Minimum Principles for Sturm–Liouville Inequalities and Applications

Phuc Ngo and Kunquan Lan ()
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Phuc Ngo: Department of Mathematics, Toronto Metropolitan University, 350 Victoria Street, Toronto, ON M5B 2K3, Canada
Kunquan Lan: Department of Mathematics, Toronto Metropolitan University, 350 Victoria Street, Toronto, ON M5B 2K3, Canada

Mathematics, 2024, vol. 12, issue 13, 1-13

Abstract: A minimum principle for a Sturm–Liouville (S-L) inequality is obtained, which shows that the minimum value of a nonconstant solution of a S-L inequality never occurs in the interior of the domain (a closed interval) of the solution. The minimum principle is then applied to prove that any nonconstant solutions of S-L inequalities subject to separated inequality boundary conditions (IBCs) must be strictly positive in the interiors of their domains and are increasing or decreasing for some of these IBCs. These positivity results are used to prove the uniqueness of the solutions of linear S-L equations with separated BCs. All of these results hold for the corresponding second-order differential inequalities (or equations), which are special cases of S-L inequalities (or equations). These results are applied to two models arising from the source distribution of the human head and chemical reactor theory. The first model is governed by a nonlinear S-L equation, while the second one is governed by a nonlinear second-order differential equation. For the first model, the explicit solutions are not available, and there are no results on the existence of solutions of the first model. Our results show that all the nonconstant solutions are increasing and are strictly positive solutions. For the second model, many results on the uniqueness of the solutions and the existence of multiple solutions have been obtained before. Our results are applied to prove that all the nonconstant solutions are decreasing and strictly positive.

Keywords: Sturm–Liouville inequalities; minimum principles; second-order differential inequalities; boundary value problems; strictly positive solutions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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