Applicable Analysis: An International Journal

Ma Fuming, ; Tassilo Küpper,

doi: 10.1080/00036819408840264pmid: N/A

In this paper we propose a numerical method to approximate the center manifold of an ordinary differential system. The approximation is determined as a projection of a center manifold on a finite dimensional subspace. The quality of the approximation depends only on the subspace and not on the distance to the equilibrium. An estimation of the error and numerical examples are included.

Chan, C. Y.; Kong, P. C.

doi: 10.1080/00036819408840265pmid: N/A

Let q and a be nonzero constants, and for some constant c such that . We show existence of a unique classical solution for the degenerate parabolic differential equation, , subject to the initial condition and the boundary conditionsu . Let . It is established that if M>∞, then the set of quenching points is in for q>0, and in for q>0. At each quenching point, it is shown that ut bows up. The critical length is proved to be the same as that for q=0. A comparison of the quenching time with that for q=0 is also given.

Kesavan, S.; Filomena, Pacella

doi: 10.1080/00036819408840266pmid: N/A

In this paper, it is proved that positive solutions of non linear equation involving the N–Laplacian in a ball in RN with Dirichlet boundary condition are radial and radially decreasing provided that the nonlinearity is a continuous function ƒ(t) (satisfying suitable growth conditions) which is strictly positive for t>0. The method generalizes that of Lions for the Laplacian in two dimensions. The method of the present paper can also be extended to an analogous mixed boundary value problem in a convex cone.

Chung, S. K.; Ha, S. N.

doi: 10.1080/00036819408840267pmid: N/A

Finite element Galerkin approximate solutions for a KdV–like Rosenau equation which models the dynamics of dense discrete systems are considered. Existence and uniqueness of exact solutions are shown and the error estimates of the continuous time Galerkin solutions are discussed. For the fully discrete time Galerkin solutions, we consider the backward Euler method which results the first order convergence in the temporal direction. For the second order convergence in time, we consider a three–level backward method and the Crank–Nicolson method which give optimal convergence in the spatial direction.

Kyril, Tintarev

doi: 10.1080/00036819408840268pmid: N/A

The paper establishes a relation between constrained minimax values of a functional g on a Hilbert space and related eigenvalues. If g is Frechet differentiable and weakly continuous, in presence of a saddle point geometry the minimax value σ(t) of g over corresponds to a set of eigenfunctions satisfying Moreover,γ has left and right hand derivatives and each of the is an eigenvalue. This abstract statement is written in mind of applications to existence of multiple nodal solutions for quasilinear elliptic equations.

Gatica, Gabriel N.; Hsiao, George C.

doi: 10.1080/00036819408840269pmid: N/A

In this paper we introduce a generalization of the usual Gårding inequality for bounded bilinear forms defined on Hilbert spaces. In addition, we give sufficient conditions for a variational problem with constraints to satisfy this generalized inequality, and show that this result induces a new procedure for the numerical treatment of such problems. More precisely, we prove that if a constrained variational problem is uniquely solvable and if its associated biliear form satisfies the generalized Gårding inequality then an asymptotically convergent nonconforming Galerkin scheme for approximating the continuous solution can be derived.

Panagiotopoulos, P. D.; Bruning, E. A. K.

doi: 10.1080/00036819408840270pmid: N/A

The problemm of minimizing a real valued function ƒ on a subset M of a reflexive Banach space E is reconsidered. Under fairly weak restrictions on the pair (ƒM) we present a necessary and sufficient condition for the existence of a minimum. This necessary and sufficient condition is a restricted monotonicity condition for the Gâteaux–derivative ƒ' of ƒ, called weak K–monotonicity of ƒ'. The set depends in a subtle way on the pair (ƒM). We give some elementary properties and examples of weakly K–monotone maps and discuss their relation with previously used notions of monotonicity. The concept of weak K–monotonicity allows to show under which circumstances the standard assumption of weak sequential lower semicontinuity of ƒ occurs as natural restriction and when various weaker assumptions suffice to ensure the existence of a minimizer.

Gilbert, R. P.; Andrea, Cianchi

doi: 10.1080/00036819408840271pmid: N/A

Let Uƒ be the solution, which vanishes at infinity, of the Poisson equation on Rn. Assume that the decreasing rearrangement of ƒ is assigned. contrary to the case , we show that, at least for p>3 does not necessarily reach its maximum when ƒ is spherically symmetric.

Butzer, P.; Zayed, Ahmed I.

doi: 10.1080/00036819408840272pmid: N/A

The aim of this paper is to show how sampling theory can play an important rol in summing up infinite series in several variables. This will be demonstrated by deriving severa summation formulas for doubly infinite series that are believed to be new. One of the interesting features of this works i that although the formulas appear to be cojmplicated, their proofs are rather eqasy and straightforward when sampling theorems are employed. The summation formulas are derived by using theorems on both uniform and non-unifrom sampling.