Sobolev type mathematical models with relatively positive operators in the sequence spaces

In the sequence spaces which are analogues of Sobolev function spaces we consider mathematical model whose prototypes are Barenblatt–Zheltov–Kochina equation and Hoff equation. One should mention that these equations are degenerate equations or Sobolev type equations. Nonexistence and nonuniqueness of the solutions is the peculiar feature of such equations. Therefore, to find the conditions for positive solution of the equations is a topical research direction. The paper highlights the conditions sufficient for positive solutions in the given mathematical model. The foundation of our research is the theory of the positive semigroups of operators and the theory of degenerate holomorphic groups of operators. As a result of merging of these theories a new theory of degenerate positive holomorphic groups of operators has been obtained. The authors believe that the results of a new theory will find their application in economic and engineering problems.