Abstract
Our main concern in this article is to investigate the existence of solution for the boundary-value problem
$$\begin{aligned}& (\phi \bigl(x'(t)\bigr)'=g_{1} \bigl(t,x(t),x'(t)\bigr),\quad \forall t\in [0,1], \\& \Upsilon _{1}\bigl(x(0),x(1),x'(0)\bigr)=0, \\& \Upsilon _{2}\bigl(x(0),x(1),x'(1)\bigr)=0, \end{aligned}$$
(
ϕ
(
x
′
(
t
)
)
′
=
g
1
(
t
,
x
(
t
)
,
x
′
(
t
)
)
,
∀
t
∈
[
0
,
1
]
,
ϒ
1
(
x
(
0
)
,
x
(
1
)
,
x
′
(
0
)
)
=
0
,
ϒ
2
(
x
(
0
)
,
x
(
1
)
,
x
′
(
1
)
)
=
0
,
where $g_{1}:[0,1]\times \mathbb{R}^{2}\rightarrow \mathbb{R}$
g
1
:
[
0
,
1
]
×
R
2
→
R
is an $L^{1}$
L
1
-Carathéodory function, $\Upsilon _{i}:\mathbb{R}^{3}\rightarrow \mathbb{R} $
ϒ
i
:
R
3
→
R
are continuous functions, $i=1,2$
i
=
1
,
2
, and $\phi :(-a,a)\rightarrow \mathbb{R}$
ϕ
:
(
−
a
,
a
)
→
R
is an increasing homeomorphism such that $\phi (0)=0$
ϕ
(
0
)
=
0
, for $0< a< \infty $
0
<
a
<
∞
. We obtain the solvability results by imposing some new conditions on the boundary functions. The new conditions allow us to ensure the existence of at least one solution in the sector defined by well ordered functions. These ordered functions do not require one to check the definitions of lower and upper solutions. Moreover, the monotonicity assumptions on the arguments of boundary functions are not required in our case. An application is considered to ensure the applicability of our results.
Existence of Solutions for Fractional Boundary Value Problems with a Quadratic Growth of Fractional Derivative
