  A sufficient condition is established for the existence of a solution to the
equation $\mathcal{T}(u,\mathcal{C}(u))=u$, by considering a class of Kannan
type equicontraction mappings $\mathcal{T}:\mathcal{A}\times
\overline{\mathcal{C}(\mathcal{A})}\to \Xi$, where $\mathcal{A}$ is a convex,
closed and bounded subset of a Banach space $\Xi$ and $\mathcal{C}$ is a
compact mapping. To fulfil the desired purpose, we engage the Sadovskii's
theorem, involving the measure of noncompactness. The relevance of the acquired
results has been illustrated by considering a certain class of initial value
problems.
