Multi-parameter second-order impulsive indefinite boundary value problems

作者：Jiao, Lishuai; Zhang, Xuemei^*

来源：Advances in Difference Equations, 2018, 2018(1): 158.

DOI：10.1186/s13662-018-1618-7

• Solvable intervals of three parameters
• Positive solutions
• Indefinite impulsive boundary value problem
• Fixed point technique

We consider the solvable intervals of three positive parameters lambda(i) (i = 1,2,3) in which the second-order impulsive boundary value problem @@@ {-x '' = a(t)xy + lambda(1)g(t)f(x), 0 < t < 1, t not equal t(k,) @@@ -y '' = lambda(2)b(t)x, 0 < t < 1, @@@ Delta x vertical bar(t=tk) = lambda I-3(k) (x(t(k))), k = 1, 2, . . . ,n, @@@ x(0) = 0, x'(1) = 0, @@@ y(0) = y(1) = 0 @@@ admits at least two positive solutions. The main interest is that the weight functions a(t), b(t), and g (t) change sign on [0,1], lambda(i), (i = 1,2,3) not equivalent to 1, and I-k not equal 0 (k = 1,2,...,n). We will obtain several interesting results: there exist positive constants lambda*, lambda(*),lambda(i)* (i = 1,3), lambda(i)** (i = 1,2,3) and alpha with alpha not equal 1 such that: (i) if alpha > 1, then for lambda(i), is an element of [lambda(i)*,+infinity) (i = 1,3) and lambda(2) is an element of [lambda(*),lambda*], the above boundary value problem admits at least two positive solutions; (ii) if 0 < alpha < 1, then for lambda(i), is an element of (0, lambda(i)**] (i = 1,2,3), the above boundary value problem admits at least two positive solutions.

• 出版日期2018-5-3
• 单位华北电力大学