Abstract:
In this paper we established some vector-valued inequalities of Gronwall type in ordered Banach spaces. Our results can be applied to investigate systems of real-valued Gronwall-type inequalities. We also show that the classical Gronwall-Bellman-Bihari integral inequality can be generalized from composition operators to a variety of operators, which include integral operators, maximal operators, geometric mean operators, and geometric maximal operators. 1. Introduction It is well known that the Gronwall-type inequalities play an important role in the study of qualitative properties of solutions to differential equations and integral equations. The Gronwall inequality was established in 1919 by Gronwall [1] and then it was generalized by Bellman [2]. In fact, if where and , and are nonnegative continuous functions on , then This result plays a key role in studying stability and asymptotic behavior of solutions to differential equations and integral equations. One of the important nonlinear generalizations of (1) and (2) was established by Bihari [3]. Assume that , and are nonnegative continuous functions on , and is a positive increasing continuous function on . Bihari showed that if and , then where , . By choosing , inequality (4) can be reduced to the form (2). Many results on the various generalizations of real-valued Gronwall-Bellman-Bihari type inequalities are established. See [4–12], [13, CH.XII], [14–16], and the references given in this literature. Another direction of generalizations is the development of the abstract Gronwall lemma. These results are closely related to the fixed points of operators. See [17, 18], [13, CH.XIV], [19], [20, Proposition 7.15], and the references given in this literature. Inequality (3) can be written in a general form where is a positive operator on continuous functions. If is a composition operator defined by , then (5) is reduced to (3). We show that if belongs to the class of operators which is defined in Section 5, then we have an upper estimate of which is similar to the form (4). It is worth pointing out that the class includes integral operators maximal operators geometric mean operators and geometric maximal operators We discuss these operators and the class in Section 5. We extend the Gronwall-Bellman-Bihari inequality (3) and (4) to the form (5) and the operator in (5) is generalized from a composition operator to the class of operators. The aim of this paper is to investigate vector-valued inequalities in ordered Banach spaces. Under suitable conditions we have estimates for which are similar to (4).

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.

Abstract:
We present several Gronwall-OuIang-type integral inequalities on time scales. Firstly, an OuIang inequality on time scales is discussed. Then we extend the Gronwall-type inequalities to multiple integrals. Some special cases of our results contain continuous Gronwall-type inequalities and their discrete analogues. Several examples are included to illustrate our results at the end.

Abstract:
In this article, some Gronwall-type integral inequalities with impulses on time scales are investigated. Our results extend some known dynamic inequalities on time scales, unify and extend some continuous inequalities and their corresponding discrete analogues. Some applications of the main results are given in the end of this article. AMS (MOS) Subject Classification 34D09; 34D99, 37M10, 35D05, 49K25, 90C46.

Abstract:
We find bounds for a Gronwall-Bihari type inequality for piecewise continuous functions. Unlike works in the prior literature, here we consider inequalities involving singular kernels in addition to functions with delays.

Abstract:
We present some new nonlinear integral inequalities Bellman-Bihari type with delay for discontinuous functions (integro-sum inequalities; impulse integral inequalities). Some applications of the results are included: conditions of boundedness (uniformly), stability by Lyapunov (uniformly), practical stability by Chetaev (uniformly) for the solutions of impulsive differential and integro-differential systems of ordinary differential equations.

Abstract:
We present some new nonlinear integral inequalities Bellman-Bihari type with delay for discontinuous functions (integro-sum inequalities; impulse integral inequalities). Some applications of the results are included: conditions of boundedness (uniformly), stability by Lyapunov (uniformly), practical stability by Chetaev (uniformly) for the solutions of impulsive differential and integro-differential systems of ordinary differential equations.

Abstract:
Some new Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales are established, which provide a handy tool in the research of qualitative and quantitative properties of solutions of delay dynamic equations on time scales. The established inequalities generalize some of the results in the work of Zhang and Meng 2008, Pachpatte 2002, and Ma 2010.