Duality theorems for functional differential equations and application to superlinear problems (Q1355806)

It is shown that the coincidence degree associated with the functional differential equation \[ x''(t)= f\bigl(t,x_t, (x')_t\bigr) \quad x_0+ Nx_1= g(x) \quad h(x)=0 \] can be obtained by the computation of the degree associated with the ordinary differential equation \[ x''(t)= \overline f\bigl(t,x(t), x'(t)\bigr) \quad x(0) + N(x_1)(0) =g(x)(0) \quad h(x)=0 \] where \(f\) and \(h\) are continuous and map bounded sets to bounded sets, \(N\) is linear and continuous, \(g\) is completely continuous and \(f(t,x,y)= \overline f(t,x(0), y(0))\). Some conditions for the existence and multiplicity for superlinear equations are also established.