M. A. Krasnoselski [4] proved that if $K$ is a nonempty closed bounded subset of a Banach space and $A,B$ are operators on $K$ such that $A$ is completely continuous, $B$ is a contraction and $Au+Bv\in K$ for all $u,v\in K$, then the equation $x=Ax+Bx$ has a solution in $K$. Many papers related to this result have been published. In particular, Melvin [5] has given conditions under which there exists a solution of the equation $x=G(x,Qx)$. We present a generalization of Melvin's theorem for the relation $Tx\in F(x,Q(Tx))$ with $F$ taking values in the family of nonempty closed convex bounded subsets of a Banach space. An application of our result to the theory of differential relations is also given.