Elliptic eigenvalue problems with highly discontinuous nonlinearities

For a family of elliptic eigenvalue problems with highly discontinuous nonlinearities, the existence of unbounded continua of positive solutions containing (0,0) is established by using techniques and results from set-valued analysis. Some special cases are then presented and discussed.

     

Existence of three solutions for a class of elliptic eigenvalue problems

In this paper, we consider a problem of the type −Δu = λ(f(u) + μg(u)) in Ω, u¦∂Ω = 0, where Ω Rⁿ is an open-bounded set, f, g are continuous real functions on R, and λ, μ ε R. As an application of a new approach to nonlinear eigenvalues problems, we prove that, under suitable hypotheses, if ¦μ¦ is small enough, then there is some λ > 0 such that the above problem has at least three distinct weak solutions in W₀^1,2(Ω).     

Sturm-Liouville operators with indefinite weight functions and eigenvalue depending boundary conditions

We consider a class of boundary value problems for Sturm-Liouville operators with indefinite weight functions. The spectral parameter appears nonlinearly in the boundary condition in the form of a function τ which has the property that λ?λτ(λ) is a generalized Nevanlinna function. We construct linearizations of these boundary value problems and study their spectral properties.     

Entire function solutions of a system of two recurrent step equations with exponential polynomial coefficients

A system of two recurrent step equations with exponential polynomial coefficients is considered. It is shown that, under certain natural conditions, its entire function solutions are exponential polynomials.     

On the well-posedness of periodic problems for the system of hyperbolic equations with finite time delay

A periodic problem for the system of hyperbolic equations with finite time delay is investigated. The investigated problem is reduced to an equivalent problem, consisting the family of periodic problems for a system of ordinary differential equations with finite delay and integral equations using the method of a new functions introduction. Relationship of periodic problem for the system of hyperbolic equations with finite time delay and the family of periodic problems for the system of ordinary differential equations with finite delay is established. Algorithms for finding approximate solutions of the equivalent problem are constructed, and their convergence is proved. Criteria of well-posedness of periodic problem for the system of hyperbolic equations with finite time delay are obtained.     

Solutions with exponential character to a linear functional equation and roots of its characteristic equation

We study the connection between solutions of the functional equation φ(x)=S_∫φ(x+M(s))σ(ds) which are comparable at infinity with functions of the form xeλx, and real roots of the characteristic equation S_∫eλM(s)σ(ds)=1.     

Parameter variation for the solution of two-point boundary-value problems and applications in the calculus of variations

In the parameter variation method, a scalar parameterk, k[0, 1], is introduced into the differential equations. The parameterk is inserted in such a way that, whenk=0, the solution of the boundary-value problem is known or readily calculated and, whenk=1, the problem is identical with the original problem. Thus, bydeforming the solution step-by-step throughk-space fromk=0 tok=1, the original problem may be solved. These solutions then provide good starting values for any convergent, iterative scheme such as the Newton-Raphson method.The method is applied to the solution of problems with various types of boundary-value specifications and is further extended to take account of situations arising in the solution of problems from variational calculus (e.g., total elapsed time not specified, optimum control not a simple function of the variables).     

A system of exponential functions with shift and the Kostyuchenko problem

We introduce a system of exponential functions with shift, study its basis properties in L ₂, and examine its connections with the Kostyuchenko problem.     

一类具非线性边界条件的高阶方程的双参数奇摄动问题

研究了一类含双参数的非线性高阶微分方程的奇摄动问题.运用合成展开法构造了问题的形式渐近解,并运用微分不等式理论证明了原问题解的存在性及所得形式渐近解的一致有效性.     

The blow-up curve of solutions of mixed problems for semilinear wave equations with exponential nonlinearities in one space dimension, II

In this paper, we consider mixed problems with a timelike boundary derivative (or a Dirichlet) condition for semilinear wave equations with exponential nonlinearities in a quarter plane. The case when the boundary vector field is tangent to the characteristic which leaves the domain in the future is also considered. We show that solutions either are global or blow up on a C¹ curve which is spacelike except at the point where it meets the boundary; at that point, it is tangent to the characteristic which leaves the domain in the future.     

A local probability exponential inequality for the large deviation of an empirical process indexed by an unbounded class of functions and its application

A local probability exponential inequality for the tail of large deviation of an empirical process over an unbounded class of functions is proposed and studied. A new method of truncating the original probability space and a new symmetrization method are given. Using these methods, the local probability exponential inequalities for the tails of large deviations of empirical processes with non-i.i.d. independent samples over unbounded class of functions are established. Some applications of the inequalities are discussed. As an additional result of this paper, under the conditions of Kolmogorov theorem, the strong convergence results of Kolmogorov on sums of non-i.i.d. independent random variables are extended to the cases of empirical processes indexed by unbounded classes of functions, the local probability exponential inequalities and the laws of the logarithm for the empirical processes are obtained.     

Optimal control problems of parabolic equations with an infinite number of variables and with equality constraints

Email: bahaa_gm{at}hotmail.com Received on December 6, 2005; Accepted on December 7, 2006 Optimal control problems of systems governed by parabolic equationswith an infinite number of variables and with additional equalityconstraints are considered. The extremum principle, as wellas sufficient condition of optimality, is formulated for theNeumann problem by using certain extensions of Dubovitskii–Milyutinmethod.     

Representation of measures with polynomial denseness in , , and its application to determinate moment problems

It has been proved that algebraic polynomials are dense in the space , , iff the measure is representable as with a finite non-negative Borel measure and an upper semi-continuous function such that is a dense subset of the space    as equipped with the seminorm . The similar representation ( ) with the same and ( , and is also a dense

subset of ) corresponds to all those measures (supported by ) that are uniquely determined by their moments on ( ). The proof is based on de Branges' theorem (1959) on weighted polynomial approximation. A more general question on polynomial denseness in a separable Fréchet space in the sense of Banach has also been examined.