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

Year 2018, Volume: 10, 121 - 125, 29.12.2018

Kadriye Aydemir , Hayati Olğar , Oktay Mukhtarov

Abstract

n 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 basic concepts 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 of physics, such as the

vibration of strings, the interaction of atomic particles motivate

to formulate and investigate the new ones, for example, a class of

Sturm-Liouville problems with nonlocal boundary conditions. Such

conditions arise when we cannot measure data directly at the

boundary. In this case, the problem is formulated, where the value

of the solution and its derivative is linked to interior points of

the considered interval. Sturm-Liouville problems together 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

consisting of the two interval Sturm-Liouville equations. This kind

of boundary value transmission problems are connected with various

physical transfer problems (for example, heat and mass transfer

problems). We define a new Hilbert space and linear differential

operator in it such a way that the considered nonlocal problem can

be interpreted as an spectral problem.  We investigate the main

spectral properties of the problem under consideration. Particularly

we present a new criteria for Sturm-Comparison theorems. Our main

result generalizes the classical comparison theorem for regular

Sturm-Liouville problems.

Keywords

Boundary value problems, Comparison theorems, Transmission conditions

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