Triple Positive Solutions For A Nonlinear Fractional Boundary Value Problem

Abstract

In this paper, we investgate the existence of three positive  solutions of a nonlinear fractional  differential equations with multi-point and multi-strip boundary conditions. The existence result is obtained by using  the Leggett-Williams fixed point theorem. An example is also given to illustrate our main results.

Keywords

• Caputo fractional derivative;,multi-strip conditions,Leggett-Williams fixed point theorem

References

1. [1] R. P. Agarwal, A. Alsaedi and A. Alsharif and B. Ahmad , On nonlinear fractional-order boundary value problems with nonlocal multi-point conditions involving Liouville-Caputo derivatives, Differ. Equ. Appl., Volume 9, Number 2 (2017), 147{160, doi:10.7153/dea-09-12.
2. [2] B. Ahmad, R. P. Agarwal, On nonlocal fractional boundary value problems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 18 (2011), 535-544.
3. [3] E. Ahmeda and A.S Elgazzar, On fractional order dierential equations model for nonlocal epidemics. Physica A 379, 607-614 (2007).
4. [4] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci. 20 (2) (2016) 763-769
5. [5] Z. Bai, W. C. Sun and W. Zhang, Positive solutions for boundary value problems of singular fractional differential equations. Abstr. Appl. Anal. 2013, Article ID 129640 (2013).
6. [6] N. Bouteraa, S. Benaicha and H. Djourdem, Positive solutions for nonlinear fractional dier- ential equation with nonlocal boundary conditions, Universal Journal of Mathematics and Applications 1 (1) (2018), 39{45.
7. [7] N. Bouteraa and S. Benaicha, Existence of solutions for three-point boundary value problem for nonlinear fractional dierential equations, Bulletin of the Transilvania University of Brasov, Series III: Mathematics, Informatics, Physics. Vol 10(59), No. 1 2017.
8. [8] A. Cabada and G. Wang, Positive solutions of nonlinear fractional dierential equations with integral boundary value conditions. J. Math. Anal. Appl. 389 (2012) 403-411.

9. [9] M. Caputo, M. Fabrizio, A new denition of fractional derivative without singular kernel, Progr. Fract. Dier. Appl. 1 (2015) 73
10. [10] Y. Cui, Q. Sun, and X. Su, Monotone iterative technique for nonlinear boundary value problems of fractional order p 2 (2; 3], Advances in Dierence Equations, vol. 2017, no. 1, article no. 248, 2017.
11. [11] Evirgen, F. and N. Ozdemir, A Fractional Order Dynamical Trajectory Approach for Op- timization Problem with Hpm, in: Fractional Dynamics and Control, (Ed. D. Baleanu, Machado, J.A.T., Luo, A.C.J.), Springer, 145{155, (2012).
12. [12] J. R. Graef, L. Kong, Q. Kong, and M. Wang, Existence and uniqueness of solutions for a fractional boundary value problem with Dirichlet boundary condition, Electron. J. Qual. Theory Differ. Equ. 2013 No.55,11 pp.
13. [13] A. Guezane-Lakoud, R. Khaldi, Solvability of fractional boundary value problem with frac- tional integral condition, Nonlinear Anal., 75 (2012), 2692-2700.
14. [14] L. Guo, L. Liu and Yonghong Wu, Existence of positive solutions for singular fractional differential equations with innite-point boundary conditions.Nonlinear Analysis: Modelling and Control, Vol. 21, No. 5, 2016, 635-650.
15. [15] J. He: Approximate analytical solution for seepage ow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167, 57-68 (1998).
16. [16] J. He: Nonlinear oscillation with fractional derivative and its applications. In: International Conference on Vibrating Engineering, Dalian, China, pp. 288-291 (1998).
17. [17] J. He: Some applications of nonlinear fractional differential equations and their approxima- tions. Bull. Am. Soc. Inf. Sci. Technol. 15, 86-90 (1999).
18. [18] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new denition of fractional derivative, J. Comput. Appl. Math. 264 (2014) 65-70.
19. [19] R. W. Leggett, L. R. Williams, Multiple positive xed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979) 673-688.
20. [20] F. Liu and K. Burrage, Novel techniques in parameter estimation for fractional dynamical models arising from biological systems. Comput. Math. Appl. 62(3), (2011), 822-833.
21. [21] K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, London, 1974.
22. [22] I. Podlubny, Fractional Dierential Equations, Academic Press, New York, 1999.
23. [23] M. Jia, X. Liu, Three nonnegative solutions for fractional differential equations with integral boundary conditions, Comput. Math. Appl., 62 (2011), 1405-1412.
24. [24] A. Keten, M. Yavuz and D. Baleanu, Nonlocal Cauchy problem via a frac- tional operator involving power kernel in Banach Spaces.Fractal Fract. 2019, 3, 27; doi:10.3390/fractalfract3020027.
25. [25] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006.
26. [26] KS. Miller and B. Ross, An Introduction to the Fractional Calculus and Dierential Equa- tions. Wiley, New York, 1993.
27. [27] S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integral and Derivatives (Theory and Applications), Gordon and Breach, Switzerland, 1993.
28. [28] T.A. Sulaiman, M. Yavuz, H. Bulut and H. M. Baskonus, Investigation of the fractional coupled viscous Burgers equation involving Mittag-Leffer kernel. Physica A 527 (2019) 121126
29. [29] Y. Sun, M. Zhao, Positive solutions for a class of fractional differential equations with integral boundary conditions. Appl. Math. Lett, 34 (2014), 17-21.
30. [30] J. Tariboon, T. Sitthiwirattham, S. K. Ntouyas, Boundary value problems for a new class of three-point nonlocal Riemann-Liouville integral boundary conditions, Adv. Dierence Equ., 2013, 2013: 213.
31. [31] F.B. Tatom, The relationship between fractional calculus and fractals, Fractals 3 (1995), pp. 217-229.
32. [32] Y. Wang, S. Liang and Q. Wang, Multiple positive solutions of fractional-order boundary value problem with integral boundary conditions. J. Nonlinear Sci. Appl., 10 (2017), 6333- 6343.
33. [33] Y. Wang and Y. Yang, Positive solutions for nonlinear Caputo fractional differential equa- tions with integral boundary conditions, J. Nonlinear Sci. Appl. 8 (2015), 99-109.
34. [34] M. Yavuz and B. Y.kran, Approximate-analytical solutions of cable equation using con- formable fractional operator. NTMSCI 5, No. 4, 209{219 (2017)
35. [35] M Yavuz1 and N. Ozdemir, A quantitative approach to fractional option pricing problems with decomposition series. Konuralp Journal of Mathematics, 6 (1) (2018) 102{109
36. [36] M. Yavuz, N. Ozdemir and Y.Y. Okur, Generalized Dierential Transform Method for Frac- tional Partial Differential Equation from Finance, Proceedings, International Conference on Fractional Differentiation and its Applications, Novi Sad, Serbia, 778{785, (2016).
37. [37] M. Yavuz, Novel solution methods for initial boundary value problems of fractional order with conformable differentiation, An Int. J. Opt. Cont.: Theo. Appl. (IJOCTA), 8 (1), 1{7, (2018).
38. [38] L. Zhou and W. Jiang, Positive solutions for fractional differential equations with multi-point boundary value problems. Journal of Applied Mathematics and Physics, 2014, 2, 108-114.