If (R[a,b]) denotes the set of Riemann integrable functional on ([a,b]); (betain R[a,b]), and x^{prime}(t)leqbeta(t cdot x t), quad forall tin[a,b], then x t)leq x(a cdot e^{int^{t}_{a}beta(s),ds},quad forall tin[a,b].
where a ( t ) and b i ( t ) are Lebesgue integrable functionals on J ( i = 1, 2, 3, 4 ) and satisfying.
where F ∈ C ( J × P × P, P ), h ∈ C ( J ). Theorem 4.1 Assume that (H2) holds, and the following conditions are satisfied: (C1) F ∈ C ( J × P 2, P ), h ∈ C ( J, P ) and ∥ F ( t, y 1, y 2 ) ∥ ≤ a ( t ) + ∑ i = 1 2 b i ( t ) ∥ y i ∥, where a ( t ) and b i ( t ) are Lebesgue integrable functionals on J ( i = 1, 2 ) and satisfying.
Definition C. Let be an integrable function, for One defines a linear functional as (2.22).
Definition B. Let be an integrable function; for one defines a linear functional as (2.1).
In the proofs of some estimations we use recent results related to the Čebyšev functional, which for two Lebesgue integrable functions (f,g:[a,b]rightarrow mathbb{R}) is defined by T f,g)=frac{1}{b-a} int_{a}^{b}f(t g(t),dt-frac{1}{b-a} int_{a}^{b}f(t),dtcdot frac{1}{b-a} int_{a}^{b}g(t),dt.
For two Lebesgue integrable functions (f,g:[alpha,beta]rightarrow mathbb{R}), we consider the Čebyšev functional: T f,g):=frac{1}{beta-alpha} int _{alpha}^{beta }f(t g(t), dt-frac {1}{beta-alpha} int _{alpha}^{beta}f(t), dtcdotfrac{1}{beta -alpha } int _{alpha}^{beta}g(t), dt.
The Dirac delta distribution is a densely defined unbounded linear functional on the Hilbert space L2 of square integrable functions.
For two Lebesgue integrable functions (f,h:[a,b]rightarrow mathbb{R}), we consider the Čebyšev functional T f,h)= frac{1}{b-a} int_{a}^{b} f(t)h(t),dt -frac{1}{b-a}int_{a}^{b} f(t),dt cdot frac{1}{b-a}int_{a}^{b} h(t),dt.
For two Lebesgue integrable functions f, g : [ a, b ] → R, consider the Čebyšev functional: C ( f, g ) : = 1 b − a ∫ a b f ( t ) g ( t ) d t − 1 ( b − a ) 2 ∫ a b f ( t ) d t ∫ a b g ( t ) d t. (1.1).
We give necessary and sufficient conditions for the minimality of generalized minimizers of linear-growth integral functionals of the formF[u]="∫Ωf x,u(x dx,u Ω⊂Rd→RN, where f:Ω×RN→R is a convex integrand and u is an integrable function satisfying a general PDE constraint.
