International Journal of Nonlinear Analysis and Applications

Solvability of infinite system of Hadamard-type fractional boundary value problem in the double sequence space

Articles in Press

Document Type : Research Paper

Authors

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

10.22075/ijnaa.2024.33215.4946

Abstract

In this work,  we are interested in representing the solution of Hadamard type fractional differential equation by introducing the concept of double sequence space $2^c(\Delta)$.  After that, we construct the Hausdorff measure of non-compactness on the space $2^c(\Delta)$.  Furthermore,  we see the existence of a solution of Hadamard-type fractional differential equation on the space $2^c(\Delta)$. After that, we demonstrate an example to see the applicability of our results.

Keywords

[1] R.P. Agarwal and D. O’Regan, Fixed Point Theory and Applications, Cambridge University Press, 2004.
[2] B. Ahmad, A. Alsaedi, S.K. Ntouyas, and J. Tariboon, Hadamard type Fractional Differential Equations, Inclusions and Inequalities, Cham, Springer International Publishing, Switzerland, 2017
[3] J. Banas and B.C. Dhage, Global asymptotic stability of solutions of a functional integral equation, Nonlinear Anal.: Theory Meth. Appl. 69 (2008), 1945–1952.
[4] J. Banas and B. Rzepka, An application of a measure of noncompactness in the study of asymptotic stability, Appl. Math. Lett. 16 (2003), 1–6.
[5] J. Banas and K. Goebel, Measure of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Vol. 60, Marcel Dekker, New York, 1980.
[6] C. Belen and M. Yildirim, the double sequence spaces, Kragujevac J. Math. 36 (2012), no. 2, 261–268.
[7] M. Berzig and E. Karapinar, Fixed point results for (αψ, βϕ)-contractive mappings for a generalized altering distance, Fixed Point Theory Appl. 2013 (2013), 205.
[8] T.J.I.A. Bromwich, An Introduction to the Theory of Infinite Series, MacMillan and Co.Ltd.New York, 1965.
[9] G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84–92. [In Italian]
[10] 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.
[11] A. Das, B. Hazarika, and P. Kumam, Some new generalization of Darbo’s fixed point theorem and its application on integral equations, Mathematics 7 (2019), no. 3, 214.
[12] A. Das, B. Hazarika, R. Arab, and M. Mursaleen, Applications of a fixed point theorem to the existence of solutions to the nonlinear functional integral equations in two variables, Rend. Circ. Mate. Palermo 68 (2019), no. 1, 139–152.
[13] A. Das, B. Hazarika, and M. Mursaleen, Application of measure of noncompactness for solvability of the infinite system of integral equations in two variables in ℓp(1 < p < ∞), Rev. Real Acad. Cien. Exactas Fıs. Natur. Ser. A. Mate. 113 (2019), 31–40.
[14] A. Das, I. Suwan, B.C. Deuri, and T. Abdeljawad, On solution of generalized proportional fractional integral via a new fixed point theorem, Adv. Differ. Equ. 2021 (2021), 427.
[15] D. Datta and B.C. Tripathy, Statistical convergence of double sequence of complex uncertain variables, J. Appl. Math. Inf. 40 (2022), no. 1-2, 191–204.
[16] B.C. Deuri and A. Das, Solvability of fractional integral equations via Darbo’s fixed point theorem, J. Pseudo-Differ. Oper. Appl. 13 (2022), no. 3, 26.
[17] R.M. Ganji, H. Jafari, and D. Baleanu, A new approach for solving multi variable orders differential equation with Mittag-Leffler kernel, Chaos Solitons Fractals 130 (2020), 109405.
[18] G.H. Hardy, On the convergence of certain multiple series, Proc. Camb. Phil. Soc. 19 (1917), 86–95.
[19] J. Harjani, B. Lopez, and K. Sadarangani, Existence and uniqueness of mild solution for a fractional differential equation under Sturm-Liouville boundary conditions when the data function is of Lipschitzian type, Demonst. Math. 53 (2020), no. 1, 167–1739.
[20] B. Hazarika, R. Arab, and M. Mursaleen, Applications of measure of noncompactness and operator type contraction for existence of solution of functional integral equations, Complex Anal. Oper. Theory 13 (2019), 3837–3851.
[21] M.D. Johansyah, A.K. Supriantna, E. Rusyaman, and J. Saputra, Application of fractional differential equation in economic growth model, AIMS Math. 6 (2021), 10266–10280.
[22] F. Khojasteh, S. Shukla, and S. Radenovic, A new approach to the study of fixed point theory for simulation functions, Filomat 26 (2015), no. 6, 1189-1194.
[23] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory an applications of Fractional Differential Equations, North[1]Holland Mathematics Studies, vol.204, Elsevier, Amsterdam, 2006.
[24] O. Kisi, Fibonacci lacunary ideal convergence of double sequence in intuitionistic fuzzy normed linear space, Math. Sci. Appl. E-notes 10 (2022), no. 3, 114–124.
[25] C. Kuratowski, Sur les espaces complets, Fund. Math. 15 (1930) 301–309.
[26] H. Mehravaran, H.A. Kayvanloo and M. Mursaleen, Solvability of infinite system of fractional differential equation in the double sequence space 2c(∆), Fractional Calc. Appl. Anal. 25 (2022), 2298–2312.
[27] F. Moricz, Extension of the spaces c and c0 from single to double sequences, Acta Math. Hungarica 57 (1991), 129–136.
[28] F. Moricz and B.E. Rhoades, Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Camb. Phil. Soc. 104 (1988), 283–294.
[29] M. Mursaleen, Differential equations in classical sequence spaces, Rev. Real Acad. Cien. Exactas F´ıs. Natur. Ser. A. Mate. 111 (2017), 587–612.
[30] H.K. Nashine and A. Das, Extension of Darbo’s fixed point theorem via shifting distance functions and its application, Nonlinear Anal.: Model. Control 27 (2022), 1–14.
[31] K. Raj, S. Sharma, and A. Choudhary, Orlicz difference triple lacunary ideal sequence space over n-normed space, Ann. Acad. Rom. Sci. Ser. Math. Appl. 14 (2022), no. 1-2, 90–106.
[32] E. Savas and R.F. Patterson, Double sequence spaces defined by a modulus, Math. Slovaca 61 (2011), no. 2, 245–256.
[33] M. Al-Smadi, O.A. Arqub, and S. Hadid, An attractive analytical technique for coupled system of fractional partial differential equations in shallow water with conformable derivative, Commun. Theor. Phys. 72 (2020), no. 8, 085001.
[34] U. Ulusu and E. Dundar, Invariant and lacunary invariant statistical convergence of order η for double set sequence, Turk. J. Sci. 7 (2022), no. 1, 14–20.
[35] J. Xu, J. Liu, and D. O’Regan, Solvability for a Hadamard-type fractional integral boundary value problem, Nonlinear Anal.: Model. Control 28 (2023), no. 4, 672–696.
[36] M. Zeltser, Investigation of Double Sequence Space by Soft and Hard Analytical Methods, Dissertationes Math[1]maticae Universitatis Tartuensis, Tartu University Press, Tartu, 2001.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 22 November 2024
  • Receive Date: 08 February 2024
  • Revise Date: 08 June 2024
  • Accept Date: 12 June 2024