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Zbl 1219.34009
Feng, Meiqiang; Liu, Xiaofang; Feng, Hanying
The existence of positive solution to a nonlinear fractional differential equation with integral boundary conditions.
(English)
[J] Adv. Difference Equ. 2011, Article ID 546038, 14 p. (2011). ISSN 1687-1847/e

Summary: We study the following boundary value problem of the fractional order differential equation $${\bold D}^\alpha_{0^+}x(t)+g(t)f(t,x)=0, \quad 0<t<1,$$ $$x(0)=0,\quad x'(1)=\int^1_0 h(t)x(t)\,dt,$$ where $1<\alpha\le 2$, $g\in C((0,1),[0,+\infty))$ and $g$ may be singular at $t=0$ or/and at $t=1$, $D^\alpha_{0^+}$ is the standard Riemann-Liouville differentiation, $h\in L^1[0,1]$ is nonnegative, and $f\in C([0,1]\times [0,+\infty),[0,+\infty))$. The expression and properties of Green's function are studied and employed to obtain some results on the existence of positive solutions by using a fixed point theorem in cones. The proofs are based on the reduction of the problem considered to the equivalent Fredholm integral equation of the second kind. The results significantly extend and improve many known results even for integer-order cases.
MSC 2000:
*34A08
34B10 Multipoint boundary value problems
34B18 Positive solutions of nonlinear boundary value problems
47N20 Appl. of operator theory to differential and integral equations

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