In this paper we present some new generalized retarded nonlinear integral inequalities of Gronwall-Bellman type. Using integral and differential skills, some new results which provide explicit bounds on unknown functions in integral inequalities are investigated. Some applications are also presented to illustrate the usefulness of some of our results.

Some new nonlinear discrete inequalities of Gronwall type for retarded functions are established. These inequalities can be used as basic tools in the study of certain classes of functional difference equations as well as discrete delay equations. AMS Subject Classifications: 26D10, 26D20, 39A10, 39A70.

Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions...

Abswuct-In this note a new robust linear controller design in the time domain is introduced for multivariable systems with linear or nonlinear time-varying model uncertainties. The Gronwall lemma is employed to investigate the robust stability conditions which are based on the upper norm-bounds of the uncertainties. The parameters of a dynamic controller are selected to satisfy the requirements...

We prove Gronwall-type estimates for the distance of integral curves of smooth vector fields on a Riemannian manifold. Such estimates are of central importance for all methods of solving ODEs in a verified way, i.e., with full control of roundoff errors. Our results may therefore be seen as a prerequisite for the generalization of such methods to the setting of Riemannian manifolds.

In this paper we will present an abstract point of view on iterative approximation schemes of fixed points for multivalued operators. More precisely, we suppose that the algorithms are convergent and we will study the impact of this hypothesis in the theory of operatorial inclusiosns: data dependence, stability and Gronwall type lemmas. Some open problems are also presented.