Abstract:

Recently, monotone iterative methods have been successfully employed to prove existence of multiple solutions and pointwise 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 (ba) 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 nvectors. 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. 