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## Über dieses Buch

This book concentrates on one- and multi-dimensional nonlinear integral and discrete Gronwall-Bellman type inequalities. It complements the author’s book on linear inequalities and serves as an essential tool for researchers interested in differential (ODE and PDE), difference, and integral equations.

The present volume is part 2 of the author’s two-volume work on inequalities.

Integral and discrete inequalities are a very important tool in classical analysis and play a crucial role in establishing the well-posedness of the related equations, i.e., differential, difference and integral equations.

## Inhaltsverzeichnis

### Chapter 1. Nonlinear One-Dimensional Continuous Integral Inequalities

Abstract
The attractive Gronwall-Bellman inequality [259] plays a vital role in studying stability and asymptotic behavior of solution of differential equations (see, e.g., [50, 69]).
Yuming Qin

### Chapter 2. Nonlinear One-Dimensional Discrete (Difference) Inequalities

Abstract
In this chapter, we shall introduce some nonlinear discrete (difference) integral inequalities.
Yuming Qin

### Chapter 3. Nonlinear One-Dimensional Discontinuous Integral Inequalities

Abstract
In this section, we shall introduce some nonlinear discontinuous Gronwall-Bellman inequalities and their generalizations.
Yuming Qin

### Chapter 4. Applications of Nonlinear One-Dimensional Continuous, Discontinuous Integral Inequalities and Discrete Inequalities

Abstract
In this chapter, we shall choose some models to apply the results in Chaps. 1–3.
Yuming Qin

### Chapter 5. Nonlinear Multi-Dimensional Continuous Integral Inequalities

Abstract
Assume that u(x,y),a(x,y),b(x,y) are non-negative continuous functions defined for all $$x,y \in \mathbb{R}_{+}$$ and $$F: \mathbb{R}_{+}^{3} \rightarrow \mathbb{R}_{+}$$ be a continuous function which satisfies the condition: for all u ≥ v ≥ 0,
$$\displaystyle\begin{array}{rcl} & & 0 \leq F(x,y,u) - F(x,y,v) \leq K(x,y,v)(u - v), {}\\ \end{array}$$
where K(x,y,v) is a non-negative continuous function defined for all $$x,y,v \in \mathbb{R}_{+}$$ .
Yuming Qin

### Chapter 6. Nonlinear Multi-Dimensional Discrete (Difference) Inequalities

Abstract
In this chapter, we collect some nonlinear discrete integral and difference inequalities.
Yuming Qin

### Chapter 7. Nonlinear Multi-Dimensional Discontinuous Inequalities

Abstract
In this chapter, we introduce some nonlinear discontinuous inequalities for multiple variables.
Yuming Qin

### Chapter 8. Applications of Nonlinear Multi-Dimensional Continuous, Discontinuous Integral Inequalities and Discrete Inequalities

Abstract
In this section, we shall give some applications of Theorems 5.1.19–5.1.21 to obtain the bounds on the solutions of some partial differential and integral equations.
Yuming Qin

### Backmatter

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