Boundary value problems for systems of nonlinear second order differential difference equations (Q1893430)

The boundary value problem for the system of nonlinear second-order differential difference equations (1) \(x''= f(t, x(t- s), x, x')\), \(0\leq t\leq 1\), \(s= \text{const}> 0\), \(x(t)= \phi(t)\), \(t\in [- s, 0]\), \(x(1)= A\), is considered, where \(x\), \(f\in \mathbb{R}^n\), \(\phi\in C([- s, 0],\mathbb{R}^n)\), \(A\in \mathbb{R}^n\). The author adopts to the above problem a new-type Nagumo condition in which the control function is a vector-valued function. This function enables the finding of a priori estimates of each component of the derivative of the solutions. In the paper the function \(f\) satisfies Nagumo or Lipschitz conditions. Under suitable assumptions by Schauder's fixed point theorem the existence of a solution to (1) is demonstrated.