The authors prove that for any given non zero vector c and any Kakutani-type point-to-set mapping on a full-dimensional compact, convex set, the set of stationary points with respect to c connects the pair of extreme sets \(X(0)\) and \(X(1)\), where \(X\) is full dimensional, compact, convex set in \(\mathbb R^n\). The authors give sufficient conditions under which there exists in \(X\) a connected set of zero points of the mappings \(\phi\) connecting \(X(0)\) and \(X(1)\), where \(\phi\) is an outer semi-continuous mapping from \(X\) to the collection of nonempty, compact convex subsets of \(\mathbb R^n\). They prove an intersection result on a convex, compact set and give an application of the result to a pure exchange economy with restricted price set. The results in the paper generalizes earlier results of L. E. J. Brouwer [Math. Ann. 71, 97–115 (1911; JFM 42.0417.01)], A. Mas-Colell [Math. Program. 6, 229–233 (1974; Zbl 0285.90068)], P. J.-J. Heringes, D. Talman and Z. Yang [SIAM J. Control Optimization 39, No. 6, 1852–1873 (2001; Zbl 1017.49014)].