## The existence of multiple positive solutions to boundary value problems of nonlinear delay differential equations with countably many singularities on infinite interval.(English)Zbl 1187.34086

Summary: We consider the existence of countably many positive solutions to a boundary value problem of a nonlinear delay differential equation with countably many singularities on infinite interval
$(\phi(x'(t)))'+a(t)f(t,x(t),x_t)=0,\quad 0<t<\infty,\quad x_0=\xi,\;\lim_{t\to\infty}x'(t)=0,$
where $$\phi:\mathbb R\to\mathbb R$$ is an increasing homeomorphism and a positive homomorphism with $$\phi(0)=0$$, $$x_t$$ is a function in $$C([-r,0],\mathbb R)$$ defined by $$x_t(\sigma)=x(t+\sigma)$$ for $$-r\leq\sigma\leq 0$$, and $$\xi\in C([-r,0],\mathbb R)$$. By using the fixed-point index theory and a new fixed-point theorem in a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem. The conclusions in this paper essentially extend and improve the known results.

### MSC:

 34K10 Boundary value problems for functional-differential equations 47N20 Applications of operator theory to differential and integral equations
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### References:

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