Sentence examples for bounded functions defined on from inspiring English sources

where ϕ i and ψ j are continuous bounded functions defined on [-σ, 0] and [-τ; 0], respectively.

where are linear continuous Volterra operators, the spectral radius of the operator is less than one, is the space of continuous functions, is the space of essentially bounded functions defined on We consider (1.1) with the following boundary condition: (1.2).

Corollary 1 Let {X t, t ∈ T} be a general infinite tree T with uniformly bounded degree defined by Definition 2. Let {g t (x, y, z), t ∈ T} be a collection of uniformly bounded functions defined on G3.

We obtain the maximum principles for the first-order neutral functional differential equation where, and are linear continuous operators, and are positive operators, is the space of continuous functions, and is the space of essentially bounded functions defined on.

This allows us to reduce problem (45) to the abstract Cauchy problem (2) in a Banach space E = C μ ( R n ) of all continuous bounded functions defined on R n satisfying the Hölder condition with the indicator μ ∈ ( 0, 1 ) with a strongly positive operator A t, x = B t, x + δ I defined by (52) (see [57, 58]).

In 1935, Grüss [1] proved the following integral inequality which gives an approximation for the integral of a product of two functions in terms of the product of integrals of the two functions: Let f and g be two bounded functions defined on [ a, b ] with γ 1 ≤ f ( x ) ≤ Γ 1 and γ 2 ≤ g ( x ) ≤ Γ 2, where γ 1, y 2, Γ 1, Γ 2 are four constants.

Let h be a continuous bounded function defined on R, and we set h ¯ = max { h ( t ) : t ∈ [ 0, w ] }. and h ̲ = min { h ( t ) : t ∈ [ 0, w ] }, where h is a continuous w-periodic function.

Let f − = inf s ∈ T f ( s ), f + = sup s ∈ T f ( s ), m ( f ) = lim T → ∞ 1 T ∫ 0 T f ( s ) d s, where f is a continuous bounded function defined on T, T is a time scale.

We first note that for every (fin C^infty _0(mathbb {R}^n times R^+)) f can be identified with a (C^infty ) and bounded function defined on (mathbb {R}^{n+ nu } times R^+) and constant in the z-variables.

So are { y n }, { T x n }, { ∇ f ( y n ) } and { ∇ f ( T x n ) }. Indeed, since f is a bounded function defined on bounded subsets of E, ∇f is also bounded on bounded subsets of E (see [[29], Proposition 1.1.11]).

Among these inequalities we have the inequality var h ( f ) ≤ ( Γ 1 − M h [ f ] ) ( M h [ f ] − γ 1 ), where var h ( f ) denotes the h-variance of f, which is a bounded function defined on [ a, b ] with γ 1 ≤ f ( x ) ≤ Γ 1, and γ 1, Γ 1 are two constants.