Operadores lineares integrais em espaços de funções contı́nuas com peso

Abstract

The classical results on the general properties of the linear integral operator of the form (Kx)(t) = (Kx)(t) = ∫ ba k(t, s)x(s)ds, t ∈ [a, b], such as action, boundedness, and complete continuity, are well known and established in the mathematical literature, especially in the spaces of continuous functions C[a, b] and in the spa- ces of integrable functions L ∞ [a, b]. In this work, we generalize these results to the weighted function spaces C α (a, b) and L ∞,α (a, b), where α is a weight function that satisfies certain re- gularity conditions. In particular, we establish criteria for the action and boundedness of the linear integral operator K in Banach spaces C α (a, b) and L ∞,α (a, b), in terms of the kernel k(t, s) and the weight function α. In addition, we obtain expressions for the norm of the operator K in terms of the kernel k(t, s) and the weight function α, and establish necessary and sufficient conditions for the complete continuity of the operator K in weighted function spaces