Necessary conditions for a minimum in variational problems with delay in the presence of degeneracie

This article examines the minimum of an extremal in the variational problem with delay under the degeneracy of the Weierstrass condition. We obtain necessary conditions of equality type and inequality type for a strong as well as for a weak local minimum. Necessary conditions of equality and inequality types are obtained for strong as well as weak local minimum. A specific example demonstrating the effectiveness of the results in this paper is provided.