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Year 2019, Volume: 1 Issue: 2, 96 - 109, 30.10.2019

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References

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  • [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 di erential 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 di er- 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 di erential 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 di erential equations with integral boundary value conditions. J. Math. Anal. Appl. 389 (2012) 403-411.
  • [9] M. Caputo, M. Fabrizio, A new de nition of fractional derivative without singular kernel, Progr. Fract. Di er. 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 Di erence 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 in nite-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 de nition 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 Di erential 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 Di erential 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. Di erence 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 Di erential 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.

Triple Positive Solutions For A Nonlinear Fractional Boundary Value Problem

Year 2019, Volume: 1 Issue: 2, 96 - 109, 30.10.2019

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 di erential 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 di er- 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 di erential 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 di erential equations with integral boundary value conditions. J. Math. Anal. Appl. 389 (2012) 403-411.
  • [9] M. Caputo, M. Fabrizio, A new de nition of fractional derivative without singular kernel, Progr. Fract. Di er. 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 Di erence 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 in nite-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 de nition 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 Di erential 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 Di erential 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. Di erence 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 Di erential 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.
There are 38 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Djourdem Habib

Slimane Benaicha This is me

Publication Date October 30, 2019
Acceptance Date August 12, 2019
Published in Issue Year 2019 Volume: 1 Issue: 2

Cite

APA Habib, D., & Benaicha, S. (2019). Triple Positive Solutions For A Nonlinear Fractional Boundary Value Problem. Maltepe Journal of Mathematics, 1(2), 96-109.
AMA Habib D, Benaicha S. Triple Positive Solutions For A Nonlinear Fractional Boundary Value Problem. Maltepe Journal of Mathematics. October 2019;1(2):96-109.
Chicago Habib, Djourdem, and Slimane Benaicha. “Triple Positive Solutions For A Nonlinear Fractional Boundary Value Problem”. Maltepe Journal of Mathematics 1, no. 2 (October 2019): 96-109.
EndNote Habib D, Benaicha S (October 1, 2019) Triple Positive Solutions For A Nonlinear Fractional Boundary Value Problem. Maltepe Journal of Mathematics 1 2 96–109.
IEEE D. Habib and S. Benaicha, “Triple Positive Solutions For A Nonlinear Fractional Boundary Value Problem”, Maltepe Journal of Mathematics, vol. 1, no. 2, pp. 96–109, 2019.
ISNAD Habib, Djourdem - Benaicha, Slimane. “Triple Positive Solutions For A Nonlinear Fractional Boundary Value Problem”. Maltepe Journal of Mathematics 1/2 (October 2019), 96-109.
JAMA Habib D, Benaicha S. Triple Positive Solutions For A Nonlinear Fractional Boundary Value Problem. Maltepe Journal of Mathematics. 2019;1:96–109.
MLA Habib, Djourdem and Slimane Benaicha. “Triple Positive Solutions For A Nonlinear Fractional Boundary Value Problem”. Maltepe Journal of Mathematics, vol. 1, no. 2, 2019, pp. 96-109.
Vancouver Habib D, Benaicha S. Triple Positive Solutions For A Nonlinear Fractional Boundary Value Problem. Maltepe Journal of Mathematics. 2019;1(2):96-109.

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