Let Ω R~n be a bounded open region with smooth boundary Ω Let φ(x, t) be a function, which may be discontinuous in t; more precisely, φ(x, t)= φ_1(x, t)-φ_2(x, t), where φ_i(x, t), i= 1,2, are Baire measurable functions on Ω × R~1, non decreasing in t for p.p. x ∈Ω, satisfying the Condition (C) of [2]. Denote by D_i the set {(x, t)|∈Ω×R~1, φ_i(x, t+0)≠φ_i(x, t-0)}, i=1,2. Suppose that D_1∩D_2=φ. We consider the second order senci-linear boundary value problem is a uniformly strongly elliptic operator with sufficiently smooth coefficients, B is theDirichlet or the Neumann boundary operator, and λ is a positive number. Suppose thatφ is optimal to (φ, L; λ)We define the set-valued mappings F_i:C(Ω)→ L~P(Ω), (p > 1),u|[φ_i(x, u(x) -0), φ_i(x, u(x) + 0)], i= 1, 2. They are upper semi-continuous and isotone. For each u, F_i(u), i = 1,2, are weakly compact convex subsets.Let K_o be the Green operator of (L, B), then K_o: L~P(Ω)C~1(Ω), (p > n), is a linear compact operator.Suppse that there are constants M_1 and M_2 such that We prove that there exists a continuous single-valued mapping K_λ: C~1(Ω)C~1(Ω) such that f∈ (I + λK_o o F_2)u is equivalent to u = K_λf.Thus, the problem (1) is equivalent to the fixed-point problem of a set-valued mapping: u ∈ K_λ o (λK_o o F_1)u.It should be noted that the values of the mapping K_λ o (λK_o o F_1) may be nonconvex subsets, since K_λ may be nonlinear. Thus the existing fixed point index theories cannot be applied. Now we treat it as the composition of a sequence of convex-set-valued mappings, and define its fixed point index to be the Leray Schauder fixed point index of the composition of the single-valued approximations of these convex-set-valued mappings.In an ordered Banach space, the fixed point index is used to extend the well known theorem of Amann about the existence of at least three solutions to the ease of sequences of isotone set-valued mappings.As an application, we make more hypotheses on φ(x, t):(1) |φ_i(x, t)| ≤M1+M_2|t|~r, r<1, i=1,2,(2) there is a continuous function d(t) ≥t 0 not identically zero, and an S ∈(0, 1) such that φ_1(x, st)-φ_2(x, t/s) ≥ d(t),(3) φ_1(x, t) = 0(t) uniformly in x as t → 0, and φ_2(x, 0) = 0. Then, there exist λ(s) and λ such that equation (1) has at least three distinct nonnegative solutions for λ>λ(s), where λ(s) ≥λ > 0, and the positive spectrum A+ = [λ, ∞). Under certain modified conditions, the results of Chang and Rabinowitz are unified.We stress that many free boundary problems can be reduced to the problem (1), and the sequence of set-valued mappings plays a role as a scheme. Our fixed point index theory seems to be a powerful tool in treating strong solutions as well as in dealing with multiplicity and bifurcation of solutions.
