Let \(X\) be a nonempty set endowed with two metrics \(d\) and \(\varrho\) such that \((X, d)\) is a complete metric space and \(d(x,y)\leq c \varrho (x,y)\), for some \(c> 0\) and for all \(x,y\in X\). In this paper, the authors establish several local and global fixed point theorems for multivalued operators \(T:X\rightarrow P(X)\) satisfying the following \(\varphi\)-contraction condition: \[ H_\varrho(T(x),T(y))\leq \varphi \big( \varrho(x,y), D_\varrho(x, T(x)), D_\varrho(y, T(y)), 2^{-1}[D_\varrho(x, T(y))+ D_\varrho(y, T(x))]\big), \] for every \(x,y\in X\). In the last part of the paper, a data dependence theorem and a homotopy result for this type of generalized \(\varphi\)-contractions are given.