Let \(X,Y\) be Banach spaces and \(F:A\rightrightarrows Y\) a set-valued mapping. A restriction of \(F\) to a subset \(B\) of \(A\) is a set-valued mapping \(F_1: B\rightrightarrows Y\) such that \(F_1(x)\subset F(x)\) for all \(x\in B\) and \(F_1(x)=\emptyset\) for all \(x\in A\setminus B\). One says that \(F\) satisfies the coacute angle condition if, for every \(y^*\in Y^*,\,y^*\neq 0\), there exists \(x\in A\) such that Re\(\langle y,y^*\rangle\geq 0\) for all \(y\in F(x)\). If this inequality is strict, then one says that \(F\) satisfies the strict coacute angle condition. The authors prove a result of the following kind. One considers a domain \(D\) in the Euclidean space \(E^n\), a set-valued mapping \(F:\bar D\to E^n\) and \(K\subset \bar D\). If there exists a restriction \(F_1\) of \(F\) to \(K\) satisfying the coacute angle condition such that \(\operatorname{conv}F_1(K)\) is compact and \(\operatorname{conv}F_1(K)\subset F(\bar D)\), then \(0\in F(\bar D)\). If there exists a restriction \(F_1\) satisfying the strict coacute angle condition, then the compactness condition can be dropped. Similar results for set-valued mappings satisfying some metric conditions (called by the authors acute angle \(\varepsilon\)-condition and coacute angle \(\delta\)-condition) are also obtained.