By Theorem 1, there exists a mapping Q :X → Y satisfying the following: (1) Q is a fixed point of J, that is, 16 Q ( x ) = Q ( 2 x ), Open image in new window (35) .
By Theorem 1.1, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e., Q ( 2 x ) = 4 Q ( x ) (2.6) .
By Theorem 1.1, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, that is, 1 6 Q ( x ) = Q ( 2 x ) (3.8) .
By Theorem 1.7, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e., 4 Q ( x ) = Q ( 2 x ) (2.18) .
Open image in new window By Theorem 1, there exists a mapping Q :X → Y, satisfying the following: (1) Q is a fixed point of J, that is, Q x 2 = 1 16 Q ( x ), Open image in new window (32) .
Open image in new window By Theorem 2, there exists a mapping Q:X→Y satisfying the following: (1) Q is a fixed point of J, that is, Q x 2 = 1 4 Q ( x ) Open image in new window (8).
Using the energy conservation theorem of Parseval, Equation (23) can be replaced by the following functions[35]: Q ( X ) = 1 2 ∫ R n D ( T ) 2 d T D ( T ) = E ∏ i = 1 n K t i − x i σ i − ∏ i = 1 n E K t i − x i σ i (25).
for all x ∈ X and t > 0. So d ( f, J f ) ≤ 1 ( 2 - c - n ) 2. By Theorem 1.1, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, that is, ( 2 - c - n ) 2 Q ( x ) = Q ( ( 2 - c - n ) x ) (22) for all x ∈ X.
So d ( f, J f ) ≤ α | 2 - c - n | 2. By Theorem 1.1, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e., Q x 2 - c - n = 1 ( 2 - c - n ) 2 Q ( x ) (10) for all x ∈ X.
This means that d ( J g, J h ) ≤ α d ( g, h ). for all g, h ∈ S. It follows from (16) that d ( f, J f ) ≤ α ( 2 - c - n ) 2. By Theorem 1.1, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, that is, Q x 2 - c - n = 1 ( 2 - c - n ) 2 Q ( x ) (17) for all x ∈ X.
This means that d ( J g, J h ) ≤ α d ( g, h ). for all g, h ∈ S. It follows from (6) that d ( f, J f ) ≤ 1 | 2 - c - n | 2. By Theorem 1.1, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e., Q ( ( 2 − c − n ) x ) = ( 2 − c − n ) 2 Q ( x ) (7) for all x ∈ X.
