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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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