Nontrivial convex solutions on a parameter of impulsive differential equation with Monge-Ampère operator

• Main Authors: Xuemei Zhang, Meiqiang Feng
• Format: Article
• Language: English
• Published: SpringerOpen 2017-11-01
• Series: Boundary Value Problems
• Subjects: continuity on a parameter; existence of nontrivial convex solutions; Monge-Ampère operator; impulsive differential equation; geometric properties
• Online Access: http://link.springer.com/article/10.1186/s13661-017-0904-8

Abstract The authors consider the impulsive differential equation with Monge-Ampère operator in the form of { ( ( u ′ ( t ) ) n ) ′ = λ n t n − 1 f ( − u ( t ) ) , t ∈ ( 0 , 1 ) , t ≠ t k , k = 1 , 2 , … , m , Δ ( u ′ ) n | t = t k = λ I k ( − u ( t k ) ) , k = 1 , 2 , … , m , u ′ ( 0 ) = 0 , u ( 1 ) = 0 , $$\textstyle\begin{cases} ( (u'(t) )^{n} )'=\lambda nt^{n-1}f (-u(t) ), \quad t\in(0,1), t\neq t_{k}, k=1, 2, \ldots, m, \\ \Delta (u' )^{n}|_{t=t_{k}}=\lambda I_{k} (-u(t_{k}) ), \quad k=1, 2, \ldots , m, \\ u'(0)=0, \quad\quad u(1)=0, \end{cases} $$ where λ is a nonnegative parameter and n ≥ 1 $n\geq1$ . We show the existence, uniqueness, and continuity results. Our approach is largely based on the eigenvalue theory and the theory of α-concave operators. The nonexistence result of a nontrivial convex solution is also studied by taking advantage of the internal geometric properties related to the problem.