Existence and Ulam-Hyers stability of the implicit fractional boundary value problem with ψ-Caputo fractional derivative

• Autorzy: Wahash Hanan A. , Abdo Mohammed S , Panchal Satish K.

Identyfikatory

DOI

10.17512/jamcm.2020.1.08

• Języki publikacji: EN

Abstrakty

EN

In this paper, we investigate the existence, uniqueness and Ulam-Hyers stability of solutions for nonlinear implicit fractional differential equations with boundary conditions involving a ψ-Caputo fractional derivative. The obtained results for the proposed problem are proved under a new approach and minimal assumptions on the function ƒ. The analysis is based upon the reduction of the problem considered to the equivalent integral equation, while some fixed point theorems of Banach and Schauder and generalized Gronwall inequality are employed to obtain our results for the problem at hand. Finally, the investigation is illustrated by providing a suitable example.

Słowa kluczowe

EN

fractional differential equations; ψ-fractional integral and derivative; existence and Ulam-Hyers stability; fixed point theorem

PL

równania różniczkowe ułamkowe; równania różniczkowe cząstkowe; pochodna ułamkowa; twierdzenie o punkcie stałym; pochodna ułamkowa Caputo

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2020
• Tom: Vol. 19, nr 1
• Strony: 89--101
• Opis fizyczny: Bibliogr. 27 poz.

Twórcy

autor

Wahash Hanan A.

• Research Scholar at Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University Aurangabad 431004 (M.S.), India

autor

Abdo Mohammed S

• Research Scholar at Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University Aurangabad 431004 (M.S.), India
• Department of Mathematics, Hodeidah University Al-Hodeidah, Yemen

autor

Panchal Satish K.

• Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University Aurangabad 431004 (M.S.), India

Bibliografia

• [1] Hilfer, R. (2000). Applications of Fractional Calculus in Physics. Singapore: World Scientific.
• [2] Kilbas, A.A., Srivastava, H.M., & Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations. Amsterdam: North-Holland Math. Stud, Elsevier.
• [3] Magin, R.L. (2006). Fractional Calculus in Bioengineering. Begell House Inc. Publisher.
• [4] Samko, S.G., Kilbas A.A., & Marichev O.I. (1993). Fractional Integrals and Derivatives, Theory and Applications. Yverdon: Gordon and Breach.
• [5] Abdo, M.S., & Panchal, S.K. (2019). Fractional integro-differential equations involving ψ-Hilfer fractional derivative. Adv. Appl. Math. Mech., 11(1), 1-22.
• [6] Abdo, M.S., & Panchal, S.K., (2018). Weighted fractional neutral functional differential equations. J. Siber. Fed. Univ. Math. Phys., 11(5), 1-15.
• [7] Agarwal, R. Hristova, S., & O’Regan, D. (2016). A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations. Fract. Calc. Appl. Anal., 19(2), 290-318.
• [8] Al-Saqabi, B., & Kiryakova, V.S. (1998). Explicit solutions of fractional integral and differential equations involving Erdelyi-Kober operators. Appl. Math. Comput., 95(1), 1-13.
• [9] Kucche, K.D., Nieto, J.J., & Venktesh, V. (2016). Theory of nonlinear implicit fractional differential equations. Diff. Equ. Dynam. Sys., 1-17.
• [10] Li, M., & Wang, J. (2015). Existence of local and global solutions for Hadamard fractional differential equations. Electron. J. Differ. Equ., 2015(166), 1-8.
• [11] Sun, Y., Zeng, Z., & Song, J. (2017). Existence and uniqueness for the boundary value problems of nonlinear fractional differential equation. Appl. Math., 8(3), 312.
• [12] Ulam, S.M. (1968). A Collection of Mathematical Problems. New York: Interscience.
• [13] Hyers, D.H. (1941). On the stability of the linear functional equation. Proc. Natl. Acad. Sci., 27, 222-224.
• [14] Andras, S. & Kolumban, J.J. (2013). On the Ulam-Hyers stability of first order differential systems with nonlocal initial conditions. Nonlinear Analysis: Theory, Methods & Applications, 82, 1-11.
• [15] Benchohraa, M., & Bouriaha, S. (2015). Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order. Moroccan. J. Pure. Appl. Anal., 1(1), 22-37.
• [16] Ibrahim, R.W. (2012). Generalized Ulam-Hyers stability for fractional differential equations. International Journal of Mathematics, 23, 1-9.
• [17] Jung, S.M. (2004). Hyers-Ulam stability of linear differential equations of first order. Applied Mathematics Letters, 17, 1135-1140.
• [18] Almeida, R. (2017). A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul., 44, 460-481.
• [19] Ardjouni, A., & Djoudi, A. (2019). Existence and uniqueness of solutions for nonlinear implicit Caputo-Hadamard fractional differential equations with nonlocal conditions. Advances in the Theory of Nonlinear Analysis and its Applications, 3(1), 46-52.
• [20] Dong, J., Feng, Y., & Jiang, J. (2017). A note on implicit fractional differential equations. Mathematica Aeterna, 7(3), 261-267.
• [21] Haoues, M., Ardjouni, A., & Djoudi, A. (2018). Existence, interval of existence and uniqueness of solutions for nonlinear implicit Caputo fractional differential equations. TJMM, 10(1), 09-13.
• [22] Nieto, J., Ouahab, A., & Venktesh, V. (2015). Implicit fractional differential equations via the Liouville-Caputo derivative. Mathematics, 3(2), 398-411.
• [23] Almeida, R. (2019). Fractional differential equations with mixed boundary conditions. Bulletin of the Malaysian Mathematical Sciences Society, 42(4), 1687-1697.
• [24] Almeida, R., Malinowska, A.B., & Monteiro, M.T. (2018). Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Math. Meth. Appl. Sci., 41(1), 336-352.
• [25] Vivek D., Elsayed E.M., & Kanagarajan K. (2018). Theory and analysis of ψ-fractional differential equations with boundary conditions. Communications in Applied Analysis, 22(3), 401-414.
• [26] Abdo, M.S., Panchal, S.K., & Abdulkafi M. Saeed (2019). Fractional boundary value problem with ψ-Caputo fractional derivative. Proc. Indian Acad. Sci. (Math. Sci.), 129, 65.
• [27] Sousa, J.V.C., & Oliveira, E.C. (2019). A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator. Diff. Equ. Appl., 11(1), 87-106

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).