2.2 Fundamental Conceptions of the Calculus of Variations
93
is called the first order distance between the functions y(x) and y 0 (x) in the interval
[a, b].
Obviously, the following inequality holds
d 0 [y, y 0 ] ≤ d 1 [y, y 0 ] ≤ · · · ≤ d n [y, y 0 ]
(2.2.5)
Equation (2.2.5) shows that the i th (0 ≤ i < j ≤ n) order distance between the
functions y(x) and y 0 (x) is small, but the j th order distance that is greater than the
i th order distance may not be small. On the other hand, for the two functions that
the j th order distance is small, their i th order distance that is less than j th order
distance must be small.
It is can be seen from the above definition that the distance is a metric of approach
degree between two functions and their derivatives.
Let a given function y 0 (x) have continuous n th derivative in the closed interval
[a, b], the n th order distance between the function y(x) and function y 0 (x) in the
closed interval [a, b] is less than a positive number δ, then the set of all the functions
y(x) is called the n th order δ neighborhood of the function y 0 (x) in the closed
interval [a, b], it is written as N n [δ, y 0 (x)], it can be expressed as
N n [δ, y 0 (x)] = { y(x)|y(x) ∈ C
n
[a, b], with d n [y(x), y 0 (x)] < δ}
(2.2.6)
According to the above definition, an arbitrary function y(x) in the n th order δ
neighborhood of the function y 0 (x) should also satisfy the following inequality
|y(x) − y 0 (x)| < δ,
y
(x) − y
0 (x)
< δ, . . . ,
y
(n)
(x) − y
(n)
0 (x)
< δ
The zero-th order δ neighborhood of the function y 0 (x) consists of all the functions
y(x) that satisfies |y(x) − y 0 (x)| < δ. Moreover the first order δ neighborhood of
the function y 0 (x) consists of all the functions y(x) that satisfies |y(x) − y 0 (x)| < δ
and
y
(x) − y
0 (x)
< δ. So the first order δ neighborhood of y 0 (x) is a part of
the zero-th order δ neighborhood of y 0 (x). The zero-th order δ neighborhood of
the function y 0 (x) is called the strong δ neighborhood or strong neighborhood
of the function. The first order δ neighborhood of the function y 0 (x) is called the
weak δ neighborhood or weak neighborhood of the function. Clearly the weak
neighborhood of the function is a part of the strong neighborhood.
The zero-th order δ neighborhood of the curve y = y(x) consists of all the curves
of the strip region that the width is 2δ located in y = y(x).
The above concepts can be applied to multivariate function.
If y(x) ∈ N n [δ, y 0 (x)], then y(x) and y 0 (x) are called having the n th order δ
approach degree or δ approach degree of n th order. If two curves have the n
th order δ approach degree, then they have any δ approach degree of less than n th
order. The higher the order of approach degree is, the closer the approach degree
of two curve is. For example, Fig. 2.3 is the two curves only with the zero-th order
93
is called the first order distance between the functions y(x) and y 0 (x) in the interval
[a, b].
Obviously, the following inequality holds
d 0 [y, y 0 ] ≤ d 1 [y, y 0 ] ≤ · · · ≤ d n [y, y 0 ]
(2.2.5)
Equation (2.2.5) shows that the i th (0 ≤ i < j ≤ n) order distance between the
functions y(x) and y 0 (x) is small, but the j th order distance that is greater than the
i th order distance may not be small. On the other hand, for the two functions that
the j th order distance is small, their i th order distance that is less than j th order
distance must be small.
It is can be seen from the above definition that the distance is a metric of approach
degree between two functions and their derivatives.
Let a given function y 0 (x) have continuous n th derivative in the closed interval
[a, b], the n th order distance between the function y(x) and function y 0 (x) in the
closed interval [a, b] is less than a positive number δ, then the set of all the functions
y(x) is called the n th order δ neighborhood of the function y 0 (x) in the closed
interval [a, b], it is written as N n [δ, y 0 (x)], it can be expressed as
N n [δ, y 0 (x)] = { y(x)|y(x) ∈ C
n
[a, b], with d n [y(x), y 0 (x)] < δ}
(2.2.6)
According to the above definition, an arbitrary function y(x) in the n th order δ
neighborhood of the function y 0 (x) should also satisfy the following inequality
|y(x) − y 0 (x)| < δ,
y
(x) − y
0 (x)
< δ, . . . ,
y
(n)
(x) − y
(n)
0 (x)
< δ
The zero-th order δ neighborhood of the function y 0 (x) consists of all the functions
y(x) that satisfies |y(x) − y 0 (x)| < δ. Moreover the first order δ neighborhood of
the function y 0 (x) consists of all the functions y(x) that satisfies |y(x) − y 0 (x)| < δ
and
y
(x) − y
0 (x)
< δ. So the first order δ neighborhood of y 0 (x) is a part of
the zero-th order δ neighborhood of y 0 (x). The zero-th order δ neighborhood of
the function y 0 (x) is called the strong δ neighborhood or strong neighborhood
of the function. The first order δ neighborhood of the function y 0 (x) is called the
weak δ neighborhood or weak neighborhood of the function. Clearly the weak
neighborhood of the function is a part of the strong neighborhood.
The zero-th order δ neighborhood of the curve y = y(x) consists of all the curves
of the strip region that the width is 2δ located in y = y(x).
The above concepts can be applied to multivariate function.
If y(x) ∈ N n [δ, y 0 (x)], then y(x) and y 0 (x) are called having the n th order δ
approach degree or δ approach degree of n th order. If two curves have the n
th order δ approach degree, then they have any δ approach degree of less than n th
order. The higher the order of approach degree is, the closer the approach degree
of two curve is. For example, Fig. 2.3 is the two curves only with the zero-th order