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Zbl 1079.34048
Bai, Zhanbing; Lü, Haishen
Positive solutions for boundary value problem of nonlinear fractional differential equation.
(English)
[J] J. Math. Anal. Appl. 311, No. 2, 495-505 (2005). ISSN 0022-247X

Summary: We investigate the existence and multiplicity of positive solutions to the boundary value problem $$D^\alpha_{0+}u(t)+f \bigl(t,u(t) \bigr)=0,\ 0<t<1, \quad u(0)=u(1)=0,$$ where $1<\alpha\le 2$ is a real number, $D_{0+}^\alpha$ is the standard Riemann-Liouville differentiation, and $f:[0,1] \times[0,\infty) \to [0,\infty)$ is continuous. By means of some fixed-point theorems in a cone, existence and multiplicity results positive solutions are obtained. The proofs are based upon the reduction of the problem considered to the equivalent Fredholm integral equation of second kind.
MSC 2000:
*34K05 General theory of functional-differential equations
34B18 Positive solutions of nonlinear boundary value problems
34B15 Nonlinear boundary value problems of ODE
26A33 Fractional derivatives and integrals (real functions)

Keywords: fractional differential equation; boundary value problem; positive solution; Green's function; fixed-point theorem

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