On existence of positive solution for initial value problem of nonlinear fractional differential equations of order $1 < \alpha\leq 2$
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Abstract
The existence of positive solution for a class of nonlinear fractionaldierential equations are investigated by method of upper and lower solutions and using Schauder and Banach xed point theorems.
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Matar, M.
(2015).
On existence of positive solution for initial value problem of nonlinear fractional differential equations of order $1 < \alpha\leq 2$.
Acta Mathematica Universitatis Comenianae, 84(1), 51-57.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/39/125
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References
[1] C. Wang, H. Zhang, Wang S. "Positive solution of a nonlinear fractional dierential equation involving Caputo derivative". Discrete Dynamics in Natural and Society, 2012, 2012: Art ID425408.
[2] C. Wang, R. Wang, S. Wang, C. Yang, "Positive Solution of Singular Boundary Value Problem for a Nonlinear Fractional Dierential Equation". Boundary Value Problems, 2011, 2011: Art ID 297026.
[3] D. Delbosco, L. Rodino, "Existence and uniqueness for a nonlinear fractional dierential equation", J. Math. Anal. Appl., 204(1996) pp. 609-625.
[4] E. Buckwar, Y. Luchko, "Invariance of a partial dierential equation of fractional order under lie group of scaling trabsformations", J. Math. Anal. Appl., 227(1998), pp. 81-97.
[5] E. Kaufmann, E. Mboumi, "Positive solutions of a boundary value problem for a nonlinear fractional dierential equation", Electronic Journal of Qualitative Theory of Dierential Equations, 3(2008), pp. 1-11.
[6] F. Mainardi, "The fundamental solutions for the fractional diusion-wave equation", Appl. Math. Letters, 9(1996), pp. 23-28.
[7] K.S. Miller, B. Ross,An introduction to the fractional calculus and fractional dierential equations, Wiley, New York, (1993).
[8] M. Matar, "Existence and uniqueness of solutions to fractional semilinear mixed Volterra-Fredholm integrodierential equations with nonlocal conditions", Electronic Journal of Dierential Equations, 155 (2009), pp.1-7.
[9] M. Matar, "Boundary value problem for fractional integro-dierential equations with nonlocal conditions", Int. J. Open Problems Compt. Math., 3 (2010), pp. 481-489.
[10] M. Matar, "On existence and uniqueness of the mild solution for fractional semilinear integro-dierential equations", Journal of Integral Equations and Applications, 23(2011), pp. 457-466.
[11] M.Matar, "On existence of solution to nonlinear fractional dierential equations for 0 < 3", Journl of Fractional Calculus and Applications, 3(2012), pp. 1-7.
[12] S. Zhang, "The existence of a positive solution for a fractional differential equation", J. Math. Anal. Appl., 252(2000), pp. 804-812.
[13] V. Ladshmikantham, A.S.Vatsals, "Basic theory of fractional differential equations", Nonlinear Anal., 60(2008), pp. 2677-2682.
[14] Z.B. Bai, T.T.Qiu, "Existence of positive solution for singular fractional differential equation", Appl. Math. Comput., 15(2009), pp. 2761-2767.
[15] Z. Zhu, G. Li, C. Cheng, "Quasi-static and dynamical analysis for viscoelastic timoshenko beam with fractional derivative constitutive relation", Appl. Math. Mech., 23(2002), pp. 1-15.
[2] C. Wang, R. Wang, S. Wang, C. Yang, "Positive Solution of Singular Boundary Value Problem for a Nonlinear Fractional Dierential Equation". Boundary Value Problems, 2011, 2011: Art ID 297026.
[3] D. Delbosco, L. Rodino, "Existence and uniqueness for a nonlinear fractional dierential equation", J. Math. Anal. Appl., 204(1996) pp. 609-625.
[4] E. Buckwar, Y. Luchko, "Invariance of a partial dierential equation of fractional order under lie group of scaling trabsformations", J. Math. Anal. Appl., 227(1998), pp. 81-97.
[5] E. Kaufmann, E. Mboumi, "Positive solutions of a boundary value problem for a nonlinear fractional dierential equation", Electronic Journal of Qualitative Theory of Dierential Equations, 3(2008), pp. 1-11.
[6] F. Mainardi, "The fundamental solutions for the fractional diusion-wave equation", Appl. Math. Letters, 9(1996), pp. 23-28.
[7] K.S. Miller, B. Ross,An introduction to the fractional calculus and fractional dierential equations, Wiley, New York, (1993).
[8] M. Matar, "Existence and uniqueness of solutions to fractional semilinear mixed Volterra-Fredholm integrodierential equations with nonlocal conditions", Electronic Journal of Dierential Equations, 155 (2009), pp.1-7.
[9] M. Matar, "Boundary value problem for fractional integro-dierential equations with nonlocal conditions", Int. J. Open Problems Compt. Math., 3 (2010), pp. 481-489.
[10] M. Matar, "On existence and uniqueness of the mild solution for fractional semilinear integro-dierential equations", Journal of Integral Equations and Applications, 23(2011), pp. 457-466.
[11] M.Matar, "On existence of solution to nonlinear fractional dierential equations for 0 < 3", Journl of Fractional Calculus and Applications, 3(2012), pp. 1-7.
[12] S. Zhang, "The existence of a positive solution for a fractional differential equation", J. Math. Anal. Appl., 252(2000), pp. 804-812.
[13] V. Ladshmikantham, A.S.Vatsals, "Basic theory of fractional differential equations", Nonlinear Anal., 60(2008), pp. 2677-2682.
[14] Z.B. Bai, T.T.Qiu, "Existence of positive solution for singular fractional differential equation", Appl. Math. Comput., 15(2009), pp. 2761-2767.
[15] Z. Zhu, G. Li, C. Cheng, "Quasi-static and dynamical analysis for viscoelastic timoshenko beam with fractional derivative constitutive relation", Appl. Math. Mech., 23(2002), pp. 1-15.