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 _ij (i,j=1...,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.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 111, pp. 63–87, 1981.