Opial Inequalities with Applications in Differential and Difference Equations

In the year 1960, Opial [6] established the following interesting integral inequality:

Beesack’s generalization of Opial’s inequality in the year 1962 is the forerunner of an enormous literature on Opial-type inequalities. In fact, over the past three decades, generalizations in various directions have been given, and Opial-type inequalities have become a subject in its own right. The purpose of this chapter is to systematically arrange these results, and to compare and contrast their merits.

Among the generalizations of Opial’s inequality (1.1.1) there is a class of inequalities which instead of the first derivative involves the n-th (n ≥ 1) derivative of the given function x(t). The first such result is due to Willett [31], published in 1968. Das’ improvements and further extensions which appeared one year later paved the way for the many subsequent results of this type. This chapter offers an up-to-date account of these results and suggests possibilities for further extensions.

In the year 1981 (April 26 – May 2) during the General Inequalities 3 meeting at Oberwolfach, Agarwal proved a two-independent variable analog of the inequality (1.1.1). This result can be stated as follows: If u(t,s) ∈ C ^(1,1)([a,T] × [c, S]), u(a, s) = u(t, c) = 0, then

The landmark 1955 paper of Fan, Taussky and Todd [8] has brought about a lively interest in discrete inequalities. Moreover, the prominence discrete analysis has gained in the past two decades makes the study of discrete inequalities even more important. For example, see [1], which contains over 400 references. Many discrete inequalities involving functions and their sums and diferences have been found. Although some results in the discrete case are similar to those already known in the continuous case, the adaptation from continuous to discrete is not always direct, but often requires special devices.

Since its discovery more than three decades ago, Opial’s inequality has found, and will certainly continue to find, many interesting applications. In fact, Opial’s inequality and its several generalizations, extensions and discretizations, play a fundamental role in establishing the existence and uniqueness of initial and boundary value problems for ordinary and partial differential equations as well as difference equations. In addition, many qualitative behaviours, such as oscillation, non-oscillation, boundedness, have been discussed in the light of Opial’s inequality. The list of topics to which Opial’s inequality can be applied keeps growing, as witnessed by the many recent publications along this line [5,6,22,23,26–28,30,34,39]. It is certain that this subject will continue to play a very important part in the future of applied mathematics.