An existence theorem for the solution to the equation is given by means of variational method where b(x)→∞, as |x|→∞ and f(x, s) has linear growth in s at infinity and sublinear growth in s at zero. For a special case, some multiplicity result is proved.
The boundedness on weighted local Hardy spaces hw1,p of the oscillatory singular integral is considered when Q(x, y)=P(x-y) for some real-valued polynomial P with its degree not less than two. Also a sufficient and necessary condition on polynomial Q on Rn×Rn such that T maps hw1,p to the weighted integrable function space Lw1 is found.
For finite rank operators in a commutative subspace lattice algebra algL we introduce the concept of correlation matrices, basing on which we prove that a finite rank operator in algL can be written as a finite sum of rank-one operators in algL, if it has only finitely many different correlation matrices. Thus we can recapture the results of J.R. Ringrose, A. Hopenwasser and R.Moore as corollaries of our theorems.
We study overdetermined systems of first order partial differential equations with singular solutions. The main result gives a characterization of such systems and asserts that the singular solution is equal to the contact singular set.
The space of continuous maps from a topological space X to topological space Y is denoted by C(X,Y) with the compact-open topology. In this paper we prove that C(X,Y) is an absolute retract if X is a locally compact separable metric space and Y a convex set in a Banach space. From the above fact we know that C(X,Y) is homomorphic to Hilbert space l2 if X is a locally compact separable metric space and Y a separable Banach space; in particular, C(Rn,Rm) is homomorphic to Hilbert space l2.
This paper is devoted to asymptotic formulae for functions related with the spectrum of the negative Laplacian in two and three dimensional bounded simply connected domains with impedance boundary conditions, where the impedances are assumed to be discontinuous functions. Moreover, asymptotic expressions for the difference of eigenvalues related to the impedance problems with different impedances are derived. Further results may be obtained.
Relative extreme values are defined by the supremum and minimum of a general jump process before its first time quitting from some state set, and relative extremum-times are defined by the first times reaching relative extreme values. The main objective of this paper is to find out the exact distributions and moments of them as the maximum of the set is up or equal to the process initial state. As especial cases, these results are applied to a general birth-death process and generalized birth-death processes.
By the using of determinantal varieties from moduli algebras of hypersurface singularites the relation of the deformation of hypersurface singularities and the deformation of their moduli algebras is studied. For a type of hypersurface singularities a weak Torelli type result is proved. This weak Torelli type result showes that for families of hypersurface singularities the moduli algebras can be used to distinguish the complex structures of singularities at least in some weak sence.
We study a system (D)x'=F(t,xt) of functional differential equations, together with a scalar equation (S)x'=-a(t)f(x)+b(t)g(x(t-h)) as well as perturbed forms. A Liapunov functional is constructed which has a derivative of a nature that has been widely discussed in the literature. On the basis of this example we prove results for (D) on asymptotic stability and equi-boundedness.
The main purpose of this paper is to investigate the existence of almost periodic solutions for the Duffing differential equation. By combining the theory of exponential dichotomies with Liapunov functions, we obtain an intersting result on the existence of almost periodic solutions.
We prove the Arnold conjecture for a product of finitely many monotone symplectic manifolds and Calabi-Yau manifolds. The key point of our proof is realized by suitably choosing perturbations of the almost complex structures and Hamiltonian functions for the product case.
In the present paper there are given a number of minimax-type inequalities for a family of functions, involving two multivalued mappings, one being strongly decomposable and the other monotone. Hence some applications to the variational-type inequality theory and to the fixed point theory.
In the present note the algebraic independence of values of several Mahler series with certain particular and different parameters at algebraic points is established by means of a typical approximation method handling the Liouville type series.
Properties of continuous solutions of a second order polynomial-like iterated functional equation are given by considering its characteristic. A useful method to discuss the general case is described indeed in this procedure.
