Abstract
We study a system of second-order dynamic equations on time scales(p1u1∇)Δ(t)-q1(t)u1(t)+λf1(t,u1(t),u2(t))=0,t∈(t1,tn),(p2u2∇)Δ(t)-q2(t)u2(t)+λf2(t,u1(t),u2(t))=0, satisfying four kinds of different multipoint boundary value conditions,fiis continuous and semipositone. We derive an interval ofλsuch that anyλlying in this interval, the semipositone coupled boundary value problem has multiple positive solutions. The arguments are based upon fixed-point theorems in a cone.
Highlights
In this paper, we consider the following dynamic equations on time scales:(p1u1∇)Δ (t) − q1 (t) u1 (t) + λf1 (t, u1 (t), u2 (t)) = 0, t ∈ (t1, tn), λ > 0, (1)(p2u2∇)Δ (t) − q2 (t) u2 (t) + λf2 (t, u1 (t), u2 (t)) = 0, satisfying one of the boundary value conditions α1u1 (t1) − β1p1 (t1) u1∇ (t1) = 0, n−2γ1u1 + δ1p1 u1∇ = ∑biu1, i=2(2) α2u2 (t1) − β2p2 (t1) u2∇ (t1) = 0, n−2γ2u2 + δ2p1 u2∇ = ∑biu2, i=2 n−2α1u1 (t1) − β1p1 (t1) u1∇ (t1) = ∑aiu1, i=2γ1u1 + δ1p1 u1∇ = 0, (3) n−2α2u2 (t1) − β2p2 (t1) u2∇ (t1) = ∑aiu2, i=2γ2u2 + δ2p1 u2∇ = 0
We study a system of second-order dynamic equations on time scales (p1u1∇)Δ(t) − q1(t)u1(t) + λf1(t, u1(t), u2(t)) = 0, t ∈ (t1, tn), (p2u2∇)Δ(t) − q2(t)u2(t) + λf2(t, u1(t), u2(t)) = 0, satisfying four kinds of different multipoint boundary value conditions, fi is continuous and semipositone
This paper shows the existence of multiple positive solutions for the boundary value problem on time scales
Summary
We consider the following dynamic equations on time scales:. (p1u1∇)Δ (t) − q1 (t) u1 (t) + λf (t, u1 (t) , u2 (t)) = 0, t ∈ (t1, tn) , λ > 0, (1). I=2 n−2 γy (tn) + δp (tn) y∇ (tn) = ∑biy (ti) , i=2 where the functions f : [0, +∞) → [0, +∞) and h : [t1, tn] → [0, +∞) are continuous. The authors discuss conditions for the existence of at least one positive solution to the second-order Sturm-Liouville-type multiple eigenvalue problem on time scales. This paper shows the existence of multiple positive solutions for the boundary value problem on time scales. In 2010, Yuan and Liu [4] study the second-order m-point boundary value problems; Yuan and Liu shows the existence of multiple positive solutions if f is semipositone and superlinear.
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