Those results are extended in [7] for Meir-Keeler cyclic contraction maps and, in general, with the p ( ≥ 2 ) -cyclic self-mapping T : ⋃ i ∈ p ¯ A i → ⋃ i ∈ p ¯ A i defined on any number of subsets of the metric space with p ¯ : = { 1, 2, …, p }.
(Where μ can be considered as a function of two variables y and i defined on C × N satisfying μ : { y } × N → S for every fixed y ∈ C and μ : C × { i } → S i for every fixed i ∈ N.) So, for every fixed i, when y varies in the chain C, from (8) we obtain a subset { μ ( y i ) : y ∈ C } ⊆ T i ( x ) ⊆ S i that is a chain in T i ( x ).
It is known that the spectral radius of the integral operator (6), considered on every finite interval t ∈ [ 0, ω ], is equal to 0 (see, for example, [1]) and the inequality sup t ≥ 0 ∫ 0 t | k i ( t, s ) | d s < 1, t ∈ [ 0, ∞ ), i = 1, …, n, implies that the spectral radii of the integral operators S i defined on the semiaxis are less than 1.
More attention has been paid to the investigation of standard contractive and Meir-Keeler-type contractive 2-cyclic self-mappings T A ∪ B → A ∪B defined on subsets A,B ⊆ X and, in general, p-cyclic self-mappings T : ⋃ i ∈ p ̄ A i → ⋃ i ∈ p ̄ A i defined on any number of subsets A i ⊂ X, i ∈ p ̄ : = 1, 2,..., p, where (X,d) is a metric space (see, for instance [13 22]).
The m potential functions ϕ i defined on unknown parameters θ reflect substantive knowledge on reaction kinetics for a given model such that regions of interest in the parameter space are prescribed prior to a posterior analysis.
In Section 3, we introduce a functional I defined on E whose critical points and weak solutions of (1.1) possess one-to-one correspondence.
Third, unlike previous studies, I explore the relationship between computer use and routine tasks, which I define on the basis of the frequency of repetitive activities that workers are asked to perform on the job.
Let for all i ∈ N the functions f ( ρ i, σ i ) be defined on Ω ( r i, s i ).
The mapping T I = (T i ) i ∈ I is defined on the closed convex domain ℵ I and onto it.
(i) Let us define on by (2.6) .
We extend each g i j to be a nonnegative continuous function, which is still denoted by g i j, defined on ℝ in the following way: if s < 0, then g i j ( s ) ≡ g i j ( 0 ).
