On a representation of the general solution of a functional-differential equation (Q1329654)

The quasilinear functional differential equation \[ (Lx)(t)= (Fx)(t),\qquad t\in [0,\infty)\tag{1} \] in which \(L\) is linear functional- differential operator, \(F\) a nonlinear superposition operator, is considered. An example of such equation is \[ (L_ 0 x)(t)= x'(t)+B(t)x'(g(t))+ A(t)x(h(t))= f(t,x(t)),\quad t\in [0,\infty),\tag{2} \] \(x'(t)= x(t)=0\) for \(t<0\). The Volterra case \(g(t)\leq t\), \(h(t)\leq t\) of (2) with \(f(t,x)= f(t)\) and its integral representation \(x(t)= \int^ t_ 0 G(t,s)f(s)ds\) is well-known. The paper is devoted to the non- Volterra case of (2), \((L_ 0 x)(t)= f(t)\) with \(g(t)\leq t+1\), \(h(t)\leq t+1\), \(t\in [0,\infty)\). The authors obtain an integral representation for the solution of the last problem by reducing it to a countable system of functional-differential equations. Green functions for such systems and kernel properties are studied.