  We provide new results of first-order necessary conditions of optimality
problem in the form of John's theorem and in the form of Karush-Kuhn-Tucker's
theorem. We establish our result in a topological vector space for problems
with inequality constraints and in a Banach space for problems with equality
and inequality constraints. Our contributions consist in the extension of the
results known for the Fr\'echet and Gateaux-differentiable functions as well as
for the Clarke's subdifferential of Lipschitz functions, to the more general
Dini-differentiable functions. As consequences, we extend the result of B.H.
Pourciau in \cite[Theorem 6, p. 445]{Po} from the convexity to the {\it
"Dini-pseudoconvexity"}.
