Year 2019,
Volume: 1 Issue: 2, 96 - 109, 30.10.2019
Djourdem Habib
,
Slimane Benaicha
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value problem with integral boundary conditions. J. Nonlinear Sci. Appl., 10 (2017), 6333-
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formable fractional operator. NTMSCI 5, No. 4, 209{219 (2017)
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with decomposition series. Konuralp Journal of Mathematics, 6 (1) (2018) 102{109
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tional Partial Differential Equation from Finance, Proceedings, International Conference on
Fractional Differentiation and its Applications, Novi Sad, Serbia, 778{785, (2016).
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with conformable differentiation, An Int. J. Opt. Cont.: Theo. Appl. (IJOCTA), 8 (1), 1{7,
(2018).
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boundary value problems. Journal of Applied Mathematics and Physics, 2014, 2, 108-114.
Triple Positive Solutions For A Nonlinear Fractional Boundary Value Problem
Year 2019,
Volume: 1 Issue: 2, 96 - 109, 30.10.2019
Djourdem Habib
,
Slimane Benaicha
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.
References
- [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] 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] E. Ahmeda and A.S Elgazzar, On fractional order dierential equations model for nonlocal
epidemics. Physica A 379, 607-614 (2007).
- [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] 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] 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] 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] 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] M. Caputo, M. Fabrizio, A new denition of fractional derivative without singular kernel,
Progr. Fract. Dier. Appl. 1 (2015) 73
- [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] 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] 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] A. Guezane-Lakoud, R. Khaldi, Solvability of fractional boundary value problem with frac-
tional integral condition, Nonlinear Anal., 75 (2012), 2692-2700.
- [14] L. Guo, L. Liu and Yonghong Wu, Existence of positive solutions for singular fractional
differential equations with innite-point boundary conditions.Nonlinear Analysis: Modelling
and Control, Vol. 21, No. 5, 2016, 635-650.
- [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] J. He: Nonlinear oscillation with fractional derivative and its applications. In: International
Conference on Vibrating Engineering, Dalian, China, pp. 288-291 (1998).
- [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] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new denition of fractional derivative,
J. Comput. Appl. Math. 264 (2014) 65-70.
- [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] 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] K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, London,
1974.
- [22] I. Podlubny, Fractional Dierential Equations, Academic Press, New York, 1999.
- [23] M. Jia, X. Liu, Three nonnegative solutions for fractional differential equations with integral
boundary conditions, Comput. Math. Appl., 62 (2011), 1405-1412.
- [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] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional
Differential Equations, Elsevier Science B.V., Amsterdam, 2006.
- [26] KS. Miller and B. Ross, An Introduction to the Fractional Calculus and Dierential Equa-
tions. Wiley, New York, 1993.
- [27] S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integral and Derivatives (Theory
and Applications), Gordon and Breach, Switzerland, 1993.
- [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] 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] 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] F.B. Tatom, The relationship between fractional calculus and fractals, Fractals 3 (1995),
pp. 217-229.
- [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] 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] 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] 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] 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] 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] 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.