Solvability of some classes of nonlinear integro-differential equations with noncompact operator (Q646848)

Several results about the existence of a solution of the integro-differential equation \[ y'(x)+qy(x)=\int_0^\infty K(x,t,y(t))dt\quad x>0 \] with initial condition \(y(0)=y_0\geq0\) and certain asymptotic behavior are obtained. More precisely, a soluion of the form \[ y(x)=y_0e^{-qx}+\int_0^xe^{-q(x-t)}\varphi(t)dt \] with nonnegative bounded \(\varphi\) is found, and in case \(y_0=0\) also with \(\lim_{x\to\infty}y(x)=\eta\). The proof uses a method of successive approximation with pointwise convergence (one of the hypotheses involves monotonicity for \(K(x,t,\cdot)\)) and a comparison with a related Wiener-Hopf equation.