It is shown, in a completely general setting, that a theorem of the alternative is logically equivalent to a duality theorem linking two constrained optimization problems. A standard technique for proving a duality theorem linking two constrained optimization problems is to apply an appropriate theorem of the alternative, sometimes called a transposition theorem (see, e.g., Mangasarian [2], Roc...

Propounding a general categorical framework for the extension of dualities, we present new proof de Vries Duality Theorem category KHaus compact Hausdorff spaces and their continuous maps, as an restricted Stone duality. Then, applying dualization to duality, give alternative duality Tych Tychonoff that was provided by Bezhanishvili, Morandi Olberding. In process doing so, obtain theorems both ...

we de ne approximate xed point in fuzzy norm spaces and prove the existence theorems, we also consider approximate pair constructive map- ping and show its relation with approximate fuzzy xed point.