Abstract
The Gronwall inequalities are of significance in mathematics and engineering. This paper generalizes the Gronwall-like inequalities from different perspectives. Using the proposed inequalities, the difficulties to discuss the controllability of integrodifferential systems of mixed type can be solved. Meanwhile, two examples as their applications are also given to show the effectiveness of our main results.
Highlights
Integral inequalities provide a powerful and important tool in the study of qualitative properties of solutions of nonlinear differential, integral, and integrodifferential equations, as well as in the modeling of science and engineering problems
Lipovan [18] proved a Gronwall-like inequality, and in order to show its applications, Lipovan applied his main results to the qualitative analysis of solutions to certain integral equations, functional differential equations, and retarded differential equations
Ye et al [19] gave a generalized Gronwall inequality with singularity which can be applied to weakly singular Volterra integral equations and fractional integral and integrodifferential equations
Summary
Integral inequalities provide a powerful and important tool in the study of qualitative properties of solutions of nonlinear differential, integral, and integrodifferential equations, as well as in the modeling of science and engineering problems (see [1]). The celebrated Gronwall inequality and its generalizations play increasingly important roles in the qualitative analysis of differential, integral, and integrodifferential equations. Lipovan [18] proved a Gronwall-like inequality, and in order to show its applications, Lipovan applied his main results to the qualitative analysis of solutions to certain integral equations, functional differential equations, and retarded differential equations.
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