International Journal of Nonlinear Analysis and Applications
Gronwall type inequalities: new fractional integral results with some applications on hybrid differential equations
Document Type : Review articles
Authors
Laboratory LPAM, UMAB, University of Mostaganem, Algeria
Abstract
In this paper, we are concerned with a class of fractional integral inequalities of Gronwall type. New integral results with some generalizations are proved. Then, some applications on hybrid differential equations with Hadamard derivative are established.
Keywords
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[15] J. Shao and F. Meng, Gronwall-Bellman type inequalities and their applications to fractional differential equations, Abst. Appl. Anal. J., Article ID 217641 (2013) 7 pages.
[16] P. Thiramanus, J. Tariboon and S. K Ntouyas, Integral inequalities with ‘maxima’ and their applications to Hadamard type fractional differential equations, J. Ineq. Appl. 398 (2014) 15 pages.
[17] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its applications to a fractional differential equation, J. Math. Anal. Appl. 328 (2007) 1075–1081.
[18] Y. Zhaoa, S. Suna, Z. Hana and Q. Li, Theory of fractional hybrid differential equations, Comput. Math. Appl. 62 (2011) 1312–1324.
[3] S. Ali Khan, K. Shah and R. Ali Kha, On coupled system of nonlinear hybrid differential equation with arbitrary order, Matrix Sci. Math. 1(2) (2017) 11–16.
[4] R. Almeida, A Gronwall inequality for a general Caputo fractional operator, arxiv: 1/ 05.100 / 9v1 [math.CA],(2017) 1–14.
[5] K. Boukerrioua, Note on some nonlinear integral inequalities and applications to differential equations, Int. J. Diff. Eq. Article ID 456216 (2011) 15 pages.
[6] C. Corduneanu, Principles of Differential and Integral Equations, Allyn & Bacon, Boston, 1971.
[7] B.C. Dhage, Quadratic perturbations Of periodic boundary value problems of second order ordinary differential equations, Diff. Equ. Appl. 2(4) (2010) 465–486.
[8] M. A. E. Herzallah and D. Baleanu , On Fractional Order Hybrid Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
[9] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier B.V., 2006.
[10] S. Lin, Generalized Gronwall inequalities and their applications to fractional differential equations, J. Ineq. Appl. 549 (2013), 9 pages .
[11] H. Lu, S. Sun, D. Yang and H. Tengi, Theory of fractional hybrid differential equations with linear perturbations of second type, Boundary Value Prob. (2013) 16 pages.
[12] K. Nouri, M. Nazari and B. Keramati, Existence results for a coupled system of fractional integro-differential equations with time-dependent delay, J. Fixed Point Theory Appl. (2017) 17 pages.
[13] I. Petr`aˇs, Fractional Derivatives, Fractional Integrals, and Fractional Differential Equations in Matlab, Engineering Education and Research Using MATLAB, Technical University of Koˇsice, Slovak Republic.
[14] D. Qian, Z. Gong and C. Li, A generalized Gronwall inequality and its application to fractional differential equations with Hadamard derivatives, http://nsc10.cankaya.edu.tr/proceedings/PAPERS/Symp2-Fractional%20 Calculus%20Applications/Paper26.pdf.
[15] J. Shao and F. Meng, Gronwall-Bellman type inequalities and their applications to fractional differential equations, Abst. Appl. Anal. J., Article ID 217641 (2013) 7 pages.
[16] P. Thiramanus, J. Tariboon and S. K Ntouyas, Integral inequalities with ‘maxima’ and their applications to Hadamard type fractional differential equations, J. Ineq. Appl. 398 (2014) 15 pages.
[17] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its applications to a fractional differential equation, J. Math. Anal. Appl. 328 (2007) 1075–1081.
[18] Y. Zhaoa, S. Suna, Z. Hana and Q. Li, Theory of fractional hybrid differential equations, Comput. Math. Appl. 62 (2011) 1312–1324.
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Volume 12, Issue 2
November 2021Pages 799-809
- Receive Date: 16 January 2020
- Revise Date: 16 March 2020
- Accept Date: 24 March 2020