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A sufficient condition is established for the existence of a solution to theequation $\mathcal{T}(u,\mathcal{C}(u))=u$, by considering a class of Kannantype 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 acompact mapping. To fulfil the desired purpose, we engage the Sadovskii'stheorem, involving the measure of noncompactness. The relevance of the acquiredresults has been illustrated by considering a certain class of initial valueproblems.

On the extended version of Krasnosel'skii's fixed point theorem for Kannan type equicontraction mappings