Multiple solutions for a class of semilinear elliptic problems with Robin boundary condition

In this paper, we show the existence of at least four nontrivial solutions for a class of semilinear elliptic problems with Robin boundary condition and jumping nonlinearities. Solvability of oscillating equations with Robin boundary condition is also investigated. We prove the conclusions by using sub-super-solution method, Fu?ík spectrum theory, mountain pass theorem in order intervals and Morse theory.     

Complete controllability of nonlinear stochastic impulsive functional systems

This paper is concerned with the controllability of a kind of nonlinear stochastic impulsive functional systems. Sufficient conditions for the complete controllability of the nonlinear stochastic impulsive functional systems are obtained by using Schauder fixed-point theorem. A numerical example is given to illustrate the effectiveness of the theoretical results.     

Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems

In this paper, we prove a mountain pass theorem in order intervals in which the position of the mountain pass point is given precisely in terms of the order structure. By using this result and constructing special flows, we deal with the existence of multiple solutions and sign-changing solutions for the following classes of elliptic Dirichlet boundary value problems: (1) nonlinear terms have concave property near zero and have superlinear but subcritical growth at infinity; (2) nonlinear terms are of the formh(x)f(u), withh(x) changing sign; (3) the asymptotically linear case. We obtain several new existence results of nodal solutions and give more comparable relations among the positive, negative and sign-changing solutions obtained. Our method is set up in an abstract setting and should be useful in other problems. Dedicated to P.H. Rabinowitz on the occasion of his 60th birthday This work was carried out while the first author was visiting Utah State University. He acknowledges the support from a NSF grant of the U.S.A. and from the Chinese National Science Foundation. Supported in part by a NSF grant.     

Preparation and characterization of nanocrystalline cellulose via low-intensity ultrasonic-assisted sulfuric acid hydrolysis

Nanocrystalline cellulose (NCC) was extracted from microcrystalline cellulose via low-intensity ultrasonic-assisted sulfuric acid hydrolysis process. NCC samples were characterized by scanning electron microscopy (SEM), transmission electron microscopy (TEM), particle size distribution (PSD) analysis, Fourier-transformed infrared spectra (FT-IR), X-ray diffraction (XRD), thermogravimetric analysis (TGA) and rheological measurement. It was found that NCC yield reached 40.4 % under the optimum process of low-intensity ultrasonic-assisted sulfuric acid hydrolysis, while it was only 33.0 % in the absence of ultrasonic treatment. Furthermore, the results showed that the two NCC samples obtained from ultrasonic-assisted hydrolysis and conventional hydrolysis were very similar in morphology, both exhibiting rod-like structures with widths and lengths of 10–20 and 50–150 nm, respectively. XRD result revealed that the NCC sample from ultrasonic-assisted hydrolysis contained a small amount of cellulose II and possessed a Segal Crystallinity Index of 90.38 % and a crystallite size of 58.99 Å, higher than those of the NCC sample from conventional hydrolysis. Moreover, PSD analysis demonstrated that the former exhibited a smaller value in average particle size than the latter. In addition, rheological measurements showed that the NCC suspensions from the ultrasonic-assisted process exhibited a lower viscosity over the range of shear rate from 0.1 to 100 s^?1 in comparison with that prepared in the absence of ultrasonic treatment.     

Thermal and flame retardant properties of novel intumescent flame retardant polypropylene composites

Novel intumescent flame retardant polypropylene (PP) composites were prepared based on a char forming agent (CFA) and silica-gel microencapsulated ammonium polyphosphate (Si-MCAPP). The thermal and flame retardancy of flame retardant PP composites were investigated by limiting oxygen index, UL-94 test, cone calorimetry, thermogravimetric analysis, scanning electron micrograph, and water resistance test. The results of cone calorimetry show that the flame retardant properties of PP with 30 wt% novel intumescent flame retardants (CFA/Si-MCAPP = 1:3) improve greatly. The peak heat release rate and total heat release decrease, respectively, from 1,140.0 to 156.8 kW m^?2 and from 96.0 to 29.5 MJ m^?2. The PP composite with CFA/Si-MCAPP = 1:3 has the excellent water resistance, and it can still obtain a UL-94 V-0 rating after 168 h soaking in water.     

ASYMPTOTICALLY CONSTANT SOLUTIONS OF SECOND ORDER DELAY SYSTEMS OF DIFFERENCE EQUATIONS

Based on the fixed point theory for a Banach space, asymptotically constant solutions are founded for a second order delay system of difference equation.     

Multiple critical points of periodic functional and some applications

In this paper we will study some existence theorems of critical points for periodic functional. As applications, we consider the multiple solutions of conservative pendulum-like equation with periodic boundary conditions and forcing term of mean value zero.