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Two existence theorems of maximal elements in <i>H</i>-spaces are obtained without compactness. More accurately, we deal with the correspondence to be of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">L</mi> </semantics> </math> </inline-formula>-majorized mappings in the setting of noncompact strategy sets but merely requiring a milder coercive condition. As applications, we obtain an equilibrium existence theorem for general abstract economies, a new fixed point theorem, and give a sufficient condition for the existence of solutions of the eigenvector problem (EIVP).