In this paper, we establish Z-eigenvalue inclusion theorems for general tensors, which reveal some crucial differences between Z-eigenvalues and H-eigenvalues. As an application, we obtain upper bounds for the largest Z-eigenvalue of a weakly symmetric nonnegative tensor, which are sharper than existing upper bounds. 2 ,. .. , x m−1 n ] T. (λ, x) is called an H-eigenpair if both of them are real. Definition 1.2. Let A be an m-order n-dimensional tensor. We say that (λ, x) ∈ C × (C n \{0}) is an E-eigenpair of A if Ax m−1 = λx and x T x = 1. (λ, x) is called a Z-eigenpair if both of them are real.

Absolute value equations, Smoothing Newton method, Global convergence, Convergence rate,

Please cite this article as: X. Zhang, Y. Wu, L. Caccetta, Nonlocal fractional order differential equations with changing-sign singular perturbation, Appl. Math. Modelling (2015), doi: http://dx.Abstract. In this paper, we study the existence of positive solutions for a class of nonlocal fractional order differential equations with changing-sign singular perturbation. By means of Schauder's fixed point theorem, the conditions for the existence of positive solutions are established respectively for the cases where the nonlinearity is positive, negative and semipositone.

No abstract

This paper considers optimal control of glycerol producing 1,3-propanediol (1,3-PD) via microbial fed-batch fermentation. The fed-batch process is formulated as a nonlinear switched time-delay system. In general, the time-delay in the fed-batch process cannot be exactly estimated. Our goal is to design an optimal switching control scheme to simultaneously maximize 1,3-PD productivity and 1,3-PD yield under time-delay uncertainty. Accordingly, we propose a robust multi-objective optimal switching control model, in which two objectives, i.e., 1,3-PD productivity and 1,3-PD yield, and their sensitivities with respect to uncertain time-delay are considered in the objective vector. The control variables in this problem are the feeding rate of glycerol, the switching instants and the terminal time of the process. By introducing an auxiliary dynamic system to calculate the objective sensitivities and performing a time-scaling transformation, we obtain an equivalent multi-objective optimal switching control problem in standard form. We then convert the equivalent multi-objective optimal control problem into a sequence of single-objective optimal switching control problems by using a modified normal boundary intersection method. A novel gradient-based single-objective solver combining control parameterization with constraint transcription technique is developed to solve these resulting singleobjective optimal control problems. Finally, numerical results are provided to verify the effectiveness of the proposed solution approach.