Abstract
In this article, sufficient conditions are obtained so that every solution of the neutral difference equation
Δ
m
(
y
n
−
p
n
L
(
y
n
−
s
)
)
+
q
n
G
(
y
n
−
k
)
=
0
,
$$\begin{equation*}\Delta^{m}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})=0,
\end{equation*}$$
or every unbounded solution of
Δ
m
(
y
n
−
p
n
L
(
y
n
−
s
)
)
+
q
n
G
(
y
n
−
k
)
−
u
n
H
(
y
α
(
n
)
)
=
0
,
n
≥
n
0
,
$$\begin{equation*}\Delta^{m}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})-u_nH(y_{\alpha(n)})=0,\quad n\geq n_0,
\end{equation*}$$
oscillates, where m=2 is any integer, Δ is the forward difference operator given by Δy
n
= y
n+1 − y
n
; Δ
m
y
n
= Δ(Δ
m−1
y
n
) and other parameters have their usual meaning. The non linear function L ∈ C (ℝ, ℝ) inside the operator Δ
m
includes the case L(x) = x. Different types of super linear and sub linear conditions are imposed on G to prevent the solution approaching zero or ±∞. Further, all the three possible cases, p
n
≥ 0, p
n
≤ 0 and p
n
changing sign, are considered. The results of this paper generalize and extend some known results.
Existence of Positive Solutions to Singular Boundary Value Problems Involving φ-Laplacian
