Abstract
The existence of at least one positive solution to a large class of both integer- and fractional-order nonlocal differential equations, of which one model case is
-
A
(
(
b
*
u
q
)
(
1
)
)
u
′′
(
t
)
=
λ
f
(
t
,
u
(
t
)
)
,
t
∈
(
0
,
1
)
,
q
≥
1
,
-A((b*u^{q})(1))u^{\prime\prime}(t)=\lambda f(t,u(t)),\quad t\in(0,1),\,q\geq 1,
is considered. Due to the coefficient
A
(
(
b
*
u
q
)
(
1
)
)
{A((b*u^{q})(1))}
appearing in the differential equation, the equation has
a coefficient containing a convolution term. By choosing the kernel b in various ways, specific nonlocal coefficients can be recovered such as nonlocal coefficients equivalent to a fractional integral of Riemann–Liouville type.
The results rely on the use of a nonstandard order cone together with topological fixed point theory. Applications to fractional differential equations are given, including a problem related to the
(
n
-
1
,
1
)
{(n-1,1)}
-conjugate problem.
A novel stability analysis for the Darboux problem of partial differential equations via fixed point theory
