Abstract
In this paper, we propose a new method to unify and extend some nonlinear dynamic double integral inequalities of two independent variables involving pairs of time scales via nabla derivative. The acquired results give explicit bounds which can be utilized to consider the subjective and quantitative properties of specific classes of dynamic conditions on time scales. An application to prove the validity of our established results is also given.
Highlights
Whenever there is a discussion about the importance of research work for boundedness, global existence, the stability of the solutions of differential and basic conditions of integral equations, the fact cannot be denied that Gronwall–Bellman inequality, Bihari inequality, and their various generalizations play a pivotal role in providing precise bounds of differential, integral, and of difference equations
Bohner and Peterson [6] examined the integral inequality on time scales of the form u x(u) ≤ a(u) + p(u) k(u, τ ) b(τ )x(τ ) + q(τ ) τ
Li [21] further studied the generalization of the nonlinear integral inequality of two independent variables on time scales as follows: uv xα(u, v) ≤ a(u, v) + b(u, v) g(τ, η)xα(τ, η) + h(τ, η)x(τ, η) η τ, u0 v0 with α > 1 being a real constant
Summary
Whenever there is a discussion about the importance of research work for boundedness, global existence, the stability of the solutions of differential and basic conditions of integral equations, the fact cannot be denied that Gronwall–Bellman inequality, Bihari inequality, and their various generalizations play a pivotal role in providing precise bounds of differential, integral, and of difference equations. Li [21] further studied the generalization of the nonlinear integral inequality of two independent variables on time scales as follows: uv xα(u, v) ≤ a(u, v) + b(u, v) g(τ , η)xα(τ , η) + h(τ , η)x(τ , η) η τ , u0 v0 with α > 1 being a real constant. Based on the works of the above-mentioned researchers and using the same context of Gronwall–Bellman type inequalities, in this paper, we explore and generalize the following nonlinear dynamic inequalities of two independent variables via nabla derivative on time scales:
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