Application of a generalization of Darbo's fixed point theorem via Mizogochi-Takahashi mappings on mixed fractional integral equations involving $(k, s)$-Riemann-Liouville and Erd\'{e}lyi-Kober fractional integrals

Document Type : Research Paper


1 Department of Mathematics, Cotton University, Panbazar, Guwahati-781001, Assam, India

2 Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran

3 Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh-791112, Arunachal Pradesh, India

4 Department of Mathematics, Azadshahr Branch, Islamic Azad University, Azadshahr, Iran


We have established the solvability of fractional integral equations with both $(k,s)$-Riemann-Liouville and Erd'{e}lyi-Kober fractional integrals using a new generalized version of the Darbo's theorem using Mizogochi-Takahashi mappings and justify the validity of our results with the help of suitable examples.


[1] M. Z. Sarikaya, Z. Dahmani, M.E. Kieis and F. Ahmad, (k, s)-Riemann-Liouville fractional integral and applications, Hacettepe J. Math. Stat. 45(1) (2016) 77–89.
[2] R.P. Agarwal, Maria Meehan and D. O’Regan, Fixed Point Theory and Applications Cambridge University Press,2001.
[3] S. Banaei, M. Mursaleen and V. Parvaneh, Some fixed point theorems via measure of noncompactness with applications to differential equations, Comput. Appl. Math. 39 (2020) 139.
[4] J. Bana´s and K. Goebel, Measure of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Vol. 60, Marcel Dekker, New York, 1980.
[5] G. Darbo, Punti uniti in trasformazioni a codominio non compatto (Italian), Rend. Sem. Mat. Univ. Padova 24 (1955) 84–92.
[6] M. A. Darwish and K. Sadarangani, On Erd´elyi-Kober type quadratic integral equation with linear modification of the argument, Appl. Math. Comput. 238 (2014) 30–42.
[7] A. Das, B. Hazarika, V. Parvaneh and M. Mursaleen, Solvability of generalized fractional order integral equations via measures of noncompactness, Math. Sci. 15 (2021) 241–251.
[8] B. Matani and J. R. Roshan, Multivariate generalized Meir-Keeler condensing operators and their applications to systems of integral equations, J. Fixed Point Theory Appl., (2020) 22:87.
[9] J.R. Roshan, Existence of solutions for a class of system of functional integral equation via measure of noncompactness, J. Comput. Appl. Math. 313 (2017) 129–141.
[10] E. Ameer, H. Aydi, M. Arshad, H. Alsamir and M.S. Noorani, Hybrid multivalued type contraction mappings in αK-complete partial b-metric spaces and applications, Symmetry 11(1) (2019) 86.
[11] E. Ameer, H. Aydi, M. Arshad and M. De la Sen, Hybrid Ciri´c type graphic ´ (Υ,Λ)-contraction mappings with applications to electric circuit and fractional differential equations, Symmetry 12(3) (2020) 467.
[12] R. Garra, E. Orsingher and F. Polito, A note on Hadamard fractional differential equations with varying coefficients and their applications in probability, Math. 8(1) 2018 4.
[13] F. Jarad, T. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of AtanganaBaleanu fractional derivative, Chaos, Solitons Fract. 117 (2018) 16–20.
[14] K. Kuratowski, Sur les espaces complets, Fund. Math. 15 (1930) 301–309.
[15] R. Mollapourasl and A. Ostadi, On solution of functional integral equation of fractional order, Appl. Math. Comput. 270 (2015) 631–643.
[16] J. Bana´s and M. Krajewska, Existence of solutions for infinite systems of differential equations in spaces of tempered sequences, Electronic J. Diff. Equ. 60 (2017) 1–28.
Volume 13, Issue 1
March 2022
Pages 859-869
  • Receive Date: 27 March 2021
  • Revise Date: 10 May 2021
  • Accept Date: 29 June 2021