Using the contractivity condition (2.4) on we have (2.55).
Using the contractivity condition (4), it follows that, for all (ninmathbb{N}), varrho bigl( d ( gz_{n+1},gx ),d ( gz_{n},gx ) bigr) = varrho bigl( d ( Tz_{n},Tx ),d ( gz_{n},gx ) bigr) >0.
Using the contractivity condition (5), psi bigl( d gz_{n+1},gx) bigr) leqalpha gz_{n},gx) psi bigl( d gz_{n+1},gx) bigr) =alpha gz_{n},gx) psi bigl( d Tz_{n},Tx) bigr) leqphi bigl( d gz_{n},gx) bigr) for all (ninmathbb{N}).
Similarly, using the contractivity condition (6), it can be proved that { g x n } is left-Cauchy in ( X, q ), so we conclude that { g x n } is a Cauchy sequence in ( X, q ).
Taking into account that (5) ⇒ (4), we can use the contractivity condition (4).
We cannot use the contractivity condition d ( Tx,Ty ) leq d ( gx,gy ) -frac{d ( gx,gy ) }{1+d ( gx,gy ) ^{2}}=frac{d ( gx,gy ) ^{3}}{1+d ( gx,gy ) ^{2}} in the previous example because the equality d ( Tx,Ty ) =frac{d ( gx,gy ) ^{3}}{1+d ( gx,gy ) ^{2}}+frac{sqrt{ d ( Tx,Ty ) d ( gx,gy ) }}{2 ( 1+d ( Tx,Ty ) d ( gx,gy ) ) } is reached for some points of the space.
Since is regular, then In any case, we can use the contractivity condition (8), which yields psi bigl( d(x_{n+1},Tz) bigr) =psi bigl( d(Tx_{n},Tz) bigr) leq theta bigl( d(x_{n},z) bigr) -varphi bigl( d z_{n},z) bigr) (9) for all (ninmathbb{N}).
Lim characterized this kind of mappings in terms of a contractivity condition using the following class of auxiliary functions.
This is an easy consequence of the contractivity of.
We firstly establish an approximate differential equation model which corresponds to the given reaction-diffusion system, and discuss the contractivity of the Picard mapping associated with the approximate model.
