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Under the bounded slope condition on the boundary values of a minimization problem for a functional of the gradient of u, we show that a continuous minimizer w is, in fact, Lipschitzian. An application of this result to prove the validity of the Euler Lagrange equation for w is presented.

where 0< T <∞, A(u) is a linear operator in X for each u in an open subset W of Y , G is a nonlinear Volterra operator defined from C([0,T ];X) into itself, and the nonlinear map f is defined from [0,T ]×Y ×Y into Y . By a strong solution to (1.1) on [0,T ′], 0< T ′ ≤ T , we mean an absolutely continuous function u from [0,T ′] into X such that u(t)∈W for almost every t ∈ [0,T ′] and satisfies ...

We show that a continuous map or a continuous flow on Rn with a certain recurrence relation must have a fixed point. Specifically, if there is a compact set W with the property that the forward orbit of every point in R n intersects W then there is a fixed point in W . Consequently, if the omega limit set of every point is nonempty and uniformly bounded then there is a fixed point. In this note...

For arbitrary w : R → [0, 1] the general form of the continuous linear functionals on the space C 0 w of all functions f continuous on the real line, lim |x|→+∞ w(x)f (x) = 0 , equipped with seminorm ||f || w := sup x∈R w(x)|f (x)| , is found. The weighted analog of the Weierstrass polynomial approximation theorem and a new version of M.G. Krein's theorem about partial fraction decomposition of...

Consider a balanced non triangular two-color Pólya-Eggenberger urn process, assumed to be large which means that the ratio σ of the replacement matrix eigenvalues satisfies 1/2 < σ < 1. The composition vector of both discrete time and continuous time models admits a drift which is carried by the principal direction of the replacement matrix. In the second principal direction, this random vector...

Consider the singular perturbation problem for εu(t; ε) + u(t; ε) = Au(t; ε) + ∫ t 0 K(t− s)Au(s; ε) ds+ f(t; ε) , where t ≥ 0, u(0; ε) = u0(ε), u (0; ε) = u1(ε), and w(t) = Aw(t) + ∫ t 0 K(t− s)Aw(s)ds+ f(t) , t ≥ 0 , w(0) = w0 , in a Banach space X when ε → 0. Here A is the generator of a strongly continuous cosine family and a strongly continuous semigroup, and K(t) is a bounded linear opera...