Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Existence of Solutions of Integral and Fractional Differential Equations Using α-type Rational F-contractions in Metric-like Spaces

  • Received : 2017.09.04
  • Accepted : 2018.08.13
  • Published : 2018.12.23
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Abstract

We present ${\alpha}$-type rational F-contractions in metric-like spaces, and respective fixed and common fixed point results for weakly ${\alpha}$-admissible mappings. Useful examples illustrate the effectiveness of the presented results. As applications, we obtain sufficient conditions for the existence of solutions of a certain type of integral equations followed by examples of nonlinear fractional differential equations that are verified numerically.

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References

  1. H. H. Alsulami, E. Karapinar and H. Piri, Fixed points of modified F-contractive mappings in complete metric-like spaces, J. Funct. Spaces, (2015), Article ID 270971, 9 pp.
  2. A. Amini-Harandi, Metric-like spaces, partial metric spaces and fixed points, Fixed Point Theory Appl., (2012), 2012:204, 10 pp. https://doi.org/10.1186/1687-1812-2012-204
  3. H. Aydi, M. Jellali and E. Karapinar, Common fixed points for generalized ${\alpha}$-implicit contractions in partial metric spaces: consequences and application, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A. Math., 109(2)(2015), 367-384.
  4. D. Baleanu, S. Rezapour and M. Mohammadi, Some existence results on nonlinear fractional differential equations, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 371(2013), Article ID 20120144, 7 pp.
  5. D. Gopal, M. Abbas, D. K. Patel and C. Vetro, Fixed points of ${\alpha}$-type F-contractive mappings with an application to nonlinear fractional differential equation, Acta Math. Sci. Ser. B, 36(3)(2016), 957-970. https://doi.org/10.1016/S0252-9602(16)30052-2
  6. P. Hitzler and A. Seda, Mathematical Aspects of Logic Programming Semantics, Chapman & Hall/CRC Studies in Informatic Series, CRC Press, 2011.
  7. N. Hussain and P. Salimi, Suzuki-Wardowski type fixed point theorems for ${\alpha}$-GF-contractions, Taiwanese J. Math., 18(6)(2014), 1879-1895. https://doi.org/10.11650/tjm.18.2014.4462
  8. E. Karapinar, M. A. Kutbi, H. Piri and D. O'Regan, Fixed points of conditionally F-contractions in complete metric-like spaces, Fixed Point Theory Appl., (2015), 2015:126, 14 pp. https://doi.org/10.1186/s13663-015-0377-3
  9. E. Karapinar and P. Salimi, Dislocated metric space to metric spaces with some fixed point theorems, Fixed Point Theory Appl., (2013), 2013:222, 19 pp. https://doi.org/10.1186/1687-1812-2013-222
  10. X.-L. Liu, A. H. Ansari, S. Chandok and S. Radenovic, On some results in metric spaces using auxiliary simulation functions via new functions, J. Comput. Anal. Appl., 24(6)(2018), 1103-1114.
  11. S. G. Malhotra, S. Radenovic and S. Shukla, Some fixed point results without monotone property in partially ordered metric-like spaces, J. Egyptian Math. Soc., 22(2014), 83-89. https://doi.org/10.1016/j.joems.2013.06.010
  12. S. G. Matthews, Partial metric topology, Proc. 8th Summer Conference on General Topology and Applications, Ann. New York Acad. Sci, 728 (1994), 183-197. https://doi.org/10.1111/j.1749-6632.1994.tb44144.x
  13. B. Samet, C. Vetro and P. Vetro, Fixed point theorems for ${\alpha}$-${\varphi}$-contractive type mappings, Nonlinear Anal., 75(2012), 2154-2165. https://doi.org/10.1016/j.na.2011.10.014
  14. N. Shobkolaei, S. Sedghi, J. R. Roshan and N. Hussain, Suzuki type fixed point results in metric-like spaces, J. Function Spaces Appl., (2013), Art. ID 143686, 9 pp.
  15. S. Shukla, S. Radenovic and V. C. Rajic, Some common fixed point theorems in 0-${\sigma}$-complete metric-like spaces, Vietnam J. Math., 41(2013), 341-352. https://doi.org/10.1007/s10013-013-0028-0
  16. W. Sintunavarat, A new approach to ${\alpha}$-${\varphi}$-contractive mappings and generalized Ulam-Hyers stability, well-posedness and limit shadowing results, Carpathian J. Math., 31(3)(2015), 395-401.
  17. W. Sintunavarat, Fixed point results in b-metric spaces approach to the existence of a solution for nonlinear integral equations, Rev. Real Acad. Cie. Exac. Fix. Nat., Ser A. Math., 110(2)(2016), 585-600. https://doi.org/10.1007/s13398-015-0251-5
  18. D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., (2012), 2012:94, 6 pp. https://doi.org/10.1186/1687-1812-2012-94