A theorem on the existence of solutions and their continuous dependence upon initial boundary conditions is proved. The method of bicharacteristics is used to transform the mixed problem into a system of integral functional equations of the Volterra type. The existence of solutions of this system is proved by the method of successive approximations using theorems on integral inequalities. Classical solutions of integral functional equations lead to generalized solutions of the original problem. Differential equations with deviated variables and differential integral problems can be obtained from the general model by specializing given operators.
We present a new class of numerical methods for the solution of quasilinear parabolic functional differential equations. The numerical methods are difference schemes which are implicit with respect to time variable. We give a complete convergence analysis for the methods and we show by an example that the new methods are considerable better that the explicit schemes. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given operators. Results obtained in the paper can be applied to differential integral problems and equations with retarded variables.
It is known that Iterated Function Systems generated by orientation preserving homeomorphisms of the unit interval admit a unique invariant measure on (0, 1). The setup for this result is the positivity of Lyapunov exponents at both fixed points and the minimality of the induced action. With the additional requirement of continuous differentiability of maps on a fixed neighborhood of {0, 1}, we present a metric in the space of such systems, which renders it complete. Using then a classical argument (and an alternative uniqueness proof), we show that almost singular invariant measures are admitted by systems lying densely in the space. This allows us to construct a residual set of systems with unique singular stationary distribution. Dichotomy between singular and absolutely continuous unique measures, is assured by taking a subspace of systems with absolutely continuous maps; the closure of this subspace is where the residual set is found.Mathematics Subject Classification (2010). 37E05; 60G30, 37C20.
Classical solutions of initial boundary value problems are approximated by solutions of associated differential difference problems. A method of lines for an unknown function for the original problem and for its partial derivatives with respect to spatial variables is constructed. A complete convergence analysis for the method is given. A stability result is proved by using differential inequalities with nonlinear estimates of the Perron type for the given operators. A discretization in time of the method of lines considered in this paper leads to new difference schemes for the original problem. It is shown by examples that the new method is considerably better than the classical schemes. 1. Introduction. For any metric spaces X and Y we denote by C(X, Y) the class of all continuous functions defined on X and taking values in Y. We will use vectorial inequalities with the understanding that the same inequalities hold between their corresponding components.
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