×

Coupled fixed point theorems with new implicit relations and an application. (English) Zbl 1356.54042

Several fixed point theorems and common fixed point theorems in metric spaces have been unified considering a general condition by an implicit relation by the reviewer [Demonstr. Math. 32, No. 1, 157–163 (1999; Zbl 0926.54030)]. I. Altun and H. Simsek [Fixed Point Theory Appl. 2010, Article ID 621469, 17 p. (2010; Zbl 1197.54053)] generalized this idea and introduced a new type of implicit relation using functions of Matkowski type in partial metric spaces.
In the present paper the authors introduce two new types of implicit relations which generalize the notions of Altun and Simsek [loc. cit.].
In the main results of the present paper, Theorems 16 and 18, the authors use these new types of new implicit relations to prove the existence of coupled fixed points in partially ordered metric spaces which generalize the results of Altun and Simsek [loc. cit.] and the results by T. G. Bhaskar and V. Lakshmikantham [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 65, No. 7, 1379–1393 (2006; Zbl 1106.47047)].
As an application of Theorem 18, the authors establish the existence for an integral equation.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
45B05 Fredholm integral equations
47H10 Fixed-point theorems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Harjani and K. Sadarangani, “Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1188-1197, 2010. · Zbl 1220.54025 · doi:10.1016/j.na.2009.08.003
[2] J. Harjani and K. Sadarangani, “Fixed point theorems for weakly contractive mappings in partially ordered sets,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3403-3410, 2009. · Zbl 1221.54058 · doi:10.1016/j.na.2009.01.240
[3] H. K. Nashine and B. Samet, “Fixed point results for mappings satisfying (\psi , \varphi )-weakly contractive condition in partially ordered metric spaces,” Nonlinear Analysis, vol. 74, pp. 2201-2209, 2011. · Zbl 1208.41014 · doi:10.1016/j.na.2010.11.024
[4] J. J. Nieto and R. Rodriguez-Lopez, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223-239, 2005. · Zbl 1095.47013 · doi:10.1007/s11083-005-9018-5
[5] J. J. Nieto and R. Rodríguez-López, “Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations,” Acta Mathematica Sinica, vol. 23, no. 12, pp. 2205-2212, 2007. · Zbl 1140.47045 · doi:10.1007/s10114-005-0769-0
[6] A. C. Ran and M. C. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435-1443, 2004. · Zbl 1060.47056 · doi:10.1090/S0002-9939-03-07220-4
[7] D. J. Guo and V. Lakshmikantham, “Coupled fixed points of nonlinear operators with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 11, no. 5, pp. 623-632, 1987. · Zbl 0635.47045 · doi:10.1016/0362-546X(87)90077-0
[8] T. Gnana Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 7, pp. 1379-1393, 2006. · Zbl 1106.47047 · doi:10.1016/j.na.2005.10.017
[9] V. Lakshmikantham and L. B. Ciric, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 12, pp. 4341-4349, 2009. · Zbl 1176.54032 · doi:10.1016/j.na.2008.09.020
[10] B. S. Choudhury and A. Kundu, “A coupled coincidence point result in partially ordered metric spaces for compatible mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 8, pp. 2524-2531, 2010. · Zbl 1229.54051 · doi:10.1016/j.na.2010.06.025
[11] N. V. Luong and N. X. Thuan, “Coupled fixed points in partially ordered metric spaces and application,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 3, pp. 983-992, 2011. · Zbl 1202.54036 · doi:10.1016/j.na.2010.09.055
[12] I. Altun and H. Simsek, “Some fixed point theorems on ordered metric spaces and application,” Fixed Point Theory and Applications, vol. 2010, Article ID 621469, 2010. · Zbl 1197.54053 · doi:10.1155/2010/621469
[13] V. Popa, “Some fixed point theorems for compatible mappings satisfying an implicit relation,” Demonstratio Mathematica, vol. 32, no. 1, pp. 157-163, 1999. · Zbl 0926.54030
[14] I. Altun and D. Turkoglu, “Some fixed point theorems for weakly compatible multivalued mappings satisfying an implicit relation,” Taiwanese Journal of Mathematics, vol. 22, no. 1, pp. 13-21, 2008. · Zbl 1199.54202 · doi:10.2298/FIL0801011A
[15] I. Altun and D. Turkoglu, “Some fixed point theorems for weakly compatible mappings satisfying an implicit relation,” Taiwanese Journal of Mathematics, vol. 13, no. 4, pp. 1291-1304, 2009. · Zbl 1194.54055
[16] M. Imdad, S. Kumar, and M. S. Khan, “Remarks on some fixed point theorems satisfying implicit relations,” Radovi Matematicki, vol. 11, no. 1, pp. 135-143, 2002. · Zbl 1033.54025
[17] V. Popa, “A general coincidence theorem for compatible multivalued mappings satisfying an implicit relation,” Demonstratio Mathematica, vol. 33, no. 1, pp. 159-164, 2000. · Zbl 0947.54023
[18] V. Popa and M. Mocanu, “Altering distance and common fixed points under implicit relations,” Hacettepe Journal of Mathematics and Statistics, vol. 38, no. 3, pp. 329-337, 2009. · Zbl 1239.47046
[19] S. Sharma and B. Deshpande, “On compatible mappings satisfying an implicit relation in common fixed point consideration,” Tamkang Journal of Mathematics, vol. 33, no. 3, pp. 245-252, 2002. · Zbl 1029.47037
[20] D. Turkoglu and I. Altun, “A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying an implicit relation,” Sociedad Matematica Mexicana Boletin, vol. 13, no. 1, pp. 195-205, 2007. · Zbl 1171.54342
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.