A gradient-like mapping of the fundamental functional of the calculus of variations in a Banach space (Q2742966)

For the fundamental functional of calculus of variations in the form \(F(x)=\int_0^1f(s,x,x') ds\), which is defined on the Banach space \(C^1_0(0,1)\), the question of construction of the gradient-like mapping \(G:C^1_0\to C^1_0\) is considered. An existence theorem under the condition \(\frac{\partial f(0,x,y)}{\partial y}=\frac{\partial f(1,x,y)}{\partial y}\) is proved.