For example, let p ( x ) = x n ∈ P n.
For example, let p ( x ) ∈ P and f ( t ) = e y t.
Figure 2 Period doubling diagram in Example 1, in ( p, x, y ) space. Figure 3 Period doubling diagram in Example 1, in ( p, x, z ) space. Figure 4 Period doubling diagram in Example 1, in ( p, y, z ) space.
For example, every polynomial p ( x ) = ∑ | α | ≤ m a α x α, where a α ∈ C, defines a tempered distribution by 〈 p ( x ), φ 〉 = ∫ R n p ( x ) φ ( x ) d x, φ ∈ S ( R n ).
For example, every polynomial p ( x ) = ∑ α ≤ m a α x α, where a α ∈ C, defines a tempered distribution by 〈 p ( x ), φ 〉 = ∫ R p ( x ) φ ( x ) d x, φ ∈ S ( R ).
Example 2 If p ( x 0, x 0 ) = 0 for some x 0 ∈ X, since E ( G ) contains all loops, it follows that the constant mapping f = x 0 is a ( p, φ ) - G -contraction for any φ ∈ Φ.
For example, P w w ctrl = P (x = w, y = w ) ctrl = N (x = w, y = w ) c t r l N ctrl.
A ( x ) ≢ 0. There exist many functions h ( x, t ) satisfying condition (H), for example, h ( x, t ) = P ( x ) | t | p − 2 t, where P ( x ) is a positive and bounded function.
We should point out that Theorem 2.3 is different from the previous results of [9], [10] in three main directions: (i) A ( x ) ≢ 0. There exist many functions h ( x, t ) satisfying condition (H), for example, h ( x, t ) = P ( x ) | t | p − 2 t, where P ( x ) is a positive and bounded function.
