. Nadler Jr., Multivalued contraction mappings, Pacific J. Math. 30 (1969), 475- 488. DOI: 10.2140/pjm.1969.30.475

M. Nazam, M. Arshad, and M. Abbas, Existence of common fixed points of improved F- contraction on partial metric spaces, Appl. Gen. Topol. 18 (2017), no. 2, 277-287. DOI: 10.4995/agt.2017.6776

I. A. Rus, A. Petruşel, and A. Sîntămărian, Data dependence of the fixed points set of multivalued weakly picard operators, Studia Univ. ``Babec{s} - Bolyai", Mathematica XLVI (2) (2001), 111--121.

I. A. Rus, A. Petruşel, and A. Sîntămărian, Data dependence of the fixed points set of some multivalued weakly Picard operators, Nonlinear Anal. 52 (2003), 1947--1959. DOI: 10.1016/S0362-546X(02)00288-2

B. Samet, C. Vetro, and P. Vetro, Fixed point theorems for α - ᴪ-contractive type mappings, Nonlinear Anal. 75 (2012), 2154--2165. DOI: 10.1016/j.na.2011.10.014

P. Semwal and R. C. Dimri, A Suzuki type coupled fixed point theorem for generalized multivalued mapping, Abstr. Appl. Anal. 2014 (2014), Article ID 820482. DOI: 10.1155/2014/820482

D. Wardowski, Fixed point of a new type of contractive mapping in complete metric spaces, Fixed Point Theory Appl. 94 (2012). DOI: 10.1186/1687-1812-2012-94




DOI: https://doi.org/10.52846/ami.v50i1.1597