Comparison Theorems for One Sturm-Liouville Problem With Nonlocal Boundary Conditions

Abstract

In this study we present a new approach for investigation of some Sturm-Liouville systems with nonlocal boundary conditions. In the theory of boundary value problems for two-order differential equations the basicconcepts and methods have been formulated studying the problems of classical mathematical physics. However,many modern problems, which arise as the mathematical modeling of some systems and processes in the fields ofphysics, such as the vibration of strings, the interaction of atomic particles motivate to formulate and investigate thenew ones, for example, a class of Sturm-Liouville problems with nonlocal boundary conditions. Such conditionsarise when we cannot measure data directly at the boundary. In this case, the problem is formulated, where the valueof the solution and its derivative is linked to interior points of the considered interval. Sturm-Liouville problemstogether with transmission conditions at some interior points is very important for solving many problems of mathematical physics. In this study we present a new approach for investigation of boundary value problems consistingof the two interval Sturm-Liouville equations. This kind of boundary value transmission problems are connectedwith various physical transfer problems (for example, heat and mass transfer problems). We define a new Hilbertspace and linear differential operator in it such a way that the considered nonlocal problem can be interpreted asan spectral problem. We investigate the main spectral properties of the problem under consideration. Particularlywe present a new criteria for Sturm-Comparison theorems. Our main result generalizes the classical comparisontheorem for regular Sturm-Liouville problems.