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Lower and Upper Solutions for General Two-Point Fractional Order Boundary Value Problems

Yıl 2015, Cilt: 5 Sayı: 1, 80 - 87, 01.06.2015

Öz

This paper establishes the existence of a positive solution of fractional order two-point boundary value problem, D q1 a+ y t + f t, y t = 0, t ∈ [a, b], y a = 0, y ′ a = 0, αDq2 a+ y b − βDq3 a+ y a = 0, where D qi a+ , i = 1, 2, 3 are the standard Riemann-Liouville fractional order derivatives, 2 < q1 ≤ 3, 0 < q2, q3 < q1, α, β are positive real numbers and b > a ≥ 0, by an application of lower and upper solution method and fixed-point theorems

Kaynakça

  • Bai, Z. and L¨u, H., (2005), Positive solutions for boundary value problems of nonlinear fractional differential equations, J. Math. Anal. Appl., 311, pp. 495–505.
  • Benchohra, M., Henderson, J., Ntoyuas, S. K. and Ouahab, A., (2008), Existence results for fractional order functional differential equations with inŞnite delay, J. Math. Anal. Appl., 338, pp. 1340-1350. [3] Guo, D. and Zhang, J., (1985), Nonlinear Fractional Analysis, Science and Technology Press, Jinan, China.
  • Habets, P. and Zanolin, F., (1994), Upper and lower solutions for a generalized Emden-Fowler equa- tion, J. Math. Anal. Appl., 181, no. 3, pp. 684-700.
  • Kauffman, E. R. and Mboumi, E., (2008), Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electron. J. Qual. Theory Differ. Equ., 3, pp. 1-11.
  • Khan, R. A., Rehman M. and Henderson, J., (2011), Existence and uniqueness of solutions for nonlin- ear fractional differential equations with integral boundary conditions, Fractional Differential Calculus, 1, pp. 29–43.
  • Kilbas, A. A., Srivasthava, H. M. and Trujillo, J. J., (2006), Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam.
  • Lee, Y. H., (1997), A multiplicity result of positive solutions for the generalized Gelfand type singular boundary value problems, Proceedings of the Second World Congress of Nonlinear Analysis, Part 6 (Athens, 1996); Nonlinear Anal., 30, no. 6, pp. 3829-3835.
  • Li, F., Sun, J. and Jia, M., (2011), Monotone iterative method for the second-order three-point boundary value problem with upper and lower solutions in the reversed order, Appl. Math. Comput., 217, no. 9, pp. 4840-4847.
  • Liang, S. and Zhang, J., (2006), Positive solutions for boundary value problems of nonlinear fractional differential equations, Elec. J. Diff. Eqns., 36, pp. 1-12.
  • Podulbny, I., (1999), Fractional Differential Equations, Academic Press, San Diego.
  • Prasad, K. R. and Krushna, B. M. B., (2013), Multiple positive solutions for a coupled system of Riemann-Liouville fractional order two-point boundary value problems, Nonlinear Stud., vol. 20, no.4, pp. 501-511.
  • Prasad, K. R. and Krushna, B. M. B., (2014), Eigenvalues for iterative systems of Sturm-Liouville fractional order two-point boundary value problems, Fract. Calc. Appl. Anal., vol. 17, no. 3, pp. 638-653, DOI: 10.2478/s13540-014-0190-4.
  • Shi, A. and Zhang, S., (2009), Upper and lower solutions method and a fractional differential equation boundary value problem, Electron. J. Qual. Theory Differ. Equ., no. 30, pp. 1-13.
  • Su, X. and Zhang, S., (2009), Solutions to boundary value problems for nonlinear differential equations of fractional order, Elec. J. Diff. Eqns., 26, pp. 1-15.
  • Zhang, S., (2006), Existence of solutions for a boundary value problem of fractional order, Acta Math. Sci., 26B, pp. 220-228.
Yıl 2015, Cilt: 5 Sayı: 1, 80 - 87, 01.06.2015

Öz

Kaynakça

  • Bai, Z. and L¨u, H., (2005), Positive solutions for boundary value problems of nonlinear fractional differential equations, J. Math. Anal. Appl., 311, pp. 495–505.
  • Benchohra, M., Henderson, J., Ntoyuas, S. K. and Ouahab, A., (2008), Existence results for fractional order functional differential equations with inŞnite delay, J. Math. Anal. Appl., 338, pp. 1340-1350. [3] Guo, D. and Zhang, J., (1985), Nonlinear Fractional Analysis, Science and Technology Press, Jinan, China.
  • Habets, P. and Zanolin, F., (1994), Upper and lower solutions for a generalized Emden-Fowler equa- tion, J. Math. Anal. Appl., 181, no. 3, pp. 684-700.
  • Kauffman, E. R. and Mboumi, E., (2008), Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electron. J. Qual. Theory Differ. Equ., 3, pp. 1-11.
  • Khan, R. A., Rehman M. and Henderson, J., (2011), Existence and uniqueness of solutions for nonlin- ear fractional differential equations with integral boundary conditions, Fractional Differential Calculus, 1, pp. 29–43.
  • Kilbas, A. A., Srivasthava, H. M. and Trujillo, J. J., (2006), Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam.
  • Lee, Y. H., (1997), A multiplicity result of positive solutions for the generalized Gelfand type singular boundary value problems, Proceedings of the Second World Congress of Nonlinear Analysis, Part 6 (Athens, 1996); Nonlinear Anal., 30, no. 6, pp. 3829-3835.
  • Li, F., Sun, J. and Jia, M., (2011), Monotone iterative method for the second-order three-point boundary value problem with upper and lower solutions in the reversed order, Appl. Math. Comput., 217, no. 9, pp. 4840-4847.
  • Liang, S. and Zhang, J., (2006), Positive solutions for boundary value problems of nonlinear fractional differential equations, Elec. J. Diff. Eqns., 36, pp. 1-12.
  • Podulbny, I., (1999), Fractional Differential Equations, Academic Press, San Diego.
  • Prasad, K. R. and Krushna, B. M. B., (2013), Multiple positive solutions for a coupled system of Riemann-Liouville fractional order two-point boundary value problems, Nonlinear Stud., vol. 20, no.4, pp. 501-511.
  • Prasad, K. R. and Krushna, B. M. B., (2014), Eigenvalues for iterative systems of Sturm-Liouville fractional order two-point boundary value problems, Fract. Calc. Appl. Anal., vol. 17, no. 3, pp. 638-653, DOI: 10.2478/s13540-014-0190-4.
  • Shi, A. and Zhang, S., (2009), Upper and lower solutions method and a fractional differential equation boundary value problem, Electron. J. Qual. Theory Differ. Equ., no. 30, pp. 1-13.
  • Su, X. and Zhang, S., (2009), Solutions to boundary value problems for nonlinear differential equations of fractional order, Elec. J. Diff. Eqns., 26, pp. 1-15.
  • Zhang, S., (2006), Existence of solutions for a boundary value problem of fractional order, Acta Math. Sci., 26B, pp. 220-228.
Toplam 15 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Research Article
Yazarlar

K. R. Prasad Bu kişi benim

B. M. B. Krushna Bu kişi benim

Yayımlanma Tarihi 1 Haziran 2015
Yayımlandığı Sayı Yıl 2015 Cilt: 5 Sayı: 1

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