The author develops the theory of pseudo differential equations and boundary value problems in nonsmooth domains. A model pseudo differential equation in a special cone is reduced to a certain integral equation. AMS Subject Classifications: 35S15, 45F05.

Consider a convex relaxation f̂ of a pseudo-boolean function f . We say that the relaxation is totally half-integral if f̂(x) is a polyhedral function with halfintegral extreme points x, and this property is preserved after adding an arbitrary combination of constraints of the form xi = xj , xi = 1 − xj , and xi = γ where γ ∈ {0, 1, 12} is a constant. A well-known example is the roof duality rela...

We shall consider generated pseudo-operations of the following form: x⊕ y = g(−1) (g(x) + g(y)) , x ̄ y = g(−1) (g(x)g(y)) , where g is a positive strictly monotone generating function and g(−1) is its pseudo-inverse. Using this type of pseudo-operations, the Riemann-Stieltjes type integral will be introduced and investigated.

There are proven generalizations of the Hölder's and Minkowski's inequalities for the pseudo-integral. There are considered two cases of the real semiring with pseudo-operations: one, when pseudo-operations are defined by monotone and continuous function g, the second semiring ([a, b], sup,), where is generated and the third semiring where both pseudo-operations are idempotent, i.e., È = sup an...

In this paper, we introduce two classes of localized integral operators on L2(Rd) with the Wiener class W and the Kurbatov class K of integral operators as their models. We show that those two classes of localized integral operators are pseudo-inverse closed non-unital subalgebra of B2, the Banach algebra of all bounded operators on L2(Rd) with usual operator norm.

We study the so-called signed discrete Choquet integral (also called non-monotonic discrete Choquet integral) regarded as the Lovász extension of a pseudo-Boolean function which vanishes at the origin. We present axiomatizations of this generalized Choquet integral, given in terms of certain functional equations, as well as by necessary and sufficient conditions which reveal desirable propertie...

The research is financed by by the National Natural Science Foundation of China (11071001) and the Research Fund for the Doctoral Program of Higher Education of China(20093401110001) Abstract In this paper, we study weighted pseudo-almost periodic solutions to general neutral integral equations. By applying the properties of weighted pseudo-almost periodic functions and the fixed point theorem,...

This work is to provide spectral and pseudo-spectral Jacobi-Galerkin approaches for the second kind Volterra integral equation. The Gauss-Legendre quadrature formula is used to approximate the integral operator and the inner product based on the Jacobi weight is implemented in the weak formulation in the numerical implementation. For some spectral and pseudo-spectral Jacobi-Galerkin methods, a ...