Let and be two functions such that.
Let and be two functions such that (2.1).
(Integration by parts) Let (f,g:[a,b]rightarrow mathbb {R}) be two functions, such that fg is differentiable.
Corollary 15 Let A and B be two nonempty closed subsets of a complete partial metric space ( X, p ) such that X = A ∪ B. Let f : A → B and g : B → A be two functions such that f ( x ) = g ( x ) for all x ∈ A ∩ B and p ( f ( x ), g ( y ) ) ≤ α p ( x, y ) ∀ x ∈ A and y ∈ B, (4).
Let ψ, φ : [ 0, ∞ ) → [ 0, ∞ ) be two functions such that ψ is an altering distance function. 1. If φ is lower semi-continuous and φ − 1 ( { 0 } ) = { 0 }, then ∈ F. 2. If φ is continuous and verifies φ − 1 ( { 0 } ) = { 0 }, then ∈ F. 3. If ψ and φ are altering distance functions, then ∈ F. .
Let F : X × X → X and g : X → X be two functions such that M ( F ( x, y ), F ( u, v ), k t ) ≥ M ( g x, g u, t ) ∗ M ( g y, g v, t ). for all x, y, u, v ∈ X, where 0 < k < 1, F ( X × X ) ⊆ g ( X ) and g is continuous and commutes with F. Then there exists a unique x ∈ X such that x = g x = F ( x, x ).
Assume that ϕ, ψ : [ 0, ∞ ) → [ 0, ∞ ) are two functions such that (a) ϕ is nondecreasing, continuous and ϕ ( 0 ) = 0 < ϕ ( t ) for every t > 0 ; (b) ψ is nondecreasing, right-continuous, and ψ ( t ) < t for every t > 0. .
Lemma 5. Let φ and ψ be two functions on I such that ψ - φ is strictly monotone increasing (resp. decreasing) on I and ψ is convex (resp. concave) on I. Then ( 1 - t ) φ ( x ) + t ψ ( y ) - ( ( 1 - t ) φ + t ψ ) ( ( 1 - t ) x + t y ) > 0 ( r e s p. < 0 ).
A function with for each, is said to be -pseudo almost periodic if there exist two functions such that, where and.
Theorem 2.2 Let 0 < ε < T be given and let τ and φ be two functions on ( ε, T ) such that φ ( t ) ≥ τ ( t ) for all t ∈ ( ε, T ).
Let ((X,d)) be a complex-valued metric space and let (f,g Xto X) be two functions and a, b, c, d, e be such that (a+b+c+2d+2e<1).
