Coupled Fixed Point for (*α*, *Ψ*)-Contractive in Partially Ordered Metric Spaces Using Compatible Mappings

ABSTRACT

In this paper, first we introduce notions of (*α*, *Ψ*)-contractive and (*α*)-admissible for a pair of map and prove a coupled coincidence point theorem for compatible mappings using these notions. Our work extends and generalizes the results of Mursaleen et al. [1]. At the end, we will provide an example in support of our result.

In this paper, first we introduce notions of (

Cite this paper

Preeti, &. and Kumar, S. (2015) Coupled Fixed Point for (*α*, *Ψ*)-Contractive in Partially Ordered Metric Spaces Using Compatible Mappings. *Applied Mathematics*, **6**, 1380-1388. doi: 10.4236/am.2015.68130.

Preeti, &. and Kumar, S. (2015) Coupled Fixed Point for (

References

[1] Mursaleen, M., Mohiuddine, S.A. and Agarwal, R.P. (2012) Coupled Fixed Point Theorems for α-ψ-Contractive Type Mappings in Partially Ordered Metric Spaces. Fixed Point Theory and Applications, 2012, 228.

[2] Turinici, M. (1986) Abstract Comparison Principles and Multivariable Gronwall-Bellman Inequalities. Journal of Mathematical Analysis and Applications, 117, 100-127.

http://dx.doi.org/10.1016/0022-247X(86)90251-9

[3] Ran, A.C.M. and Reurings, M.C.B. (2004) A Fixed Point Theorem in Partially Ordered Sets and Some Applications to Matrix Equations. Proceedings of the American Mathematical Society, 132, 1435-1443.

http://dx.doi.org/10.1090/S0002-9939-03-07220-4

[4] Nieto, J.J. and Rodríguez-López, R. (2005) Contractive Mapping Theorems in Partially Ordered Sets and Applications to Ordinary Differential Equations. Order, 22, 223-239.

http://dx.doi.org/10.1007/s11083-005-9018-5

[5] Nieto, J.J. and Rodriguez-López, R. (2007) Existence and Uniqueness of Fixed Point in Partially Ordered Sets and Applications to Ordinary Differential Equations. Acta Mathematica Sinica, English Series, 23, 2205.

http://dx.doi.org/10.1007/s10114-005-0769-0

[6] Bhaskar, T.G. and Lakshmikantham, V. (2006) Fixed Point Theorems in Partially Ordered Metric Spaces and Applications. Nonlinear Analysis, 65, 1379-1393.

http://dx.doi.org/10.1016/j.na.2005.10.017

[7] Guo, D. and Lakshmikantham, V. (1987) Coupled Fixed Points of Nonlinear Operators with Applications. Nonlinear Analysis, 11, 623-632.

http://dx.doi.org/10.1016/0362-546X(87)90077-0

[8] Lakshmikantham, V. and Ciric, L. (2009) Coupled Fixed Point Theorems for Nonlinear Contractions in Partially Ordered Metric Spaces. Nonlinear Analysis, 70, 4341-4349.

http://dx.doi.org/10.1016/j.na.2008.09.020

[9] Choudhury, B.S. and Kundu, A. (2010) A Coupled Coincidence Point Result in Partially Orderedmetric Spaces for Compatible Mappings. Nonlinear Analysis, 73, 2.

http://dx.doi.org/10.1016/j.na.2010.06.025

[10] Samet, B., Vetro, C. and Vetro, P. (2012) Fixed Point Theorems for α-ψ-Contractive Type Mappings. Nonlinear Analysis, 75, 2154-2165.

http://dx.doi.org/10.1016/j.na.2011.10.014

[11] Shatanawi, W., Samet, B. and Abbas, M. (2012) Coupled Fixed Point Theorems for Mixed Monotone Mappings in Ordered Partial Metric Spaces. Mathematical and Computer Modelling, 55, 680-687.

[12] Abdeljawad, T. (2012) Coupled Fixed Point Theorems for Partially Contractive Type Mappings. Fixed Point Theory and Applications, 2012, 148.

[13] Aydi, H., Samet, B. and Vetro, C. (2011) Coupled Fixed Point Results in Cone Metric Spaces for W-Compatible Mappings. Fixed Point Theory and Applications, 2011, 27.

[14] Choudhury, B.S., Das, K. and Das, P. (2012) Coupled Coincidence Point Results for Compatible Mappings in Partially Ordered Fuzzy Metric Spaces. Fuzzy Sets and Systems, 222, 84-97.

[15] Amini-Harandi, A. (2013) Coupled and Tripled Fixed Point Theory in Partially Ordered Metric Spaces with Application to Initial Value Problem. Mathematical and Computer Modelling, 57, 2343-2348.

[16] Jungck, G. (1996) Compatible Mappings and Common Fixed Points. International Journal of Mathematics and Mathematical Sciences, 9, 771-779.

[17] Karapinar, E. and Samet, B. (2012) Generalized α-ψ-Contractive Type Mappings and Related Fixed Point Theorems with Applications. Abstract and Applied Analysis, 2012, Article ID: 793486.

[18] Luong, N.V. and Thuan, N.X. (2011) Coupled Fixed Points in Partially Ordered Metric Spaces and Application. Nonlinear Analysis: Theory, Methods & Applications, 74, 983-992.

http://dx.doi.org/10.1016/j.na.2010.09.055

[19] Nashine, H.K., Samet, B. and Vetro, C. (2012) Coupled Coincidence Points for Compatible Mappings Satisfying Mixed Monotone Property. The Journal of Nonlinear Science and Applications, 5, 104-114.

[20] Nashine, H.K., Kadelburg, Z. and Radenović, S. (2012) Coupled Common Fixed Point Theorems for W*-Compatible Mappings in Ordered Cone Metric Spaces. Applied Mathematics and Computation, 218, 5422-5432.

http://dx.doi.org/10.1016/j.amc.2011.11.029

[21] Samet, B. and Vetro, C. (2010) Coupled Fixed Point, F-Invariant Set and Fixed Point of N-Order. Annals of Functional Analysis, 1, 46-56.

http://dx.doi.org/10.15352/afa/1399900586

[22] Samet, B. and Vetro, C. (2011) Coupled Fixed Point Theorems for Multi-Valued Nonlinear Contraction Mappings in Partially Ordered Metric Spaces. Nonlinear Analysis, 74, 4260-4268.

http://dx.doi.org/10.1016/j.na.2011.04.007

[23] Sintunavarat, W., Cho, Y.J. and Kumam, P. (2012) Coupled Fixed Point Theorems for Nonlinear Contractions without Mixed Monotone Property. Fixed Point Theory and Applications, 2012, 170.

[1] Mursaleen, M., Mohiuddine, S.A. and Agarwal, R.P. (2012) Coupled Fixed Point Theorems for α-ψ-Contractive Type Mappings in Partially Ordered Metric Spaces. Fixed Point Theory and Applications, 2012, 228.

[2] Turinici, M. (1986) Abstract Comparison Principles and Multivariable Gronwall-Bellman Inequalities. Journal of Mathematical Analysis and Applications, 117, 100-127.

http://dx.doi.org/10.1016/0022-247X(86)90251-9

[3] Ran, A.C.M. and Reurings, M.C.B. (2004) A Fixed Point Theorem in Partially Ordered Sets and Some Applications to Matrix Equations. Proceedings of the American Mathematical Society, 132, 1435-1443.

http://dx.doi.org/10.1090/S0002-9939-03-07220-4

[4] Nieto, J.J. and Rodríguez-López, R. (2005) Contractive Mapping Theorems in Partially Ordered Sets and Applications to Ordinary Differential Equations. Order, 22, 223-239.

http://dx.doi.org/10.1007/s11083-005-9018-5

[5] Nieto, J.J. and Rodriguez-López, R. (2007) Existence and Uniqueness of Fixed Point in Partially Ordered Sets and Applications to Ordinary Differential Equations. Acta Mathematica Sinica, English Series, 23, 2205.

http://dx.doi.org/10.1007/s10114-005-0769-0

[6] Bhaskar, T.G. and Lakshmikantham, V. (2006) Fixed Point Theorems in Partially Ordered Metric Spaces and Applications. Nonlinear Analysis, 65, 1379-1393.

http://dx.doi.org/10.1016/j.na.2005.10.017

[7] Guo, D. and Lakshmikantham, V. (1987) Coupled Fixed Points of Nonlinear Operators with Applications. Nonlinear Analysis, 11, 623-632.

http://dx.doi.org/10.1016/0362-546X(87)90077-0

[8] Lakshmikantham, V. and Ciric, L. (2009) Coupled Fixed Point Theorems for Nonlinear Contractions in Partially Ordered Metric Spaces. Nonlinear Analysis, 70, 4341-4349.

http://dx.doi.org/10.1016/j.na.2008.09.020

[9] Choudhury, B.S. and Kundu, A. (2010) A Coupled Coincidence Point Result in Partially Orderedmetric Spaces for Compatible Mappings. Nonlinear Analysis, 73, 2.

http://dx.doi.org/10.1016/j.na.2010.06.025

[10] Samet, B., Vetro, C. and Vetro, P. (2012) Fixed Point Theorems for α-ψ-Contractive Type Mappings. Nonlinear Analysis, 75, 2154-2165.

http://dx.doi.org/10.1016/j.na.2011.10.014

[11] Shatanawi, W., Samet, B. and Abbas, M. (2012) Coupled Fixed Point Theorems for Mixed Monotone Mappings in Ordered Partial Metric Spaces. Mathematical and Computer Modelling, 55, 680-687.

[12] Abdeljawad, T. (2012) Coupled Fixed Point Theorems for Partially Contractive Type Mappings. Fixed Point Theory and Applications, 2012, 148.

[13] Aydi, H., Samet, B. and Vetro, C. (2011) Coupled Fixed Point Results in Cone Metric Spaces for W-Compatible Mappings. Fixed Point Theory and Applications, 2011, 27.

[14] Choudhury, B.S., Das, K. and Das, P. (2012) Coupled Coincidence Point Results for Compatible Mappings in Partially Ordered Fuzzy Metric Spaces. Fuzzy Sets and Systems, 222, 84-97.

[15] Amini-Harandi, A. (2013) Coupled and Tripled Fixed Point Theory in Partially Ordered Metric Spaces with Application to Initial Value Problem. Mathematical and Computer Modelling, 57, 2343-2348.

[16] Jungck, G. (1996) Compatible Mappings and Common Fixed Points. International Journal of Mathematics and Mathematical Sciences, 9, 771-779.

[17] Karapinar, E. and Samet, B. (2012) Generalized α-ψ-Contractive Type Mappings and Related Fixed Point Theorems with Applications. Abstract and Applied Analysis, 2012, Article ID: 793486.

[18] Luong, N.V. and Thuan, N.X. (2011) Coupled Fixed Points in Partially Ordered Metric Spaces and Application. Nonlinear Analysis: Theory, Methods & Applications, 74, 983-992.

http://dx.doi.org/10.1016/j.na.2010.09.055

[19] Nashine, H.K., Samet, B. and Vetro, C. (2012) Coupled Coincidence Points for Compatible Mappings Satisfying Mixed Monotone Property. The Journal of Nonlinear Science and Applications, 5, 104-114.

[20] Nashine, H.K., Kadelburg, Z. and Radenović, S. (2012) Coupled Common Fixed Point Theorems for W*-Compatible Mappings in Ordered Cone Metric Spaces. Applied Mathematics and Computation, 218, 5422-5432.

http://dx.doi.org/10.1016/j.amc.2011.11.029

[21] Samet, B. and Vetro, C. (2010) Coupled Fixed Point, F-Invariant Set and Fixed Point of N-Order. Annals of Functional Analysis, 1, 46-56.

http://dx.doi.org/10.15352/afa/1399900586

[22] Samet, B. and Vetro, C. (2011) Coupled Fixed Point Theorems for Multi-Valued Nonlinear Contraction Mappings in Partially Ordered Metric Spaces. Nonlinear Analysis, 74, 4260-4268.

http://dx.doi.org/10.1016/j.na.2011.04.007

[23] Sintunavarat, W., Cho, Y.J. and Kumam, P. (2012) Coupled Fixed Point Theorems for Nonlinear Contractions without Mixed Monotone Property. Fixed Point Theory and Applications, 2012, 170.