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.
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.
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.
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.
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]).
Adopting the basic assumptions of Yang [11], we assume the following: Assumption (A1) (i) g : A → R is a bounded function defined on the compact subset A of R d ; (ii) { ξ t : t = 0, ± 1, … } is a strictly stationary and LNQD time series with E ξ 1 = 0, Var ( ξ 1 ) = σ 2 ∈ ( 0, ∞ ) ; (iii) For each n, the joint distribution of { ε n i : 1 ≤ i ≤ n } is the same as that of { ξ 1, …, ξ n }.
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.
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).
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]).
