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In this paper, some necessary and sufficient conditions for the oscillation of a differential equation with fractional delay and piecewise constant argument are obtained.

In this paper, we will first study the existence and uniqueness of the solution for a one-dimensional nonlinear inverse diffusion problem via an auxiliary problem and the Schauder fixed-point theorem. Furthermore, a numerical algorithm based on the finite-difference method and the least-squares scheme for solving a nonlinear inverse problem is proposed. At the beginning of the numerical algorithm, Taylor's series expansion is employed to linearize nonlinear terms and then the finite-difference method is used to discretize the problem domain. The present approach is to rearrange the matrix forms of the differential governing equations and estimate the unknown diffusion coefficient. The least-squares method is adopted to find the solution. It is assumed that no prior information is available on the functional form of the unknown diffusion coefficient in the present study, and thus, it is classified as the function estimation in inverse calculation. Results show that an excellent estimation on the diffusion coefficient can be obtained within a couple of minutes of CPU time on a Pentium IV-2.4GHz PC.

This paper deals with a degenerate parabolic system with nonlocal nonlinear sources, subject to homogeneous Dirichlet condition. We establish the critical exponent as well as the blow-up rate and profile for the system. It is interesting to observe that the blow-up exponent is determined by the interaction between the nonlinear diffusion and the nonlinear reaction, while the blow-up rate is independent of the nonlinear diffusion. The effects from the nonlocality of the sources are substatial.

In most of existing principal components analysis (PCA) learning algorithms, the learning rates are required to approach zero as learning step increases. However, in many practical applications, due to computational round-off limitations and tracking requirements, constant learning rates must be used. This paper proposes a PCA learning algorithm with a constant learning rate. It will prove via DDT (Deterministic Discrete Time) method that this PCA learning algorithm is globally convergent. Simulations are carried out to illustrate the theory.

The existence of multiple positive solutions to superlinear repulsive singular impulsive differential equations is discussed in this paper. The proof relies on a nonlinear alternative of Leray–Schauder type and on Krasnoselskii fixed-point theorem on compression and expansion of cone.