In this article, we study a class of evolution variational inequalities with p(x, t)-growth conditions on bounded domains. By means of the penalty method and Galerkin’s approximation, we obtain the existence of weak solutions. Moreover, the boundedness of weak solutions is also investigated by applying Moser’s iterative method.

In the present paper, we rst modify the concepts of weakly fuzzy boundedness, strongly fuzzy boundedness, fuzzy continuity, strongly fuzzy continuity and weakly fuzzy continuity. Then, we try to nd some relations by making a comparative study of the fuzzy norms of linear operators.

This paper is concerned with the existence and multiplicity of weak solutions for a p(x)-Kirchhoff problem by using variational method genus theory. We prove simplicity boundedness principal eigenvalue.

We consider a class of stochastic partial differential equations arising as a model for amorphous thin film growth. Using a spectral Galerkin method, we verify the existence of stationary mild solutions, although the specific nature of the nonlinearity prevents us from showing the uniqueness of the solutions as well as their boundedness (in time).

We provide existence and uniqueness of global (and local) mild solutions for a general class of semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures under local Lipschitz and linear growth (or local boundedness, resp.) conditions. The socalled “method of the moving frame” allows us to reduce the SPDE problems to SDE problems.

This paper focuses on the phase transitions of a 2×2 system of mixed type for viscosity-capillarity with periodic initial-boundary condition in a viscoelastic material. By the Liapunov functional method, we prove the existence, uniqueness, regularity and uniform boundedness of the solution. The results are correct even for large initial data.

The existence of weak solutions to the continuous coagulation equation with multiple fragmentation is shown for a class of unbounded coagulation and fragmentation kernels, the fragmentation kernel having possibly a singularity at the origin. This result extends previous ones where either boundedness of the coagulation kernel or no singularity at the origin for the fragmentation kernel was assumed.

We consider a periodically forced singular oscillator in which the potential has subquadratic growth at infinity and admits a singularity. Using Moser’s twist theorem of invariant curves, we show the existence of quasi-periodic solutions. This solves the Littlewood problem on the boundedness of all solutions for such a system.

In this paper we characterize the existence and uniqueness of periodic solutions of inhomogeneous abstract delay equations and establish maximal regularity results for strong solutions. The conditions are obtained in terms of R-boundedness of linear operators determined by the equations and LFourier multipliers. Periodic mild solutions are also studied and characterized.

We provide sufficient conditions for the existence and uniqueness of solutions to a stochastic differential equation which arises in the price impact model developed in [1] and [2]. These conditions are stated as smoothness and boundedness requirements on utility functions or Malliavin differentiability of payoffs and endowments.