The existence of solutions for boundary value problem of fractional hybrid differential equations. (English) Zbl 1352.34011

Sun, Shurong; Zhao, Yige; Han, Zhenlai; Li, Yanan

Commun. Nonlinear Sci. Numer. Simul. 17, No. 12, 4961-4967 (2012).

Summary: In this paper, we study the existence of solutions for the boundary value problem of fractional hybrid differential equations
\[ D_{0^+}^\alpha \left[\frac{x(t)}{f(t,x(t))}\right]+g(t,x(t))=0,\quad 0<t<1,\quad x(0)=x(1)=0, \]
where \(1<\alpha\leq 2\) is a real number, \(D_{0^+}^\alpha\) is the Riemann-Liouville fractional derivative. By a fixed point theorem in Banach algebra due to Dhage, an existence theorem for fractional hybrid differential equations is proved under mixed Lipschitz and Carathéodory conditions. As an application, examples are presented to illustrate the main results.

MSC:

34A08	Fractional ordinary differential equations
26A33	Fractional derivatives and integrals
34B15	Nonlinear boundary value problems for ordinary differential equations

Keywords:

fractional differential equation; boundary value problem; fractional Green’s function; fixed point theorem

References:

[1]	Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equation, (1993), John Wiley New York · Zbl 0789.26002
[2]	Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004
[3]	Podlubny, I., Fractional differential equations, Mathematics in science and engineering, (1999), Academic Press New York/Lindon/Toronto
[4]	Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integral and derivative, Theory and applications, (1993), Gordon and Breach Switzerland · Zbl 0818.26003
[5]	Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, J math anal appl, 204, 609-625, (1996) · Zbl 0881.34005
[6]	Li, Q.; Sun, S., On the existence of positive solutions for initial value problem to a class of fractional differential equation, (), 886-889
[7]	Li, Q.; Sun, S.; Zhang, M.; Zhao, Y., On the existence and uniqueness of solutions for initial value problem of fractional differential equations, J univ jinan, 24, 312-315, (2010)
[8]	Li, Q.; Sun, S.; Han, Z.; Zhao, Y., On the existence and uniqueness of solutions for initial value problem of nonlinear fractional differential equations, (), 452-457
[9]	Jafari, H.; Gejji, V.D., Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method, Appl math comput, 180, 700-706, (2006) · Zbl 1102.65136
[10]	Zhang, M.; Sun, S.; Zhao, Y.; Yang, D., Existence of positive solutions for boundary value problems of fractional differential equations, J univ jinan, 24, 205-208, (2010)
[11]	Zhao, Y.; Sun, S., On the existence of positive solutions for boundary value problems of nonlinear fractional differential equations, (), 682-685
[12]	Zhao, Y.; Sun, S.; Han, Z.; Zhang, M., Existence on positive solutions for boundary value problems of singular nonlinear fractional differential equations, (), 480-485
[13]	Zhao, Y.; Sun, S.; Han, Z.; Li, Q., The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations, Commun nonlinear sci numer simulat, 16, 2086-2097, (2011) · Zbl 1221.34068
[14]	Zhao, Y.; Sun, S.; Han, Z.; Li, Q., Positive solutions to boundary value problems of nonlinear fractional differential equations, Abst appl anal, 2011, 1-16, (2011)
[15]	Qiu, T.; Bai, Z., Existence of positive solutions for singular fractional equations, Electron J differ eqns, 146, 1-9, (2008)
[16]	Bai, Z.; Lü, H., Positive solutions for boundary value problem of nonlinear fractional differential equation, J math anal appl, 311, 495-505, (2005) · Zbl 1079.34048
[17]	Zhao, Y.; Sun, S.; Han, Z.; Li, Q., Theory of fractional hybrid differential equations, Comput math appl, 62, 1312-1324, (2011) · Zbl 1228.45017
[18]	Dhage, B.C., On α-condensing mappings in Banach algebras, Math student, 63, 146-152, (1994) · Zbl 0882.47033
[19]	Dhage, B.C.; Lakshmikantham, V., Basic results on hybrid differential equations, Nonlinear anal, 4, 414-424, (2010) · Zbl 1206.34020
[20]	Dhage, B.C., A nonlinear alternative in Banach algebras with applications to functional differential equations, Nonlinear funct anal appl, 8, 563-575, (2004) · Zbl 1067.47070
[21]	Dhage, B.C., Fixed point theorems in ordered Banach algebras and applications, Panam math J, 9, 93-102, (1999) · Zbl 0964.47026
[22]	Kilbas, A.A.; Srivastava, H.H.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier Science B.V. Amsterdam · Zbl 1092.45003
[23]	Dhage, B.C., On a fixed point theorem in Banach algebras with applications, Appl math lett, 18, 273-280, (2005) · Zbl 1092.47045