Nonlinear boundary value problems of the calculus of variations (Q1422590)

The author considers nonlinear boundary value problems (BVP) of the calculus of variations. The integral functional \[ \int^b_a L(t,x(t),x'(t))\,dt\tag{1} \] is minimized in the set of functions satisfying the boundary conditions \[ x(a)=A,\quad x(b)=B.\tag{2} \] This problem is equivalent to the BVP \[ \frac{d}{dt}\,L_x,(t,x,x')=L_x(t,x,x'),\tag{3} \] \[ x(a)=A,\quad x(b)=B,\tag{2} \] where equation (3) is the equation of Euler-Lagrange for the functional (1). The existence of extremals of the variational problem (1), (2) or the existence of a solution to the BVP (3), (2) is proved by the method of upper and lower solutions.