The type system is defined in Fig. 8, as a set of rules for deriving judgements (Gamma,vdash,e : tau ), meaning that term (e) has type (tau ), in typing context (Gamma ) (which contains type assumptions for the free variables in (e)).
For the connections between our assumptions and Namah Roquejoffre type assumptions, we refer to Sect. 4.
The similar quasi-periodic type assumptions are common in the asymptotic fixed point theory, see, e.g., [2, 25, 28].
Stronger type assumptions can be made about the local data than about external data, hence our type system mixes static and dynamic typing.
In many cases, modular type conditions are much more natural as modular type assumptions can be more easily verified than their metric or norm counterparts.
The weak, strong and strict converse duality theorems are proved for these programs under the second-order generalized convexity type assumptions.
In this paper, weak, strong and strict converse duality theorems have been discussed for nondifferentiable minimax fractional programming problems in the framework of generalized convexity type assumptions.
Liu and Wu [7, 8] and Ahmad [9] obtained sufficient optimality conditions and duality for minimax fractional programming under generalized convex type assumptions.
In the present paper, we are concerned with second-order duality for nondifferentiable minimax fractional programming under the second-order generalized convexity type assumptions.
For the case (Vequiv0), Chang and Wang in [3] obtained the existence of a positive ground state under the general Berestycki-Lions type assumptions.
