Positive solutions of operator equations on half-line. (English) Zbl 1086.47035

The paper is devoted to the proof of an existence of positive solutions to the operator equation \[ x(t) = A x(t),\quad t\in {\mathbb R}_{+}, \tag{1} \] where \(A\) is a special kind of nonlinear Hammerstein type integral operator. The proof is based on Krasnosel’skii’s theorem on the existence of a fixed point for a completely continuous operator on a cone in a Banach space. The non-uniqueness of positive solutions is discussed, too.


47J05 Equations involving nonlinear operators (general)
45P05 Integral operators
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
47H10 Fixed-point theorems
45M20 Positive solutions of integral equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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