On bounded lower \(\Lambda\)-variation (Q465426)

The authors introduce a new notion of the lower \(\Lambda\)-variation of a real-valued function \(f\) on an interval \(I\) NEWLINE\[NEWLINE\underline{\text{var}}_{\Lambda}f:=\inf\{\text{var}_{\Lambda}g:f=g\;\text{ a. e. on } I\},NEWLINE\]NEWLINE where \(\text{var}_{\Lambda}g\) stands for the \(\Lambda\)-variation of \(g\). They prove that the vector space NEWLINE\[NEWLINE\underline{\Lambda\text{BV}}[0,1]:=\{f\in L^1[0,1]:\underline{\text{var}}_{\Lambda}f<+\infty\}NEWLINE\]NEWLINE endowed with the norm \(|f|_{\Lambda}:=\|f\|_1+\underline{\text{var}}_{\Lambda}f\) forms a Banach space. It is proved that in each equivalence class \([f]\) there is a representative, whose \(\Lambda\)-variation coincides with \(\underline{\text{var}}_{\Lambda}f\). Such representatives of the class are called good ones. Several topological and geometrical properties of the space \(\underline{\Lambda\text{BV}}[0,1]\) are provided. Applications to the linear differential equation of first order and nonlinear integral equations are given.