Multi-parameter second-order impulsive indefinite boundary value problems

• Main Authors: Lishuai Jiao, Xuemei Zhang
• Format: Article
• Language: English
• Published: SpringerOpen 2018-05-01
• Series: Advances in Difference Equations
• Subjects: Solvable intervals of three parameters; Positive solutions; Indefinite impulsive boundary value problem; Fixed point technique
• Online Access: http://link.springer.com/article/10.1186/s13662-018-1618-7

Abstract We consider the solvable intervals of three positive parameters λi $\lambda _{i}$ ( i=1,2,3 $i=1,2,3$) in which the second-order impulsive boundary value problem {−x″=a(t)xy+λ1g(t)f(x),0<t<1,t≠tk,−y″=λ2b(t)x,0<t<1,Δx|t=tk=λ3Ik(x(tk)),k=1,2,…,n,x(0)=0,x′(1)=0,y(0)=y(1)=0 $$\textstyle\begin{cases} -x''=a(t)xy+\lambda_{1}g(t)f(x),& 0< t< 1, t\neq t_{k},\\ -y''=\lambda_{2}b(t)x,& 0< t< 1,\\ \Delta x|_{t=t_{k}}=\lambda_{3}I_{k}(x(t_{k})),& k=1,2,\dots,n,\\ x(0)=0,\qquad x'(1)=0,\\ y(0)=y(1)=0 \end{cases} $$ admits at least two positive solutions. The main interest is that the weight functions a(t) $a(t)$, b(t) $b(t)$, and g(t) $g(t)$ change sign on [0,1] $[0,1]$, λi $\lambda_{i}$ (i=1,2,3)≢1 $(i=1,2,3)\not\equiv1$, and Ik≠0 $I_{k}\neq0$ ( k=1,2,…,n $k=1,2,\ldots,n$). We will obtain several interesting results: there exist positive constants λ∗ $\lambda^{*}$, λ∗ $\lambda_{*}$, λi∗ $\lambda_{i}^{*}$ ( i=1,3 $i=1,3$), λi∗∗ $\lambda_{i}^{**}$ ( i=1,2,3 $i=1,2,3$) and α with α≠1 $\alpha\neq1$ such that: (i) if α>1 $\alpha>1$, then for λi∈[λi∗,+∞) $\lambda_{i}\in[\lambda _{i}^{*},+\infty)$ ( i=1,3 $i=1,3$) and λ2∈[λ∗,λ∗] $\lambda_{2}\in[\lambda_{*}, \lambda^{*} ]$, the above boundary value problem admits at least two positive solutions; (ii) if 0<α<1 $0<\alpha<1$, then for λi∈(0,λi∗∗] $\lambda_{i}\in(0,\lambda_{i}^{**}]$ ( i=1,2,3 $i=1,2,3$), the above boundary value problem admits at least two positive solutions.