chapter  12
57 Pages

Chapter 12

Sturm-Liouville Problems and Related Inequalities

Related Inequalities This chapter is devoted to special type of boundary value problems

which lead to the concepts of eigenvalues and eigenfunctions, orthogonality, and finite Fourier series. While in relation to differential equations these notions play a fundamental role in the study of mathematical physics and engineering, and have resulted in a vast amount of advanced mathematies, in the discrete ca..o;e their importance is not fully explored, except that most of these problems arc equivalent to some special matrix eigenvalue problems. We shall exploit this equivalence to derive Wirtinger and Opial type inequalities. Next, in this chapter we shall touch upon cone theory and use it to prove the existence and the comparison theorems for the least positive eigenvalues of the (p, n-p) discrete boundary value problems. Finally, as a further application to cone theory we shall discuss positive solutions and nonlinear eigenvalue problems for third order difference equations.