We consider the system of equations of trigonometric functions
$$ \textstyle\begin{cases} \sin(x_{1}+x_{2}+x_{3})=x_{1}, \\ \cos(x_{1}+x_{2}+x_{3})=x_{2}, \\ \sin^{2}(x_{1}+x_{2}+x_{3})=2x_{3}. \end{cases} $$
(4.4)
Let
$$\textstyle\begin{cases} T_{1}(x_{1},x_{2},x_{3})=\sin(x_{1}+x_{2}+x_{3}), \\ T_{2}(x_{1},x_{2},x_{3})=\cos(x_{1}+x_{2}+x_{3}), \\ T_{3} (x_{1},x_{2},x_{3})=\frac{1}{2}\sin^{2}(x_{1}+x_{2}+x_{3}). \end{cases} $$
Then
\(T_{1}\),
\(T_{2}\),
\(T_{3}\) are three nonlinear mappings from
\([-\pi, +\pi]^{3}\) into
\([-\pi, +\pi]\). On the other hand, we have that
$$\begin{aligned}& \bigl\vert T_{1}(x_{1},x_{2},x_{3})-T_{1}(y_{1},y_{2},y_{3}) \bigr\vert \\& \quad = \bigl\vert \sin(x_{1}+x_{2}+x_{3})- \sin(y_{1}+y_{2}+y_{3}) \bigr\vert \\& \quad \leq \bigl\vert (x_{1}+x_{2}+x_{3})-(y_{1}+y_{2}+y_{3}) \bigr\vert \\& \quad \leq \vert x_{1}-y_{1} \vert + \vert x_{2}-y_{2} \vert + \vert x_{3}-y_{3} \vert \\& \quad \leq\sqrt{ \frac{1}{3}\bigl( \vert x_{1}-y_{1} \vert ^{2}+ \vert x_{2}-y_{2} \vert ^{2}+ \vert x_{3}-y_{3} \vert ^{2} \bigr)}, \\& \bigl\vert T_{2}(x_{1},x_{2},x_{3})-T_{2}(y_{1},y_{2},y_{3}) \bigr\vert \\& \quad = \bigl\vert \cos(x_{1}+x_{2}+x_{3})- \cos(y_{1}+y_{2}+y_{3}) \bigr\vert \\& \quad \leq \bigl\vert (x_{1}+x_{2}+x_{3})-(y_{1}+y_{2}+y_{3}) \bigr\vert \\& \quad \leq \vert x_{1}-y_{1} \vert + \vert x_{2}-y_{2} \vert + \vert x_{3}-y_{3} \vert \\& \quad \leq\sqrt{ \frac{1}{3}\bigl( \vert x_{1}-y_{1} \vert ^{2}+ \vert x_{2}-y_{2} \vert ^{2}+ \vert x_{3}-y_{3} \vert ^{2} \bigr)}, \\& \bigl\vert T_{3}(x_{1},x_{2},x_{3})-T_{3}(y_{1},y_{2},y_{3}) \bigr\vert \\& \quad =\frac{1}{2} \bigl\vert \sin^{2}(x_{1}+x_{2}+x_{3})- \sin^{2}(y_{1}+y_{2}+y_{3}) \bigr\vert \\& \quad \leq\frac{1}{2} \bigl\vert \sin(x_{1}+x_{2}+x_{3})+ \sin(y_{1}+y_{2}+y_{3}) \bigr\vert \bigl\vert \sin (x_{1}+x_{2}+x_{3})-\sin(y_{1}+y_{2}+y_{3}) \bigr\vert \\& \quad = \bigl\vert \sin(x_{1}+x_{2}+x_{3})- \sin(y_{1}+y_{2}+y_{3}) \bigr\vert \\& \quad \leq \bigl\vert (x_{1}+x_{2}+x_{3})-(y_{1}+y_{2}+y_{3}) \bigr\vert \\& \quad \leq \vert x_{1}-y_{1} \vert + \vert x_{2}-y_{2} \vert + \vert x_{3}-y_{3} \vert \\& \quad \leq\sqrt{ \frac{1}{3}\bigl( \vert x_{1}-y_{1} \vert ^{2}+ \vert x_{2}-y_{2} \vert ^{2}+ \vert x_{3}-y_{3} \vert ^{2} \bigr)} \end{aligned}$$
for all
\((x_{1},x_{2},x_{3}), (y_{1},y_{2}, y_{3}) \in[-\pi, +\pi]^{3}\). Hence
\(T_{1}\),
\(T_{2}\),
\(T_{3}\) are three-variable nonexpansive mappings from
\([-\pi, +\pi]^{3}\) into
\([-\pi, +\pi]\) in the uniformly convex Banach space
\(R=(-\infty ,+\infty)\). Since
\(R^{3}\) with norm
$$\bigl\Vert (x_{1},x_{2},x_{3}) \bigr\Vert = \sqrt{ |x_{1}|^{2}+|x_{2}|^{2}+|x_{3}|^{2}} $$
is a uniformly convex Banach space, by Theorem
4.5 the system of operator equations
$$\textstyle\begin{cases} T_{1}(x_{1},x_{2},x_{3})=x_{1}, \\ T_{2}(x_{1},x_{2},x_{3})=x_{2}, \\ T_{3}(x_{1},x_{2},x_{3})=x_{3} \end{cases} $$
has a solution
\((p_{1}, p_{2}, p_{3})\), and the iterative sequences
\(\{x_{1,n}\}, \{x_{2,n}\}, \{x_{3,n}\}\subset X\) defined by
$$\begin{aligned}& x_{1,n+1}=\alpha_{n} x_{1,n}+(1-\alpha_{n})T_{1}(x_{1,n},x_{2,n}, x_{3,n}), \\& x_{2,n+1}=\alpha_{n} x_{2,n}+(1-\alpha_{n})T_{2}(x_{1,n}, x_{2,n}, x_{3,n}), \\& x_{3,n+1}=\alpha_{n} x_{3,n}+(1-\alpha_{n})T_{3}(x_{1,n}, x_{2,n},x_{3,n}) \end{aligned}$$
converge to elements
\(p_{1}\),
\(p_{2}\),
\(p_{3}\), respectively, where
\(0< a\leq \alpha_{n}\leq b<1\) for two constants
a,
b. Then the system of equations of trigonometric functions (
4.4) has a solution
\((p_{1}, p_{2}, p_{3})\), and the iterative sequences
\(\{x_{1,n}\}, \{x_{2,n}\}, \{x_{3,n}\}\subset X\) defined by
$$\begin{aligned}& x_{1,n+1}=\alpha_{n} x_{1,n}+(1-\alpha_{n}) \sin(x_{1,n}+x_{2,n}+x_{3,n}), \\& x_{2,n+1}=\alpha_{n} x_{2,n}+(1-\alpha_{n}) \cos(x_{1,n}+x_{2,n}+x_{3,n}), \\& x_{3,n+1}=\alpha_{n} x_{3,n}+(1-\alpha_{n}) \frac{1}{2}\sin ^{2}(x_{1,n}+x_{2,n}+x_{3,n}) \end{aligned}$$
converge to elements
\(p_{1}\),
\(p_{2}\),
\(p_{3}\), respectively.