The existence of positive solution to a nonlinear fractional differential equation with integral boundary conditions.
(English) Zbl 1219.34009
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 RiemannLiouville 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 Greenâ€™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 integerorder cases.
MSC:
34A08  Fractional differential equations 
34B10  Nonlocal and multipoint boundary value problems for ODE 
34B18  Positive solutions of nonlinear boundary value problems for ODE 
47N20  Applications of operator theory to differential and integral equations 
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