On existence of solutions for some nonlinear fractional differential equations via Wardowski-Mizoguchi-Takahashi type contractions

Document Type : Research Paper

Authors

1 Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran.

2 Department of Mathematics, Marand Branch, Islamic Azad University, Marand, Iran

3 Institute of Research and Development of Processes, Leioa 48940, Spain

4 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore-632014, TN, India

Abstract

Using the concept of extended Wardowski-Mizoguchi-Takahashi contractions, we investigate the existence of solutions for three type of nonlinear fractional differential equations. To patronage our main results, some examples of nonlinear fractional differential equations are given.

Keywords

[1] I. Podlubny, Fractional Differential Equations, NY, New York, Academic Press, 1999.
[2] S.G. Samko, A.A. Kilbas and O.I. Maricev, Fractional Integral and Derivative, UK. London, Gordon and Breach, 1993.
[3] MU. Ali, T. Kamran, W. Sintunavarat and PH. Katchang, Mizoguchi-Takahashi’s fixed point theorem with α, η functions, Abstr. Appl. Anal.   2013 (2013), Article ID 418798, 4 pages.
[4] 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 Anal. 72 (2010) 2238–2242.
[5] A. Aghajani, M. Abbas and J.R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca  64(4) (2014) 941–960.
[6] A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and application of fractional differential equations, North Holland Math. Stud. 204 (2006).
[7] Z. Mustafa, J.R. Roshan, V. Parvaneh and Z. Kadelburg, Fixed point theorems for weakly T-Chatterjea and weakly T-Kannan contractions in b-metric spaces, J. Ineq. Appl. 2014 (2014) 46.
[8] J.R. Roshan, V. Parvaneh and I. Altun, Some coincidence point results in ordered b-metric spaces and applications in a system of integral equations, Appl. Math. Comput. 226 (2014) 725-737.
[9] G. Mınak and I. Altun, Some new generalizations of Mizoguchi-Takahashi type fixed point theorem, J. Ineq. Appl. 2013 (2013) 493.
[10] M.E. Gordji and M. Ramezani, A generalization of Mizoguchi and Takahashi’s theorem for single-valued mappings in partially ordered metric spaces, Nonlinear Anal. 74(13) (2011) 4544-4549.
[11] E. Karapınar and B. Samet, Generalized (α−ψ)- contractive type mappings and related fixed point theorems with applications, Abst. Appl. Anal. 2012 (2012) Article ID: 793486.
[12] N. Hussain, E. Karapınar, P. Salimi and P. Vetro, Fixed point results for Gm-Meir-Keeler contractive and G- (α, ψ)-Meir-Keeler contractive mappings, Fixed Point Theory Appl. 34 (2013).
[13] B. Samet, C. Vetro and P. Vetro, Fixed point theorem for α − ψ-contractive type mappings. Nonlinear Anal. 75 (2012) 2154–2165.
[14] N. Mizoguchi and W. Takahashi, Fixed point theorems for multi-valued mappings on complete metric space, J. Math. Anal. Appl. 141 (1989) 177-188.
[15] B. Mohammadi, V. Parvaneh, H. Aydi and H. Isik, Extended Mizoguchi-Takahashi type fixed point theorems and their application, Math. 7 (2019) 575.
[16] D. Wardowski, Fixed points of a new type of contractive mappings in MbMSs, Fixed Point Theory Appl. 94 (2012).
Volume 12, Issue 1
May 2021
Pages 893-902
  • Receive Date: 15 October 2018
  • Revise Date: 11 June 2019
  • Accept Date: 28 June 2020