From Tables 1 and 2, it is easy to verify that mappings and satisfy condition (2.1).
We give an alternative proof of Theorem 17 by showing that mappings satisfying (F1) and (F4) are CJM contractions.
Communication is type safe, the middleware ensuring that mappings only occur between compatible endpoints, i.e. that schema and interaction modes agree, and messages correspond to the schema for that mapping.
A first integrability detector was proposed based on the observation that mappings integrable through spectral methods have confined singularities [11], that is, any singularity spontaneously appearing due to the choice of initial conditions disappears after a few iteration steps.
(2) If in the previous example we take F ( x, y ) = 1 2 x - 1 3 y, gx = x, ϕ ( t ) = 5 6 t, then we obtain that mappings F, g and ϕ satisfy all conditions of the Theorem 2.1, but they do not satisfy the conditions of corresponding Theorem from [7].
Assume that mappings g : L l o c 1 ( a, b ] → A R [ a, b ] and c : L l o c 1 ( a, b ] → ℝ are increasing and order-bounded, that p : [a, b] → ℝ+, and that 1 p ∈ L l o c 1 ( a, b ]. Then, the IVP (3.1) has in S the smallest and greatest solutions that are increasing with respect to g and c.
