It involves an interesting generalization of the class of bounded functions on a group or semigroup and may be stated as follows.
If ({f_{n}}) is a sequence of (L^{p} -uniformly bounded functions on a measure space, and (f_{n} rightarrow f) aL^{p} -uniformly, then liminf_{n rightarrowinfty}|f_{n} |^{p} = liminf _{ n rightarrow infty} |f_{n} - f |^{p} + |f |^{p}, for all (p in(0,infty)).
If ({f_{n}}) is a sequence of (L_{p} -uniformly bounded functions on a measure space, and (f_{n} rightarrow f) aL_{p} -uniformly, then liminf_{n}Vert f_{n}Vert ^{p} = liminf_{n}Vert f_{n}-fVert ^{p}+ Vert fVert ^{p}, for all (pin(0,infty)).
Theorem 6 Let f, u : [ a, b ] → R be bounded functions on [ a, b ]. (i) If there exist constants n, N such that n ≤ u ( t ) ≤ N for any t ∈ [ a, b ], u is Riemann integrable and f is K-Lipschitzian ( K > 0 ), then | T g ( f ; u ) | ≤ 1 2 K ( N − n ) ( b − a ).
Let k and h be bounded functions on (I=[0,1]) and s an integrable bounded function on I with (M_{1}=sup_{tin I}|k(t)|), (M_{2}=sup_{tin I}|s(t)|
Let f be a bounded functions on ({mathbb R}) whose Fouriesr transform is a compact subset of ([0,infty )).
Lemma 1 Let p, v : [ a, b ] → C two bounded functions on the compact interval [ a, b ]. (i) If p is continuous and v is of bounded variation, then the Riemann-Stieltjes integral ∫ a b p ( t ) d v ( t ) exists and | ∫ a b p ( t ) d v ( t ) | ≤ max t ∈ [ a, b ] | p ( t ) | ⋁ a b ( v ), (3.1) .
It generates a linear semigroup acting on differentiable bounded functions on : (1.1).
[26] (Integrability criterion) Let f be a bounded function on I = [ a, b ] T, a, b ∈ T.
[26] Let f be a bounded function on I = [ a, b ] T, a, b ∈ T, m ≤ f (t) ≤ M for all t ∈ I, and g be a function defined and monotonically increasing on I. Then m ( g ( b ) - g ( a ) ) ≤ ∫ a b f ( t ) Δ g ( t ) ≤ ∫ a b f ¯ ( t ) Δ g ( t ) ≤ M ( g ( b ) - g ( a ) ).
(3.1) Conversely, if f is a bounded function on ℝ and (z(cdot)) is a solution of (3.1), then the function (x : mathbb {R}to X) given by x t)= biggl[ Phi z(t) + lim_{n toinfty} int_{- infty}^{t} V^{Q}(t - s) Pi^{ Q bigl(Gamma^{n} f(s)bigl),ds biGamma^{nquad t in mathbb {R}, (3.2) is a mild solution of (1.1).
