Abstract: Some authors employed the method and technique of differential inequalities to obtain fairly general results concerning the existence and asymptotic behavior, as ε→n⁺, of the solutions of scalar boundary value problems εu^u=h(t,y),a<t<b,y(a,ε)=A,y(b,ε)=B. In this paper, we extend these results to vector boundary value problems, under analogous stability conditions on the solution u=xu(t) of the reduced equation 0=h(t, u) Two types of asymptotic behavior are studied, depending on whether the reduced solution u(f) has or does not have a con tinuous first derivative in (a, b) leading to the phenomena of boundary and angular layers.

Key words: Z-C-X space, separable real Banach space, random semiclosed 1-set-contractive operator, random solution, random continuous operator