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We are concerned with a kind of three-dimensional system of rational difference equations, given by Kurbanli (2011). A new expression of solution of the system is presented, and the asymptotical behavior is described. At the same time, we also consider a different system and obtain some results, which expand the study of such a kind of difference equations and the method can be applied to other systems.

Porous silicon nitride ceramics with various porosities were fabricated by liquid phase sintering of mixtures containing fibrous and equiaxed α- Si3 N4 powder with a various content ratios. The effects of the contents of the fibrous α- Si3 N4 powder (0%-100%) on the microstructure and mechanical properties of porous Si3 N4 ceramics were studied. As the increase of the fibrous α- Si3 N4 powder content, both the density of green bodies and the linear shrinkage decreased, resulting in increased porosity due to the inhibited densification by the fibrous Si3 N4 particle. XRD analysis proved the complete formation of single β- Si3 N4 phase. SEM analysis revealed that the microstructure of the low content of fibrous α- Si3 N4 porous ceramics was almost composed of fine elongated β- Si3 N4 grains with high aspect ratio while numerous coarse elongated β- Si3 N4 grains with low aspect ratio surrounding fine grains were formed as the content of the fibrous α- Si3 N4 powder increased. With the increase in content of the fibrous α- Si3 N4 powder from 0% to 100%, the porosity changed from 47.8% to 56.6%, and the flexural strength decreased from 146 to 62 MPa correspondingly, indicating a flexural adjustment of the porosity and mechanical properties. [ABSTRACT FROM AUTHOR]