Solvability of infinite systems of fractional differential equations in the space of tempered sequence space $m^\beta(\phi)$

Document Type : Research Paper


Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.


The purpose of this article, is to establish the existence of solution of infinite systems of fractional differential equations in space of tempered sequence $m^\beta(\phi)$ by using techniques associated with Hausdorff measures of noncompactness. Finally, we provide an example to highlight and establish the importance of our main result.


[1] A. Aghajani, J. Bana´s and Y. Jalilian, Existence of solution for a class of nonlinear Volterra singular integral equation, Comput. Math. Appl. , 62 (2011) 1215-1227.
[2] A. Aghajani, M. Mursaleen and A. Shole Haghighi, Fixed point theorems for Meir–Keeler condensing operators via measure of noncompactness, Acta Math. Sci. , 35B(3) (2015) 552–566.
[3] A. Aghajani, E. Pourhadi and J.J. Trujillo, Application of measure of noncompactness to a Cauchy problem for fractional differential equation in Banach spaces, Fract. Calc. Appl. Anal., 16(4) (2003) 962—977.
[4] R. Arab, R. Allahyari and A. Shole Haghighi, Existence of solutions of infinite systems of integral equations in two variables via measure of noncompactness, Appl. Math. Comput., 246 (2014) 283-291.
[5] J. Bana´s and K. Goebel, Measure of noncompactness in Banach spaces, Lecture notes in pure and applied mathematics. vol. 60. New York: Marcel Dekker, (1980).
[6] J. Bana´s and D. O’Regan, On existence and local attractivity of solutions of a quadratic Volterra integral equation of fraction order, J. Math. Anal. Appl. , 345 (2008) 573-582.
[7] J. Bana´s M. and Mursaleen, Sequence spaces and measures of noncompactness with applications to differential and integral equations, New Delhi: Springer, 2014.
[8] E. Cuesta and J.F Codes, Image processing by means of a linear integro differential equation Visualization imaging and image processing (2003), paper 91, Clagary, (2003). Hamza MH, editor. Acta Press.
[9] G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend Sem Mat Univ Padova., 24 (1955) 84–92.
[10] W. Deng, Short memory principal and a predictor-corrector approach for fractional differential equations, J. Comput. Appl. Math., 206 (2007) 174–188.
[11] K. Diethelm, The analysis of fractional differential equations An application-oriented exposition using differential operators of Caputo type, Springer Science Business Media (2010).
[12] L.S. Goldenˇstein, L.T. Gohberg and A.S. Murkus, Investigations of some properties of bounded linear operators with their q-norms, Uˇcen. Zap. Kishinevsk. Uni. , 29 (1957) 29-36.
[13] L.S. Goldenˇstein and A.S. Murkus, On a measure of noncompactness of bounded sets and linear operators, Studies in Algebra and Math. Anal. Kishinev, (1965) 45-54.
[14] B. Hazarika, E. Karapınar, R. Arab and M. Rabbani, Metric-like spaces to prove existence of solution for nonlinear quadratic integral equation and numerical method to solve it, J. Comput. Appl. Math., 328 (15)(2018) 302–313.
[15] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science Publishers, vol. 204, 2006.
[16] K. Kuratowski, Sur les espaces complets, Fund. Math., 15 (1930) 301-309.
[17] K. Maleknejad, P. Torabi and R. Mollapourasl, Fixed point method for solving nonlinear quadratic Volterra integral equations, Comput. Math. Appl., 62 (2011) 2555-2566.
[18] A. Meir and E.A. Keeler, Theorem on contraction mappings, J. Math. Anal. Appl., 28 (1969) 326–329.
[19] M. Mursaleen, Application of measure of noncompactness to infinite system of differential equations, Canad. Math. Bull., 56 (2013) 388-394.
[20] M. Mursaleen, Some geometric properties of a sequence space related to lp, Bull. Austral. Math. Soc. , 67 (2003) 343-347.
[21] M. Mursaleen, Differential equations in classical sequence spaces, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 111(2) (2017) 587–612.
[22] M. Mursaleen and A. Alotaibi, Infinite system of differential equations in some BK spaces, Abstract Appl. Anal., Volume 2012, Article ID 863483, 20 pages.
[23] M. Mursaleen, B. Bilalov and S.M.H. Rizvi, Applications of measure of noncompactness to infinite system of fractional differential equations, Filomat. 31 (11) (2017) 3421–3432.
[24] I. Podlubny, Fractional order systems and fractional order controllers, Technical report of-03-94. Institute of Experimental Physics, Slovak Acad. of Sci.; (1994).
[25] I. Podlubny, Fractional Differential Equations An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, vol. 198, 1998.
[26] M. Rabbani, A. Das, B. Hazarika and R. Arab, Measure of noncompactness of a new space of tempered sequences and its application on fractional differential equations, Chaos, Solitons and Fractals 140 (2020) 110221.
[27] W.L.C Sargent, Some sequence spaces related to the HP spaces, J. London Math. Soc. 35 (1960) 161-171.
Volume 13, Issue 1
March 2022
Pages 1023-1034
  • Receive Date: 29 April 2021
  • Revise Date: 06 September 2021
  • Accept Date: 15 September 2021