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Multiple solutions of singular boundary value problems for differential systems. (English) Zbl 1055.34041
The authors study the solvability of the following singular boundary value problem $$\cases (p(t) x'(t))'=\lambda f(t,x(t), y(t)),\quad & t\in(0, 1),\\ (p(t) y'(t))'=\lambda g(t,x(t), y(t)),\endcases$$ $$\alpha x(0)-\beta p(0) x'(0)=\gamma x(1)+\delta p(1)x'(1)= 0,\ \alpha y(0)-\beta p(0)y'(0)=\gamma y(1)+\delta p(1)y'(1)= 0,$$ where $\lambda\in \bbfR^+= [0,\infty)$ is a parameter, the constants $\alpha,\beta,\gamma,\delta\ge 0$ are such that $\alpha\gamma+ \alpha\delta+ \beta\gamma> 0$, and $p:[0,1]\to (0,\infty)$, $f: (0,1)\times (0,\infty)\times \bbfR\to \bbfR^+$ and $g: (0,1)\times (0,\infty)\times \bbfR\to\bbfR$ are continuous, but $f(t,x,y)$ and $g(t,x,y)$ may be singular at $t= 0$, $t= 1$ and $x= 0$. The existence of at least two solutions is proved under assumptions that there exist $N> 0$ and suitable continuous functions $h: (0,1)\to \bbfR^+$, $w: (0,\infty)\to \bbfR^+$, $e: \bbfR\to \bbfR^+$ such that $$\gather \vert g(t,x,y)\vert\le Nf(t,x,y)\text{ for }(t,x,y)\in (0,1)\times (0,\infty)\times \bbfR,\\ f(t,x,y)\le h(t) w(x) e(y)\text{ for }t\in (0,1)\text{ and }(x,y)\in (0,\infty)\times \bbfR,\endgather$$ and for every constant $r > 0$, there exists a function $\phi_{r}\ge 0$ such that $0< \int^1_0 \phi_{r}(t)\,dt< \infty$ and $$f(t,x,y)\ge \phi_{r}(t)\text{ for }t\in (0,1)\text{ and }(x,y)\in (0,r]\times [-Nr,Nr].$$ Moreover, the authors construct a special cone and use the fixed-point index theory in a cone.

##### MSC:
 34B16 Singular nonlinear boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE
Full Text:
##### References:
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