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

an analogous way, we define the Henstock ‐integrability of .

      Definition 1.41: We say that is Henstock ‐integrable (or Henstock variationally integrable with respect to ), if there exists a function (called the associate function of ) such that for every , there is a gauge on such that for every ‐fine ,

      We write in this case.

      Next, we define indefinite vector integrals.

      Definition 1.42 (Indefinite Vector Integrals):

      1 Given and , we define the indefinite integral of with respect to byIf, in addition, , then

      2 Given and , we define the indefinite integral of with respect to byIf, moreover, , then

      Note that Definition 1.42 yields the inclusions

      If in item (ii) of Definition 1.42, we consider the particular case, where , and for every , is the identity in , then instead of , , and , we write simply

      respectively, where

      If in item (i) of Definition 1.42, we have , then one can identify the isomorphic spaces and and, hence, the spaces , , , and can also be identified, because one has

      Indeed. It is clear that