we show that the variational inequality $vi(c,a)$ has aunique solution for a relaxed $(gamma , r)$-cocoercive,$mu$-lipschitzian mapping $a: cto h$ with $r>gamma mu^2$, where$c$ is a nonempty closed convex subset of a hilbert space $h$. fromthis result, it can be derived that, for example, the recentalgorithms given in the references of this paper, despite theirbecoming more complicated, are not...

In this paper, we investigate the existence of positive solutions for the ellipticequation $Delta^{2},u+c(x)u = lambda f(u)$ on a bounded smooth domain $Omega$ of $R^{n}$, $ngeq2$, with Navier boundary conditions. We show that there exists an extremal parameter$lambda^{ast}>0$ such that for $lambda< lambda^{ast}$, the above problem has a regular solution butfor $lambda> lambda^{ast}$, the probl...

We consider the linear program rain { c'x: Ax ~< b } and the associated exponential penalty function fr(x) = c'x + rEexp [ ( A ~ bi)/r]. For r close to 0, the unconstrained minimizer x(r) offr admits an asymptotic expansion of the form x (r) = x* + rd* + *l(r) where x* is a particular optimal solution of the linear program and the error term ~(r) has an exponentially fast decay. Using duality t...

It is known that, for given integers s ≥ 0 and j > 0, the nested recursion R(n) = R(n−s−R(n− j))+R(n−2j−s−R(n−3j)) has a closed form solution for which a combinatorial interpretation exists in terms of an infinite, labeled tree. For s = 0, we show that this solution sequence has a closed form as the sum of ceiling functions C(n) = ∑j−1 i=0 ⌈

In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Urysohn integral equation. We prove that the approximated solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the same orders, O(n−r) in L2-norm and O(n 1 2 −r) in infinity norm, and the iterated Legendre Galerkin ...

Here Ω is the unit disk centered at the origin, containing a small hole of radius ε centered at x = 0, i.e. Ωε = {x | |x| ≤ ε}. Calculate the exact solution, and from it determine an approximation to the solution in the outer region |x| ≫ O(ε). Can you re-derive this result from singular perturbation theory in the limit ε → 0? (Hint: the leading-order outer problem for Case I is different from ...

The present article is devoted to the study of the well-posedness, blow-up criterion and lower semicontinuity of the existence time for the full compressible Euler equations in R , N ≥ 1. First, let π = p(γ−1)/2γ and B 2,r ↪→ C, we show that the solution v = (π, u, S) is well-posed in B 2,r to the full compressible Euler equations, provided there exists a c > 0 such that the initial data entrop...

Solution Sketch. This is similar to something done in Lecture 7. Let R be the relation that defines an NP search problem. Define the language L that contains all the pairs (x, z) such that z is the prefix of a solution zz′ such that (x, zz′) ∈ R. Under the assumption that P = NP, there is a polynomial time algorithm A that decides L. Now, given x, we can construct a solution y such that (x, y) ...

We study the following nonlinear elliptic problem −∆u = F (u) in R where F (u) is a periodic function. Moser (1986) showed that for any minimal and nonself-intersecting solution, there exist α ∈ R and C > 0 such that (∗) |u− α · x| ≤ C. He also showed the existence of solutions with any prescribed α ∈ R. In this note, we first prove that any solution satisfying (*) with nonzero vector α must be...

Let g : R 7→ R be a locally Lipschitz continuous function and BR(0) the open N -ball (N ≥ 2) of center 0 and radius R > 0. A classical result due to Gidas, Ni and Nirenberg [4] asserts that any positive solution u of (1) −∆u+ g(u) = 0 in BR(0) which vanishes on ∂BR(0) is radial. A conjecture proposed by H. Brezis is that any large solution of (1), that is a solution which verifies (2) lim |x|→R...