Asymmetrical variational problems for functionals with deviating argument (Q752457)

For the following variational problem with deviating argument \[ (1)\quad \int^{x_ 1}_{x_ 0}F(x,y(x),y(x-\tau),y'(x),y'(x-\tau))dx\to extr,\quad \tau >0, \] two types of boundary conditions are considered: (2) \(y(x)=\phi (x)\) for \(x\in [x_ 0-\tau,x_ 0]\), \(y(x_ 1)=y_ 1\), where \(\phi\) is given function; and (3) \(y(x)=\phi (x)\) for \(x\in [x_ 0-\tau,x_ 0]\) and \(y(x)=\psi (x)\) for \(x\in [x_ 1-\tau,x_ 1]\), where \(\phi\), \(\psi\) are given. Problem (1), (2) is the asymmetric variational problem, problem (1), (3) is the symmetric variational problem with deviating argument. In the present paper asymmetric variational problems are studied, cases of several variable deviations and unknown vector functions being included; analogues of the Weierstrass-Erdmann and the Legendre conditions are derived. Finally, a Bolza problem with deviating argument is considered.