Let \(\beta \) be a bornology on a Banach space. Necessary optimality conditions are derived for a mathematical programming problem in \(\beta\)-smooth Banach spaces with finite lower semi-continuous objective, inequality and continuous equality constraints as well as a set constraint. A lower semicontinuous function \(f\) is (rowly spoken) \(\beta\) subdifferentiable at \(x\) if it can be supported at \(x\) by a \(\beta\)-smooth Lipschitzian function \(g.\) 1.) Using the well-known fuzzy sum rule for \( \beta\)-subdifferentials several fuzzy multiplier rules for the above problem are derived. 2.) Elements \(\left( \bar{x}^{*},0\right) \) of the \( \beta -\)normal cone of the epigraph of \(f\) at \(\bar{x}\) can be approximated in an arbitrary close weak* neighbourhood by \(\beta\)-subgradients of \(f\) at \(x\) with \((x,f(x)) \) in an arbitrary ball around \((\bar x,f(\bar x)) \). 3.) In case of Fréchet bornologies the weak* neighbourhood can be replaced by the norm ball. 4.) The multiplier rule implies a chain rule and other subdifferential calculus rules as sum rules, product rules and quotient rules. 5.) The above multiplier rule gives fuzzy and limiting necessary optimality conditions for MPEC with l.s.c. objective.