Abstract
In this article we discuss some new generalized nonlinear Gronwall-Bellman-Type integral inequalities with two variables, which include a non-constant term outside the integrals. We use our result to deal with the estimate on the solutions of partial differential equations with the initial and boundary conditions. Mathematics Subject Classification 2000: 26D10; 26D15; 26D20; 34A40.
Highlights
Various generalizations of Gronwall inequality [1,2] are fundamental tools in the study of existence, uniqueness, boundedness, stability and other qualitative properties of solutions of differential equations, integral equations, and differential-integral equations
There are a lot of articles investigating its generalizations such as [3-23]
Pachpatte [19] provided the explicit estimations of following integral inequalities: n αi(t) up(t) ≤ c + p
Summary
Various generalizations of Gronwall inequality [1,2] are fundamental tools in the study of existence, uniqueness, boundedness, stability and other qualitative properties of solutions of differential equations, integral equations, and differential-integral equations. Agarwal et al [3] obtained the explicit bounds to the solutions of the following retarded integral inequalities: n αi(t) φ(u(t)) ≤ c + In this article, motivated mainly by the works of Agarwal et al [3] and Chen et al [6], Cheung [7], Pachpatte [19], we discuss more general forms of following integral inequalities: n αi(x) βi(y) ψ(u(x, y)) ≤ a(x, y) + b(x, y) w(u(s, t))[fi(s, t)φ(u(s, t))
Sign-in/Register to view highlights & summary 
Published Version (
Free)
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 call