Several New Types of Fixed Point Theorems and Their Applications to Two-Point Ordinary Differential Equations

Affiliation(s)

School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, Jiangsu, People’s Republic of China.

Department of Mathematics, Shawnee State University, Portsmouth, Ohio, USA.

Department of Mathematics, Nanjing University, Nanjing, Jiangsu, People’s Republic of China.

School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, Jiangsu, People’s Republic of China.

Department of Mathematics, Shawnee State University, Portsmouth, Ohio, USA.

Department of Mathematics, Nanjing University, Nanjing, Jiangsu, People’s Republic of China.

ABSTRACT

The present paper is mainly concerned with several new types of fixed point theorems in different spaces such as cone metric spaces and fuzzy metric spaces. By using these obtained fixed point theorems, we then prove the existence and uniqueness of the solutions to two classes of two-point ordinary differential equation problems.

The present paper is mainly concerned with several new types of fixed point theorems in different spaces such as cone metric spaces and fuzzy metric spaces. By using these obtained fixed point theorems, we then prove the existence and uniqueness of the solutions to two classes of two-point ordinary differential equation problems.

KEYWORDS

Expansive Mapping; Cone Metric Space; Fuzzy Metric Space; Two-Point Ordinary Differential Equations

Expansive Mapping; Cone Metric Space; Fuzzy Metric Space; Two-Point Ordinary Differential Equations

Cite this paper

C. Zhang, J. Li, Y. Zhang and X. Feng, "Several New Types of Fixed Point Theorems and Their Applications to Two-Point Ordinary Differential Equations,"*Applied Mathematics*, Vol. 3 No. 10, 2012, pp. 1109-1116. doi: 10.4236/am.2012.310163.

C. Zhang, J. Li, Y. Zhang and X. Feng, "Several New Types of Fixed Point Theorems and Their Applications to Two-Point Ordinary Differential Equations,"

References

[1] K. Deimling, “Nonlinear Functional Analysis,” SpringerVerlag, Berlin, 1985. doi:10.1007/978-3-662-00547-7

[2] C. J. Zhang, “Set-Valued Analysis and Its Applications to Economics,” The Science Press, Beijing, 2004.

[3] W. Walter, “Remarks on a Paper by F. Browder about Contraction,” Nonlinear Analysis, Vol. 5, 1981, pp. 21-25. doi:10.1016/0362-546X(81)90066-3

[4] T. Suzuki, “A New Type of Fixed Point Theorem in Metric Spaces,” Nonlinear Analysis, Vol. 71, 2009, pp. 53135317. doi:10.1016/j.na.2009.04.017

[5] A. Meir and E. Keeler, “A Theorem on Contraction Mappings,” Journal of Mathematical Analysis and Applications, Vol. 28, 1969, pp. 326-329. doi:10.1016/0022-247X(69)90031-6

[6] T. Cardinali and P. Rubbioni, “An Extension to Multifunctions of the Keeler-Meir’s Fixed Point Theorem,” Fixed Point Theory, Vol. 7, No. 1, 2006, pp. 23-36.

[7] L.-G. Huang and X. Zhang, “Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 332, No. 2, 2007, pp. 1468-1476.

[8] A. George and P. Veeramani, “On Some Result in Fuzzy Metric Space,” Fuzzy Sets and Systems, Vol. 64, 1994, pp. 395-399. doi:10.1016/0165-0114(94)90162-7

[9] S. Sedghi, I. Altunb and N. Shobe, “Coupled Fixed Point Theorems for Contractions in Fuzzy Metric Spaces,” Nonlinear Analysis, Vol. 72, 2010, pp. 1298-1304. doi:10.1016/j.na.2009.08.018

[10] X.-H. Zhu and J.-Z. Xiao, “Note on ‘Coupled Fixed Point Theorems for Contractions in Fuzzy Metric Spaces’,” Nonlinear Analysis, Vol. 74, 2011, pp. 5475-5479. doi:10.1016/j.na.2011.05.034

[11] S. S. Zhang, “Fixed Point Theorems of Mappings on Probabilistic Metric Spaces with Applications,” Scientia Sinca (Series A), Vol. 11, 1983.

[12] J. Rodriguez-Lopez and S. Romaguera, “The Hausdorff Fuzzy Metric on Compact Sets,” Fuzzy Sets and Systems, Vol. 147, No. 2, 2004, pp. 273-283.

[13] A. Amini-Harandi and H. Emami, “A Fixed Point Theorem for Contraction Type Maps in Partially Ordered Metric Spaces and Application to Ordinary Differential Equations,” Nonlinear Analysis, Vol. 72, 2010, pp. 2238-2242. doi:10.1016/j.na.2009.10.023

[14] T. G. Bhaskar and V. Lakshmikantham, “Fixed Point Theorems in Partially Ordered Metric Spaces and Applications,” Nonlinear Analysis, Vol. 65, 2006, pp. 1379-1393. doi:10.1016/j.na.2005.10.017

[15] J. Harjani and K. Sadarangani, “Generalized Contractions in Partially Ordered Metric Spaces and Applications to Ordinary Differential Equations,” Nonlinear Analysis, Vol. 72, 2010, pp. 1188-1197. doi:10.1016/j.na.2009.08.003

[16] J. Harjani and K. Sadarangani, “Fixed Point Theorems for Weakly Contractive Mappings in Partially Ordered Sets,” Nonlinear Analysis, Vol. 71, 2009, pp. 3403-3410. doi:10.1016/j.na.2009.01.240

[1] K. Deimling, “Nonlinear Functional Analysis,” SpringerVerlag, Berlin, 1985. doi:10.1007/978-3-662-00547-7

[2] C. J. Zhang, “Set-Valued Analysis and Its Applications to Economics,” The Science Press, Beijing, 2004.

[3] W. Walter, “Remarks on a Paper by F. Browder about Contraction,” Nonlinear Analysis, Vol. 5, 1981, pp. 21-25. doi:10.1016/0362-546X(81)90066-3

[4] T. Suzuki, “A New Type of Fixed Point Theorem in Metric Spaces,” Nonlinear Analysis, Vol. 71, 2009, pp. 53135317. doi:10.1016/j.na.2009.04.017

[5] A. Meir and E. Keeler, “A Theorem on Contraction Mappings,” Journal of Mathematical Analysis and Applications, Vol. 28, 1969, pp. 326-329. doi:10.1016/0022-247X(69)90031-6

[6] T. Cardinali and P. Rubbioni, “An Extension to Multifunctions of the Keeler-Meir’s Fixed Point Theorem,” Fixed Point Theory, Vol. 7, No. 1, 2006, pp. 23-36.

[7] L.-G. Huang and X. Zhang, “Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 332, No. 2, 2007, pp. 1468-1476.

[8] A. George and P. Veeramani, “On Some Result in Fuzzy Metric Space,” Fuzzy Sets and Systems, Vol. 64, 1994, pp. 395-399. doi:10.1016/0165-0114(94)90162-7

[9] S. Sedghi, I. Altunb and N. Shobe, “Coupled Fixed Point Theorems for Contractions in Fuzzy Metric Spaces,” Nonlinear Analysis, Vol. 72, 2010, pp. 1298-1304. doi:10.1016/j.na.2009.08.018

[10] X.-H. Zhu and J.-Z. Xiao, “Note on ‘Coupled Fixed Point Theorems for Contractions in Fuzzy Metric Spaces’,” Nonlinear Analysis, Vol. 74, 2011, pp. 5475-5479. doi:10.1016/j.na.2011.05.034

[11] S. S. Zhang, “Fixed Point Theorems of Mappings on Probabilistic Metric Spaces with Applications,” Scientia Sinca (Series A), Vol. 11, 1983.

[12] J. Rodriguez-Lopez and S. Romaguera, “The Hausdorff Fuzzy Metric on Compact Sets,” Fuzzy Sets and Systems, Vol. 147, No. 2, 2004, pp. 273-283.

[13] A. Amini-Harandi and H. Emami, “A Fixed Point Theorem for Contraction Type Maps in Partially Ordered Metric Spaces and Application to Ordinary Differential Equations,” Nonlinear Analysis, Vol. 72, 2010, pp. 2238-2242. doi:10.1016/j.na.2009.10.023

[14] T. G. Bhaskar and V. Lakshmikantham, “Fixed Point Theorems in Partially Ordered Metric Spaces and Applications,” Nonlinear Analysis, Vol. 65, 2006, pp. 1379-1393. doi:10.1016/j.na.2005.10.017

[15] J. Harjani and K. Sadarangani, “Generalized Contractions in Partially Ordered Metric Spaces and Applications to Ordinary Differential Equations,” Nonlinear Analysis, Vol. 72, 2010, pp. 1188-1197. doi:10.1016/j.na.2009.08.003

[16] J. Harjani and K. Sadarangani, “Fixed Point Theorems for Weakly Contractive Mappings in Partially Ordered Sets,” Nonlinear Analysis, Vol. 71, 2009, pp. 3403-3410. doi:10.1016/j.na.2009.01.240