带有积分边值条件的高阶奇异脉冲微分方程正解-数学、应用数学专业论文.docxVIP

曲阜 曲阜师范大学硕士学位论文 i i Abstract Existence of positive solutions for a nonlinear singular impulsive boundary value problems with integral boundary conditions are important subjects in the theory of differential equations. In this article, we investigate a class of the existence positive solutions for a nonlinear fourth-order singular impulsive boundary value problems with inte- gral boundary conditions and a class of the existence of positive solutions for a nonlinear 2n-order singular impulsive boundary value problems with integral boundary conditions. By using the fixed point index theory, we get the results of the existence of the problems. The thesis is divided into two chapters. In Chapter 1, we study the existence positive solutions for a nonlinear fourth- order singular impulsive boundary value problems with integral boundary condi- tions: ? ?x(4)(t)) = f (t, x(t), y(t), xll(t), yll(t)), ? ??y(4)(t) = g(t, x(t), y(t), xll(t), yll(t)), ? ? ? |??xlll | ? ? ? t=tk = ?Ik (x(tk )), k = 1, 2, . . . , m, ??ylll|t=tk = ?Ik (y(tk )), k = 1, 2, . . . , m, ? ? 1 ?a1x(0) ? b1xl(0) = 0 m1(s)x(s)ds, ? ? 1 ??a2xll(0) ? b2xlll(0) = 0 m2(s)xll(s)ds, 1?c1x(1) + d1xl(1) = 0 n1(s)x(s)ds, 1 ? ? 1?c2 ? 1 ? ? (1) + d2x lll (1) = 0 n2(s)xll 1 (s)ds, ??a3y(0) ? b3yl(0) = 0 m3(s)y(s)ds, ? ? ??a4yll ? ? ? (0) ? b4y lll (0) = 1 0 1 0 m4(s)yll  (s)ds, ?c y(1) + d yll(1) = 0 n (s)y(s)ds, 3 3 3 ? ? 1 ??c4yll(1) + d4ylll(1) = 0 n4(s)yll(s)ds, where J = [0, 1], J l = J \{t1, t2, . . . , tm}, 0 t1 t2 · · · tm 1, f, g ∈ C((0, 1) × R+ × R+ × R? × R?, R), f, g may be singular at t = 0 or/and t = 1 and may take negative values, in which R+ = [0, +∞), R? = (?∞, 0], ai, bi, ci, di ≥ 0 and ρi = aici + aidi + bici 0 and mi, ni ∈ L1[0, 1], i = 1, 2, 3, 4, are ii ii nonnegative, ?xl|t=tk denotes the jump of xl(t) at t = tk , i.e., + l ? ?xl|t=tk = xl(tk ) ? x (tk ), where x(n)(t+) and x(n)(t?) represent the right-land limit and left-hand limit of k k x(n) at t = tk , r