## Monotone Method for Nonlinear Boundary Value Problems Arising in Transport Process

##### Abstract

Recently, monotone iterative methods have been successfully employed to prove existence of multiple solutions and point-wise bounds on solutions
of nonlinear boundary value problems for both ordinary and partial differential equations (see, [1], [3]-[6], [9]). In the transport process [10] of n different
types of particles in a finite rod of length (b-a) the equation governing the particle's density is given by the following linear system of equations
[see pdf for notation]
where Ai, Bi (i = 0,1,2) are n x n matrices and x,y,p,q are n-vectors. The components x1,...,xn of the vector x represent the n distinct type of particles moving in the forward direction along the rod while the componants
y1,...,yn of y are the ones moving in the backward direction. When the end of the rod are subjected to incident fluxes, the boundary conditions becomes
[see pdf for notation]
where the vectors xa, yb are given. Physical reasons demand that A0,B0
are diagonal matrices and all the elements in the matrices Ai, Bi (i = 0,1,2) are nonnegative functions on [a,b]. This specific boundary value problem then investigated by the method of successive approximations in [2,8] and by monotone method in [7]. Because of the importance of this problem in other physical applications, we extend in this paper the monotone technique to a general class of nonlinear boundary value problem which includes the tre”-;, problem treated in [2,7] as a special case.