Open AccessSpecial cases of critical linear difference equations
Author(s) -
Jan Jekl
Publication year - 2021
Publication title -
electronic journal on the qualitative theory of differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.524
H-Index - 33
ISSN - 1417-3875
DOI - 10.14232/ejqtde.2021.1.79
Subject(s) - mathematics , converse , independent equation , criticality , order (exchange) , linear equation , interlacing , simultaneous equations , equivalence (formal languages) , class (philosophy) , mathematical analysis , pure mathematics , differential equation , physics , geometry , finance , nuclear physics , economics , artificial intelligence , computer science , operating system
In this paper, we investigate even-order linear difference equationsand their criticality. However, we restrict our attention only to severalspecial cases of the general Sturm–Liouville equation. We wish toinvestigate on such cases a possible converse of a known theorem. Thistheorem holds for second-order equations as an equivalence; however, onlyone implication is known for even-order equations. First, we show theconverse in a sense for one term equations. Later, we show an upper boundon criticality for equations with nonnegative coefficients as well.Finally, we extend the criticality of the second-order linear self-adjointequation for the class of equations with interlacing indices. In this way,we can obtain concrete examples aiding us with our investigation.