The Existence of Positive Solutions for the Caputo-Fabrizio Fractional Boundary Value Problems at Resonance

Abstract

This paper deals with a class of nonlinear fractional boundary value problems at resonance with Caputo-Fabrizio fractional derivative. We establish some new necessary conditions for the existence of positive solutions for the fractional boundary value problems at resonance by using the Leggett-Williams norm-type theorem for coincidences due to O' Regan and Zima. Some examples are constructed to support our results.

Keywords

• Fractional Caputo-Fabrizio differential equation
• positive solution
• resonance
• necessary condition

References

1. Abdeljawad, T., Baleanu, D., On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80(1)(2017), 11–27.
2. Ahmad, B., Luca, R., Existence of solutions for sequential fractional integro-differential equations and inclusions with nonlocal boundary conditions, Appl. Math. Comput., 339(2018), 516–534.
3. Al-Refai, M., Abdeljawad, T., Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel, Adv. Difference Equ. 2017(1)(2017), 1–12.
4. Alkahtani, B., Atangana, A., Controlling the wave movement on the surface of shallow water with the Caputo–Fabrizio derivative with fractional order, Chaos Solitons Fractals, 89(2016), 539–546.
5. Atangana, A., On the new fractional derivative and application to nonlinear Fisher’s reaction-diusion equation, Appl. Math. Comput., 273(2016), 948–956.
6. Atangana, A., Alkahtani, B.S.T., Extension of the resistance, inductance, capacitance electrical circuit to fractional derivative without singular kernel, Adv. Mech. Math., 7(6)(2015), 1687814015591937.
7. Atangana, A., Alqahtani, R.T., Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation, Adv. Dierence Equ. 2016(1)(2016), 1–13.
8. Baleanu, D., Mousalou, A., Rezapour, S., On the existence of solutions for some infinite coeffcient-symmetric Caputo-Fabrizio fractional integro-differential equations, Bound. Value Probl., 2017(1)(2017), 1–9.

9. Cabada, A., Hamdi, Z., Nonlinear fractional differential equations with integral boundary value conditions, Appl. Math. Comput., 228(2014), 251–257.
10. Cabada, A., Wang, G., Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl., 389(1)(2012), 403–411.
11. Caputo, M., Fabrizio, M., A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl.,1(2)(2015), 73–85.
12. Cui, Y., Uniqueness of solution for boundary value problems for fractional differential equations, Appl. Math. Lett., 51(2016), 48–54.
13. Cui, Y., Ma, W., Sun, Q., Su, X., New uniqueness results for boundary value problem of fractional differential equation, Nonlinear Anal. Model. Control, 23(1) (2018), 31–39.
14. Doungmo Goufo, E.F., Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Burgers equation, Math. Model. Anal., 21(2) (2016), 188–198.
15. Gomez-Aguilar, J.F., Yepez-Martinez, H., Calder Onramon, C., Cruzorduna, I., Escobar-Jimenez, R.F. et all., Modeling of a mass-springdamper system by fractional derivatives with and without a singular kernel, Entropy, 17(9)(2015), 6289–6303.
16. Heymans, N., Podlubny, I., Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheologica Acta, 45 (2006), 765–771.
17. Infante, G., Zima, M., Positive solutions of multi-point boundary value problems at resonance, Nonlinear Anal., 69(8)(2008), 2458–2465.
18. Jiang, W., The existence of solutions to boundary value problems of fractional differential equations at resonance, Nonlinear Anal., 74(5)(2011), 1987–1994.
19. Kaczorek, T., Borawski, K., Fractional descriptor continuous-time linear systems described by the Caputo-Fabrizio derivative, Int. J. Appl. Math. Comput. Sci., 26(3)(2016), 533–541.
20. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations, vol. 204. Elsevier, 2006.
21. Kosmatov, N., A boundary value problem of fractional order at resonance, Electron. J. Differential Equations, 2010(2010), Paper–No.197.
22. Kosmatov, N., Jiang, W., Resonant functional problems of fractional order, Chaos Solitons Fractals,91(2016), 573–579.
23. Losada, J., Nieto, J.J., Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1(2)(2015), 87–92.
24. Orsingher, E., Beghin, L., Time-fractional telegraph equations and telegraph processes with Brownian time, Probab. Theory and Related Fields, 128(1)(2004), 141–160.
25. O’Regan, D., Zima, M., Leggett-williams norm-type theorems for coincidences, Arch. Math., 87(3)(2006), 233–244.
26. Toprakseven, S., The existence and uniqueness of initial-boundary value problems of the fractional Caputo-Fabrizio differential equations, Universal Journal of Mathematics and Applications, 2(2)(2019), 100–106.
27. Toprakseven, S., The existence of positive solutions and a Lyapunov type inequality for boundary value problems of the fractional Caputo-Fabrizio differential equations, Sigma Journal of Engineering and Natural Sciences, 37(4)(2019), 1129–1137.
28. Wang, Y., Liu, L., Positive solutions for a class of fractional 3-point boundary value problems at resonance, Adv. Difference Equ., 2017(1)(2017), 1–13.
29. Wang, Y., Liu, L., Wu, Y., Positive solutions for a nonlocal fractional differential equation, Nonlinear Anal., 74(11 )(2011), 3599–3605.
30. Yang, L., Shen, C., On the existence of positive solution for a kind of multi-point boundary value problem at resonance, Nonlinear Anal., 72(11)(2010), 4211–4220.
31. Zhang, X., Zhong, Q., Uniqueness of solution for higher-order fractional differential equations with conjugate type integral conditions, Fract. Calc. Appl. Anal., 20(6)(2017), 1471–1484.
32. Zhang, X., Zhong, Q., Uniqueness of solution for higher-order fractional differential equations with conjugate type integral conditions, Fract. Calc. Appl. Anal., 20(6)(2017), 1471–1484.