Journal of Applied Mathematics and Computing
https://doi.org/10.1007/s12190-019-01285-8
ORIGINAL RESEARCH
Successive iteration technique for singular nonlinear
system with four-point boundary conditions
Amit K. Barnwal
1
· Priti Pathak
1
Received: 18 June 2019
© Korean Society for Informatics and Computational Applied Mathematics 2019
Abstract
In this paper, the existence of at least one positive solution of the system of singular
differential equations with four-point coupled boundary conditions is discussed. A
constructive monotonic iterative technique on the equivalent completely continuous
nonlinear operator is used to establish the result. This method produces an approximate
solution in the form of series which is very helpful in developing a numerical scheme
for the positive solution of the system. It is demonstrated through the examples.
Keywords Positive solution · Coupled singular boundary value problems ·
Monotonic iterative technique · Coupled boundary conditions
Mathematics Subject Classification 34B15 · 34B16 · 34B18
1 Introduction
We aim in this work to discuss the existence of positive solution (PS) for the following
singular nonlinear system of differential equations with four-point coupled boundary
conditions (BCs)
⎧
⎨
⎩
− ( py
′
)
′
= qf
1
(t , y , z ), t ∈ (0, 1),
− ( pz
′
)
′
= qf
2
(t , y , z ), t ∈ (0, 1),
y (0) = z (0) = 0, y (1) = μ
1
z (ν
1
), z (1) = μ
2
y (ν
2
).
(1)
Here parameters μ
1
,ν
1
,μ
2
and ν
2
satisfy 0 <μ
1
μ
2
h (ν
1
)h (ν
2
)< h (1)
2
, h (t ) =
t
0
1
p(x )
dx and ν
1
,ν
2
∈ (0, 1). Boundary value problems (BVPs) (1) is singular in
B Amit K. Barnwal
amit.bhu1@gmail.com
Priti Pathak
pritiipathak@gmail.com
1
Department of Mathematics and Scientific Computing, MMM University of Technology, Gorakhpur
273010, India
123