Consider the equation \[ \bigl(p(t) y^\Delta(t)\bigr)^\Delta+ q(t)y\bigl(\sigma(t) \bigr)= f(t),\tag{*} \] where \((\cdot)^\Delta\) is so-called delta derivative, \(\sigma(t)= \inf\{\tau >t,\;t\in\mathbb{T}\}\) and \(\mathbb{T}\) is a closed subset of \(\mathbb{R}\). Here \(p,q\) and \(f\) are right dense continuous functions on \(\mathbb{T}\) and \(p\) is a positive function.
On the basis of the Banach fixed-point theorem it is proved that under some assumptions, equation (*) has a solution which converges to zero as \(t\to\infty\). Moreover for each number \(C\) there is a unique bounded on \([a,\infty)\) solution with \(y(a)=C\).