Let $A$ be the following set of 2-dimensional functions of one variable with strictly increasing and continuous components:
$$A = \{(x_1(\cdot), x_2(\cdot)): x_i: [0,1] \rightarrow \mathbb{R}, x_i(\cdot) \text{ is strictly increasing and continuous} , \qquad 0 \leq x_2(t)-x_1(t) \leq \delta, \\
x_1(0)=0, \, x_2(0)=\delta, \, x_2(1)=a, \, x_1(c)=a-\delta \}$$
for some known $\delta >0$, $a>\delta$ and $c \in (0,1)$.
I have a system the maps 2-dimensional functions $(x_1(\cdot) ,x_2(\cdot))$ defined on $[0,1]$ into 2-dimensional functions $(y_1(\cdot), y_2 (\cdot))$:
\begin{align*}
y_1(t)&=F_1(x_1(t),x_2(t)), \\
y_2(t)&=F_2(x_1(t),x_2(t)),
\end{align*}
for $t \in [0,1]$. I can show that $(F_1,F_2)$ maps $A$ into $A$.
Is there a fixed point theorem I can use here to show that $(F_1, F_2)$ has a fixed point?
P.S. I can show that even if I take non-decreasing continuous functions $x_1$, $x_2$, their respective images $y_1$ and $y_2$ are going to be strictly increasing continuous functions. However, I cannot relax the continuity condition.