On multiple fixed-sign solutions of a discrete system with Hermite boundary conditions

We consider the discrete system Δmui(k) = Pi k,u1(k),u2(k), . . . , un(k) , k∈ {0, 1, . . . , N}, 1 i n, together with Hermite boundary conditions, Δj ui(kν ) = 0, j= 0, . . . , mν − 1, ν = 1, . . . , r, 1 i n, where r 2, mν 1 for ν = 1, . . . , r, r ν=1mν = m, and kν’s are integers such that 0 = k1 < k1 + m1 < k2 < k2 + m2 < ··· < kr kr + mr − 1 = N + m. By using the Leggett–Williams fixed point theorem and the Five functionals fixed point theorem, we establish two different sets of criteria for the existence of three solutions of the system which are of ‘fixed-signs.’ Examples are also included to illustrate the results obtained.  2004 Elsevier Inc. All rights reserved.