Difference equations of finite order arise very often in various problems in mathematics and applied sciences, for example in mathematical physics and biology.
These processes are encountered, for example, in mathematical models in population dynamics as well as in models of propagation of perturbation in matter with memory.
For example, in mathematical morphology, lattice appeals to integral geometry, stereolgy, and random set models; it is mainly its algebraic facet which has become popular.
A special case of this problem (k= 2 and j= 1) was recently studied by Hirvensalo and Karhumaki [M.Hirvensalo, J.Karhumaki, Computing partial information out of uncomputable one The first digit of 2n to base 3 as an example, in: Mathematical Foundations of Computer Science, in: Lecture Notes in Computer Science, Springer-Verlag, Inc., 2002] for which they presented a polynomial time algorithm.
Examples include many famous equations in mathematical physics, such as the Korteweg-de Vries equation [4].
Lefèvre's mathematical examples in natural philosophy are all the more remarkable because he elsewhere disparaged the calculatores who followed in the tradition of Thomas Bradwardine (Lefèvre 1510: 2r; Rice 1972: 227).
For example, they can be found in mathematical analysis and in other theories like mathematical physics, mathematical biology, ordinary differential equations, probability theory and statistics, chemistry, economics.
Usually the payoff function of the various options (for example, European and American options) in mathematical finance is convex and this property leads to the corresponding value function to be convex with respect to the underlying stock price (see for details El Karoui et al. [5] and Hobson [6]).
SDEs (see (17, 18) for references) have long been used in mathematical finance, for example to model the uncertainty in an asset (19).
Variational inequalities arise in optimal stochastic control [29] as well as in other problems in mathematical physics, for examples, deformation of elastic bodies stretched over solid obstacles, elastoplastic torsion, and so forth, [30].
