On solvability of a class of nonlinear second order integro-differential equations in the space \(W_1^2(\mathbb{R}^+)\) (Q260365)

On the infinite interval \([0,\infty)\) the nonlinear integro-differential equation \[ -y''+\mu y=\int_0^\infty K(x-t)H(t,y(t))dt \] with boundary conditions \(y(0)=0\) and \(y\in W_1^2([0,\infty))\) is studied. Assuming that \(K\) is bounded and continuous with finite second moment, \(K(-\tau)>K(\tau)\geq0\) for \(\tau>0\), and assuming for \(H\) some monotonicity and certain upper and lower estimates, the existence of a solution \(y\) satisfying \(y(x)>0\) for \(x>0\) is obtained. The proof consists in the reduction to an integral equation and the application of an iteration procedure, using bounds obtained from associated Wiener-Hopf equations.