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

, the function is increasing.

      3 (SB3) Given and , .

      Definition 1.24: Consider the BT . Then, instead of and , we write simply and , respectively. Hence,

      and we call any element of a function of bounded semivariation.

      Definition 1.25: Given a function , a normed space, and a division , we define

      and the variation of is given by

      If , then is called a function of bounded variation, in which case, we write .

      It is not difficult to prove that and .

      Moreover (see [127, Corollary I.3.4]), .

      The space is complete when equipped with the variation norm, , given by

for When there is no room for misunderstanding, we may use the notation instead of .

      Remark 1.26: Consider a BT . The definition of variation of a function , where is a normed space, can also be considered as a particular case of the ‐variation of in two different ways.

      1 Let , or and . By the definition of the norm in , we haveThus, when we consider the BT , we write and instead of and , respectively.

      2 Let , or and . By the Hahn–Banach Theorem, we haveand, hence,

      Definition 1.27: Given , we define the spaces