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Positive solutions of an initial value problem for nonlinear fractional differential equations. (English) Zbl 1242.35215
Summary: We investigate the existence and multiplicity of positive solutions for the nonlinear fractional differential equation initial value problem \(D^\alpha_{0+} u(t) + D^\beta_{0+} u(t) = f(t, u(t)), u(0) = 0\), \(0 < t < 1\), where \(0 < \beta < \alpha < 1\), \(D^\alpha_{0+}\) is the standard Riemann-Liouville differentiation and \(f : [0, 1] \times [0, \infty) \rightarrow [0, \infty)\) is continuous. By using some fixed-point results on cones, some existence and multiplicity results of positive solutions are obtained.

MSC:
35R11 Fractional partial differential equations
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[1] R. P. Agarwal, D. O’Regan Donal, and S. Stan\vek, “Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 371, no. 1, pp. 57-68, 2010. · Zbl 1206.34009
[2] K. Balachandran, S. Kiruthika, and J. J. Trujillo, “Remark on the existence results for fractional impulsive integrodifferential equations in Banach spaces,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2244-2247, 2012. · Zbl 1256.34065
[3] F. Chen, J. J. Nieto, and Y. Zhou, “Global attractivity for nonlinear fractional differential equations,” Nonlinear Analysis. Real World Applications, vol. 13, no. 1, pp. 287-298, 2012. · Zbl 1238.34011
[4] D. B\ualeanu, R. P. Agarwal, O. G. Mustafa, and M. Co\csulschi, “Asymptotic integration of some nonlinear differential equations with fractional time derivative,” Journal of Physics A, vol. 44, no. 5, Article ID 055203, 2011. · Zbl 1238.26008
[5] D. B\ualeanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, Complexity, Nonlinearity and Chaos, World Scientific, 2012. · Zbl 1248.26011
[6] A. A. Kilbas, O. I. Marichev, and S. G. Samko, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993. · Zbl 0818.26003
[7] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Application of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, 2006. · Zbl 1092.45003
[8] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0789.26002
[9] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[10] R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), vol. 378 of CISM Courses and Lectures, pp. 223-276, Springer, Vienna, Austria, 1997. · Zbl 0916.34011
[11] R. Gorenflo and F. Mainardi, “Fractional relaxation of distributed order,” in Complexus Mundi, pp. 33-42, World Scientific, Hackensack, NJ, USA, 2006. · Zbl 1167.26303
[12] F. Mainardi, A. Mura, G. Pagnini, and R. Gorenflo, “Sub-diffusion equations of fractional order and their fundamental solutions,” in Proceedings of the International Symposium on Mathematical Methods in Engineering, J. A. Tenreiro-Machado and D. Baleanu, Eds., pp. 23-55, Springer, Ankara, Turkey, 2006. · Zbl 1135.35004
[13] F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the space-time fractional diffusion equation,” Fractional Calculus & Applied Analysis, vol. 4, no. 2, pp. 153-192, 2001. · Zbl 1054.35156
[14] A. V. Chechkin, R. Gorenflo, I. M. Sokolov, and V. Yu. Gonchar, “Distributed order time fractional diffusion equation,” Fractional Calculus & Applied Analysis, vol. 6, no. 3, pp. 259-279, 2003. · Zbl 1089.60046
[15] A. Kochubei, “Distributed order calculus and equations of ultraslow diffusion,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 252-281, 2008. · Zbl 1149.26014
[16] K. S. Miller, “Fractional differential equations,” Journal of Fractional Calculus, vol. 3, pp. 49-57, 1993. · Zbl 0789.26002
[17] V. Daftardar-Gejji and A. Babakhani, “Analysis of a system of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp. 511-522, 2004. · Zbl 1058.34002
[18] S. Q. Zhang, “The existence of a positive solution for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 804-812, 2000. · Zbl 0972.34004
[19] S. Q. Zhang, “Existence of positive solution for some class of nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 278, no. 1, pp. 136-148, 2003. · Zbl 1026.34008
[20] Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495-505, 2005. · Zbl 1079.34048
[21] M. Stojanović, “Existence-uniqueness result for a nonlinear n-term fractional equation,” Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 244-255, 2009. · Zbl 1195.34014
[22] M. A. Krasnoselski, Positive Solutions of Operator Equations, P. Noordhoff, Groningen, The Netherlands, 1964.
[23] R. W. Leggett and L. R. Williams, “Multiple positive fixed points of nonlinear operators on ordered Banach spaces,” Indiana University Mathematics Journal, vol. 28, no. 4, pp. 673-688, 1979. · Zbl 0421.47033
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