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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 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.

34A08Fractional differential equations
34B10Nonlocal and multipoint boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
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