We analyse a nonlinear stochastic partial differential equation that corresponds to viscous shallow water (of the Camassa--Holm type) perturbed by convective, position-dependent noise term. establish existence of weak solutions in $H^m$ ($m\in\mathbb{N}$) using Galerkin approximations and compactness method. derive series priori estimates combine model-specific energy law with non-standard regu...

<abstract><p>The momentous objective of this work is to discuss some qualitative properties solutions such as the estimate solutions, continuous dependence on initial conditions and existence uniqueness extremal a new class fractional differential equations involving two derivatives in sense Caputo derivative with respect another function $ \Phi $. Firstly, using generalized Laplace...

Given an evolution equation, a standard way to prove the well posedness of the Cauchy problem is to establish a Gronwall type estimate, bounding the distance between any two trajectories. There are important cases, however, where such estimates cannot hold, in the usual distance determined by the Euclidean norm or by a Banach space norm. In alternative, one can construct different distance func...

We derive explicit stability conditions for delay difference equations in C n (the set of n complex vectors) and estimates for the size of the solutions are derived. Applications to partial difference equations, which model diffusion and reaction processes, are given. 1. Introduction. Stability of systems of difference equations with delays has been discussed by many authors, for example, see G...

We establish some nonlinear integral inequalities for functions defined on a time scale. The results extend some previous Gronwall and Bihari type inequalities on time scales. Some examples of time scales for which our results can be applied are provided. An application to the qualitative analysis of a nonlinear dynamic equation is discussed.

Some new nonlinear dynamic integral 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 dynamic equations as well as dynamic delay equations. AMS Subject Classification: 39A10.

Some new nonlinear retarded integral inequalities of Gronwall-like type are established, which mainly generalized some results given by Cho, Dragomir and Kim (J. Korean Math. Soc. 43 (2006), No. 3, pp. 563–578) and can be used in the analysis of various problems in the theory of certain classes of differential equations and integral equations. Applications examples are also indicated.