Positivity
https://doi.org/10.1007/s11117-018-0583-4 Positivity
Positive solutions of nonlinear multi-point boundary
value problems
Abdulkadir Dogan
1
Received: 19 April 2017 / Accepted: 7 May 2018
© Springer International Publishing AG, part of Springer Nature 2018
Abstract This paper deals with the existence of positive solutions of nonlinear dif-
ferential equation
u
′′
(t ) + a(t ) f (u (t )) = 0, 0 < t < 1,
subject to the boundary conditions
u (0) =
m−2
i =1
a
i
u (ξ
i
), u
′
(1) =
m−2
i =1
b
i
u
′
(ξ
i
),
where ξ
i
∈ (0, 1) with 0 <ξ
1
<ξ
2
< ··· <ξ
m−2
< 1, and a
i
, b
i
satisfy a
i
, b
i
∈
[0, ∞), 0 <
∑
m−2
i =1
a
i
< 1, and
∑
m−2
i =1
b
i
< 1. By using Schauder’s fixed point
theorem, we show that it has at least one positive solution if f is nonnegative and
continuous. Positive solutions of the above boundary value problem satisfy the Harnack
inequality
inf
0≤t ≤1
u (t ) ≥ γ ‖u ‖
∞
.
Keywords Differential equation · Nonlinear boundary value problems · Positive
solutions · Fixed point theorem
Mathematics Subject Classification 34B15 · 34B18
B Abdulkadir Dogan
abdulkadir.dogan@agu.edu.tr
1
Department of Applied Mathematics, Faculty of Computer Sciences, Abdullah Gul University,
38039 Kayseri, Turkey