Let \(E\) be a Banach space ordered by a convex closed reproducing acute cone \(E_+\) and \(X\) be a space which is complete with respect to a vector-valued metric \(\mathcal{P} : X \times X \to E_+\). Let \(Cl(X)\) denote the collection of all nonempty closed (w.r.t.\ \(\mathcal{P}\)) subsets of \(X\) and \(Q : E \to E\) be a positive (i.e., \(Q(E_+) \subset E_+\)) linear operator with a spectral radius \(\rho (Q) < 1\). The main result is the following generalization of the Nadler fixed point theorem. Let a multimap \(\Phi : X \to Cl(X)\) be a \(Q\)-contraction, i.e., \[ \forall x,\tilde x \in X \quad \forall y \in \Phi (x) \quad \exists \tilde y \in \Phi (\tilde x): \quad \mathcal{P}(\tilde y,y) \leq Q\,\mathcal{P}(\tilde x,x). \] Then there exists a fixed point \(x \in \Phi (x)\) satisfying the estimate \[ \mathcal{P}(x,x_0) \leq (I - Q)^{-1}\mathcal{P}(x_1,x_0) \] for all \(x_0 \in X,\) \(x_1 \in \Phi (x_0)\). As application, the solvability of an integral inclusion and a boundary value problem for a functional differential inclusion are considered.