几类带有积分边界条件非线性微分方程组正解存在性.doc

i ? + b1(t)g1(t, u(t), v(t), u′′(t), v′′(t)), ? (t) = a2(t)f2(t, u(t), v(t), u′′(t), v′′(t)) ? + b2(t)g2(t, u(t), v(t), u′′(t), v′′(t)), t ∈ (0, 1), 1 1 u(0) = k1(s)u(s)ds, u(1) = h1(s)u(s)ds, 0 0 1 1 m1(s)u′′(s)ds, u′′(1) = n1(s)u′′(s)ds, 0 0 1 1 k2(s)v(s)ds, v(1) = h2(s)v(s)ds, 0 0 1 1 m2(s)v′′(s)ds, v′′(1) = n2(s)v′′(s)ds, 0 0 ai, bi ∈ C((0, 1), [0, +∞)), ai(t) bi(t) fi, gi : [0, 1] × [0, +∞) × [0, +∞) × (?∞, 0] × (?∞, 0] → [0, +∞)  (1.1.1) t = 0, 1 ki, hi, mi, ni ∈ L1[0, 1] (i = 1, 2)  i [6] Sturm-Liouville  ? 2 0 t +∞,  1 1 1 1 2 t→+∞ t→+∞ t→+∞ t→+∞  +∞ 0 +∞ 0 fi L1-Carath′odory mi, ni ∈ L1[0, +∞)  1 m1(s)u1(s)ds, +∞ n1(s)u1(s)ds, 2 2 2 m2(s)u2(s)ds, +∞ n2(s)u2(s)ds,  (2.1.1) pi ∈ C(R+, R+) ∩ C1(0, +∞) , pi (0, +∞) +∞ 0 1  +∞, αi, βi, γi, δi ≥ 0 ρi = βiγi + αiδi + +∞ 1  0, i = 1, 2.  Banach  ? (t) = μg(t, u(t), v(t)), 0 t 1, 1 k1(s)u(s)ds, u′′(0) = u′′(1) = 0 1 k2(s)v(s)ds, v′′(0) = v′′(1) = 0 λ, μ 0, ki, hi ∈ L1[0, 1] J = [0, 1], J ′ = (0, 1), i = 1, 2. Banach ii  1 h1(s)u′′(s)ds, 0 1 h2(s)v′′(s)ds, 0 f, g ∈ C(J ′×P ×P, P ), Abstract With the development of modern physics and applied mathematics, it de- mands the mathematical ability of analyzing and controling the objective phe- nomena toward to the overall high and precision level , which made the results of the nonlinear analysis was accumulated, and gradually formed an important branch subject of the present analysis mathematics - Nonlinear functional analy- sis. Nonlinear functional analysis is a research discipline in analysis mathematics both to have the profound theory and to have the widespread application. It takes the nonlinear problems appearing in mathematics and the natural sciences as background to establish some general theories and methods to handle nonlinear problem. The boundary value problem of nonlinear stems from