Abstract
In this paper, we prove some new generalizations of dynamic Opial-type inequalities on time scales. From these inequalities, as special cases, we formulate some integral and discrete inequalities proved in the literature and also extend some obtained dynamic inequalities on time scales. The main results are proved by using some algebraic inequalities, Hölder’s inequality, and a simple consequence of Keller’s chain rule on time scales.
Highlights
Yang [27] simplified Beesack’s proof and extended inequality (1.3) and proved that: If x is an absolutely continuous function on (a, b) with x(a) = 0, b q(t) x(t)
In 1960, Opial [24] proved the following inequality: b x(t)x (t) dt ≤ b – a b x (t) 2 dt, (1.1) a4a where x is absolutely continuous on [a, b] and x(a) = x(b)0, and the constant b–a 4 is the best possible.Equality holds in (1.1) if and only if x(t) = c(t – a), for a ≤ x ≤ b – a, 2 and x(t) = c(b – t), for b – a ≤ x ≤ b, 2 where c is a constant
Opial’s inequality together with its numerous generalizations, extensions, and discretizations has been playing a fundamental role in the study of the existence and uniqueness properties of solutions of initial and boundary value problems for differential equations as well as difference equations [3, 20]
Summary
Yang [27] simplified Beesack’s proof and extended inequality (1.3) and proved that: If x is an absolutely continuous function on (a, b) with x(a) = 0, b q(t) x(t) Hua [15] extended inequality (1.2) and proved that: If x is an absolutely continuous function with x(a) = 0, b x(t) p x (t) dt ≤ (b – a)p b x (t) p+1 dt, a p+1 a where p is a positive integer. If the time scale equals the real (or the integers), the results represent the classical results for differential (or difference) inequalities.
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