Acknowledgement
The authors thankful to the reviewers for valuable suggestion to improve the manuscript.
References
- R.P. Agarwal, D. O'Regan, S. Stanek, Positive solutions for mixed problems of singular fractional differential equations, Math. Nachr. 285 (2012), 27-41.
- L.C. Becker, T.A. Burton, I.K. Purnaras, Complementary equations: a fractional differential equation and a Volterra integral equation, Electron. J. Qual. Theory Differ. Equ. 12 (2015), 1-24.
- A Ayari, K Boukerrioua, Some new GronwallL-Bihari type inequalities associated with generalized fractionnal operators and applications, Rad HAZU, Matematicke znanosti 26 (2022), 127-138.
- B. Ben Nasser, K. Boukerrioua, M. Defoort, M. Djemai and M.A. Hammami, Refinements of some Pachpatte and Bihari inequalities on time scales, Nonlinear Dyn. Syst. Theory 2 (2017), 807-825.
- K. Boukerrioua, D. Diabi, B. Kilani, Some new Gronwall-bihari type inequalities and its application in the analysis for solutions to fractional differential equations, International Journal of Mathematical and Computational Methods 5 (2020), 60-68.
- D. Henry, Geometric theory of semilinear parabolic equations, Springer-Verlag, Berlin, Heidelberg, New York, 1981.
- Q.H. Ma and E.H. Yang, Estimates on solutions of some weakly singular Volterra integral inequalities, Acta Math. Appl. Sin. 25 (2002), 505-515.
- M. Medved, A new approach to an analysis of Henry type integral inequalities and their Bihari type versions, J. Math. Anal. Appl. 214 (1997), 349-366.
- M. Medved, Integral inequalities and global solutions of semilinear evolution equations,, J. Math. Anal. Appl. 267 (2002), 634-650.
- M. Mekki, K. Boukerrioua, B. Kilani, M.L. Sahari, New explicit bounds on Gronwall- Bellman-Bihari-Gamidov integral inequalities and their weakly singular analogues with applications, Kragujevac Journal of Mathematics 44 (2020), 603-615.
- I. Podlubny, Fractional differential equations. An introduction to fractional derivatives, fractional differentialequations,to methods of the solution and some of their applications, Academic Press, San Diego, 1999.
- S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives, Theory and applications, Gordon and Breach Science Publishers, Yverdon, 1993.
- J.R.L. Webb Weakly singular Gronwall inequalities and applications to fractional differential equations, J. Math. Anal. Appl. 471 (2019), 692-711.
- D. Willett, Weakly Nonlinear vector integral equations as contraction mappings, J. Math. Anal. Appl. Arch. Ration. Mech. Anal. 15 (1964), 79-86.
- T. Zhu, Existence and uniqueness of positive solutions for fractional differential equations, Bound. Value. Probl. 22 (2019), 1-11.
- Tao Zhu, Fractional integral inequalities and global solutions of fractional differential equations, Electronic Journal of Qualitative Theory of Differential Equations 5 (2020), 1-16.
- Tao Zhu, Weakly Singular Integral Inequalities and Global Solutions for Fractional Differential Equations of Riemann-Liouville, Mediterranean Journal of Mathematics 18 (2021).