Abadie condition for infinite programming problems under Relaxed Constant Rank Constraint Qualification Plus

We consider infinite programming problems with constraint sets defined by systems of infinite number of inequalities and equations given by continuously differentiable functions defined on Banach spaces. In the approach proposed here we represent these systems with the help of coefficients in a given Schauder basis. We prove the Abadie condition under the new infinite-dimensional Relaxed Constant Rank Constraint Qualification Plus and we discuss the existence of Lagrange multipliers. The main tools are: Rank Theorem and Ljusternik Theorem.