COUPLED FIXED POINT THEOREMS FOR CONTRACTIVE TYPE CONDITION IN PARTIAL METRIC SPACES WITH APPLICATIONS

Gurucharan Singh Saluja

DOI Number
https://doi.org/10.22190/FUMI230204037S
First page
577
Last page
597

Abstract


This paper deals with a coupled fixed point theorem for a mapping satisfying contractive type condition in the setting of partial metric spaces. Furthermore, we give some consequences of the established result. Also, we give an example to validate the result and state some applications to the main result of a self mapping which is involved in an integral type contraction. Our results extend and generalize several previously published results from the existing literature. Specially, our results generalize the results of {\it Aydi} \cite{A11}.

Keywords

Coupled fixed point, contractive type condition, partial metric space

Full Text:

PDF

References


bibitem{AKR10} {sc M. Abbas, M. Ali Khan {rm and} S. Radenovi$acute{c}$}: emph{Common coupled fixed point theorems in cone metric spaces for $w$-compatible mappings}, Appl. Math. Comput. {bf 217} (2010), 195--202.

bibitem{AEO08} {sc R. P. Agarwal, M. A. El-Gebeily {rm and} D. O'Regan}: emph{Generalized contraction in partially ordered metric spaces}, Applicable Anal. {bf 87(1)} (2008), 109--116.

bibitem{AAV13} {sc J. Ahmad, M. Arshad {rm and} C. Vetro}: emph{On a theorem of Khan in a generalized metric space}, Int. J. Anal. (2013)(2). DOI:10.1155/2013/852727.

bibitem{ASS10} {sc I. Altun, F. Sola {rm and} H. Simsek}: emph{Generalized contractions on partial metric spaces}, Topology and its Appl. {bf 157} (2010), 2778--2785.

bibitem{A11} {sc H. Aydi}: emph{Some coupled fixed point results on partial metric spaces}, International J. Math. Math. Sci. 2011, Article ID 647091, 11 pages.

bibitem{AAV12} {sc H. Aydi, M. Abbas {rm and} C. Vetro}: emph{Partial Hausdorff metric and Nadler's fixed point theorem on partial metric spaces}, Topology and Its Appl. {bf 159} (2012), No. 14, 3234--3242.

bibitem{APS12} {sc H. Aydi, M. Postolache {rm and} W. Shatanawi}: emph{Coupled fixed point results for $(psi,phi)$-weakly contractive mappings in ordered $G$-metric spaces}, Comput. Math. Appl. {bf 63} (2012), 298--309.

bibitem{B22} {sc S. Banach}: emph{Sur les operation dans les ensembles abstraits et leur application aux equation integrals}, Fund. Math.

{bf 3}(1922), 133--181.

bibitem{BL06} {sc T. Gnana Bhaskar {rm and} V. Lakshmikantham}: emph{Fixed point theorems in partially ordered metric spaces and applications}, Nonlinear Analysis: TMA, {bf 65(7)} (2006), 1379--1393.

bibitem{CL09} {sc L. Ciric {rm and} V. Lakshmikantham}: emph{Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces}, Nonlinear Analysis: TMA, {bf 70(12)} (2009), 4341--4349.

bibitem{CMP13} {sc S. Chandok, Z. Mustafa {rm and} M. Postolache}: emph{Coupled common fixed point theorems for mixed $g$-monotone mappings in partially ordered $G$-metric spaces}, U. Politech. Buch.: Ser. A {bf 75(4)} (2013), 13--26.

bibitem{C13} {sc S. Chandok}: emph{Coupled common fixed point theorems for a contractive condition of rational type in ordered metric spaces}, J. Appl. Math. Informatics {bf 31} (2013), No. 5-6, 643--649.

bibitem{CKK15} {sc S. Chandok, D. Kumar {rm and} M. S. Khan}: emph{Some results in partial metric space using auxiliary functions}, Applied Math. E-Notes {bf 15} (2015), 233--242.

bibitem{CNT15} {sc S. Chandok, T. D. Narang {rm and} M. A. Taoudi}: emph{Some coupled fixed point theorems for mappings satisfying a generalized contractive condition of rational type}, Palestine J. Math. {bf 4(2)} (2015), 360--366.

bibitem{C15} {sc S. Chandok}: emph{Some fixed point theorems for $(alpha,beta)$-admisible Geraghty type contractive mappings and related results}, Math. Sci. {bf 9} (2015), 127--135.

bibitem{CK10} {sc B. S. Choudhury {rm and} A. Kundu}: emph{A coupled coicidence point result in partially ordered metric spaces for compatible mappings}, Nonlinear Anal.: TMA, {bf 73} (2010), 2524--2531.

bibitem{CMP13a} {sc B. S. Choudhury, N. Metiya {rm and} M. Postolache}: emph{A generalized weak contraction principle with applications to coupled coincidence point problems}, Fixed Point Theory Appl. (2013), {bf 2013:152}.

bibitem{E62} {sc M. Edelstein}: emph{On fixed points and periodic points under contraction mappings}, J. Lond. Math. Soc. {bf 37} (1962), 74--79.

bibitem{GL87} {sc D. Guo {rm and} V. Lakshmikantham}: emph{Coupled fixed point of nonlinear operator with application}, Nonlinear Anal. TMA., {bf 11} (1987), 623--632.

bibitem{HR73} {sc G. C. Hardy {rm and} T. Rogers}: emph{A generalization of fixed point theorem of S. Reich}, Can. Math. Bull. {bf 16} (1973), 201--206.

bibitem{H99} {sc R. Heckmann}: emph{Approximation of metric spaces by partial metric spaces}, Appl. Categ. Structures, {bf 7}, No. 1-2, (1999), 71--83.

bibitem{H10} {sc S. Hong}: emph{Fixed points of multivalued operators in ordered metric spaces with applications}, Nonlinear Anal., TMA., {bf 72(11)} (2010), 3929--3942.

bibitem{K69} {sc R. Kannan}: emph{Some results on fixed points-II}, Amer. Math. Mon. {bf 76} (1969), 71--76.

bibitem{KST13} {sc E. Karapinar, W. Shatanawi {rm and} K. Tas}: emph{Fixed point theorems on partial metric spaces involving rational expressions}, Miskolc Math. Notes {bf 14} (2013), 135--142.

bibitem{KR18} {sc B. Khomdaram {rm and} Y. Rohen}: emph{Some common coupled fixed point theorems in $S_{b}$-metric spaces}, Fasciculi Math. {bf 60} (2018), 79--92.

bibitem{KC13} {sc J. K. Kim {rm and} S. Chandok}: emph{Coupled common fixed point theorems for generalized nonlinear contraction mappings with the mixed monotone property in partially ordered metric spaces}, Fixed Point Theory Appl. (2013), {bf 2013:307}.

bibitem{KOL17} {sc J. K. Kim, G. A. Okeke {rm and} W. H. Lim}: emph{Common coupled fixed point theorems for $w$-compatible mappings in partial metric spaces}, Global J. Pure Appl. Math. {bf 13(2)} (2017), 519--536.

bibitem{K22} {sc K. S. Kim}: emph{Coupled fixed point theorems under new coupled implicit relation in Hilbert spaces}, Demonstratio Math. {bf 55} (2022), 81--89.

bibitem{KCDD21} {sc P. Konar, S. Chandok, S. Dutta {rm and} M. De la sen}: emph{Coupled optimal results with an application to integral equations}, Axioms {bf 2021}, 10, 73. (https//:doi.org/10.3390/axioms 10020073)

bibitem{K01} {sc H. P. A. K$ddot{u}$nzi}: emph{Nonsymmetric distances and their associated topologies about the origins of basic ideas in the area of asymptotic topology}, Handbook of the History Gen. Topology (eds. C.E. Aull and R. Lowen), Kluwer Acad. Publ., {bf 3} (2001), 853--868.

bibitem{LT11a} {sc N. V. Luong {rm and} N. X. Thuan}: emph{Coupled fixed points in partially ordered metric spaces and application}, Nonlinear Anal. {bf 74} (2011), 983--992.

bibitem{LT11b} {sc N. V. Luong {rm and} N. X. Thuan}: emph{Fixed point theorems for generalized weak contractions satisfying rational expressions in ordered metric spaces}, Fixed Point Theory Appl., {bf 2011}(46), (2011).

bibitem{LT11} {sc N. V. Luong {rm and} N. X. Thuan}: emph{Coupled fixed points theorems for mixed monotone mappings and an application to integral equations}, Comput. Math. Appl. {bf 62} (2011), 4238--4248.

bibitem{MSS13} {sc H. P. Masiha, F. Sabetghadam {rm and} N. Shahzad}: emph{Fixed point theorems in partial metric spaces with an application}, Filomat {bf 27(4)}(2013), 617--624.

bibitem{M92} {sc S. G. Matthews}: emph{Partial metric topology}, Research report 2012, Dept. Computer Science, University of Warwick, 1992.

bibitem{M94} {sc S. G. Matthews}: emph{Partial metric topology}, Proceedings of the 8th summer conference on topology and its applications, Annals of the New York Academy of Sciences, {bf 728} (1994), 183--197.

bibitem{M81} {sc B. Monjardet}: emph{Metrics on partially ordered sets: A survey}, Discrete Math. {bf 35} (1981), 173--184.

bibitem{NS11} {sc H. K. Nashine {rm and} W. Shatanawi}: emph{Coupled common fixed point theorems for pair of commuting in partially ordered complete metric spaces}, Comput. Math. Appl. {bf 62} (2011), 1984--1993.

bibitem{NK12} {sc H. K. Nashine {rm and} Z. Kadelburg}: emph{Partially ordered metric spaces, rational contractive expressios and couple fixed points}, Nonlinear Funct. Anal. Appl. {bf 17(4)} (2012), 471--489.

bibitem{N12} {sc H. K. Nashine}: emph{Coupled common fixed point results in ordered $G$-metric spaces}, J. Nonlinear Sci. Appl. {bf 1} (2012), 1--13.

bibitem{NKR13} {sc H. K. Nashine, Z. Kadelburg {rm and} S. Radenovi$acute{c}$}: emph{Common fixed point theorems for weakly isotone increasing mappings in ordered partial metric spaces}, Math. Comput. Model. {bf 57(9-10)} (2013), 2355--2365.

bibitem{NKSS16} {sc H. K. Nashine, J. K. Kim, A. K. Sharma {rm and} G. S. Saluja}: emph{Some coupled fixed point without mixed monotone mappings}, Nonlinear Funct. Anal. Appl. {bf 21(2)} (2016), 235--247.

bibitem{NL07} {sc J. J. Nieto {rm and} R. R. Lopez}: emph{Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equation}, Acta Math. Sinica, English Series, {bf 23} (2007), No. 12, 2205--2212.

bibitem{OOA12} {sc J. O. Olaleru, G. A. Okeke {rm and} H. Akewe}: emph{Coupled fixed point theorems for generalized $varphi$-mappings satisfying contractive condition of integral type on cone metric spaces}, Int. J. Math. Model. Comput. {bf 2(2)} (2012), 87--98.

bibitem{PSNP17} {sc R. Pant, R. Shukla, H. K. Nashine {rm and} R. Panicker}: emph{Some new fixed point theorems in partial metric spaces with applications}, J. Function spaces, {bf 2017}, Article ID 1072750, 13 pages.

bibitem{R14} {sc S. Radenovi$acute{c}$}: emph{${it Bhaskar-Lakshmikantham}$ type results for monotone mappings in partially ordered metric spaces}, Int. J. Nonlinear Anal. {bf 5} (2014), 37--49.

bibitem{RR04} {sc A. C. M. Ran {rm and} M. C. B. Reurings}: emph{A fixed point theorem in partially ordered sets and some application to matrix equations}, Proc. Amer. Math. Soc. {bf 132} (2004), 1435--1443.

bibitem{RK20} {sc N. Seshagiri Rao {rm and} K. Kalyani}: emph{Coupled fixed point theorems in partially ordered metric spaces}, Fasciculi Math. {bf 64} (2020), 77--89.

bibitem{R71} {sc S. Reich}: emph{Some remarks concerning contraction mappings}, Can. Math. Bull. {bf 14} (1971), 121--124.

bibitem{RZ01} {sc S. Reich {rm and} A. J. Zaslavski}: emph{Well posedness of fixed point problem}, Far East J. Math. special volume part III (2001), 393--401.

bibitem{SMS09} {sc F. Sabetghadam, H. P. Mashiha {rm and} A. H. Sanatpour}: emph{Some coupled fixed point theorems in cone metric spaces}, Fixed Point Theory Appl. (2009), Article ID 125426, 8 pages.

bibitem{S10} {sc B. Samet}: emph{Coupled fixed point theorems for a generalized ${it Meir-Keeler}$ contractions in a partially ordered metric spaces}, Nonlinear Anal. {bf 72(12)} (2010), 4508--4517.

bibitem{SY10} {sc B. Samet {rm and} H. Yazidi}: emph{Coupled fixed point theorems in a partial order $epsilon$-chainable metric spaces}, J. Math. Comput. Sci. {bf 1(3)} (2010), 142--151.

bibitem{SY} {sc B. Samet {rm and} H. Yazidi}: emph{Coupled fixed point theorems for contraction involving rational expressions in partially ordered metric spaces}, priprint, arXiv:1005.3142v1 [math GN].

bibitem{SG17} {sc S. Sedghi {rm and} A. Gholidehneh}: emph{Coupled fixed point theorems in $S_{b}$-metric spaces}, Nonlinear Funct. Anal. Appl. {bf 22(2)} (2017), 217--228.

bibitem{SC21} {sc S. Sharma {rm and} S. Chandok}: emph{Existence of best proximity point with an application to nonlinear intigral equations}, J. Math. (2021), Article ID 3886659, 7 pages. DOI: 10.1155/2021/3886659.

bibitem{SSA12} {sc W. Shatanawi, B. Samet {rm and} M. Abbas}: emph{Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces}, Math. Comput. Modell. {bf 55} (2012), Nos. 3-4, 680--687.

bibitem{SP13} {sc W. Shatanawi {rm and} M. Posolache}: emph{Common fixed point results of mappings for nonlinear contractions of cyclic form in ordered metric spaces}, Fixed Point Theory Appl. (2013), {bf 2013:60}. DOI: 10.1186/1687-1812-2013-60.

bibitem{SC88} {sc M. R. Singh {rm and} A. K. Chatterjee}: emph{Fixed point theorems}, Commun. Fac. Sci. Univ. Ank., Series A1 {bf 37} (1988), 1--4.

bibitem{S74} {sc D. R. Smart}: emph{Fixed point theorems}, Cambridge University Press, Cambridge, 1974.

bibitem{V05} {sc O. Valero}: emph{On Banach fixed point theorems for partial metric spaces}, Appl. Gen. Topl. {bf 6(2)} (2005), 229--240.

bibitem{W75} {sc E. S. Wolk}: emph{Continuous convergence in partially ordered sets}, Gen. Topol. Appl. {bf 5} (1975), 221--234.

bibitem{W73} {sc C. S. Wong}: emph{Common fixed points of two mappings}, Pac. J. Math. {bf 48} (1973), 299--312.




DOI: https://doi.org/10.22190/FUMI230204037S

Refbacks

  • There are currently no refbacks.




© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)