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

Document Type : Research Paper


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


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.


[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