Generalized Ordinary Differential Equations in Abstract Spaces and Applications. Группа авторов

and , we have

      since for every , we have

      The proof is then complete.

      A proof of the next result, borrowed from [72, Theorem 8], follows from the definitions of the integrals.

      

      Theorem 1.59: Let and . If is bounded, then and

      (1.5)

       If, in addition, , then .

      Corollary 1.60: Suppose with and . Then, and (1.5) holds.

Proof. By Theorem 1.49, . Then, the result follows from Theorem 1.47, since .

      The next corollaries follow from Theorems 1.49 and 1.53.

      Corollary 1.61: Suppose with and . Then, , and we have

(1.6)

       and the following integration by parts formula holds

(1.7)

Corollary 1.62: Consider functions and . Then, and equalities (1.6) and (1.7) hold.

      The next two theorems generalize Corollary 1.62. For their proofs, the reader may want to consult [72].

      Theorem 1.63: Consider . If respectively, , then respectively, and both (1.6) and