# Asymptotic Behavior of Solutions for a Class of Fractional Integro-differential Equations

Article

## Abstract

In this paper, we study the asymptotic behavior of solutions for a general class of fractional integro-differential equations. We consider the Caputo fractional derivative. Reasonable sufficient conditions under which the solutions behave like power functions at infinity are established. For this purpose, we use and generalize some well-known integral inequalities. It was found that the solutions behave like the solutions of the associated linear differential equation with zero right hand side. Our findings are supported by examples.

## Keywords

Asymptotic behavior fractional integro-differential equation Caputo fractional derivative integral inequalities nonlocal source

## Mathematics Subject Classification

34A08 35B40 26D10

## References

1. 1.
Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109(3), 973–1033 (2010)
2. 2.
Agarwal, R.P., Ntouyas, S.K., Ahmad, B., Alhothuali, M.S.: Existence of solutions for integro-differential equations of fractional order with nonlocal three-point fractional boundary conditions. Adv. Differ. Equ. 2013(1), 1–9 (2013)
3. 3.
Aghajani, A., Jalilian, Y., Trujillo, J.: On the existence of solutions of fractional integro-differential equations. Fract. Calc. Appl. Anal. 15(1), 44–69 (2012)
4. 4.
Ahmad, A.M., Furati, K.M., Tatar, N.E.: Asymptotic power type behavior of solutions to a nonlinear fractional integro-differential equation. Electron. J. Differ. Equ. 2017(134), 1–16 (2017)
5. 5.
Alsaedi, A.: Existence of solutions for integrodifferential equations of fractional order with antiperiodic boundary conditions. Int. J. Differ. Equ. 2009 (2009)Google Scholar
6. 6.
Anguraj, A., Karthikeyan, P., Trujillo, J.: Existence of solutions to fractional mixed integrodifferential equations with nonlocal initial condition. Adv. Differ. Equ. 2011(1), 690,653 (2011)
7. 7.
Băleanu, D., Mustafa, O.G., Agarwal, R.P.: Asymptotic integration of (1+ $$\alpha$$)-order fractional differential equations. Comput. Math. Appl. 62(3), 1492–1500 (2011)
8. 8.
Baleanu, D., Nazemi, S.Z., Rezapour, S.: The existence of positive solutions for a new coupled system of multiterm singular fractional integrodifferential boundary value problems. Abstr. Appl. Anal. 2013, 15 (2013). (Article ID 368659)
9. 9.
Brestovanska, E., Medved’, M.: Asymptotic behavior of solutions to second-order differential equations with fractional derivative perturbations. Electron. J. Differ. Equ. 2014(201), 1–10 (2014)
10. 10.
Cheng, Y.W., Ding, H.S.: Asymptotic behavior of solutions to a linear Volterra integrodifferential system. Abstr. Appl. Anal. 2013, 5 (2013). (Article ID 245905)
11. 11.
Dannan, F.M.: Integral inequalities of Gronwall–Bellman–Bihari type and asymptotic behavior of certain second order nonlinear differential equations. J. Mat. Anal. Appl. 108(1), 151–164 (1985)
12. 12.
De Barra, G.: Measure Theory and Integration. Elsevier, Oxford (2003)
13. 13.
Furati, K.M., Tatar, N.: Behavior of solutions for a weighted Cauchy-type fractional differential problem. J. Fract. Calc. 28, 23–42 (2005)
14. 14.
Grace, S.R.: On the oscillatory behavior of solutions of nonlinear fractional differential equations. Appl. Math. Comput. 266, 259–266 (2015)
15. 15.
Halanay, A.: On the asymptotic behavior of the solutions of an integro-differential equation. J. Math. Anal. Appl. 10(2), 319–324 (1965)
16. 16.
Kendre, S., Jagtap, T., Kharat, V.: On nonlinear fractional integrodifferential equations with non local condition in Banach spaces. Nonlinear Anal. Differ. Equ. 1, 129–141 (2013)
17. 17.
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. Elsevier, Oxford (2006)
18. 18.
Kirane, M., Medved, M., Tatar, N.: Semilinear Volterra integrodifferential problems with fractional derivatives in the nonlinearities. Abstr. Appl. Anal. 2011, 11 (2011). (Article ID 510314)
19. 19.
Kusano, T., Trench, W.F.: Existence of global solutions with prescribed asymptotic behavior for nonlinear ordinary differential equations. Ann. Mat. Pura Appl. 142(1), 381–392 (1985)
20. 20.
Lang, S.: Fundamentals of Differential Geometry. Volume 191 of Graduate Texts in Mathematics. Springer, New York (1999)
21. 21.
Levin, J.: The asymptotic behavior of the solution of a Volterra equation. Proc. Am. Math. Soc. 14(4), 534–541 (1963)
22. 22.
Medved’, M., Pospíšil, M.: Asymptotic integration of fractional differential equations with integrodifferential right-hand side. Math. Model. Anal. 20(4), 471–489 (2015)
23. 23.
Medved, M.: On the asymptotic behavior of solutions of nonlinear differential equations of integer and also of non-integer order. Electron. J. Qual. Theory Differ. Equ. 10, 1–9 (2012)
24. 24.
Medved, M.: Asymptotic integration of some classes of fractional differential equations. Tatra Mt. Math. Publ. 54(1), 119–132 (2013)
25. 25.
Medved, M., Moussaoui, T.: Asymptotic integration of nonlinear $$\phi$$-Laplacian differential equations. Nonlinear Anal. Theory Methods Appl. 72(3), 2000–2008 (2010)
26. 26.
Mustafa, O.G., Baleanu, D.: On the Asymptotic Integration of a Class of Sublinear Fractional Differential Equations. arXiv:0904.1495 (2009) (arXiv preprint)
27. 27.
Mustafa, O.G., Rogovchenko, Y.V.: Asymptotic integration of a class of nonlinear differential equations. Appl. Math. Lett. 19(9), 849–853 (2006)
28. 28.
Pinto, M.: Integral inequalities of Bihari-type and applications. Funkc. Ekvacioj 33(3), 387–403 (1990)
29. 29.
Royden, H.L., Fitzpatrick, P.: Real Analysis, vol. 198. Macmillan, New York (1988)
30. 30.
Tatar, N.: Existence results for an evolution problem with fractional nonlocal conditions. Comput. Math. Appl. 60(11), 2971–2982 (2010)

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