a characteriation of continuity of the $p$-$lambda$-variation function is given and the helly's selection principle for $lambda bv^{(p)}$ functions is established. a characterization of the inclusion of waterman-shiba classes into classes of functions with given integral modulus of continuity is given. a useful estimate on modulus of variation of functions of class $lambda bv^{(p)}$ is found.

the notion of generalized locally bounded $i$-topological vectorspaces is introduced. some of their important properties arestudied. the relationship between this kind of spaces and thelocally bounded $i$-topological vector spaces introduced by wu andfang [boundedness and locally bounded fuzzy topological vectorspaces, fuzzy math. 5 (4) (1985) 87$-$94] is discussed. moreover, wealso use the fam...

The notion of generalized locally bounded $I$-topological vectorspaces is introduced. Some of their important properties arestudied. The relationship between this kind of spaces and thelocally bounded $I$-topological vector spaces introduced by Wu andFang [Boundedness and locally bounded fuzzy topological vectorspaces, Fuzzy Math. 5 (4) (1985) 87$-$94] is discussed. Moreover, wealso use the fam...

A way to measure the lower growth rate of φ : Ω × [ 0 , ∞ ) → $$ \varphi :\Omega \times \left[0,\infty \right)\to \right) is require t ↦ ( x − r t\mapsto \left(x,t\right){t}^{-r} be increasing in \left(0,\infty . If this condition holds with = 1 r=1 then inf u ∈ f + W ∫ | ∇ d \underset{u\in f+{W}_0^{1,\varphi}\left(\Omega \right)}{\operatorname{inf}}{\int}_{\Omega}\varphi \left(x,|\nabla u|\rig...

A characteriation of continuity of the $p$-$Lambda$-variation function is given and the Helly's selection principle for $Lambda BV^{(p)}$ functions is established. A characterization of the inclusion of Waterman-Shiba classes into classes of functions with given integral modulus of continuity is given. A useful estimate on modulus of variation of functions of class $Lambda BV^{(p)}$ is found.