For given
\(\varsigma , \bar{\varsigma } \in \varSigma \), let
\(m(\varsigma )\) and
\(m(\bar{\varsigma })\) be two solutions of problem (
1), then for any
\(\tau > 0\), we have
$$\begin{aligned}& \begin{aligned} &\begin{aligned} m(\varsigma ) &= \mathcal{F}\bigl(m(\varsigma ),\varsigma \bigr) \\ &=\tau \bigl[\mathcal{M}\bigl(m(\varsigma ),\varsigma \bigr)+\mathcal{A}\bigl(g \bigl(m( \varsigma ),\varsigma \bigr),\varsigma \bigr)\oplus \mathcal{F}\bigl(h \bigl(m(\varsigma ), \varsigma \bigr),\varsigma \bigr) \bigr] \\ &\quad {} \wedge \mathcal{B}\bigl(m(\varsigma ),\varsigma \bigr)+ I\bigl(m(\varsigma ), \varsigma \bigr), \end{aligned} \\ &\begin{aligned} m(\bar{\varsigma }) &= \mathcal{F}\bigl(m(\bar{\varsigma }),\bar{\varsigma } \bigr) \\ &=\tau \bigl[\mathcal{M}\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr)+ \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\bar{\varsigma }\bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\bar{\varsigma } \bigr),\bar{ \varsigma }\bigr) \bigr] \\ &\quad {} \wedge \mathcal{B}\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr)+ I\bigl(m( \bar{ \varsigma }),\bar{\varsigma }\bigr). \end{aligned} \end{aligned} \end{aligned}$$
(6)
As per conditions that
\(\mathcal{M}\),
\(\mathcal{A}\),
\(\mathcal{B}\),
\(( \mathcal{M}+\mathcal{A}\oplus \mathcal{F})\),
g,
h, and
\((\mathcal{M}+ \mathcal{A}\oplus \mathcal{F})\wedge \mathcal{B}\) are comparison mappings regarding the second slot
ς with each other, respectively, and Lemma
2.2, we obtain
$$\begin{aligned} \theta &= m(\varsigma )\oplus m(\bar{\varsigma }) \\ &= \mathcal{F}\bigl(m(\varsigma ),\varsigma \bigr)\oplus \mathcal{F}\bigl(m( \bar{ \varsigma }),\bar{\varsigma }\bigr) \\ &\leq \mathcal{F}\bigl(m(\varsigma ),\varsigma \bigr)\oplus \theta \oplus \mathcal{F}\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr) \\ &=\bigl[\mathcal{F}\bigl(m(\varsigma ),\varsigma \bigr)\oplus \mathcal{F} \bigl(m(\bar{ \varsigma }),\varsigma \bigr)\bigr]\oplus \bigl[\mathcal{F}\bigl(m( \bar{\varsigma }), \varsigma \bigr)\oplus \mathcal{F}\bigl(m(\bar{\varsigma }), \bar{\varsigma }\bigr)\bigr]. \end{aligned}$$
(7)
Further, since
\((\mathcal{M}+\mathcal{A}\oplus \mathcal{F})\) is a
B-restricted-accretive mapping with constants
\(\alpha _{1}\),
\(\alpha _{2}\),
\(\mathcal{M}\) is
\(\lambda _{M}\)-ordered compression,
\(\mathcal{A}\) is
\(\lambda _{A}\)-ordered compression,
\(\mathcal{F}\) is
\(\lambda _{F}\)-ordered compression,
\(\mathcal{B}\) is
\(\lambda _{B}\)-ordered compression,
g is
\(\lambda _{g}\)-ordered compression,
h is
\(\lambda _{h}\)-ordered compression, regarding the slot
ς, respectively, so by Theorem
4.1, we obtain
$$\begin{aligned}& \mathcal{F}\bigl(m(\varsigma ),\varsigma \bigr)\oplus \mathcal{F}\bigl(m(\bar{ \varsigma }),\varsigma \bigr) \\& \quad \leq \bigl[\tau \bigl(\mathcal{M}\bigl(m(\varsigma ),\varsigma \bigr)+\mathcal{A}\bigl(g\bigl(m(\varsigma ),\varsigma \bigr), \varsigma \bigr)\oplus \mathcal{F}\bigl(h\bigl(m(\varsigma ),\varsigma \bigr), \varsigma \bigr) \bigr) \\& \qquad {}\wedge \mathcal{B}\bigl(m(\varsigma ),\varsigma \bigr)+I\bigl(m(\varsigma ), \varsigma \bigr) \bigr]\oplus \bigl[\tau \bigl(\mathcal{M}\bigl(m(\bar{\varsigma }),\varsigma \bigr)+ \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr) \\& \qquad {} \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr) \bigr) \wedge \mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr) + I\bigl(m(\bar{ \varsigma }),\varsigma \bigr) \bigr] \\& \quad \leq \alpha _{1} \bigl[ \bigl(\tau \bigl[\mathcal{M} \bigl(m(\varsigma ),\varsigma \bigr)+\mathcal{A}\bigl(g\bigl(m(\varsigma ), \varsigma \bigr),\varsigma \bigr)\oplus \mathcal{F}\bigl(h\bigl(m(\varsigma ), \varsigma \bigr),\varsigma \bigr) \bigr]\wedge \mathcal{B}\bigl(m(\varsigma ), \varsigma \bigr) \bigr) \\& \qquad {}\times \bigl\{ \tau \bigl[\mathcal{M}\bigl(m(\bar{\varsigma }),\varsigma \bigr)+ \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr) \bigr]\wedge \mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr) \bigr\} \bigr] \\& \qquad {}+\alpha _{2}\bigl(m( \varsigma )\oplus m(\bar{\varsigma })\bigr) \\& \quad \leq \varPsi \bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr), \end{aligned}$$
(8)
where
\(\varPsi =\alpha _{1} [\tau (\lambda _{M}+(\lambda _{A}\lambda _{g} \oplus \lambda _{F}\lambda _{h})) \vee \lambda _{B} ]+\alpha _{2} < 1\) for condition (
5), and
$$\begin{aligned}& \mathcal{F}\bigl(m(\bar{\varsigma }),\varsigma \bigr)\oplus \mathcal{F}\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr) \\& \quad \leq \bigl[\tau \bigl(\mathcal{M}\bigl(m(\bar{ \varsigma }),\varsigma \bigr)+\mathcal{A}\bigl(g \bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr)\oplus \mathcal{F} \bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr), \varsigma \bigr) \bigr) \\& \qquad {}\wedge \mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr)+I\bigl(m(\bar{ \varsigma }),\varsigma \bigr) \bigr] \\& \qquad {}\oplus \bigl[\tau \bigl(\mathcal{M}\bigl(m(\bar{\varsigma }),\bar{\varsigma } \bigr)+\mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\bar{ \varsigma }\bigr)\oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\bar{ \varsigma }\bigr) \bigr) \\& \qquad {} \wedge \mathcal{B}\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr)+I\bigl(m( \bar{ \varsigma }),\bar{\varsigma }\bigr) \bigr] \\& \quad \leq \alpha _{1} \bigl[ \bigl(\tau \bigl[\mathcal{M}\bigl(m(\bar{ \varsigma }), \varsigma \bigr)+\mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }), \varsigma \bigr),\varsigma \bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr)\bigr] \\& \qquad {} \wedge \mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr) \bigr) \\& \qquad {} \oplus \bigl(\tau \bigl[\mathcal{M}\bigl(m(\bar{\varsigma }),\bar{\varsigma } \bigr)+ \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\bar{ \varsigma }\bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\bar{ \varsigma }\bigr),\bar{ \varsigma }\bigr)\bigr] \\& \qquad {} \wedge \mathcal{B}\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr) \bigr) \bigr]+ \alpha _{2}\bigl(m(\bar{\varsigma })\oplus m(\bar{\varsigma }) \bigr) \\& \quad \leq \alpha _{1} \bigl[\tau \bigl(\bigl[\mathcal{M}\bigl(m(\bar{ \varsigma }), \varsigma \bigr)+\mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }), \varsigma \bigr),\varsigma \bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr)\bigr] \\& \qquad {} \oplus \bigl[\mathcal{M}\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr)+ \mathcal{A}\bigl(g\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr),\bar{\varsigma }\bigr)\oplus \mathcal{F}\bigl(h\bigl(m(\bar{ \varsigma }),\bar{\varsigma } \bigr),\bar{\varsigma }\bigr)\bigr] \bigr) \bigr] \\& \qquad {} \vee \bigl[\mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr) \oplus \mathcal{B}\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr)\bigr] \\& \quad \leq \alpha _{1} \bigl[\tau \bigl(\bigl[\mathcal{M}\bigl(m(\bar{ \varsigma }), \varsigma \bigr)\oplus \mathcal{M}\bigl(m(\bar{\varsigma }),\bar{ \varsigma }\bigr)\bigr]+\bigl[ \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }), \varsigma \bigr),\varsigma \bigr) \\& \qquad {} \oplus \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr), \bar{ \varsigma }\bigr)\bigr]\oplus \bigl[\mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr), \varsigma \bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr),\bar{ \varsigma }\bigr)\bigr] \bigr) \bigr] \\& \qquad {} \vee \bigl[\mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr) \oplus \mathcal{B}\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr)\bigr]. \end{aligned}$$
(9)
Combining equations (
7), (
8), and (
9), and by making use of Lemma
2.2, we obtain
$$\begin{aligned}& \begin{aligned} m(\varsigma )\oplus m(\bar{\varsigma }) &\leq \bigl[\varPsi \bigl(m(\varsigma ) \oplus m(\bar{\varsigma })\bigr)\bigr]\oplus \alpha _{1} \bigl[\tau \bigl(\bigl( \mathcal{M}\bigl(m(\bar{\varsigma }),\varsigma \bigr)\oplus \mathcal{M}\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr)\bigr) \\ &\quad {}+\bigl(\mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr)\oplus \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr), \bar{\varsigma }\bigr)\bigr) \\ &\quad {}\oplus \bigl(\mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr), \varsigma \bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\bar{ \varsigma }\bigr)\bigr) \bigr) \\ &\quad {}\vee \bigl(\mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr)\oplus \mathcal{B}\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr)\bigr) \bigr], \end{aligned} \\& \begin{aligned} (1\oplus \varPsi ) \bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) &\leq \alpha _{1} \bigl[\tau \bigl(\bigl(\mathcal{M}\bigl(m(\bar{\varsigma }), \varsigma \bigr)\oplus \mathcal{M}\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr) \bigr) \\ &\quad {}+\bigl(\mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr)\oplus \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr), \bar{\varsigma }\bigr)\bigr) \\ &\quad {}\oplus \bigl(\mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr), \varsigma \bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\bar{ \varsigma }\bigr)\bigr) \bigr) \\ &\quad {}\vee \bigl(\mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr)\oplus \mathcal{B}\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr)\bigr) \bigr], \end{aligned} \\& \bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) \leq \biggl( \frac{\alpha _{1}}{1 \oplus \varPsi } \biggr) \bigl[\tau \bigl(\bigl(\mathcal{M}\bigl(m(\bar{\varsigma }), \varsigma \bigr)\oplus \mathcal{M}\bigl(m(\bar{\varsigma }),\bar{\varsigma } \bigr)\bigr) \\& \hphantom{\bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) \leq{}}+\bigl(\mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr)\oplus \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr), \bar{\varsigma }\bigr)\bigr) \\& \hphantom{\bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) \leq{}}\oplus \bigl(\mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr), \varsigma \bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\bar{ \varsigma }\bigr)\bigr) \bigr) \\& \hphantom{\bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) \leq{}}\vee \bigl(\mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr)\oplus \mathcal{B}\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr)\bigr) \bigr] \\& \hphantom{\bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr)}\leq \biggl(\frac{\alpha _{1}}{1\oplus \varPsi } \biggr) \bigl[\tau (\bigl( \mathcal{M}\bigl(m( \bar{\varsigma }),\varsigma \bigr)\oplus \mathcal{M}\bigl(m(\bar{ \varsigma }), \bar{\varsigma }\bigr)\bigr) \\& \hphantom{\bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) \leq{}}+\bigl(\mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr)\oplus \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr), \varsigma \bigr)\oplus \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\varsigma \bigr) \\& \hphantom{\bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) \leq{}}\oplus \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr), \bar{ \varsigma }\bigr)\bigr) \oplus \bigl(\mathcal{F}\bigl(h\bigl(m(\bar{ \varsigma }),\varsigma \bigr), \varsigma \bigr)\oplus \mathcal{F}\bigl(h\bigl(m( \bar{\varsigma }),\bar{\varsigma }\bigr), \varsigma \bigr) \\& \hphantom{\bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) \leq{}}\oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr), \varsigma \bigr)\oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\bar{ \varsigma }\bigr)\bigr) \big) \\& \hphantom{\bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) \leq{}}\vee \bigl(\mathcal{B}\bigl(m(\bar{ \varsigma }),\varsigma \bigr) \oplus \mathcal{B}\bigl(m(\bar{\varsigma }),\bar{ \varsigma }\bigr)\bigr) \bigr]. \end{aligned}$$
(10)
Using the continuity of the parametric mappings regarding the second slot
\(\varsigma \in \varSigma \), we have
$$\begin{aligned}& \lim_{\varsigma \rightarrow \bar{\varsigma }} \bigl\Vert g\bigl(m(\bar{\varsigma }), \varsigma \bigr)-g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr) \bigr\Vert = 0, \\& \lim_{\varsigma \rightarrow \bar{\varsigma }} \bigl\Vert h\bigl(m(\bar{\varsigma }), \varsigma \bigr)-h\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr) \bigr\Vert =0, \\& \lim_{\varsigma \rightarrow \bar{\varsigma }} \bigl\Vert \mathcal{B}\bigl(m(\bar{ \varsigma }), \varsigma \bigr)-\mathcal{B}\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr) \bigr\Vert =0, \\& \lim_{\varsigma \rightarrow \bar{\varsigma }} \bigl\Vert \mathcal{A}\bigl(g\bigl(m(\bar{ \varsigma }),\varsigma \bigr),\varsigma \bigr)-\mathcal{A}\bigl(g\bigl(m(\bar{ \varsigma }), \varsigma \bigr),\bar{\varsigma }\bigr) \bigr\Vert =0, \\& \lim_{\varsigma \rightarrow \bar{\varsigma }} \bigl\Vert \mathcal{F}(\cdot , \varsigma )- \mathcal{F}(\cdot , \bar{\varsigma }) \bigr\Vert =0, \\& \lim_{\varsigma \rightarrow \bar{\varsigma }} \bigl\Vert \mathcal{M}(\cdot , \varsigma )- \mathcal{M}(\cdot , \bar{\varsigma }) \bigr\Vert =0. \end{aligned}$$
From Proposition
3, we have
$$ \lim_{\varsigma \rightarrow \bar{\varsigma }}\bigl(m(\varsigma )\oplus m(\bar{ \varsigma }) \bigr)=\theta $$
and
$$ \lim_{\varsigma \rightarrow \bar{\varsigma }} \bigl\Vert m(\varsigma )-m(\bar{ \varsigma }) \bigr\Vert =0. $$
(11)
It ensures that the answer
\(m(\varsigma )\) of problem (
1) is continuous at
\(\varsigma =\bar{\varsigma }\). □