Existence and the optimal choice of the values of parameters in inequalities guaranteeing the validity of embedding theorems

For systems of linear algebraic equations with matrix of coefficientsa $a_{ij}$ ($i,j=1,\dots,n$) where under certain assumptions regarding the form of the matrix and for nonnegative right-hand sides, one establishes necessary and sufficient conditions for the existence of strictly positive solutions. One considers applications in the imbedding theory of functional spaces.