×

zbMATH — the first resource for mathematics

The existence of positive solutions for boundary value problem of the fractional Sturm-Liouville functional differential equation. (English) Zbl 1295.34086
Summary: We study boundary value problems for the following nonlinear fractional Sturm-Liouville functional differential equations involving the Caputo fractional derivative \[ \begin{gathered} {^CD^\beta}(p(t)^\beta(p(t)^C D^\alpha u(t))+ f(t,u(t-\tau), u(t+\beta))= 0,\;t\in (0,1),\\ {^CD^\alpha} u(1)= ({^CD^\alpha} u(0))''= 0,\\ au(t)- bu'(t)= \eta(t),\;t\in [-\tau,0]\;cu(t)+ du'(t)= \xi(t),\;t\in [1,1+\theta],\end{gathered} \] where \({^CD^\alpha}\), \({^CD^\beta}\) denote the Caputo fractional derivatives, \(f\) is a nonnegative continuous functional defined on \(C([-\tau,1+\theta],\mathbb{R})\), \(1<\alpha\leq 2\), \(2< \beta\leq 3\), \(0<\tau\), \(\theta< 1/4\) are suitably small, \(a,b,c,d>0\), and \(\eta\in C([-\tau,0], [0,\infty))\), \(\xi\in C([1,1+\theta], [0,\infty))\). By means of the Guo-Krasnoselskii fixed point theorem and the fixed point index theorem, some positive solutions are obtained, respectively. As an application, an example is presented to illustrate our main results.

MSC:
34K37 Functional-differential equations with fractional derivatives
34K10 Boundary value problems for functional-differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Oldham, K. B.; Spanier, J., The Fractional Calculus, (1974), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0428.26004
[2] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equation, (1993), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0789.26002
[3] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integral and Derivative, Theory and Applications, (1993), Yverdon, Switzerland: Gordon and Breach, Yverdon, Switzerland · Zbl 0818.26003
[4] Podlubny, I., Fractional Differential Equations, (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0918.34010
[5] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations, (2006), Amsterdam, The Netherlands: Elsevier Science, Amsterdam, The Netherlands · Zbl 1092.45003
[6] Diethelm, K., The Analysis of Fractional Differential Equations, (2010), Berlin, Germany: Springer, Berlin, Germany · Zbl 1215.34001
[7] Agarwal, R. P.; Zhou, Y.; Wang, J.; Luo, X., Fractional functional differential equations with causal operators in Banach spaces, Mathematical and Computer Modelling, 54, 5-6, 1440-1452, (2011) · Zbl 1228.34124
[8] Agarwal, R. P.; Benchohra, M.; Hamani, S., Boundary value problems for fractional differential equations, Georgian Mathematical Journal, 16, 3, 401-411, (2009) · Zbl 1198.26004
[9] Agarwal, R. P.; Zhou, Y.; He, Y., Existence of fractional neutral functional differential equations, Computers & Mathematics with Applications, 59, 3, 1095-1100, (2010) · Zbl 1189.34152
[10] Lakshmikantham, V.; Leela, S.; Devi, J. V., Theory of Fractional Dynamic Systems, (2009), Cambridge, UK: Cambridge Academic, Cambridge, UK · Zbl 1188.37002
[11] Zhao, Y.; Sun, S.; Han, Z.; Li, Q., The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations, Communications in Nonlinear Science and Numerical Simulation, 16, 4, 2086-2097, (2011) · Zbl 1221.34068
[12] Zhao, Y.; Sun, S.; Han, Z.; Li, Q., Positive solutions to boundary value problems of nonlinear fractional differential equations, Abstract and Applied Analysis, 2011, (2011) · Zbl 1210.34009
[13] Zhao, Y.; Sun, S.; Han, Z.; Zhang, M., Positive solutions for boundary value problems of nonlinear fractional differential equations, Applied Mathematics and Computation, 217, 16, 6950-6958, (2011) · Zbl 1227.34011
[14] Feng, W.; Sun, S.; Han, Z.; Zhao, Y., Existence of solutions for a singular system of nonlinear fractional differential equations, Computers & Mathematics with Applications, 62, 3, 1370-1378, (2011) · Zbl 1228.34018
[15] Pan, Y.; Han, Z.; Sun, S.; Zhao, Y., The existence of solutions to a system of discrete fractional boundary value problems, Abstract and Applied Analysis, 2012, (2012)
[16] Pan, Y.; Han, Z.; Sun, S.; Huang, Z., The existence and uniqueness of solutions to boundary value problems of fractional difference equations, Mathematical Sciences, 6, 7, 1-10, (2012) · Zbl 1264.39005
[17] Ahmad, B.; Nieto, J. J., Riemann-Liouville fractional differential equations with fractional boundary conditions, Fixed Point Theory on Fixed Point Theory, Computation and Applications, 13, 2, 329-336, (2012) · Zbl 1315.34006
[18] Ding, X.; Feng, Y.; Bu, R., Existence, nonexistence and multiplicity of positive solutions for nonlinear fractional differential equations, Journal of Applied Mathematics and Computing, 40, 1-2, 371-381, (2012) · Zbl 1302.34005
[19] O’Regan, D.; Staněk, S., Fractional boundary value problems with singularities in space variables, Nonlinear Dynamics of Nonlinear Dynamics and Chaos in Engineering Systems, 71, 4, 641-652, (2013) · Zbl 1268.34023
[20] Maraaba, T.; Baleanu, D.; Jarad, F., Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, Journal of Mathematical Physics, 49, 8, (2008) · Zbl 1152.81550
[21] Li, X.; Song, L.; Wei, J., Positive solutions for boundary value problem of nonlinear fractional functional differential equations, Applied Mathematics and Computation, 217, 22, 9278-9285, (2011) · Zbl 1223.34107
[22] Zhao, Y.; Chen, H.; Huang, L., Existence of positive solutions for nonlinear fractional functional differential equation, Computers & Mathematics with Applications, 64, 10, 3456-3467, (2012) · Zbl 1277.34111
[23] Benchohra, M.; Henderson, J.; Ntouyas, S. K.; Ouahab, A., Existence results for fractional order functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications, 338, 2, 1340-1350, (2008) · Zbl 1209.34096
[24] Zhou, Y.; Jiao, F.; Li, J., Existence and uniqueness for p-type fractional neutral differential equations, Nonlinear Analysis: Theory, Methods and Applications A, 71, 7-8, 2724-2733, (2009) · Zbl 1175.34082
[25] Zhou, Y.; Jiao, F.; Li, J., Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear Analysis: Theory, Methods and Applications A, 71, 7-8, 3249-3256, (2009) · Zbl 1177.34084
[26] Wang, J.; Zhou, Y.; Wei, W., A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces, Communications in Nonlinear Science and Numerical Simulation, 16, 10, 4049-4059, (2011) · Zbl 1223.45007
[27] Babakhani, A.; Baleanu, D.; Agarwal, R. P., The existence and uniqueness of solutions for a class of nonlinear fractional differential equations with infinite delay, Abstract and Applied Analysis, 2013, (2013) · Zbl 1277.34110
[28] Zhou, Y.; Tian, Y.; He, Y.-Y., Floquet boundary value problem of fractional functional differential equations, Electronic Journal of Qualitative Theory of Differential Equations, 50, 1-13, (2010)
[29] Zhang, L.; Ahmad, B.; Wang, G.; Agarwal, R. P., Nonlinear fractional integro-differential equations on unbounded domains in a Banach space, Journal of Computational and Applied Mathematics, 249, 51-56, (2013) · Zbl 1302.45019
[30] Li, Y.; Sun, S.; Yang, D.; Han, Z., Three-point boundary value problems of fractional functional differential equations with delay, Boundary Value Problems, 2013, article 38, (2013) · Zbl 1301.47102
[31] Bai, C., Existence of positive solutions for a functional fractional boundary value problem, Abstract and Applied Analysis, 2010, (2010) · Zbl 1197.34156
[32] Ouyang, Z.; Chen, Y.; Zou, S., Existence of positive solutions to a boundary value problem for a delayed nonlinear fractional differential system, Boundary Value Problems, 2011, (2011) · Zbl 1219.34103
[33] Norkin, S. B., Differential Equations of the Second Order with Retarded Argument. Differential Equations of the Second Order with Retarded Argument, Translation of Mathematical Monographs, 31, (1972), Providence, RI, USA: American Mathematical Society, Providence, RI, USA
[34] Bai, C.; Ma, J., Eigenvalue criteria for existence of multiple positive solutions to boundary value problems of second-order delay differential equations, Journal of Mathematical Analysis and Applications, 301, 2, 457-476, (2005) · Zbl 1081.34061
[35] Guo, D. J.; Lakshmikantham, V., Nonlinear Problems in Abstract Cone, (1988), Orlando, Fla, USA: Academic Press, Orlando, Fla, USA
[36] Deimling, K., Nonlinear Functional Analysis, (1985), Berlin, Germany: Springer, Berlin, Germany · Zbl 0559.47040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.