[Paper Review] Some linear and nonlinear integral inequalities on time scales in two independent variables

Rui A. C. Ferreira, Delfim F. M. Torres|ArXiv.org|May 24, 2008

Nonlinear Differential Equations Analysis10 references3 citations

TL;DR

This paper establishes new linear and nonlinear integral inequalities of Gronwall-Bellman-Bihari type for functions of two independent variables on general time scales, using delta calculus and exponential function properties. The key contribution is a generalized estimate for solutions of partial delta dynamic equations, with explicit bounds derived via time-scale calculus and applied to initial boundary value problems.

ABSTRACT

We establish some linear and nonlinear integral inequalities of Gronwall-Bellman-Bihari type for functions with two independent variables on general time scales. The results are illustrated with examples, obtained by fixing the time scales to concrete ones. An estimation result for the solution of a partial delta dynamic equation is given as an application.

Motivation & Objective

• To develop integral inequalities for functions of two independent variables on arbitrary time scales.
• To extend classical Gronwall-Bellman-Bihari inequalities to the time-scale setting with two variables.
• To provide explicit bounds for solutions of partial delta dynamic equations using the derived inequalities.
• To unify and generalize existing results in differential and difference forms by using a unified time-scale framework.

Proposed method

• The authors use delta calculus on time scales to derive bounds for double integrals involving functions of two variables.
• They apply the exponential function $ e_p $ on time scales to estimate solutions of integral inequalities.
• The proof technique involves delta-differentiation, integration, and recursive bounding using the regressive function class $ \mathcal{R} $.
• A key step is transforming the inequality into a form amenable to application of known time-scale exponential function properties.
• The method is validated by applying it to a partial delta dynamic equation with initial boundary conditions.
• Specific time scales such as $ \mathbb{R} $, $ \mathbb{Z} $, and hybrid sequences $ \mathbb{T}^\alpha $ are used to illustrate the results.

Experimental results

Research questions

• RQ1 How can Gronwall-Bellman-Bihari-type inequalities be extended to functions of two independent variables on general time scales?
• RQ2 What are the necessary conditions on the functions and time scales to ensure boundedness of solutions to double integral inequalities?
• RQ3 How do the derived inequalities specialize to classical differential and difference forms on $ \mathbb{R} $ and $ \mathbb{Z} $?
• RQ4 Can the results be applied to estimate solutions of partial delta dynamic equations on time scales?
• RQ5 What is the role of the exponential function $ e_p $ in deriving sharp bounds for such inequalities?

Key findings

• The paper establishes a linear integral inequality bound: $ u(t_1,t_2) \leq a(t_1,t_2) e_{\int_{a_2}^{t_2} f(t_1,s_2)\Delta_2 s_2}(t_1,a_1) $, valid for nondecreasing $ a $ and nonnegative $ f $.
• For $ \mathbb{T}_1 = \mathbb{T}_2 = \mathbb{R} $, the bound reduces to $ u(x,y) \leq a(x,y) \exp\left(\int_{x_0}^x \int_{y_0}^y f(t,s) dt ds\right) $, matching classical results.
• For $ \mathbb{T}_1 = \mathbb{T}_2 = \mathbb{Z} $, the bound becomes $ u(m,n) \leq a(m,n) \prod_{s=m_0}^{m-1} \left[1 + \sum_{t=n_0}^{n-1} f(s,t) \right] $, consistent with difference equation bounds.
• For a hybrid time scale $ \mathbb{T}^\alpha \times \mathbb{T}^\beta $, the bound is expressed via the product formula $ e_p(t_k^\alpha, t_0^\alpha) = \prod_{n=1}^k (1 + \alpha_n p(t_{n-1})) $.
• An application to a partial delta dynamic equation yields the solution estimate $ u(t_1,t_2) \leq \sqrt{g(t_1)+h(t_2)} \left[ e_{\int_0^{t_2} s_2 (g(t_1)+h(s_2))^{-1/2} \Delta_2 s_2}(t_1,0) \right]^{1/2} $.
• The results are sharp and generalize known inequalities in both continuous and discrete settings through a unified time-scale framework.