The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
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APPLICATION OF CONTRACTION MAPPING PRINCIPLE IN PERIODIC BOUNDARY VALUE PROBLEMS

  • Amrish Handa (Department of Mathematics, Govt. P. G. Arts and Science College)
  • Received : 2022.10.04
  • Accepted : 2023.07.10
  • Published : 2023.08.31
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Abstract

We prove some common fixed point theorems for β-non-decreasing mappings under contraction mapping principle on partially ordered metric spaces. We study the existence of solution for periodic boundary value problems and also give an example to show the degree of validity of our hypothesis. Our results improve and generalize various known results.

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References

  1. S.A. Al-Mezel, H. Alsulami, E. Karapinar & A. Roldan: Discussion on multidimensional coincidence points via recent publications. Abstr. Appl. Anal. 2014 (2014), Article ID 287492. https://doi.org/10.1155/2014/287492
  2. A. Alotaibi & S.M. Alsulami: Coupled coincidence points for monotone operators in partially ordered metric spaces. Fixed Point Theory Appl. 2011 (2011), Paper No. 44. https://doi.org/10.1186/1687-1812-2011-44
  3. S.M. Alsulami: Some coupled coincidence point theorems for a mixed monotone operator in a complete metric space endowed with a partial order by using altering distance functions. Fixed Point Theory Appl. 2013 (2013), Paper No. 194. https://doi.org/10.1186/1687-1812-2013-194
  4. T.G. Bhaskar & V. Lakshmikantham: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65 (2006), no. 7, 1379-1393. https://doi.org/10.1016/j.na.2005.10.017
  5. B.S. Choudhury & A. Kundu: A coupled coincidence point results in partially ordered metric spaces for compatible mappings. Nonlinear Anal. 73 (2010), 2524-2531. https://doi.org/10.1016/j.na.2010.06.025
  6. B.S. Choudhury, N. Metiya & M. Postolache: A generalized weak contraction principle with applications to coupled coincidence point problems. Fixed Point Theory Appl. 2013 (2013), Paper No. 152. https://doi.org/10.1186/1687-1812-2013-152
  7. B. Deshpande & A. Handa: Coincidence point results for weak ψ - ϕ contraction on partially ordered metric spaces with application. Facta Universitatis Ser. Math. Inform. 30 (2015), no. 5, 623-648.
  8. B. Deshpande, A. Handa & C. Kothari: Existence of coincidence point under generalized nonlinear contraction on partially ordered metric spaces. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 23 (2016), no. 1, 35-51. http://dx.doi.org/10.7468/jksmeb.2016.23.1.35
  9. B. Deshpande, A. Handa & S.A. Thoker: Existence of coincidence point under generalized nonlinear contraction with applications. East Asian Math. J. 32 (2016), no. 3, 333-354. http://dx.doi.org/10.7858/eamj.2016.025
  10. B. Deshpande & A. Handa: On coincidence point theorem for new contractive condition with application. Facta Universitatis Ser. Math. Inform. 32 (2017), no. 2, 209-229. https://doi.org/10.22190/FUMI1702209D
  11. I.M. Erhan, E. Karapinar, A. Roldan & N. Shahzad: Remarks on coupled coincidence point results for a generalized compatible pair with applications. Fixed Point Theory Appl. 2014 (2014), Paper No. 207. http://dx.doi.org/10.1186/1687-1812-2014-207
  12. D. Guo & V. Lakshmikantham: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 11 (1987), no. 5, 623-632. https://doi.org/10.1016/0362-546X(87)90077-0
  13. A. Handa: Multidimensional coincidence point results for contraction mapping principle. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 26 (2019), no. 4, 277-288. https://doi.org/10.7468/jksmeb.2019.26.4.277
  14. A. Handa: Utilizing isotone mappings under Mizoguchi-Takahashi contraction to prove multidimensional fixed point theorems with application. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 26 (2019), no. 4, 289-303. https://doi.org/10.7468/jksmeb.2019.26.4.289
  15. A. Handa: Existence of coincidence point under generalized Geraghty-type contraction with application. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 27 (2020), no. 3, 109-124. https://doi.org/10.7468/jksmeb.2020.27.3.109
  16. A. Handa: Multidimensional fixed point results for contraction mapping principle with application. Facta Univ. Ser. 35 (2020), no. 4, 919-928. https://doi.org/10.22190/FUMI2004919H
  17. J. Harjani, B. Lopez & K. Sadarangani: Fixed point theorems for mixed monotone operators and applications to integral equations. Nonlinear Anal. 74 (2011), 1749-1760. https://doi.org/10.1016/j.na.2010.10.047
  18. J. Harjani & K. Sadarangani: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. 72 (2010), 1188-1197. https://doi.org/10.1016/j.na.2009.08.003
  19. G. Jungck: compatible mappings and common fixed points. Int. J. Math. Math. Sci. 9 (1986), no. 4, 771-779. https://doi.org/10.1155/S0161171286000935
  20. G. Jungck & B.E. Rhoades: Fixed point for set-valued functions without continuity. Indian J. Pure Appl. Math. 29 (1998), no. 3, 227-238.
  21. V. Lakshmikantham & L. Ciric: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 70 (2009), no. 12, 4341-4349. https://doi.org/10.1016/j.na.2008.09.020
  22. N.V. Luong & N.X. Thuan: Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal. 74 (2011), 983-992. https://doi.org/10.1016/j.na.2010.09.055
  23. J.J. Nieto & R. Rodriguez-Lopez: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22 (2005), 223-239. https://doi.org/10.1007/s11083-005-9018-5
  24. A.C.M. Ran & M.C.B. Reurings: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Amer. Math. Soc. 132 (2004), 1435-1443. https://doi.org/10.1090/S0002-9939-03-07220-4
  25. A. Razani & V. Parvaneh, Coupled coincidence point results for (ψ, α, β)-weak contractions in partially ordered metric spaces. J. Appl. Math. 2012, Article ID 496103. https://doi.org/10.1155/2012/496103
  26. B. Samet, E. Karapinar, H. Aydi & V.C. Rajic: Discussion on some coupled fixed point theorems. Fixed Point Theory Appl. 2013 (2013), Paper No. 50. https://doi.org/10.1186/1687-1812-2013-50
  27. F. Shaddad, M.S.M. Noorani, S.M. Alsulami & H. Akhadkulov: Coupled point results in partially ordered metric spaces without compatibility. Fixed Point Theory Appl. 2014 (2014), Paper No. 204. https://doi.org/10.1186/1687-1812-2014-204
  28. Y. Su: Contraction mapping principle with generalized altering distance function in ordered metric spaces and applications to ordinary differential equations. Fixed Point Theory Appl. 2014 (2014), Paper No. 227. https://doi.org/10.1186/1687-1812-2014-227