New generalizations for Gronwall type inequalities involving a $ \psi $-fractional operator and their applications

Jehad Alzabut; Yassine Adjabi; Weerawat Sudsutad; Mutti-Ur Rehman; Jehad Alzabut; Yassine Adjabi; Weerawat Sudsutad; Mutti-Ur Rehman

doi:10.3934/math.2021299

AIMS Mathematics

2021, Volume 6, Issue 5: 5053-5077. doi: 10.3934/math.2021299

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New generalizations for Gronwall type inequalities involving a $ \psi $-fractional operator and their applications

1.
Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 12435, Saudi Arabia
2.
Department of Mathematics, Faculty of Sciences, University of M'hamed Bougara, Boumerdes, Dynamic Systems Laboratory, Faculty of Mathematics, U.S.T.H.B., Algeria
3.
Department of Applied Statistics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand
4.
Department of Mathematics, Sukkur IBA University, 65200, Sukkur-Pakistan

Received: 02 January 2021 Accepted: 22 February 2021 Published: 09 March 2021
MSC : 26A24, 26A33, 26D10, 34A08, 34A12, 34D20, 35A23

In this paper, we provide new generalizations for the Gronwall's inequality in terms of a $ \psi $-fractional operator. The new forms of Gronwall's inequality are obtained within a general platform that includes several existing results as particular cases. To apply our results and examine their validity, we prove the existence and uniqueness of solutions for $ \psi $-fractional initial value problem. Further, the Ulam-Hyers stability of solutions for $ \psi $-fractional differential equations is discussed. For the sake of illustrating the proposed results, we give some particular examples.
- $ \psi $-fractional operators,
- generalized Gronwall's inequality,
- $ \psi $-fractional initial value problem,
- existence and uniqueness,
- Ulam-Hyers stability
Citation: Jehad Alzabut, Yassine Adjabi, Weerawat Sudsutad, Mutti-Ur Rehman. New generalizations for Gronwall type inequalities involving a $ \psi $-fractional operator and their applications[J]. AIMS Mathematics, 2021, 6(5): 5053-5077. doi: 10.3934/math.2021299

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Abstract

In this paper, we provide new generalizations for the Gronwall's inequality in terms of a $ \psi $-fractional operator. The new forms of Gronwall's inequality are obtained within a general platform that includes several existing results as particular cases. To apply our results and examine their validity, we prove the existence and uniqueness of solutions for $ \psi $-fractional initial value problem. Further, the Ulam-Hyers stability of solutions for $ \psi $-fractional differential equations is discussed. For the sake of illustrating the proposed results, we give some particular examples.

References

[1]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives theroy and applications, Gordon and Breach Science Publishers, 1993.
[2]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier Science, 2006.
[3]	V. Kiryakova, Generalized fractional calculus and applications, New York: J. Wiley & Sons, Inc., 1994.
[4]	T. J. Osler, The fractional derivative of a composite function, SIAM J. Math. Anal., 1 (1970), 288-293.
[5]	V. Kiryakova, A brief story about the operators of generalized fractional calculus, Fract. Calc. Appl. Anal., 11 (2008), 203-220.
[6]	J. V. da C. Sousa, E. C. de Oliveira, On the $\psi $ -Hilfer fractional derivative, Commun. Nonlinear Sci., 60 (2018), 72-91.
[7]	A. Seemab, M. Ur Rehman, J. Alzabut, A. Hamdi, On the existence of positive solutions for generalized fractional boundary value problems, Bound. Value Probl., 2019 (2019), 186.
[8]	R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci., 44 (2017), 460-481.
[9]	R. Almeida, A. B. Malinowska, M. T. T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Math. Methods Appl. Sci., 41 (2018), 336-352.
[10]	U. N. Katugampola, New Approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865.
[11]	D. S. Oliveira, E. C. de Oliveira, Hilfer-Katugampola fractional derivatives, Comp. Appl. Math., 37 (2018), 3672-3690.
[12]	J. V. da C. Sousa, E. C. de Oliveira, A Gronwall inequality and the Cauchy-type problem by means of $\psi $ -Hilfer operator, Diff. Equations Appl., 11 (2019), 87-106.
[13]	B. G. Pachpatte, Inequalities for differential and integral equations, San Diego: Academic Press, 1998.
[14]	T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. Math., 2 (1919), 292-296.
[15]	R. Bellman, Asymptotic series for the solutions of linear differential-difference equations, Rend. Circ. Mat. Palermo, 7 (1958), 261-269.
[16]	R. P. Agarwal, S. F. Deng, W. N. Zhang, Generalization of a retarded Gronwall-like inequality and its applications, Appl. Math. Comput., 165 (2005), 599-612.
[17]	B. G. Pachpatte, Integral and finit difference inequalities and applications, North Holland: Elsevier Science, 205 (2006).
[18]	S. D. Kender, S. G. Latpate, S. S. Ranmal, Some nonlinear integral inequalies for volterra-fredholm integral equations. Adv. Inequal. Appl., 2014 (2014).
[19]	F. C. Jiang, F. W. Meng, Explicit bounds on some new nonlinear integral inequalities with delay, J. Comput. Appl. Math. 205 (2007), 479-486.
[20]	H. P. Ye, J. M. Gao, Y. S. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.
[21]	W. M. Wang, Y. Q. Feng, Y. Y. Wang, Nonlinear Gronwall-Bellman type inequalities and their applications, Mathematics, 5 (2017), 31.
[22]	K. S. Nisar, G. Rahman, J. Choi, S. Mubeen, M. Arshad, Certain Gronwall type inequalities associated with Riemann-Liouville k- and Hadamard k-fractional derivatives and their applications, East Asian Math. J., 34 (2018), 249-263.
[23]	G. Rahman, T. Abdeljawad, F. Jarad, A. Khan, K. S. Nisar, Certain inequalities via generalized proportional Hadamard fractional integral operators, Adv. Differ. Equ., 2019 (2019), 454.
[24]	Y. Adjabi, F. Jarad, T. Abdeljawad, On generalized fractional operators and a Gronwall type inequality with applications, 17 (2017), 5457-5473.
[25]	R. Almeida, A Gronwall inequality for a general Caputo fractional operator, Math. Inequal. Appl., 20 (2017), 1089-1105.
[26]	J. Alzabut, T. Abdeljawad, F. Jarad, W. Sudsutad, A Gronwall inequality via the generalized proportional fractional derivative with applications, J. Inequal. Appl. 2019 (2019), 101.
[27]	J. Alzabut, W. Sudsutad, Z. Kayar, H. Baghani, A new Gronwall-Bellman inequality in frame of generalized proportional fractional derivative, Mathematics, 7 (2019), 747.
[28]	R. Almeida, A. B. Malinowska, T. Odzijewicz, An extension of the fractional Gronwall inequality, In: A. B. Malinowska, D. Mozyrska, L. Sajewski. (Eds), Advances in non-integer order calculus and its applications, RRNR 2018. Lecture Notes in Electrical Engineering, 559, Springer, Cham.
[29]	D. Willett, A linear generalization of Gronwall's inequality, P. Am. Math. Soc., 16 (1965), 774-778.
[30]	S. Y. Lin, Generalized Gronwall inequalities and their applications to fractional differential equations, J. Inequal. Appl., 2013 (2013), 549.
[31]	S. Y. Lin, New results for generalized Gronwall inequalities and their applications, Abstr. Appl. Anal., 2014 (2014), 168594.
[32]	S. S. Dragomir, Some Gronwall type inequalities and applications, Hauppauge: Nova Science Pub Inc, 2003.
[33]	Z. F. Zhang, Z. Z. Wei, A generalized Gronwall inequality and its application tofractional neutral evolution inclusions, J. Inequal. Appl., 2016 (2016), 45.
[34]	J. Alzabut, T. Abdeljawad, A generalized discrete fractional Gronwall inequality and its application on the uniqueness of solutions for nonlinear delay fractional difference system, Appl. Anal. Discrete Math., 12 (2018), 36-48.
[35]	T. Abdeljawad, J. Alzabut, D. Baleanu, A generalized $q$-fractional Gronwall inequality and its applications to nonlinear delay $q$-fractional difference systems, J. Inequal. Appl., 2016 (2016), 240.
[36]	H. D. Liu, F. W. Meng, Some new nonlinear integral inequalities with weakly singular kernel and their applications to FDEs, J. Inequal. Appl., 2015 (2015), 209.
[37]	X. Liu, A. Peterson, B. G. Jia, L. Erbe, A generalized h-fractional Gronwall's inequality and its applications for nonlinear h-fractional difference systems with 'maxima', J. Differ. Equ. Appl., 25 (2019), 815-836.
[38]	O. P. Agrawal, Some generalized fractional calculus operators and their applications in integral equations, Fract. Calc. Anal. Appl., 15 (2012), 700-711.
[39]	L. C. Evans, Partial differential equations, vol. 19 of Graduate Studies in Mathematics, Providence, RI, USA: American Mathematical Society, 1998.
[40]	R. Gorenflo, A. A. Kilbas, F. Mainardi, S. Rogosin, Mittag-Leffler functions, related topics and applications, Berlin: Springer, 2014.

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