A Unified Framework for Generalized Hardy-type Inequalities on Time Scales
E. E. Aribike
Department of Mathematical Sciences, Lagos State University of Science and Technology, Ikorodu, Nigeria.
O. K. Adewale
Department of Mathematical Sciences, Tai Solarin University of Education, Ogun State, Nigeria.
K. Rauf *
Department of Mathematics, University of Ilorin, Ilorin, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
This paper focuses on the study of weighted Hardy-type inequalities within the broad and unifying concept of time scale calculus. By applying H¨older’s inequality as a major analytical tool, the study develops refined versions of Hardy-type inequalities that extend a number of well-known classical results to a more general mathematical context. The work further builds on suitable adaptations of Opial-type inequalities on time scales, making it possible to treat both continuous and discrete cases together under a single mathematical approach. In addition, the derived inequalities also provide useful techniques for the qualitative analysis of dynamic equations on time scales. In particular, they contribute to the investigation of important solution properties such as boundedness, oscillation, and stability. In essence, this study broadens the applicability of classical Hardy-type inequalities by incorporating them into the flexible and comprehensive structure of time scale calculus.
Keywords: Hardy-type inequalities, continuous functions, weighted functions, H¨older′s inequality