Abstract
This paper is committed to introducing some Bihari type inequalities for scalar functions of one independent variable under an initial condition associated with an arbitrary time scale mathbb{T}. The integrals involve the maximum of an unknown function over a past time interval. We not only solve some new estimated bounds of a specific class of retarded and nonlinear dynamic inequalities but also derive and unify continuous inequalities along with the corresponding discrete analogs of some known results with ‘maxima’ on time scales. We illustrate some applications of the considered inequalities to represent the advantages of our work. The main results will be proved by utilizing some examination procedures and the basic technique of Keller’s chain rule on time scales.
Highlights
1 Introduction Integral and differential inequalities have turned out to be important devices in the investigation of the differential and integral equations that happen in nature or are built by several mathematicians
M where r is a continuous function defined on the interval M = [m, m + n] and m, l, k, n are nonnegative constants
For j ∈ Crd(T, R), the composition of two functions on time scales is defined by j(u) ◦ ρ–1(λ) = j ρ–1(λ), u ∈ T, λ ∈ T
Summary
Integral and differential inequalities have turned out to be important devices in the investigation of the differential and integral equations that happen in nature or are built by several mathematicians (see [1,2,3]). The above mentioned inequalities are not directly applicable in the study of certain nonlinear retarded differential and integral equations. These kinds of inequalities have many applications, when one wants to study the existence and uniqueness of the solutions of a differential equation (see [33,34,35]).
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
