We first introduce notions of strong t-quasiconvexity and strong t-quasiconvex-likeness along with relaxed Hölder continuity with respect to an ordering cone of a set-valued map. Then, these new conditions are utilized to establish sufficient conditions for the non-emptiness of solution sets and the stability in the sense of Hölder continuity of solution maps to parametric set-valued Ky Fan inequalities. Our approach is different from the existing ones. At the end of the paper, we present an application of the main results to set-valued Nash equilibrium in games.