ABSTRACT
We considered at first the problem of minimization of a function on one variable f = f ( σ ) $ f=f(\sigma ) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315152448/5968d418-c0e2-45a5-91c4-37245a5e7358/content/inline-math3_1.tif"/> on the set of real numbers. Suppose this function is differentiable. If τ $ \tau $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315152448/5968d418-c0e2-45a5-91c4-37245a5e7358/content/inline-math3_2.tif"/> is a point of its minimum, then f ′ ( τ ) = 0 $ f^{\prime }(\tau )=0 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315152448/5968d418-c0e2-45a5-91c4-37245a5e7358/content/inline-math3_3.tif"/> . Then we had the minimization problem for a functional I on a Banach space V. The point u is its solution whenever the function of one variable f = f ( σ ) = I ( u + σ h ) $ f=f(\sigma )=I(u+\sigma h) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315152448/5968d418-c0e2-45a5-91c4-37245a5e7358/content/inline-math3_4.tif"/> has the minimum at the zero point for all h ∈ V $ h\in V $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315152448/5968d418-c0e2-45a5-91c4-37245a5e7358/content/inline-math3_5.tif"/> . If the functional I is G a ^ $ \hat{\mathrm{a}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315152448/5968d418-c0e2-45a5-91c4-37245a5e7358/content/inline-math3_6.tif"/> teaux differentiable at a point u, then we have the stationary condition I ′ ( u ) = 0 $$ \begin{aligned} I^{\prime }(u)\,=\,0 \end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315152448/5968d418-c0e2-45a5-91c4-37245a5e7358/content/math3_1.tif"/>
