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We present a characterization of inclusion among Riesz−Medvedev bounded variation spaces, i.e., we shall present necessary and sufficient conditions for the Young functions ${\varphi }_1$ and ${\varphi }_2$ so that $R{V}_{{\varphi }_1}[a,b]\subset R{V}_{\varphi {}_2}[a,b]$ or $R{V}_{{\varphi }_1}^{*}[a,b]\subset R{V}_{{\varphi }_2}^{*}[a,b]$.

In this paper we study existence and uniqueness of solutions for the Hammerstein equation $$u\left(x\right)=v\left(x\right)+\lambda {\int }_{I_a^b}K(x,y)f(y,u\left(y\right))dy$$ in the space of function of bounded total $\varphi$-variation in the sense of Hardy-Vitali-Tonelli, where $\lambda \in ℝ$, $K:I_a^b\times I_a^b⟶ℝ$ and $f:I_a^b\times ℝ⟶ℝ$ are suitable functions. The existence and uniqueness of solutions are proved by means of the Leray-Schauder nonlinear alternative and the Banach contraction mapping principle.

In this paper we extend the well known Riesz lemma to the class of bounded $\varphi$-variation functions in the sense of Riesz defined on a rectangle $I_a^b\subset {ℝ}^2$. This concept was introduced in [2], where the authors proved that the space $B{V}_{\varphi }^R(I_a^b;ℝ$ of such functions is a Banach Algebra. Moreover, they characterized also the Nemytskii operator acting in this space. Thus our result creates a continuation of the paper [2].