On the other hand, the same argument presented in the continuous case can be transposed here: the integration of the two Riccati equations in cascade can always be performed through reduction to linear second-order mappings.
The main result of this subsection is as follows, which extends the conclusion of the Banach fixed point theorem (Theorem 1.2) to higher-order contraction mappings: (Higher-order contraction mapping theorem).
Let T be a family of order preserving mappings defined on X. Assume any two mappings from T form a symmetric Banach operator pair.
Now, as in the first-order case, we classify higher-order Lipschitz mappings into three cases, thus T is an rth-order contraction mapping if the polynomial (p z):=z^{r}-sum_{k=0}^{r-1}c_{k}z^{k}) is stable, that is, (|lambda|<1) if (p lambda)=0 ).
In this paper we introduce the concept of fuzzy order -contractive mappings and give two fixed point theorems on ordered non-Archimedean fuzzy metric spaces for fuzzy order -contractive type mappings.
For order preserving mappings T, which are nonexpansive in L1 and L∞, two principal, mutually equivalent results are shown.
It is remarked that the notion of weakly order contractive mappings purely relies on order structure.
We also prove some common fixed point results for order preserving mappings.
Then, any commuting family of order preserving mappings (T i )i∈I, T i : X → X, has a common fixed point.
In this section, we investigate the existence of a common fixed point of a commuting family of order preserving mappings defined on a complete lattice.
We establish the existence of common fixed points for weakly order contractive mappings by using only order-theoretic properties as follows: Theorem 24 Let (E, ≤) be a complete vector lattice and K its positive cone.
