Solutions of certain integral equations of the Hammerstein type (Q1382897)

The author studies the problem of nontrivial solutions of the scalar nonlinear equation \[ G(x)= \int^\infty_0k(x-t) Y\bigl(G(t) \bigr) dt,\;x\in R^+ =[0,\infty) \] with respect to the unknown function \(G\). The kernel \(K\) is an even function, satisfying the conservativity condition and the function \(Y\) defined on \(R_\tau= [\tau,\infty)\), \(\tau\geq 0\), has the form \(Y(x)= x-\omega (x)\), \(\omega\in L_1 (R_\tau) \cap C(R_\tau)\). He obtains as the main result the approximate construction of a one-parameter family of positive solutions \(G\) with the asymptotic property \(G(x)=O(x)\), \(x\to \infty\), of this equation.