1,n\rrbracket,i=1,2}$ starting and ending at $(a_{1},a_{2})$ and
$(b_{1},b_{2})$ respectively. We denote its law by
$\widetilde{\mathbb{P}}^{(n;p,q);(a_{1},a_{2}),(b_{1},b_{2})}$. The following
lemma asserts $\mathsf{Gap}_{\beta}$ event is very likely under non-
intersection.

######  Lemma C.11.

Fix $\varepsilon,\rho\in(0,1)$ and $M>0$. There exist
$\beta(\varepsilon,\rho,M)>0$, $n_{0}(\varepsilon,\rho,M)>0$, such that for
all $n\geq n_{0}$

$\displaystyle\widetilde{\mathbb{P}}^{(n;p,q);(a_{1},a_{2}),(b_{1},b_{2})}\left(\mathsf{Gap}_{\beta}\mid\mathsf{NI}\right)\geq
1-\varepsilon.$

###### Proof.

Recall that $\mathsf{Gap}_{\beta}$ event is intersection of six smaller ‘Gap’
events: $\mathsf{Gap}_{i,\beta}$ defined around (4.22). For simplicity write
$\widetilde{\mathbb{P}}$ for
$\widetilde{\mathbb{P}}^{(n;p,q);(a_{1},a_{2}),(b_{1},b_{2})}$. We now analyze
each ‘Gap’ event separately.

$\mathsf{Gap}_{1,\beta}$ and $\mathsf{Gap}_{2,\beta}$. Note that for
$k\in\llbracket 1,p\rrbracket$, $S_{1}(k)-S_{2}(k)$ is itself a random walk.
The $\mathsf{NI}$ event corresponds to the event of this random walk being
non-negative. By classical result about growth of random walks conditioned to
stay non-negative (see [Rit81, Theorem 2]) it follows that one can choose
$\beta$ small enough such that
$\widetilde{\mathbb{P}}(\mathsf{Gap}_{1,\beta}\mid\mathsf{NI})\geq
1-\frac{\varepsilon}{6}$. By the same argument one has
$\widetilde{\mathbb{P}}(\mathsf{Gap}_{2,\beta}\mid\mathsf{NI})\geq
1-\frac{\varepsilon}{6}$ for all large enough $n$ by choosing $\beta$ small
enough.

$\mathsf{Gap}_{3,\beta}$. Note that combining (C.2), (C.3), and (C.4) from
Lemma C.5 we have tightness of the endpoint of random walks conditioned on
non-intersection. Combining this with (C.1), one can choose $\gamma$ small
enough such that

$\displaystyle\widetilde{\mathbb{P}}\big{(}(S_{1}(p),S_{2}(p)),(S_{1}(n-q),S_{2}(n-q)\mid\mathsf{NI})\in\mathcal{P}_{n,\gamma}\big{)}\geq
1-\tfrac{\varepsilon}{12},$ (C.23)

where

$\displaystyle\mathcal{P}_{n,\gamma}:=\\{(z_{1},z_{2})\in\mathbb{R}^{2}:|z_{i}|\leq\gamma^{-1}\sqrt{n},z_{1}-z_{2}\geq\gamma\sqrt{n}\\}.$
(C.24)

In other words, with probability $1-\frac{\varepsilon}{12}$, the endpoints of
the middle portions of the modified random bridges are in
$\mathcal{P}_{\gamma}$ when conditioned upon non-intersection . Thus,

$\displaystyle\widetilde{\mathbb{P}}(\mathsf{Gap}_{3,\beta}\mid\mathsf{NI})\geq(1-\tfrac{\varepsilon}{12})\cdot\inf_{(a_{1},a_{2}),(b_{1},b_{2})\in\mathcal{P}_{n,\gamma}}\mathbb{P}^{n-p-q+1;(a_{1},a_{2}),(b_{1},b_{2})}(\mathsf{Gap}_{3,\beta}\mid\mathsf{NI}).$
(C.25)

Since the increments are drawn from a smooth density, for each fixed $n$, the
probability

$\mathbb{P}^{n-p-q+1;(a_{1},a_{2}),(b_{1},b_{2})}(\mathsf{Gap}_{3,\beta}\mid\mathsf{NI})$

is jointly continuous with respect to the starting and ending points of the
random bridge. As $\mathcal{P}_{n,\gamma}$ is closed, the infimum in (C.25) is
attained at some point
$(a_{1}^{*},a_{2}^{*}),(b_{1}^{*},b_{2}^{*})\in\mathcal{P}_{n,\gamma}$. Take
any subsequential limit of
$\frac{1}{\sqrt{n}}(a_{1}^{*},a_{2}^{*}),\frac{1}{\sqrt{n}}(b_{1}^{*},b_{2}^{*})$
say $(u_{1},u_{2}),(v_{1},v_{2})$. Then $|u_{i}|,|v_{i}|\leq\gamma^{-1}$ and
$u_{1}-u_{2},v_{1}-v_{2}\geq\gamma$. By invariance principle for Brownian
bridges, this conditional law under diffusive scaling converges to non-
intersecting Brownian bridges (with variance $\int x^{2}\mathsf{f}(x)dx$)
$(B_{1},B_{2})$ starting at $(u_{1},u_{2})$ ending at $(v_{1},v_{2})$. We have
$\mathbb{P}(\inf_{x\in[0,1]}(B_{1}(x)-B_{2}(x))>0)=1$. This implies along this
subsequence the limit of
$\mathbb{P}^{n-p-q+1;(a_{1}^{*},a_{2}^{*}),(b_{1}^{*},b_{2}^{*})}(\mathsf{Gap}_{3,\beta}\mid\mathsf{NI})$
is $1$. Since this holds for all subsequences, we thus see that for all large
enough $n$, r.h.s. (C.25) can be made at least $1-\tfrac{\varepsilon}{6}$.

$\mathsf{Gap}_{4,\beta}$ and $\mathsf{Gap}_{5,\beta}$. We shall first show
$\mathsf{Gap}_{4,\beta}$ happens with high probability under non-intersection.
Note that this event only depends on the first part of the modified random
bridge (which is just two independent pure random walk) independent of the
other two parts. Hence

$\displaystyle\widetilde{\mathbb{P}}(\neg\mathsf{Gap}_{4,\beta}\mid\mathsf{NI})$
(C.26)
$\displaystyle=\widetilde{\mathbb{P}}\bigg{(}\bigcap_{k=2}^{p}\\{S_{1}(k)-S_{1}(k-1)\geq\beta^{-1}k^{1/8}\\}\mid\bigcap_{k=2}^{p}\\{S_{1}(k)\geq
S_{2}(k)\\}\bigg{)}$ (C.27)
$\displaystyle\leq\sum_{k=2}^{p}\frac{\widetilde{\mathbb{P}}\bigg{(}\\{S_{1}(k)-S_{1}(k-1)\geq\beta^{-1}k^{1/8}\\}\cap\bigcap_{j\in\\{2\\}\cup\llbracket
k+2,p\rrbracket}\\{S_{1}(k)\geq
S_{2}(k)\\}\bigg{)}}{\widetilde{\mathbb{P}}\bigg{(}\bigcap_{k=2}^{p}\\{S_{1}(k)\geq
S_{2}(k)\\}\bigg{)}},$ (C.28)

where the above inequality follows via an union bound. Since under
$\widetilde{\mathbb{P}}$, $(S_{1}(\ell),S_{2}(\ell))_{\ell\in\llbracket
1,p\rrbracket}$ are two independent random walks starting from
$(a_{1},a_{2})$. From (C.6) we get that

$\displaystyle\widetilde{\mathbb{P}}\bigg{(}\bigcap_{k=2}^{p}\\{S_{1}(k)\geq
S_{2}(k)\\}\bigg{)}\geq\frac{\mathrm{C}^{-1}}{\sqrt{p}}\cdot\widetilde{\mathbb{P}}(S_{1}(2)\geq
S_{2}(2)).$ (C.29)
$\displaystyle\widetilde{\mathbb{P}}\bigg{(}\\{S_{1}(k)-S_{1}(k-1)\geq\beta^{-1}k^{1/8}\\}\cap\bigcap_{j\in\\{2\\}\cup\llbracket
k+2,p\rrbracket}\\{S_{1}(k)\geq S_{2}(k)\\}\bigg{)}$ (C.30)
$\displaystyle\leq\widetilde{\mathbb{E}}\bigg{[}\mathbf{1}_{S_{1}(2)\geq
S_{2}(2)}\mathbf{1}_{S_{1}(k)-S_{1}(k-1)\geq\beta^{-1}k^{\frac{1}{8}}}\\!\cdot\\!\widetilde{\mathbb{E}}\bigg{[}\prod_{j=k+2}^{p}\mathbf{1}_{S_{1}(j)\geq
S_{2}(j)}\mid\sigma\big{(}(S_{1}(\ell),S_{2}(\ell))_{\ell\in\llbracket
1,k+1\rrbracket}\big{)}\bigg{]}\bigg{]}$ (C.31)

By Lemma C.3, we have the following bound for the interior conditional
expectation above:

$\displaystyle\widetilde{\mathbb{E}}\bigg{[}\prod_{j=k+2}^{p}\mathbf{1}_{S_{1}(j)\geq
S_{2}(j)}\mid\sigma\big{(}(S_{1}(\ell),S_{2}(\ell))_{\ell\in\llbracket
1,k+1\rrbracket}\big{)}\bigg{]}\leq\frac{\mathrm{C}\\!\cdot\\!\max\\{S_{1}(k+1)-S_{2}(k+1),1\\}}{\sqrt{p-k+1}}.$
(C.32)

Under $\widetilde{\mathbb{P}}$, the increments of $S_{1}(\cdot)$ and
$S_{2}(\cdot)$ are independent and distributed as $\mathsf{f}$ which has
exponential tails by assumption. We now claim that

$\displaystyle\widetilde{\mathbb{E}}\bigg{[}\mathbf{1}_{S_{1}(2)\geq
S_{2}(2)}\mathbf{1}_{S_{1}(k)-S_{1}(k-1)\geq\beta^{-1}k^{1/8}}\\!\cdot\\!\max\\{S_{1}(k+1)-S_{2}(k+1),1\\}\bigg{]}$
(C.33) $\displaystyle\hskip 56.9055pt\leq\mathrm{C}\cdot k\cdot
e^{-\frac{1}{\mathrm{C}}\beta^{-1}k^{1/8}}\cdot\widetilde{\mathbb{P}}(S_{1}(2)\geq
S_{2}(2))$

We shall prove (C.33) later. Assuming it, combining the estimates from (C.31),
(C.32), and (C.33) we get

$\displaystyle\eqref{b31}\leq\mathrm{C}^{2}\sum_{k=2}^{p}\sqrt{\tfrac{p}{p-k+1}}\cdot
k\cdot e^{-\frac{1}{\mathrm{C}}\beta^{-1}k^{1/8}}.$

Taking $\beta$ small enough, the right-hand side can be made smaller than
$\varepsilon/6$. Thus,
$\widetilde{\mathbb{P}}(\mathsf{Gap}_{4,\beta}\mid\mathsf{NI})\geq
1-\frac{\varepsilon}{6}$ for all small enough $\beta$. An exact same argument
leads to $\widetilde{\mathbb{P}}(\mathsf{Gap}_{5,\beta}\mid\mathsf{NI})\geq
1-\frac{\varepsilon}{6}$ for all small enough $\beta$ as well.

To prove Eq. (C.33), we start by writing $X(k):=S_{1}(k)-S_{1}(k-1)$ and
$Y(k):=S_{2}(k)-S_{2}(k-1)$. For $k=2$, observe that

l.h.s. (C.33)
$\displaystyle\leq\widetilde{\mathbb{E}}\bigg{[}\mathbf{1}_{S_{1}(2)\geq
S_{2}(2)}\\!\cdot\\!\max\\{S_{1}(3)-S_{2}(3),1\\}\bigg{]}$
$\displaystyle\leq\widetilde{\mathbb{E}}\bigg{[}\mathbf{1}_{S_{1}(2)\geq
S_{2}(2)}\\!\cdot\\!\max\\{S_{1}(2)-S_{2}(2),1\\}\bigg{]}+\widetilde{\mathbb{E}}\bigg{[}\mathbf{1}_{S_{1}(2)\geq
S_{2}(2)}\\!\cdot\\!\max\\{X(3)-Y(3),1\\}\bigg{]}.$

By (C.9), the first expectation above is less than
$\mathrm{C}^{\prime}\cdot\widetilde{\mathbb{P}}(S_{1}(2)\geq S_{2}(2))$. For
the second expectation by independence we get

$\widetilde{\mathbb{E}}\bigg{[}\mathbf{1}_{S_{1}(2)\geq
S_{2}(2)}\cdot\max\\{X(3)-Y(3),1\\}\bigg{]}=\widetilde{\mathbb{P}}(S_{1}(2)\geq
S_{2}(2))\cdot\widetilde{\mathbb{E}}[\max\\{X(3)-Y(3),1\\}].$

Since $X(3)$ and $Y(3)$ have exponential tails, by adjusting the constant
$\mathrm{C}^{\prime}$ we get
$\mbox{l.h.s.~{}\eqref{b34}}\leq\mathrm{C}^{\prime}\cdot\widetilde{\mathbb{P}}(S_{1}(2)\geq
S_{2}(2))$. This proves (C.33) for $k=2$ upon adjusting $\mathrm{C}$. For
$k\geq 3$, using the fact that
$\max\\{\sum_{i}A_{i},1\\}\leq\sum_{i}\max\\{A_{i},1\\}$, we get

l.h.s. (C.33)
$\displaystyle\leq\sum_{i=3}^{k+1}\widetilde{\mathbb{E}}\bigg{[}\mathbf{1}_{S_{1}(2)\geq
S_{2}(2)}\mathbf{1}_{X(k)\geq\beta^{-1}k^{1/8}}\\!\cdot\\!\max\\{X(i)-Y(i),1\\}\bigg{]}$
(C.34) $\displaystyle\hskip
56.9055pt+\widetilde{\mathbb{E}}\bigg{[}\mathbf{1}_{S_{1}(2)\geq
S_{2}(2)}\mathbf{1}_{X(k)\geq\beta^{-1}k^{1/8}}\\!\cdot\\!\max\\{S_{1}(2)-S_{2}(2),1\\}\bigg{]}$
$\displaystyle\leq\sum_{i=3}^{k+1}\widetilde{\mathbb{P}}(S_{1}(2)\geq
S_{2}(2))\cdot\widetilde{\mathbb{E}}\bigg{[}\mathbf{1}_{X(k)\geq\beta^{-1}k^{1/8}}\\!\cdot\\!\max\\{X(i)-Y(i),1\\}\bigg{]}$
$\displaystyle\hskip
56.9055pt+\widetilde{\mathbb{E}}\bigg{[}\mathbf{1}_{S_{1}(2)\geq
S_{2}(2)}\\!\cdot\\!\max\\{S_{1}(2)-S_{2}(2),1\\}\bigg{]}\cdot\widetilde{\mathbb{P}}\big{(}X(k)\geq\beta^{-1}k^{1/8}\big{)}.$

Using (C.9) again, we have
$\widetilde{\mathbb{E}}\bigg{[}\mathbf{1}_{S_{1}(2)\geq
S_{2}(2)}\\!\cdot\\!\max\\{S_{1}(2)-S_{2}(2),1\\}\bigg{]}\leq\mathrm{C}^{\prime}\cdot\widetilde{\mathbb{P}}(S_{1}(2)\geq
S_{2}(2))$. Using exponential tail estimates for $X(\ell),Y(\ell)$ we obtain
that

$\displaystyle\widetilde{\mathbb{E}}\bigg{[}\mathbf{1}_{X(k)\geq\beta^{-1}k^{1/8}}\\!\cdot\\!\max\\{X(i)-Y(i),1\\}\bigg{]}\leq\mathrm{C}\exp\big{(}-\tfrac{1}{\mathrm{C}}\beta^{-1}k^{1/8}\big{)},$
$\displaystyle\widetilde{\mathbb{P}}\big{(}X(k)\geq\beta^{-1}k^{1/8}\big{)}\leq\mathrm{C}\exp\big{(}-\tfrac{1}{\mathrm{C}}\beta^{-1}k^{1/8}\big{)}.$

Putting this estimates back in r.h.s. (C.34) we arrive at (C.33).

$\mathsf{Gap}_{6,\beta}$. From (C.23), we get that the endpoints of the middle
part of the modified random walk are in $\mathcal{P}_{n,\gamma}$ (defined in
(C.24)) with probability $1-\frac{\varepsilon}{12}$. Whenever the endpoints
are in $\mathcal{P}_{n,\gamma}$, by Lemma C.7, the probability of non-
intersection of the middle portion of the walk is lower bounded by some
constant $\phi>0$. Under this event, we may use the KMT coupling [DW21a] on
the middle portion bridge of the modified random bridge to deduce that

$\mathbb{P}^{n-p-q+1;(c_{1},c_{2}),(d_{1},d_{2})}(|S_{1}(k)-S_{1}(k-1)|\geq\beta^{-1}\log
n)\leq\tfrac{1}{n^{2}}.$

for all small enough $\beta$ and for all
$(c_{1},c_{2}),(d_{1},d_{2})\in\mathcal{P}_{n,\gamma}$. Combining all these
estimates, by a union bound we have the desired result. ∎

We end this section with a modulus of continuity estimate for non-intersecting
random walks.

######  Lemma C.12.

Fix $M,\gamma>0$. There exists $n_{0}(M,\gamma)>0$ and $\delta(M,\gamma)>0$
such that for all $n\geq n_{0}$ and for all $0\leq a_{1}-a_{2}\leq M+2\log\log
n$ we have (recall the modulus of continuity $\omega_{\delta}$ from (5.1))

$\displaystyle\sum_{i=1}^{2}\mathbb{P}^{n;(a_{1},a_{2})}\bigg{(}\omega_{\delta}(S_{i}(\cdot),\llbracket
1,n\rrbracket)\geq\gamma\sqrt{n}\mid\mathsf{NI}_{0}\llbracket
2,n\rrbracket\bigg{)}\leq\varepsilon,$

###### Proof.

Fix $\gamma>0$. We write $\mathbb{P}$ for $\mathbb{P}^{n;(a_{1},a_{2})}$. By
Corollary C.6 one can choose $\rho$ such that

$\displaystyle\mathbb{P}\bigg{(}\sup_{i=1,2}\omega_{\delta}(S_{i}(\cdot),\llbracket
1,n\rho\rrbracket)\geq\gamma\sqrt{n}\mid\mathsf{NI}_{0}\llbracket
2,n\rrbracket\bigg{)}\leq\varepsilon.$

Thus it suffices to control the modulus of continuity away from zero: on the
interval $\llbracket n\rho/2,n\rrbracket$ (assuming $\delta<\rho/2$). Towards
this end let $I_{v}:=\\{(x_{1},x_{2}):|x_{i}|\leq
v^{-1}\sqrt{n},x_{1}-x_{2}\geq v\sqrt{n}\\}$. By Lemma C.5, one can choose $v$
small enough to get $\mathbb{P}(\mathsf{A}_{v}\mid\mathsf{NI}_{0}\llbracket
2,n\rrbracket)\geq 1-\varepsilon$ where
$\mathsf{A}_{v}:=\\{(S_{1}(n\rho/2),S_{2}(n\rho/2))\in I_{v}\\}$. Let
$\mathcal{F}:=\sigma\big{(}S_{1}(n\rho/2),S_{2}(n\rho/2)\big{)}$. Note that

$\mathbb{P}\Big{(}\omega_{\delta}(S_{i}(\cdot),\llbracket
n\rho/2,n\rrbracket)\geq\gamma\sqrt{n}\mid\mathsf{NI}_{0}\llbracket
2,n\rrbracket\Big{)}\leq\varepsilon+\frac{\mathbb{P}\big{(}\mathsf{A}_{v}\cap\\{\omega_{\delta}(S_{i}(\cdot),\llbracket
n\rho/2,n\rrbracket)\geq\gamma\sqrt{n}\\}\cap\mathsf{NI}_{0}\llbracket
2,n\rho/2\rrbracket\big{)}}{\mathbb{P}(\mathsf{NI}_{0}\llbracket
2,n\rrbracket)}.$ (C.35)

Note that $\\{\omega_{\delta}(S_{i}(\cdot),\llbracket
n\rho/2,n\rrbracket)\geq\gamma\sqrt{n}\\}$ is independent of $\mathcal{F}$. By
Lemma C.7, we have
$\mathbf{1}_{\mathsf{A}_{v}}\cdot\mathbb{E}[\mathbf{1}_{\mathsf{NI}_{0}\llbracket
n\rho/2,n\rrbracket}\mid\mathcal{F}]\geq\mathbf{1}_{\mathsf{A}_{v}}\phi$ for
some $\phi>0$. Combining these two facts we get

$\displaystyle\mathbb{P}\left(\mathsf{A}_{v}\cap\\{\omega_{\delta}(S_{i}(\cdot),\llbracket
n\rho/2,n\rrbracket)\geq\gamma\sqrt{n}\\}\cap\mathsf{NI}_{0}\llbracket
2,n\rho/2\rrbracket\right)$
$\displaystyle=\mathbb{P}\left(\mathsf{A}_{v}\cap\mathsf{NI}_{0}\llbracket
2,n\rho/2\rrbracket\right)\mathbb{P}\left(\omega_{\delta}(S_{i}(\cdot),\llbracket
n\rho/2,n\rrbracket)\geq\gamma\sqrt{n}\right)$
$\displaystyle\leq\phi^{-1}\cdot\mathbb{E}\left[\mathbf{1}_{\mathsf{A}_{v}\cap\mathsf{NI}_{0}\llbracket
2,n\rho/2\rrbracket}\mathbb{E}[\mathbf{1}_{\mathsf{NI}_{0}\llbracket
n\rho/2,n\rrbracket}\mid\mathcal{F}]\right]\cdot\mathbb{P}\left(\omega_{\delta}(S_{i}(\cdot),\llbracket
n\rho/2,n\rrbracket)\geq\gamma\sqrt{n}\right)$
$\displaystyle\leq\phi^{-1}\cdot\mathbb{P}(\mathsf{A}_{v}\cap\mathsf{NI}_{0}\llbracket
2,n\rrbracket)\cdot\mathbb{P}\left(\omega_{\delta}(S_{i}(\cdot),\llbracket
n\rho/2,n\rrbracket)\geq\gamma\sqrt{n}\right)$
$\displaystyle\leq\phi^{-1}\cdot\mathbb{P}(\mathsf{NI}_{0}\llbracket
2,n\rrbracket)\cdot\mathbb{P}\left(\omega_{\delta}(S_{i}(\cdot),\llbracket
n\rho/2,n\rrbracket)\geq\gamma\sqrt{n}\right).$ (C.36)

Invoking the modulus of continuity of random walks we can choose $\delta$
small enough such that
$\mathbb{P}\left(\omega_{\delta}(S_{i}(\cdot),\llbracket
n\rho/2,n\rrbracket)\geq\gamma\sqrt{n}\right)$ is at most $\varepsilon\phi$
for all large enough $n$. This, implies

$\eqref{b55}\leq\varepsilon\cdot\mathbb{P}(\mathsf{NI}_{0}\llbracket
2,n\rrbracket).$

Using this inequality we see that r.h.s. (C.35) is at most $2\varepsilon$.
Hence combining the near zero and away zero modulus of continuity we get the
desired result by adjusting $\gamma$ and $\varepsilon$. ∎

##  Appendix D Supporting calculations

In this section we provide a detailed verification of various tedious
calculations. We first show how to go from (2.11) to (2.12)-(2.14) under the
change of variables $u_{i,j}=\log\big{(}t_{N+\lfloor j/2\rfloor-i+1,N-\lceil
j/2\rceil-i+2}\big{)}$ for $(i,j)\in K_{N}$. This follows from the fact that
the factor $\prod t_{i,j}^{-1}$ in (2.11) is absorbed as the Jacobian of the
change of variables, as well as the following four relations:

$\displaystyle\quad\prod_{j=1}^{N}\left(\frac{\tau_{2N-2j+2}\tau_{2N-2j}}{\tau_{2N-2j+1}^{2}}\right)^{\theta_{j}}=\prod_{i=1}^{N}\left(e^{-\theta_{i}u_{i,2N-2i+2}}\prod_{j=1}^{N-i+1}e^{\theta_{N-j+1}(u_{i,2j-1}-u_{i,2j})}\prod_{j=1}^{N-i}e^{\theta_{N-j+1}(u_{i,2j+1}-u_{i,2j})}\right)$
(D.1)
$\displaystyle\quad\sum_{i>j}\frac{t_{i-1,j}}{t_{i,j}}=\sum_{i=1}^{N}\sum_{j=1}^{N-i+1}e^{u_{i,2j-1}-u_{i,2j}}+\sum_{i=1}^{N-1}\sum_{j=1}^{N-i}e^{u_{i+1,2j}-u_{i,2j+1}}$
(D.2) $\displaystyle\quad\sum_{i\geq
j>1}\frac{t_{i,j-1}}{t_{i,j}}=\sum_{i=1}^{N-1}\sum_{j=1}^{N-i}e^{u_{i,2j+1}-u_{i,2j}}+\sum_{i=1}^{N-1}\sum_{j=1}^{N-i}e^{u_{i+1,2j}-u_{i,2j-1}}$
(D.3)
$\displaystyle\quad\prod_{j=1}^{N}t_{j,j}^{(-1)^{N-j+1}\alpha}=\prod_{j=1}^{N}t_{N-j+1,N-j+1}^{(-1)^{j}\alpha}=\prod_{i=1}^{N}e^{(-1)^{i}u_{i,1}\alpha}.$
(D.4)

While (D.4) is obvious, (D.1), (D.2) and (D.3) are shown below. We continue
with the same notations as in the proof of Theorem 1.3.

Verification of (D.1). Note that from the transformation we have

$\displaystyle e^{u_{j-i+1,2N-2j+1}}=t_{i+2N-2j,i},\qquad
e^{u_{j-i+1,2N-2j+2}}=t_{i+2N-2j+1,i}.$

This yields

$\displaystyle\tau_{2N-2j}^{\theta_{j}}=\prod_{i=1}^{j}t_{i+2N-2j,i}^{\theta_{j}}=\prod_{i=1}^{j}e^{\theta_{j}u_{j-i+1,2N-2j+1}}=\prod_{i=1}^{j}e^{\theta_{j}u_{i,2N-2j+1}}.$

Similarly we have

$\displaystyle\tau_{2N-2j+2}^{\theta_{j}}=e^{-\theta_{j}u_{j,2N-2j+3}}\prod_{i=1}^{j}e^{\theta_{j}u_{i,2N-2j+3}},\quad\tau_{2N-2j+1}^{\theta_{j}}=\prod_{i=1}^{j}e^{\theta_{j}u_{i,2N-2j+2}}.$

Thus,

$\displaystyle\prod_{j=1}^{N}\left(\frac{\tau_{2N-2j+2}\tau_{2N-2j}}{\tau_{2N-2j+1}^{2}}\right)^{\theta_{j}}$
$\displaystyle=\prod_{j=1}^{N}\left(e^{-\theta_{j}u_{j,2N-2j+3}}\prod_{i=1}^{j}e^{\theta_{j}(u_{i,2N-2j+1}+u_{i,2N-2j+3}-2u_{i,2N-2j+2})}\right)$
$\displaystyle=\left(\prod_{i=1}^{N}e^{-\theta_{i}u_{i,2N-2i+3}}\right)\cdot\left(\prod_{i=1}^{N}\prod_{j=i}^{N}e^{\theta_{j}(u_{i,2N-2j+1}+u_{i,2N-2j+3}-2u_{i,2N-2j+2})}\right)$
$\displaystyle{=}\left(\prod_{i=1}^{N}e^{-\theta_{i}u_{i,2N-2i+3}}\right)\cdot\left(\prod_{i=1}^{N}\prod_{j=1}^{N-i+1}e^{\theta_{N-j+1}(u_{i,2j-1}+u_{i,2j+1}-2u_{i,2j})}\right),$

where the last equality follows by changing the dummy variable $j$ has been
changed to $N-j+1$. The last term above is clearly equal to the right-hand
side of (D.1).

Verification of (D.2). Let us write

$\displaystyle\sum_{i>j}\frac{t_{i-1,j}}{t_{i,j}}=\sum_{j=1}^{N}\sum_{i=j+1}^{2N-j+1}\frac{t_{i-1,j}}{t_{i,j}}$
$\displaystyle=\sum_{j=1}^{N}\sum_{r=1}^{2N-2j+1}\frac{t_{j+r-1,j}}{t_{j+r,j}}$
$\displaystyle=\sum_{j=1}^{N}\sum_{r=1}^{N-j}\frac{t_{j+2r-1,j}}{t_{j+2r,j}}+\sum_{j=1}^{N}\sum_{r=1}^{N-j+1}\frac{t_{j+2r-2,j}}{t_{j+2r-1,j}}.$
(D.5)

Observe that

$\displaystyle e^{u_{N-r-j+1,2r+1}}=t_{j+2r,j},\quad
e^{u_{N-r-j+2,2r}}=t_{j+2r-1,j}.$ (D.6)

Thus we have

$\displaystyle\eqref{ettb}$
$\displaystyle=\sum_{j=1}^{N}\sum_{r=1}^{N-j}e^{u_{N-r-j+2,2r}-u_{N-r-j+1,2r+1}}+\sum_{j=1}^{N}\sum_{r=1}^{N-j+1}e^{u_{N-r-j+2,2r-1}-u_{N-r-j+2,2r}}$
$\displaystyle=\sum_{j=1}^{N}\sum_{r=1}^{j-1}e^{u_{j-r+1,2r}-u_{j-r,2r+1}}+\sum_{j=1}^{N}\sum_{r=1}^{j}e^{u_{j-r+1,2r-1}-u_{j-r+1,2r}}\quad(j\mapsto
N-j+1)$
$\displaystyle=\sum_{r=1}^{N-1}\sum_{j=r+1}^{N}e^{u_{j-r+1,2r}-u_{j-r,2r+1}}+\sum_{r=1}^{N}\sum_{j=r}^{N}e^{u_{j-r+1,2r-1}-u_{j-r+1,2r}}$
$\displaystyle=\sum_{r=1}^{N-1}\sum_{i=1}^{N-r}e^{u_{i+1,2r}-u_{i,2r+1}}+\sum_{r=1}^{N}\sum_{i=1}^{N-r+1}e^{u_{i,2r-1}-u_{i,2r}},$

where $(j\mapsto N-j+1)$ means the dummy variable $j$ has been changed to
$N-j+1$ to obtain the equality in the second step. The last equality follows
by setting $j-r\mapsto i$ and $j-r\mapsto i-1$ in the first and second sum
respectively. A final interchange of sum in each of the two terms leads to the
right hand side of (D.2).

Verification of (D.3). We follow the same above strategy and write

$\displaystyle\sum_{i\geq
j>1}\frac{t_{i,j-1}}{t_{i,j}}=\sum_{j=2}^{N}\sum_{i=j}^{2N-j+1}\frac{t_{i,j-1}}{t_{i,j}}$
$\displaystyle=\sum_{j=2}^{N}\sum_{r=1}^{2N-2j+2}\frac{t_{j+r-1,j-1}}{t_{j+r-1,j}}$
$\displaystyle=\sum_{j=2}^{N}\sum_{r=1}^{N-j+1}\frac{t_{j+2r-1,j-1}}{t_{j+2r-1,j}}+\sum_{j=2}^{N}\sum_{r=1}^{N-j+1}\frac{t_{j+2r-2,j-1}}{t_{j+2r-2,j}}$
(D.7)

Due to (D.6) we have

$\displaystyle\eqref{ettc}$
$\displaystyle=\sum_{j=2}^{N}\sum_{r=1}^{N-j+1}e^{u_{N-r-j+2,2r+1}-u_{N-r-j+2,2r}}+\sum_{j=2}^{N}\sum_{r=1}^{N-j+1}e^{u_{N-r-j+3,2r}-u_{N-r-j+2,2r-1}}$
$\displaystyle=\sum_{j=1}^{N-1}\sum_{r=1}^{j}e^{u_{j-r+1,2r+1}-u_{j-r+1,2r}}+\sum_{j=1}^{N-1}\sum_{r=1}^{j}e^{u_{j-r+2,2r}-u_{j-r+1,2r-1}}\quad(j\mapsto
N-j+1)$
$\displaystyle=\sum_{r=1}^{N-1}\sum_{j=r}^{N-1}e^{u_{j-r+1,2r+1}-u_{j-r+1,2r}}+\sum_{r=1}^{N-1}\sum_{j=r}^{N-1}e^{u_{j-r+2,2r}-u_{j-r+1,2r-1}}$
$\displaystyle=\sum_{r=1}^{N-1}\sum_{i=1}^{N-r}e^{u_{i,2r+1}-u_{i,2r}}+\sum_{r=1}^{N-1}\sum_{i=1}^{N-r}e^{u_{i+1,2r}-u_{i,2r-1}}\quad(j-r\mapsto
i-1).$

A final interchange of sum in each of the two terms leads to the right hand
side of (D.3). This completes the verification of all three equalities.

Verification of (3.22). Note that $\theta_{c}$ is a function of $p$ defined as
a solution of the equation
$\Psi^{\prime}(\theta_{c})-p\Psi^{\prime}(2\theta-\theta_{c})=0$. Set
$g(p)=\theta_{c}$. Note that $g(1)=\theta$. By differentiating the equation
with respect to $p$ we get

$g^{\prime}(p)\Psi^{\prime\prime}(g(p))-\Psi^{\prime}(2\theta-g(p))+p\Psi^{\prime\prime}(2\theta-g(p))g^{\prime}(p)=0$

This implies
$g^{\prime}(1)=\Psi^{\prime}(\theta)/2\Psi^{\prime\prime}(\theta)$. Since
$p-1=O(N^{-1/3})$. By Taylor expansion around $1$ to first three terms we get

$\displaystyle(N-k)f_{\theta,p}$
$\displaystyle=-(N-k)\big{(}\Psi(g(p))+p\Psi(2\theta-g(p))\big{)}$
$\displaystyle=-(N-k)\Big{(}2\Psi(\theta)+(p-1)\Psi(\theta)+(p-1)^{2}\big{(}\Psi^{\prime\prime}(\theta)(g^{\prime}(1))^{2}-g^{\prime}(1)\Psi^{\prime}(\theta)\big{)}+O(N^{-1})\Big{)}$
$\displaystyle=-2N\Psi(\theta)+\frac{k^{2}(\Psi^{\prime}(\theta))^{2}}{\Psi^{\prime\prime}(\theta)(N-k)}+O(1),$

where in the final line we used the fact that $p-1=2k/(N-k)$ and the formula
for $g^{\prime}(1)$ derived above. Taking $k=MN^{2/3}$ we arrive at the
leading orders claimed in the first part of (3.22). The second part follows by
observing that by Taylor expansion up to first order we have

$\log\sigma_{\theta,p}=\log\sigma_{\theta,1}+O(p-1)=\log\sigma_{\theta,1}+O(N^{-1/3}).$

Thus,
$\sigma_{\theta,p}/\sigma_{\theta,1}=\sigma_{\theta,p}/(-\Psi^{\prime\prime}(\theta))^{1/3}\stackrel{{\scriptstyle
N\to\infty}}{{\to}}1,$ proving the second claim in (3.22).

## Glossary

General notation used throughout the text
---
$\mathpzc{HSLG}$ |  half-space log-gamma | Sec. 1.1
$\mathbb{Z}_{\geq k}$ |  set of all integers $\geq k$ | Sec. 1.1
$\mathbb{Z}^{\operatorname{half}}$ |  $\big{\\{}(i,j)\in(\mathbb{Z}_{\geq 1})^{2}:j\leq i\big{\\}}$ | Sec. 1.1
$W_{i,j}$ |  inverse-gamma polymer weights | Eq. (1.1)
$\Pi_{m,n}$ |  set of all directed paths from $(1,1)$ to $(m,n)$ in $\mathbb{Z}^{\operatorname{half}}$ | Sec. 1.1
$w(\pi)$ |  weight of path $\pi$ | Eq. (1.2)
$Z_{(\alpha,\vec{\theta})}(m,n)=Z(m,n)$ |  point-to-point $\mathpzc{HSLG}$ polymer partition function | Eq. (1.2)
$\mathcal{F}_{N}^{\alpha}(s)$ |  centered and scaled $\mathpzc{HSLG}$ free energy process | Eq. (1.3)
$\Psi(z)$ |  digamma function | Eq. (1.4)
$Z_{N}^{\operatorname{line}}(k)$ |  point-to-line $\mathpzc{HSLG}$ polymer partition function | Eq. (1.8)
$Z_{\operatorname{sym}}^{(r)}(m,n)$ |  multipath point-to-point symmetrized log-gamma polymer partition function | Eq. (2.10)
$\mathcal{L}^{N}$ |  $\mathpzc{HSLG}$ line ensemble | Def. 2.7
$\mathcal{K}_{k,T}$ and $\mathcal{K}_{k,T}^{\prime}$ |  two important domains for $\mathpzc{HSLG}$ Gibbs measures | Eq. (2.4)
$\alpha_{1}$ and $\alpha_{2}$ |  scalings for the boundary parameter | Eq. (3.11)
$\mathsf{Gap}_{\beta}$ |  gap event | Eq. (4.22)
$\omega_{\delta}^{N}(f;\llbracket 1,U\rrbracket)$ |  modulus of continuity | Eq. (5.1)
Basic probability densities and distributions
$\operatorname{Gamma}^{-1}(\beta)$ |  inverse-gamma distribution with density against Lebesgue given by $\mathbf{1}\\{x>0\\}\Gamma^{-1}(\beta)x^{-\beta-1}e^{-1/x}$ | Sec. 1.1
$W_{e}(x)$ |  weight function for edges | Eq. (1.5)
$W(a;b,c):=\exp(-e^{a-b}-e^{a-c}),\ \ a,b,c\in\mathbb{R}$ | Eq. (2.1)
$G_{\theta,(-1)^{m}}(x):=e^{\theta(-1)^{m}x-e^{(-1)^{m}y}}/\Gamma(\theta),\ \ \theta\in\mathbb{R},m\in\mathbb{Z}_{\geq 0},y\in\mathbb{R}$ | Eq. (2.2)
$\mathsf{f}_{\theta}(x)=\int_{\mathbb{R}}G_{\theta,+1}(y)G_{\theta,-1}(x-y)dy,\ \ \theta\in\mathbb{R},x\in\mathbb{R}$ | Eq. (2.3)
$\mathsf{g}_{\zeta}(x)=G_{\zeta,+1}(x)$ | Eq. (2.3)
$\xi_{\theta_{1},\theta_{2};\pm 1}^{(a,b)}(x)$ |  | Eq. (4.5)
Probability distributions on random walks and bridges
$f_{k,T}^{\vec{y},\vec{z}}(\mathbf{u})$ |  density of the $\mathpzc{HSLG}$ Gibbs measure on the domain $\mathcal{K}_{k,T}$ with boundary condition $(\vec{y},\vec{z})$ | Eq. (2.5)
$\mathbb{P}_{\alpha}^{\vec{y},\vec{z};k,T}$ |  $\mathpzc{HSLG}$ Gibbs measure on $\mathcal{K}_{k,T}$ with boundary condition $(\vec{y},\vec{z})$ (the $\alpha$ subscript is sometimes dropped when clear) | Def. 2.3
$\mathbb{P}_{\alpha}^{\vec{y};(-\infty)^{T};k,T}$ |  bottom free $\mathpzc{HSLG}$ Gibbs measure on $\mathcal{K}_{k,T}$ with boundary condition $(\vec{y}$ (the $\alpha$ subscript is sometimes dropped when clear) | Def. 2.4
$Q_{k,T}^{\vec{y}^{\prime},\vec{z}}(\mathbf{u})$ |  $\mathpzc{HSLG}$ Gibbs measure on the domain $\mathcal{K}_{k,T}^{\prime}$ with boundary condition $(\vec{y},\vec{w})$ | Eq. (2.6)
$\mathbb{P}_{\operatorname{WPRW}}^{n;(a_{1},a_{2})}$ and $\mathbb{P}_{\operatorname{PRW}}^{n;(a_{1},a_{2})}$ |  law of weighted paired random walk and paired random walk of length $n$ started from $(a_{1},a_{2})$ | Def. 1.7
$\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}$ |  law of two independent random bridges of length $n$ started from $(a_{1},a_{2})$ and ending at $(b_{1},b_{2})$ with increments drawn from $\mathsf{f}_{\theta}$ defined in (2.3) | Def. 3.6
$\widetilde{\mathbb{P}}^{(n;p,q);(a_{1},a_{2}),(b_{1},b_{2})}$ |  law of two independent $(n;p,q)$-modified random bridges of length $n$ started from $(a_{1},a_{2})$ ending at $(b_{1},b_{2})$ with increments drawn from $\mathsf{f}_{\theta}$ | Def. 4.9
$\mathbb{P}^{n;(a_{1},a_{2})}$ |  law of two independent random walks of length $n$ started from $(a_{1},a_{2})$ with increments drawn from $\mathsf{f}_{\theta}$ | Def. 3.6

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# Existence of solution to a new class of coupled variational-hemivariational
inequalities

Yunru Bai1,∗, Stanislaw Migórski2, Van Thien Nguyen3, Jianwen Peng4

1School of Science, Guangxi University of Science and Technology, Liuzhou
545006, Guangxi, China.

2College of Applied Mathematics, Chengdu University of Information Technology,
Chengdu 610225, Sichuan, P.R. China, and Jagiellonian University in Krakow,
Faculty of Mathematics and Computer Science,
ul. Lojasiewicza 6, 30348 Krakow, Poland.

3Departement of Mathematics, FPT University, Education zone, Hoa Lac High Tech
Park,
Km29 Thang Long highway, Thach That ward, Hanoi, Vietnam.

4School of Mathematics Science, Chongqing Normal University, Chongqing 401331,
P.R. China.

Abstract. The objective of this paper is to introduce and study a complicated
nonlinear system, called coupled variational-hemivariational inequalities,
which is described by a highly nonlinear coupled system of inequalities on
Banach spaces. We establish the nonemptiness and compactness of the solution
set to the system. We apply a new method of proof based on a multivalued
version of the Tychonoff fixed point principle in a Banach space combined with
the generalized monotonicity arguments, and elements of the nonsmooth
analysis. Our results improve and generalize some earlier theorems obtained
for a very particular form of the system.

Keywords. Coupled variational-hemivariational inequalities; existence; the
Tychonoff fixed point theorem; the Clarke subgradient; compactness.

††footnotetext:  ∗Corresponding author. E-mail addresses<EMAIL_ADDRESS>(Y.
Bai<EMAIL_ADDRESS>(S. Migórski<EMAIL_ADDRESS>(V. T.
Nguyen<EMAIL_ADDRESS>(J. Peng) Received February 10, 2022; Accepted
March 7, 2022.

##  1\. Introduction

In this paper we study the existence of solution to a new class of systems of
two nonlinear coupled variational-hemivariational inequalities with
constraints. Each inequality involves a nonlinear operator, the generalized
(Clarke) directional derivative of a locally Lipschitz function, a convex
potential, and a constraint set. The main feature of the system is a strong
coupling which appears in the nonlinear operators and the generalized
directional derivatives. Our results concern existence and compactness of the
solution set to the system, and generalize the results obtained very recently
in [11] by using a different method.

To introduce the problem we need the following functional framework which will
be used throughout the paper. Let $(V,\|\cdot\|_{V})$ and $(E,\|\cdot\|_{E})$
be real reflexive Banach spaces, and $C\subset V$ and $D\subset E$ be
nonempty, closed and convex sets. We are given two nonlinear operators
$A\colon E\times V\to V^{*}$ and $B\colon V\times E\to E^{*}$, two convex
functions $\psi\colon V\to\overline{\mathbb{R}}:=\mathbb{R}\cup\\{+\infty\\}$
and $\theta\colon E\to\overline{\mathbb{R}}$, two nonlinear functions $J\colon
Z_{1}\times X\to\mathbb{R}$ and $H\colon Z_{2}\times Y\to\mathbb{R}$ (which
are locally Lipschitz continuous with respect to their second variables), four
linear operators $\gamma_{1}\colon V\to X$, $\gamma_{2}\colon E\to Y$,
$\delta_{1}\colon E\to Z_{1}$ and $\delta_{2}\colon V\to Z_{2}$, and two
elements $h\in V^{*}$ and $l\in E^{*}$. The system of two coupled nonlinear
variational-hemivariational inequalities reads as follows.

######  Problem 1.1.

Find $u\in C$ and $w\in D$ satisfying the following inequalities

$\langle
A(w,u),v-u\rangle_{V}+J^{0}(\delta_{1}w,\gamma_{1}u;\gamma_{1}(v-u))+\psi(v)-\psi(u)\geq\langle
h,v-u\rangle_{V}$ (1.1)

for all $v\in C$, and

$\langle
B(u,w),z-w\rangle_{E}+H^{0}(\delta_{2}u,\gamma_{2}w;\gamma_{2}(z-w))+\theta(z)-\theta(w)\geq\langle
l,z-w\rangle_{E}$ (1.2)

for all $z\in D$.

It should be mentioning that Problem 1.1 is new and contains many challenging
and important problems as its special cases. We point out below several
interesting particular cases of Problem 1.1.

  * (i)

If $J$ and $H$ are independent of their first variables, respectively, then
Problem 1.1 reduces to the following coupled system: find $(u,w)\in C\times D$
such that

$\langle
A(w,u),v-u\rangle_{V}+J^{0}(\gamma_{1}u;\gamma_{1}(v-u))+\psi(v)-\psi(u)\geq\langle
h,v-u\rangle_{V}$ (1.3)

for all $v\in C$, and

$\langle
B(u,w),z-w\rangle_{E}+H^{0}(\gamma_{2}w;\gamma_{2}(z-w))+\theta(z)-\theta(w)\geq\langle
l,z-w\rangle_{E}$ (1.4)

for all $z\in D$. This kind of coupled inequalities (1.3)–(1.4) has not been
studied in the literature.

  * (ii)

When $\psi=\theta\equiv 0$, then Problem 1.1 takes the following form of two
coupled hemivariational inequalities: find $(u,w)\in C\times D$ satisfying

$\langle
A(w,u),v-u\rangle_{V}+J^{0}(\delta_{1}w,\gamma_{1}u;\gamma_{1}(v-u))\geq\langle
h,v-u\rangle_{V}$ (1.5)

for all $v\in C$, and

$\langle
B(u,w),z-w\rangle_{E}+H^{0}(\delta_{2}u,\gamma_{2}w;\gamma_{2}(z-w))\geq\langle
l,z-w\rangle_{E}$ (1.6)

for all $z\in D$. To the best of our knowledge, there is no results available
in the literature for the system (1.5)–(1.6).

  * (iii)

If $J\equiv 0$ and $H\equiv 0$, then Problem 1.1 becomes the following system
of coupled variational inequalities: find $(u,w)\in C\times D$ satisfying

$\langle A(w,u),v-u\rangle_{V}+\psi(v)-\psi(u)\geq\langle h,v-u\rangle_{V}$
(1.7)

for all $v\in C$, and

$\langle B(u,w),z-w\rangle_{E}+\theta(z)-\theta(w)\geq\langle
l,z-w\rangle_{E}$ (1.8)

for all $z\in D$. This system has been considered and investigated in [11] in
which the authors applied the Kakutani-Ky Fan fixed point theorem for
multivalued operators to prove the existence of solutions to system
(1.7)–(1.8). In this paper, in contrast to [11], we give a new proof which is
based on a multivalued version of the Tychonoff fixed point principle in a
Banach space combined with the theory of nonsmoth analysis, generalized
monotonicity arguments and the Minty approach.

  * (iv)

When $\theta\equiv 0$, $H\equiv 0$ and $D=E$, then Problem 1.1 can be
reformulated as the following variational-hemivariational inequality subjected
to a nonlinear equation constraint: find $(u,w)\in C\times D$ such that

$\displaystyle\left\\{\begin{array}[]{lll}\langle
A(w,u),v-u\rangle_{V}+J^{0}(\delta_{1}w,\gamma_{1}u;\gamma_{1}(v-u))+\psi(v)-\psi(u)\geq\langle
h,v-u\rangle_{V},\\\\[2.84526pt] \hskip 227.62204pt\mbox{for all}\ v\in C,\\\
B(u,w)=l.\end{array}\right.$

  * (v)

Assume that $\psi\equiv 0$, $\theta\equiv 0$, $J\equiv 0$, $H\equiv 0$, $C=V$
and $D=E$. Then Problem 1.1 is equivalent to the following nonlinear system of
two coupled equations: find $(u,w)\in C\times D$ such that

$\displaystyle\left\\{\begin{array}[]{lll}A(w,u)=h,\\\\[5.69054pt]
B(u,w)=l.\end{array}\right.$

  * (vi)

Suppose that $\theta\equiv 0$, $H\equiv 0$, $D=E$ and $B$ is independent of
its first variable. Then Problem 1.1 can be reformulated as the following
parameter control system driven by a variational-hemivariational inequality:
find $u\in C$ and $w\in W$ such that

$\langle
A(w,u),v-u\rangle_{V}+J^{0}(\delta_{1}w,\gamma_{1}u;\gamma_{1}(v-u))+\psi(v)-\psi(u)\geq\langle
h,v-u\rangle_{V}$

for all $v\in C$, where the admissible set $W$ is defined by $W:=\\{w\in E\mid
B(w)=l\\}$.

  * (vii)

If $A$ and $J$ are independent of their first variables, $B\equiv 0$,
$\theta\equiv 0$, $H\equiv 0$ and $l=0$, then Problem 1.1 reduces to the
following elliptic variational-hemivariational inequality: find $x\in C$ such
that

$\langle
Au,v-u\rangle_{V}+J^{0}(\gamma_{1}(u);\gamma_{1}(v-u))+\psi(v)-\psi(u)\geq\langle
h,v-u\rangle_{V}$ (1.11)

for all $v\in C$.

The variational-hemivariational inequalities of the form (1.11) have been
studied from various perspectives. For example, the results on noncoercive
hemivariational inequalities can be found in [2] where the equilibrium
problems have been employed, and in [12] where an application to contact
problems in mechanics were treated. Several classes of variational-
hemivariational and hemivariational inequalities that model problems in
contact mechanics have been also studied in [9, 18]. The nonconvex star-shaped
constraints sets in evolution hemivariational inequalities have been studied
in [5], and singular perturbations of inequality problems were analyzed in
[8]. Optimal control problems and inverse problems for the aforementioned
inequalities have been investigated in [17, 25]. The elliptic variational-
hemivariational inequalities have been treated in [13, 14], differential
hemivariational inequalities in [19], and related double phase obstacle
problems were considered in [23, 24]. For other recent results on
hemivariational inequalities, we refer, for example, to [16, 15, 20, 21, 22,
26, 27] and the references therein.

The rest of the paper is organized as follows. In Section 2 we recall a
preliminary material needed in the sequel. Section 3 is devoted to state the
hypotheses on the data of Problem 1.1, and to deliver the main results of this
paper which contain the nonemptiness and compactness of the solution set to
Problem 1.1.

##  2\. Mathematical Background

In this section, we recall a necessary preliminary material which will be used
throughout the paper. More details can be found in [1, 3, 4, 6, 18].

Let $(E,\|\cdot\|_{E})$ be a Banach space, $E^{*}$ be its dual space, and
$\langle\cdot,\cdot\rangle_{E}$ denote the duality brackets between $E^{*}$
and $E$. We adopt the symbols ”$\ \stackrel{{\scriptstyle
w}}{{\longrightarrow}}\ $” and ”$\to$” to symbolize the weak convergence and
the strong convergence in various spces, respectively.

We recall definitions and properties of upper semicontinuous multivalued
operators.

######  Definition 2.1.

Let $Y$ and $Z$ be topological spaces, $D\subset Y$ be a nonempty set, and
$G\colon Y\to 2^{Z}$ be a multivalued map.

(i):

The map $G$ is called upper semicontinuous (u.s.c., for short) at $y\in Y,$ if
for each open set $O\subset Z$ such that $G(y)\subset O$, there exists a
neighborhood $N(y)$ of $y$ satisfying $G(N(y)):=\cup_{z\in N(y)}G(z)\subset
O$. If it holds for each $y\in D$, then $G$ is called to be upper
semicontinuous in $D$.

(ii):

The map $G$ is closed at $y\in Y$, if for every sequence
$\\{(y_{n},z_{n})\\}\subset\mbox{\rm Gr}(G)$ satisfying
$(y_{n},z_{n})\to(y,z)$ in $Y\times Z$, it holds $(y,z)\in\mbox{\rm Gr}(G)$,
where $\mbox{\rm Gr}(G)$ is the graph of the map $G$ defined by

$\displaystyle\mbox{\rm Gr}(G):=\\{(y,z)\in Y\times Z\mid z\in G(y)\\}.$

If it holds for each $y\in Y$, then $G$ is called to be closed (or $G$ has a
closed graph).

Let $X_{1}$ and $X_{2}$ be two Banach spaces. A multivalued map $F\colon
X_{1}\to 2^{X_{2}}$ is called sequentially weakly-weakly closed, if $F$ is
sequentially closed from $X_{1}$ endowed with the weak topology into the
subsets of $X_{2}$ with the weak topology.

The following result provides two useful criteria for the upper semicontinuity
of a multivalued map.

######  Proposition 2.1.

Let $F\colon X\to 2^{Y}$ with $X$ and $Y$ topological spaces. The following
conditions are equivalent:

(i):

$F$ is upper semicontinuous.

(ii):

For each closed set $C\subset Y$, $F^{-}(C):=\\{x\in X\mid F(x)\cap
C\neq\emptyset\\}$ is closed in $X$.

(iii):

For each open set $O\subset Y$, $F^{+}(O):=\\{x\in X\mid F(x)\subset O\\}$ is
open in $X$.

The following definitions provide useful notions from the theory of nonsmooth
analysis.

######  Definition 2.2.

Let $V$ be a reflexive Banach space, $\psi\colon V\to\overline{\mathbb{R}}$ be
a proper, convex and l.s.c. function, and $A\colon V\to 2^{V^{*}}$ be a
multivalued operator. The operator $A$ is called to be

  1. (i)

$\psi$-pseudomonotone, if for any $u$, $v\in V$ fixed, there exists an element
$u^{*}\in Au$ such that

$\langle u^{*},v-u\rangle_{X}+\psi(v)-\psi(u)\geq 0,$

then we have

$\langle v^{*},v-u\rangle_{X}+\psi(v)-\psi(u)\geq 0$

for all $v^{*}\in A(v)$.

  2. (ii)

stable $\psi$-pseudomonotone with respect to the set $W\subset V^{*}$, if $A$
and $V\ni u\mapsto Au-w\subset V^{*}$ are $\psi$-pseudomonotone for each $w\in
W$.

######  Definition 2.3.

Let $X$ be a Banach space with the dual space $X^{*}$ and norm
$\|\cdot\|_{X}.$ A function $J\colon X\to\mathbb{R}$ is called to be a locally
Lipschitz continuous at $u\in X$, if there are a neighborhood $N(u)$ of $u$
and a constant $L_{u}>0$ with

$|J(w)-J(v)|\leq L_{u}\|w-v\|_{X}\ \ \mbox{for all}\ \ w,v\in N(u).$

Given a locally Lipschitz function $J\colon X\to\mathbb{R}$, the generalized
(Clarke) directional derivative of $J$ at the point $u\in X$ in the direction
$v\in X$, denoted by $J^{0}(u;v)$, is defined by

$J^{0}(u;v)=\limsup\limits_{\lambda\to 0^{+},\,w\to u}\frac{J(w+\lambda
v)-J(w)}{\lambda}.$

Further, the generalized subgradient of $J\colon X\to\mathbb{R}$ at $u\in X$
is given by

$\partial J(u)=\\{\,\xi\in X^{*}\mid
J^{0}(u;v)\geq\langle\xi,v\rangle_{X^{*}\times X}\ \ \mbox{\rm for all}\ \
v\in X\,\\}.$

The generalized subgradient and generalized directional derivative of a
locally Lipschitz function enjoy nice properties and rich calculus. Here, we
summarize some basic results (see for example [18, Proposition 3.23]).

######  Proposition 2.2.

Let $J\colon X\to\mathbb{R}$ be a locally Lipschitz function. Then

(i):

For every $x\in X$, the function $X\ni v\mapsto J^{0}(x;v)\in\mathbb{R}$ is
positively homogeneous and subadditive, i.e., $J^{0}(x;\lambda v)=\lambda
J^{0}(x;v)$ for all $\lambda\geq 0$, $v\in X$, and $J^{0}(x;v_{1}+v_{2})\leq
J^{0}(x;v_{1})+J^{0}(x;v_{2})$ for all $v_{1}$, $v_{2}\in X$, respectively.

(ii):

For every $x\in X$, the set $\partial J(x)$ is a nonempty, convex, and weakly*
compact subset of $X^{*}$, which is bounded by the Lipschitz constant
$L_{x}>0$ of $J$ near $x$.

(iii):

The graph of the generalized subgradient operator $\partial J$ of $J$ is
closed in $X\times(w^{*}\mbox{--}X^{*})$ topology, i.e., if
$\\{x_{n}\\}\subset X$ and $\\{\xi_{n}\\}\subset X^{*}$ are sequences such
that $\xi_{n}\in\partial J(x_{n})$ and $x_{n}\to x$ in $X$, $\xi_{n}\
\stackrel{{\scriptstyle w}}{{\longrightarrow}}\ \xi$ in $X^{*}$, then
$\xi\in\partial J(x)$, where, recall, $(w^{*}\mbox{--}X^{*})$ denotes the
space $X^{*}$ equipped with weak* topology.

We end the section by recalling a multivalued version of the Tychonoff fixed
point principle in a Banach space, its proof can be found in [7, Theorem 8.6].

######  Theorem 1.

Let $C$ be a bounded, closed and convex subset of a reflexive Banach space
$E$, and $S\colon C\to 2^{C}$ be a multivalued map such that

(i):

$S$ has bounded, closed and convex values,

(ii):

$S$ is weakly-weakly u.s.c..

Then $S$ has a fixed point in $C$.

##  3\. Main Results

This section is devoted to the main results of the paper which include the
nonemptiness and compactness of the solution set to Problem 1.1.

Before we state and prove the results, we make the following hypotheses on the
data of Problem 1.1.

${\underline{H(0)}}$: $C$ and $D$ are nonempty, closed and convex subsets of
$V$ and $E$, respectively.

${\underline{H(1)}}$: $h\in V^{*}$ and $l\in E^{*}$.

${\underline{H(2)}}$: $\gamma_{1}\colon V\to X$, $\gamma_{2}\colon E\to Y$,
$\delta_{1}\colon E\to Z_{1}$ and $\delta_{2}\colon V\to Z_{2}$ are bounded,
linear and compact.

${\underline{H(\psi)}}$: $\psi\colon V\to\overline{\mathbb{R}}$ is a convex
and lower semicontinuous function such that

dom$(\psi):=\\{u\in V\mid\psi(u)<+\infty\\}\cap C\neq\emptyset$.

${\underline{H(\theta)}}$: $\theta\colon E\to\overline{\mathbb{R}}$ is a
convex and lower semicontinuous function such that

dom$(\theta):=\\{w\in E\mid\theta(w)<+\infty\\}\cap D\neq\emptyset$.

${\underline{H(J)}}$: $J\colon Z_{1}\times X\to\mathbb{R}$ is such that

  * (i)

for every $w\in Z_{1}$, $X\ni u\mapsto J(w,u)\in\mathbb{R}$ is locally
Lipschitz continuous.

  * (ii)

there exists a constant $c_{J}\geq 0$ such that

$\|\xi\|_{X^{*}}\leq c_{J}\left(1+\|u\|_{X}+\|w\|_{Z_{1}}\right)$

for all $\xi\in\partial J(w,u)$, $u\in X$ and $w\in Z_{1}$.

  * (iii)

the following inequality is valid

$\limsup_{n\to\infty}J^{0}(w_{n},u;v)\leq J^{0}(w,u;v),$

whether $u$, $v$ and sequence $\\{w_{n}\\}\subset Z_{1}$ is such that

$w_{n}\to w\mbox{ in $Z_{1}$ as $n\to\infty$}$

for some $w\in Z_{1}$.

${\underline{H(H)}}$: $H\colon Z_{2}\times Y\to\mathbb{R}$ is such that

  * (i)

for every $u\in Z_{2}$, $Y\ni w\mapsto H(u,w)\in\mathbb{R}$ is locally
Lipschitz continuous.

  * (ii)

there exists a constant $c_{H}\geq 0$ such that

$\|\eta\|_{Y^{*}}\leq c_{H}\left(1+\|u\|_{Z_{2}}+\|w\|_{Y}\right)$

for all $\eta\in\partial H(u,w)$, $u\in Z_{2}$ and $w\in Y$.

  * (iii)

the following inequality is valid

$\limsup_{n\to\infty}H^{0}(u_{n},w;z)\leq H^{0}(u,w;z),$

whether $w$, $z\in Y$ and sequence $\\{w_{n}\\}\subset Z_{2}$ is such that

$w_{n}\to w\mbox{ in $Z_{2}$ as $n\to\infty$}$

for some $w\in Z_{2}$.

${\underline{H(A)}}$: $A\colon E\times V\to V^{*}$ satisfies the following
conditions:

  * (i)

for any $w\in E$ and $v$, $u\in V$, the following inequality holds

$\limsup_{\lambda\to 0}\langle A(w,tv+(1-t)u),v-u\rangle_{V}\leq\langle
A(w,u),v-u\rangle_{V}.$

  * (ii)

for any $w\in E$ fixed, the multivalued mapping $V\ni u\mapsto
A(w,u)+\gamma_{1}^{*}\partial J(\delta_{1}w,\gamma_{1}u)\subset V^{*}$ is
stable $\psi$-pseudomonotone with respect to $\\{h\\}$.

  * (iii)

if $\\{w_{n}\\}\subset E$ and $\\{u_{n}\\}\subset V$ are such that

$w_{n}\ \stackrel{{\scriptstyle w}}{{\longrightarrow}}\ w\mbox{ in $E$ and
$u_{n}\ \stackrel{{\scriptstyle w}}{{\longrightarrow}}\ u$ in $V$ as
$n\to\infty$}$

for some $(w,u)\in E\times V$, then we have

$\limsup_{n\to\infty}\langle A(w_{n},v),v-u_{n}\rangle_{V}\leq\langle
A(w,v),v-u\rangle_{V}.$

  * (iv)

the following growth condition is satisfied

$\|A(w,u)\|_{V^{*}}\leq b_{A}(1+\|u\|_{V}+\|w\|_{E})$

for all $(w,u)\in E\times V$ with some $b_{A}>0$.

  * (v)

there exists a function
$r_{A}\colon\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}$ such that

$\langle A(w,u),u\rangle_{V}-J^{0}(\delta_{1}w,\gamma_{1}u;-\gamma_{1}u)\geq
r_{A}(\|u\|_{V},\|w\|_{E})\|u\|_{V}\mbox{ for all $u\in V$ and $w\in E$},$

and

    * $\bullet$

for every nonempty and bounded set $O\subset\mathbb{R}_{+}$, we have
$r_{A}(t,s)\to+\infty$ as $t\to+\infty$ for all $s\in O$,

    * $\bullet$

for any constants $c_{1}$, $c_{2}\geq 0$, it holds
$r_{A}(t,c_{1}t+c_{2})\to+\infty$ as $t\to+\infty$,

    * $\bullet$

for sequences $\\{s_{n}\\}\subset\mathbb{R}_{+}$ and $\\{t_{n}\\}\subset$ such
that

$s_{n}\to+\infty,\mbox{ $t_{n}\to+\infty$ and $\frac{t_{n}}{s_{n}}\to 0$ as
$n\to\infty$},$

we have

$r_{A}(s_{n},t_{n})\to+\infty\mbox{ as $n\to\infty$}.$

${\underline{H(B)}}$: $B\colon V\times E\to E^{*}$ satisfies the following
conditions:

  * (i)

for any $u\in V$ and $z$, $w\in E$, it holds

$\limsup_{\lambda\to 0}\langle B(u,tz+(1-t)w),z-w\rangle_{E}\leq\langle
B(u,w),z-w\rangle_{E}.$

  * (ii)

for each $u\in V$ fixed, the multivalued mapping $E\ni w\mapsto
B(u,w)+\gamma_{2}^{*}\partial H(\delta_{2}u,\gamma_{2}w)\subset E^{*}$ is
stable $\theta$-pseudomonotone with respect to $\\{l\\}$.

  * (iii)

if $\\{w_{n}\\}\subset E$ and $\\{u_{n}\\}\subset V$ are such that

$w_{n}\ \stackrel{{\scriptstyle w}}{{\longrightarrow}}\ w\mbox{ in $E$ and
$u_{n}\ \stackrel{{\scriptstyle w}}{{\longrightarrow}}\ u$ in $V$ as
$n\to\infty$}$

for some $(w,u)\in E\times V$, then we have

$\limsup_{n\to\infty}\langle B(u_{n},z),z-w_{n}\rangle_{E}\leq\langle
B(u,z),z-w\rangle_{E}.$

  * (iv)

the following growth condition is satisfied

$\|B(u,w)\|_{E^{*}}\leq b_{B}(1+\|u\|_{V}+\|w\|_{E})$

for all $(w,u)\in E\times V$ with some $b_{B}>0$.

  * (v)

there exists a function
$r_{B}\colon\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}$ such that

$\langle B(u,w),w\rangle_{E}-H^{0}(\delta_{2}u,\gamma_{2}w;-\gamma_{2}w)\geq
r_{B}(\|w\|_{E},\|u\|_{V})\|w\|_{E}\mbox{ for all $u\in V$ and $w\in E$},$

and

    * $\bullet$

for every nonempty and bounded set $O\subset\mathbb{R}_{+}$, we have
$r_{B}(t,s)\to+\infty$ as $t\to+\infty$ for all $s\in O$,

    * $\bullet$

for any constants $c_{1}$, $c_{2}\geq 0$, it holds
$r_{B}(t,c_{1}t+c_{2})\to+\infty$ as $t\to+\infty$,

    * $\bullet$

for sequences $\\{s_{n}\\}\subset\mathbb{R}_{+}$ and $\\{t_{n}\\}\subset$ such
that

$s_{n}\to+\infty,\mbox{ $t_{n}\to+\infty$ and $\frac{t_{n}}{s_{n}}\to 0$ as
$n\to\infty$},$

we have

$r_{B}(s_{n},t_{n})\to+\infty\mbox{ as $n\to\infty$}.$

######  Theorem 2.

Under the assumptions $H(A)$, $H(B)$, $H(0)$, $H(1)$, $H(2)$, $H(J)$, $H(H)$,
$H(\psi)$ and $H(\theta)$, the set of solutions to Problem 1.1, denoted by
${\mathbb{S}}(h,l)$, is nonempty and weakly compact in $V\times E$.

###### Proof.

The proof of this theorem is divided into five steps.

Step 1. If the set ${\mathbb{S}}(h,l)$ of solutions to Problem 1.1 is
nonempty, then ${\mathbb{S}}(h,l)$ is bounded.

Assume that ${\mathbb{S}}(h,l)$ is nonempty. Let $(u,w)\in{\mathbb{S}}(h,l)$,
$u_{0}\in\mbox{dom}\psi\cap C$ and $w_{0}\in\mbox{dom}\theta\cap D$ be
arbitrary fixed. Then, we have

$\langle
A(w,u),u_{0}-u\rangle_{V}+J^{0}(\delta_{1}w,\gamma_{1}u;\gamma_{1}(u_{0}-u))+\psi(u_{0})-\psi(u)\geq\langle
h,u_{0}-u\rangle_{V},$

and

$\langle
B(u,w),w_{0}-w\rangle_{E}+H^{0}(\delta_{2}u,\gamma_{2}w;\gamma_{2}(w_{0}-w))+\theta(w_{0})-\theta(w)\geq\langle
l,w_{0}-w\rangle_{E}.$

From the subadditivity of $x\mapsto J^{0}(w,u;x)$ and $x\mapsto H^{0}(u,w;x)$,
we have

$\displaystyle\langle
A(w,u),u\rangle_{V}-J^{0}(\delta_{1}w,\gamma_{1}u;-\gamma_{1}u)$
$\displaystyle\leq$ $\displaystyle\langle
A(w,u),u_{0}\rangle_{V}+J^{0}(\delta_{1}w,\gamma_{1}u;\gamma_{1}u_{0})+\psi(u_{0})-\psi(u)-\langle
h,u_{0}-u\rangle_{V},$

and

$\displaystyle\langle
B(u,w),w\rangle_{E}-H^{0}(\delta_{2}u,\gamma_{2}w;-\gamma_{2}w)$
$\displaystyle\leq$ $\displaystyle\langle
B(u,w),w_{0}\rangle_{E}+H^{0}(\delta_{2}u,\gamma_{2}w;\gamma_{2}w_{0})+\theta(w_{0})-\theta(w)-\langle
l,w_{0}-w\rangle_{E}.$

Let $\xi\in X^{*}$ and $\eta\in Y^{*}$ be such that

$\displaystyle\xi\in\partial J(\delta_{1}w,\gamma_{1}u)\ \mbox{ and }\
\langle\xi,\gamma_{1}u_{0}\rangle_{X}=J^{0}(\delta_{1}w,\gamma_{1}u;\gamma_{1}u_{0}),$
$\displaystyle\eta\in\partial H(\delta_{2}u,\gamma_{2}w)\ \mbox{ and }\
\langle\eta,\gamma_{2}w_{0}\rangle_{Y}=H^{0}(\delta_{2}u,\gamma_{2}w;\gamma_{1}w_{0}).$

Recall that $\psi$ and $\theta$ are convex and l.s.c., so, we invoke [3,
Proposition 5.2.25] to find constants $\alpha_{\psi}$, $\alpha_{\theta}$,
$\beta_{\psi}$, $\beta_{\theta}\geq 0$ such that

$\psi(u)\geq-\alpha_{\psi}\|u\|_{V}-\beta_{\psi}\ \mbox{ and }\
\theta(w)\geq-\alpha_{\theta}\|w\|_{E}-\beta_{\theta}$

for all $(u,w)\in V\times E$. We apply the inequalities above and hypotheses
$H(A)$(iv)–(v) and $H(B)$(iv)–(v) to infer that

$\displaystyle r_{A}(\|u\|_{V},\|w\|_{E})\|u\|_{V}$ $\displaystyle\leq$
$\displaystyle\langle
A(w,u),u\rangle_{V}-J^{0}(\delta_{1}w,\gamma_{1}u;-\gamma_{1}u)$
$\displaystyle\leq$ $\displaystyle\langle
A(w,u),u_{0}\rangle_{V}+\langle\xi,\gamma_{1}u_{0}\rangle_{X}+\psi(u_{0})-\psi(u)-\langle
h,u_{0}-u\rangle_{V}$ $\displaystyle\leq$ $\displaystyle
b_{A}(1+\|u\|_{V}+\|w\|_{E})\|u_{0}\|_{V}+c_{J}(1+\|\gamma_{1}u\|_{X}+\|\delta_{1}w\|_{Z_{1}})\|\gamma_{1}u_{0}\|_{X}$
$\displaystyle+\psi(u_{0})-\psi(u)+\|h\|_{V^{*}}\left(\|u_{0}\|_{V}+\|u\|_{V}\right)$
$\displaystyle\leq$ $\displaystyle
b_{A}(1+\|u\|_{V}+\|w\|_{E})\|u_{0}\|_{V}+c_{J}(1+\|\gamma_{1}\|\|u\|_{V}+\|\delta_{1}\|\|w\|_{E})\|\gamma_{1}\|\|u_{0}\|_{V}$
$\displaystyle+\psi(u_{0})+\alpha_{\psi}\|u\|_{V}+\beta_{\psi}+\|h\|_{V^{*}}\left(\|u_{0}\|_{V}+\|u\|_{V}\right),$

and

$\displaystyle r_{B}(\|w\|_{E},\|u\|_{V})\|w\|_{E}$ $\displaystyle\leq$
$\displaystyle\langle
B(u,w),w\rangle_{E}-H^{0}(\delta_{2}u,\gamma_{2}w;-\gamma_{2}w)$
$\displaystyle\leq$ $\displaystyle\langle
B(u,w),w_{0}\rangle_{E}+H^{0}(\delta_{2}u,\gamma_{2}w;\gamma_{w}w_{0})+\theta(w_{0})-\theta(w)-\langle
l,w_{0}-w\rangle_{E}$ $\displaystyle\leq$ $\displaystyle
b_{B}(1+\|u\|_{V}+\|w\|_{E})\|w_{0}\|_{E}+c_{H}(1+\|\delta_{2}\|\|u\|_{V}+\|\gamma_{2}\|\|w\|_{E})\|\gamma_{2}\|\|w_{0}\|_{E}$
$\displaystyle+\theta(w_{0})+\alpha_{\theta}\|w\|_{E}+\beta_{\theta}+\|l\|_{E^{*}}\left(\|w_{0}\|_{E}+\|w\|_{E}\right).$

Hence, we have

$\displaystyle r_{A}(\|u\|_{V},\|w\|_{E})$ $\displaystyle\leq$
$\displaystyle\frac{b_{A}(1+\|u\|_{V}+\|w\|_{E})\|u_{0}\|_{V}+c_{J}(1+\|\gamma_{1}\|\|u\|_{V}+\|\delta_{1}\|\|w\|_{E})\|\gamma_{1}\|\|u_{0}\|_{V}}{\|u\|_{V}}$
$\displaystyle+\frac{\psi(u_{0})+\alpha_{\psi}\|u\|_{V}+\beta_{\psi}+\|h\|_{V^{*}}\left(\|u_{0}\|_{V}+\|u\|_{V}\right)}{\|u\|_{V}},$
(3.1)

and

$\displaystyle r_{B}(\|w\|_{E},\|u\|_{V})$ $\displaystyle\leq$
$\displaystyle\frac{b_{B}(1+\|u\|_{V}+\|w\|_{E})\|w_{0}\|_{E}+c_{H}(1+\|\delta_{2}\|\|u\|_{V}+\|\gamma_{2}\|\|w\|_{E})\|\gamma_{2}\|\|w_{0}\|_{E}}{\|w\|_{E}}$
$\displaystyle+\frac{\theta(w_{0})+\alpha_{\theta}\|w\|_{E}+\beta_{\theta}+\|l\|_{E^{*}}\left(\|w_{0}\|_{E}+\|w\|_{E}\right)}{\|w\|_{E}}.$
(3.2)

Assume that ${\mathbb{S}}(h,l)$ is unbounded. Without any loss of generality,
we may suppose that there exists a sequence
$\\{(u_{n},w_{n})\\}\subset{\mathbb{S}}(h,l)$ satisfying
$\|u_{n}\|_{V}+\|w_{n}\|_{E}\to\infty$ as $n\to\infty$, namely, one of the
following conditions holds:

$\|u_{n}\|_{V}\uparrow+\infty\mbox{ as $n\to\infty$ and $\\{w_{n}\\}$ is
bounded in $E$},$ (3.3)

or

$\|w_{n}\|_{E}\uparrow+\infty\mbox{ as $n\to\infty$ and $\\{u_{n}\\}$ is
bounded in $V$},$ (3.4)

or

$\|w_{n}\|_{E}\uparrow+\infty\mbox{ and $\|u_{n}\|_{V}\uparrow+\infty$ as
$n\to\infty$}.$ (3.5)

If (3.3) is true, then from (3) we obtain

$\displaystyle r_{A}(\|u_{n}\|_{V},\|w_{n}\|_{E})$ $\displaystyle\leq$
$\displaystyle\frac{b_{A}(1+\|u_{n}\|_{V}+\|w_{n}\|_{E})\|u_{0}\|_{V}+c_{J}(1+\|\gamma_{1}\|\|u_{n}\|_{V}+\|\delta_{1}\|\|w_{n}\|_{E})\|\gamma_{1}\|\|u_{0}\|_{V}}{\|u_{n}\|_{V}}$
$\displaystyle+\frac{\psi(u_{0})+\alpha_{\psi}\|u_{n}\|_{V}+\beta_{\psi}+\|h\|_{V^{*}}\left(\|u_{0}\|_{V}+\|u_{n}\|_{V}\right)}{\|u_{n}\|_{V}}.$

Taking the limit as $n\to\infty$ in the inequality above and using assumption
$H(A)$(v), it yields

$\displaystyle+\infty=\lim_{n\to\infty}r_{A}(\|u_{n}\|_{V},\|w_{n}\|_{E})$
$\displaystyle\leq$
$\displaystyle\lim_{n\to\infty}\bigg{[}\frac{b_{A}(1+\|u_{n}\|_{V}+\|w_{n}\|_{E})\|u_{0}\|_{V}+c_{J}(1+\|\gamma_{1}\|\|u_{n}\|_{V}+\|\delta_{1}\|\|w_{n}\|_{E})\|\gamma_{1}\|\|u_{0}\|_{V}}{\|u_{n}\|_{V}}$
$\displaystyle+\frac{\psi(u_{0})+\alpha_{\psi}\|u_{n}\|_{V}+\beta_{\psi}+\|h\|_{V^{*}}\left(\|u_{0}\|_{V}+\|u_{n}\|_{V}\right)}{\|u_{n}\|_{V}}\bigg{]}$
$\displaystyle=$
$\displaystyle\,b_{A}\|u_{0}\|_{V}+c_{J}\|\gamma_{1}\|^{2}\|u_{0}\|_{V}+\alpha_{\psi}+\|h\|_{V^{*}}.$

This leads to a contradiction. So, we conclude that ${\mathbb{S}}(h,l)$ is
bounded. Likewise, we can employ the same argument to obtain a contradiction
when (3.4) occurs. If (3.5) holds, we distinguish further the following cases:

  * i)

$\frac{\|u_{n}\|_{V}}{\|w_{n}\|_{E}}\to+\infty$ or
$\frac{\|w_{n}\|_{E}}{\|u_{n}\|_{E}}\to+\infty$ as $n\to\infty$.

  * ii)

there exists $n_{0}\in\mathbb{N}$ such that
$0<c_{0}\leq\frac{\|u_{n}\|_{V}}{\|w_{n}\|_{E}}\leq c_{1}$ for all $n\geq
n_{0}$ for some $c_{0},c_{1}>0$.

Concerning the case i), we only examine the situation if
$\frac{\|u_{n}\|_{V}}{\|w_{n}\|_{E}}\to+\infty$, because the same conclusion
can be obtained by using a similar proof when
$\frac{\|w_{n}\|_{E}}{\|u_{n}\|_{E}}\to+\infty$ as $n\to\infty$. Keeping in
mind (3), one has

$\displaystyle r_{A}(\|u_{n}\|_{V},\|w_{n}\|_{E})$ $\displaystyle\leq$
$\displaystyle\frac{b_{A}(1+\|u_{n}\|_{V}+\|w_{n}\|_{E})\|u_{0}\|_{V}+c_{J}(1+\|\gamma_{1}\|\|u_{n}\|_{V}+\|\delta_{1}\|\|w_{n}\|_{E})\|\gamma_{1}\|\|u_{0}\|_{V}}{\|u_{n}\|_{V}}$
$\displaystyle+\frac{\psi(u_{0})+\alpha_{\psi}\|u_{n}\|_{V}+\beta_{\psi}+\|h\|_{V^{*}}\left(\|u_{0}\|_{V}+\|u_{n}\|_{V}\right)}{\|u_{n}\|_{V}}.$

By virtue of (3.5) and the condition i), we infer

$\displaystyle+\infty=\lim_{n\to\infty}r_{A}(\|u_{n}\|_{V},\|w_{n}\|_{E})$
$\displaystyle\leq$
$\displaystyle\lim_{n\to\infty}\bigg{[}\frac{b_{A}(1+\|u_{n}\|_{V}+\|w_{n}\|_{E})\|u_{0}\|_{V}+c_{J}(1+\|\gamma_{1}\|\|u_{n}\|_{V}+\|\delta_{1}\|\|w_{n}\|_{E})\|\gamma_{1}\|\|u_{0}\|_{V}}{\|u_{n}\|_{V}}$
$\displaystyle+\frac{\psi(u_{0})+\alpha_{\psi}\|u_{n}\|_{V}+\beta_{\psi}+\|h\|_{V^{*}}\left(\|u_{0}\|_{V}+\|u_{n}\|_{V}\right)}{\|u_{n}\|_{V}}\bigg{]}$
$\displaystyle=$
$\displaystyle\,b_{A}\|u_{0}\|_{V}+c_{J}\|\gamma_{1}\|^{2}\|u_{0}\|_{V}+\alpha_{\psi}+\|h\|_{V^{*}}.$

This leads to a contradiction too. This implies that ${\mathbb{S}}(h,l)$ is
bounded. Moreover, if the condition ii) holds, then we can use hypothesis
$H(B)$(v) to get

$\displaystyle+\infty=\lim_{n\to\infty}r_{B}(\|w_{n}\|_{E},\|u_{n}\|_{V})$
$\displaystyle\leq$
$\displaystyle\frac{b_{B}(1+\|u_{n}\|_{V}+\|w_{n}\|_{E})\|w_{0}\|_{E}+c_{H}(1+\|\delta_{2}\|\|u_{n}\|_{V}+\|\gamma_{2}\|\|w_{n}\|_{E})\|\gamma_{2}\|\|w_{0}\|_{E}}{\|w_{n}\|_{E}}$
$\displaystyle+\frac{\theta(w_{0})+\alpha_{\theta}\|w_{n}\|_{E}+\beta_{\theta}+\|l\|_{E^{*}}\left(\|w_{0}\|_{E}+\|w_{n}\|_{E}\right)}{\|w_{n}\|_{E}}$
$\displaystyle\leq$
$\displaystyle\,b_{B}(c_{1}+1)\|w_{0}\|_{E}+c_{H}(\|\delta_{2}\|c_{1}+\|\gamma_{2}\|)\|\gamma_{2}\|\|w_{0}\|_{E}+\theta(w_{0})+\alpha_{\theta}+\|l\|_{E^{*}}.$

This implies that the set ${\mathbb{S}}(h,l)$ is bounded.

Step 2. For each $w\in E$ (resp. $u\in V$) fixed, the solution set of
inequality problem $(\ref{eqn1})$ (resp. $(\ref{eqn2})$) is nonempty, bounded
and closed.

Let $w\in E$ be arbitrary fixed. By the definition of generalized subgradient
in the sense of Clarke, we have

$\displaystyle\langle A(w,u)+\gamma^{*}_{1}\xi,u\rangle_{V}=\langle
A(w,u),u\rangle_{V}-\langle\xi,-\gamma_{1}u\rangle_{X}$ $\displaystyle\geq$
$\displaystyle\langle
A(w,u),u\rangle_{V}-J^{0}(\delta_{1}y,\gamma_{1}u;-\gamma_{1}u)$

for all $\xi\in\partial J(\delta_{1}w,\gamma_{1}u)$. Taking into account
hypothesis $H(A)$(v) and the inequality above, it gives

$\displaystyle\frac{\inf_{\xi\in\partial J(\delta_{1}w,\gamma_{1}u)}\langle
A(w,u)+\gamma^{*}\xi,u\rangle_{V}}{\|u\|_{V}}\geq\frac{\langle
A(w,u),u\rangle_{V}-J^{0}(\delta_{1}y,\gamma_{1}u;-\gamma_{1}u)}{\|u\|_{V}}$
$\displaystyle\geq$ $\displaystyle\,r_{A}(\|u\|_{V},\|w\|_{E})$

for all $u\in V$. This means that multivalued operator

$V\ni u\mapsto A(w,u)+\gamma_{1}^{*}\partial J(\delta_{1}w,\gamma_{1}u)\subset
V^{*}$

is coercive. In an analogous way, we can verify that for every $u\in V$ fixed,
multivalued operator

$E\ni w\mapsto B(u,w)+\gamma_{2}^{*}\partial H(\delta_{2}u,\gamma_{2}w)\subset
E^{*}$

is coercive as well. Therefore, we can invoke the same arguments as in the
proof of [12, Theorem 3] to conclude that the solution set of inequality
problem (1.1) (resp. (1.2)) is nonempty, bounded and closed.

Next, we introduce the multivalued map $\Gamma\colon C\times D\to 2^{C\times
D}$ defined by

$\Gamma(u,w):=(\mathcal{P}(w),\mathcal{Q}(u))\ \mbox{ for all }\ (u,w)\in
C\times D,$ (3.6)

where $\mathcal{P}\colon E\to 2^{C}$ and $\mathcal{Q}\colon V\to 2^{D}$ stand
for the solution mappings problems (1.1) and (1.2), respectively, namely,
$\mathcal{P}(w)$ and $\mathcal{Q}(u)$ are the solution sets of problems (1.1)
and (1.2) corresponding to $w\in E$ and $u\in V$, respectively. From the
definition of $\Gamma$, it is not difficult to show that $(u,w)\in C\times D$
is a fixed point of $\Gamma$ if and only if it is a solution of Problem 1.1.
Based on this property, we are going to verify that $\Gamma$ has at least one
fixed point in $C\times D$.

Step 3. There exists a bounded, closed and convex subset $\mathcal{X}$ of
$C\times D$ such that $\Gamma$ maps $\mathcal{X}$ into itself.

Indeed, it is sufficient to show that there exists a constant $m_{0}>0$ such
that

$\Gamma(\mathcal{O}(m_{0}))\subset\mathcal{O}(m_{0}),$ (3.7)

where $\mathcal{O}(m_{0})$ is defined by

$\mathcal{O}(m_{0}):=\\{(u,w)\in C\times D\,\mid\,\|u\|_{V}\leq m_{0}\mbox{
and }\|w\|_{E}\leq m_{0}\\}.$

Arguing by contradiction, there is no $m_{0}>0$ such that (3.7) holds. So, for
each $n\in\mathbb{N}$, there exist sequences $(u_{n},w_{n})$,
$(v_{n},z_{n})\in C\times D$ satisfying

$(u_{n},w_{n})\in\mathcal{O}(n),\mbox{ $(v_{n},z_{n})\in\Gamma(u_{n},w_{n})$
and $\|v_{n}\|_{V}>n$ or $\|z_{n}\|_{E}>n$}.$

Now, we suppose that $\|v_{n}\|_{V}>n$. It follows from (3) that

$\displaystyle r_{A}(\|v_{n}\|_{V},\|w_{n}\|_{E})$ $\displaystyle\leq$
$\displaystyle\frac{b_{A}(1+\|v_{n}\|_{V}+\|w_{n}\|_{E})\|u_{0}\|_{V}+c_{J}(1+\|\gamma_{1}\|\|v_{n}\|_{V}+\|\delta_{1}\|\|w_{n}\|_{E})\|\gamma_{1}\|\|u_{0}\|_{V}}{\|v_{n}\|_{V}}$
$\displaystyle+\frac{\psi(u_{0})+\alpha_{\psi}\|v_{n}\|_{V}+\beta_{\psi}+\|h\|_{V^{*}}\left(\|u_{0}\|_{V}+\|v_{n}\|_{V}\right)}{\|v_{n}\|_{V}}.$

Recalling that $\|w_{n}\|_{E}\leq n<\|v_{n}\|_{V}$, we apply hypothesis
$H(A)$(v) to find that

$\displaystyle+\infty=\lim_{n\to\infty}r_{A}(\|v_{n}\|_{V},\|w_{n}\|_{E})$
$\displaystyle\leq$
$\displaystyle\lim_{n\to\infty}\bigg{[}\frac{b_{A}(1+\|v_{n}\|_{V}+\|w_{n}\|_{E})\|u_{0}\|_{V}+c_{J}(1+\|\gamma_{1}\|\|v_{n}\|_{V}+\|\delta_{1}\|\|w_{n}\|_{E})\|\gamma_{1}\|\|u_{0}\|_{V}}{\|v_{n}\|_{V}}$
$\displaystyle+\frac{\psi(u_{0})+\alpha_{\psi}\|v_{n}\|_{V}+\beta_{\psi}+\|h\|_{V^{*}}\left(\|u_{0}\|_{V}+\|v_{n}\|_{V}\right)}{\|v_{n}\|_{V}}\bigg{]}$
$\displaystyle\leq$
$\displaystyle\,2b_{A}\|u_{0}\|_{V}+c_{J}(\|\gamma_{1}\|+\|\delta_{1}\|)\|\gamma_{1}\|\|u_{0}\|_{V}+\alpha_{\psi}+\|h\|_{V^{*}}.$

Hence we get a contradiction. As we did before, it can also lead to a
contraction when the case $\|z_{n}\|_{E}>n$ occurs. This means that there
exists a constant $m_{0}>0$ such that (3.7) is valid. Therefore, we conclude
that there exists a bounded, closed and convex subset $\mathcal{X}$ of
$C\times D$ such that $\Gamma$ maps $\mathcal{X}$ into itself.

Step 4. $\Gamma$ is weakly-weakly upper semicontinuous.

Let $\mathcal{M}\subset C\times D$ be an arbitrary weakly closed set such that
$\Gamma^{-}(\mathcal{M})\neq\emptyset$. From Proposition 2.1, it is sufficient
to verify that $\Gamma^{-}(\mathcal{M})$ is weakly closed in $V\times E$. Let
$\\{(v_{n},z_{n})\\}\subset\Gamma^{-}(\mathcal{M})$ be such that

$(u_{n},w_{n})\ \stackrel{{\scriptstyle w}}{{\longrightarrow}}\ (u,w)\mbox{ in
$V\times E$ as $n\to\infty$}$ (3.8)

for some $(u,w)\in V\times E$. Hence, for every $n\in\mathbb{N}$, we are able
to find $(u_{n},w_{n})\in V\times E$ satisfying

$(v_{n},z_{n})\in\Gamma(u_{n},w_{n})\cap\mathcal{M}.$

From (3) and (3), we have

$\displaystyle r_{A}(\|v_{n}\|_{V},\|w_{n}\|_{E})$ $\displaystyle\leq$
$\displaystyle\frac{b_{A}(1+\|v_{n}\|_{V}+\|w_{n}\|_{E})\|u_{0}\|_{V}+c_{J}(1+\|\gamma_{1}\|\|v_{n}\|_{V}+\|\delta_{1}\|\|w_{n}\|_{E})\|\gamma_{1}\|\|u_{0}\|_{V}}{\|v_{n}\|_{V}}$
$\displaystyle+\frac{\psi(u_{0})+\alpha_{\psi}\|v_{n}\|_{V}+\beta_{\psi}+\|h\|_{V^{*}}\left(\|u_{0}\|_{V}+\|v_{n}\|_{V}\right)}{\|v_{n}\|_{V}},$

and

$\displaystyle r_{B}(\|z_{n}\|_{E},\|u_{n}\|_{V})$ $\displaystyle\leq$
$\displaystyle\frac{b_{B}(1+\|u_{n}\|_{V}+\|z_{n}\|_{E})\|w_{0}\|_{E}+c_{H}(1+\|\delta_{2}\|\|u_{n}\|_{V}+\|\gamma_{2}\|\|z_{n}\|_{E})\|\gamma_{2}\|\|w_{0}\|_{E}}{\|z_{n}\|_{E}}$
$\displaystyle+\frac{\theta(w_{0})+\alpha_{\theta}\|z_{n}\|_{E}+\beta_{\theta}+\|l\|_{E^{*}}\left(\|w_{0}\|_{E}+\|z_{n}\|_{E}\right)}{\|z_{n}\|_{E}}.$

Combining the latter with $H(A)$(v), $H(B)$(v), we infer that sequence
$\\{(v_{n},$ $z_{n})\\}$ is bounded in $V\times E$. Passing to a relabeled
subsequence if necessary, we may suppose that

$(v_{n},z_{n})\ \stackrel{{\scriptstyle w}}{{\longrightarrow}}\ (v,z)\mbox{ in
$V\times E$ as $n\to\infty$}$ (3.9)

for some $(v,z)\in V\times E$.

Next, for each $n\in\mathbb{N}$, let $\xi_{n}\in X^{*}$ and $\eta_{n}\in
Y^{*}$ be such that

$\displaystyle\langle\xi_{n},\gamma_{1}(x-v_{n})\rangle_{X}=J^{0}(\delta_{1}w_{n},\gamma_{1}v_{n};\gamma_{1}(x-v_{n})),$
$\displaystyle\langle\eta_{n},\gamma_{2}(y-z_{n})\rangle_{Y}=H^{0}(\delta_{2}u_{n},\gamma_{2}z_{n};\gamma_{2}(y-z_{n})).$

Hence, we have

$\langle
A(w_{n},v_{n})+\gamma_{1}^{*}\xi_{n},x-v_{n}\rangle_{V}+\psi(x)-\psi(v_{n})\geq\langle
h,x-v_{n}\rangle_{V}$

for all $x\in C$, and

$\langle
B(u_{n},z_{n})+\gamma_{2}^{*}\eta_{n},y-z_{n}\rangle_{E}+\theta(y)-\theta(z_{n})\geq\langle
l,y-z_{n}\rangle_{E}$

for all $y\in D$. Then, we use the hypotheses $H(A)$(ii) and $H(B)$(ii) to
find that

$\displaystyle\langle h,x-v_{n}\rangle_{V}$ $\displaystyle\leq$
$\displaystyle\langle
A(w_{n},x)+\gamma_{1}^{*}\alpha_{n},x-v_{n}\rangle_{V}+\psi(x)-\psi(v_{n})$
$\displaystyle\leq$ $\displaystyle\langle
A(w_{n},x),x-v_{n}\rangle_{V}+J^{0}(\delta_{1}w_{n},\gamma_{1}x;\gamma_{1}(x-v_{n}))+\psi(x)-\psi(v_{n})$

for all $\alpha_{n}\in\partial J(\delta_{1}w_{n},\gamma_{1}x)$ and $x\in C$,
and

$\displaystyle\langle l,y-z\rangle_{E}$ $\displaystyle\leq$
$\displaystyle\langle
B(u_{n},y)+\gamma_{2}^{*}\beta_{n},y-z_{n}\rangle_{E}+\theta(y)-\theta(z_{n})$
$\displaystyle\leq$ $\displaystyle\langle
B(u_{n},y),y-z_{n}\rangle_{E}+H^{0}(\delta_{2}u_{n},\gamma_{2}y;\gamma_{2}(y-z_{n}))+\theta(y)-\theta(z_{n})$

for all $\beta_{n}\in\partial H(\delta_{2}u_{n},\gamma_{2}y)$ and $y\in D$.
Passing to the upper limit as $n\to\infty$ in the inequalities above and using
hypotheses $H(J)$(iii), $H(H)$(iii), $H(A)$(iii) and $H(B)$(iii), we obtain

$\displaystyle\langle h,x-v\rangle_{V}$ $\displaystyle=$
$\displaystyle\limsup_{n\to\infty}\langle h,x-v_{n}\rangle_{V}$
$\displaystyle\leq$ $\displaystyle\limsup_{n\to\infty}\left[\langle
A(w_{n},x),x-v_{n}\rangle_{V}+J^{0}(\delta_{1}w_{n},\gamma_{1}x;\gamma_{1}(x-v_{n}))+\psi(x)-\psi(v_{n})\right]$
$\displaystyle\leq$ $\displaystyle\limsup_{n\to\infty}\langle
A(w_{n},x),x-v_{n}\rangle_{V}+\limsup_{n\to\infty}J^{0}(\delta_{1}w_{n},\gamma_{1}x;\gamma_{1}(x-v_{n}))$
$\displaystyle+\psi(x)-\liminf_{n\to\infty}\psi(v_{n})$ $\displaystyle\leq$
$\displaystyle\langle
A(w,x),x-v\rangle_{V}+J^{0}(\delta_{1}w,\gamma_{1}x;\gamma_{1}(x-v))+\psi(x)-\psi(v),$

and

$\displaystyle\langle l,y-z_{n}\rangle_{E}$ $\displaystyle=$
$\displaystyle\limsup_{n\to\infty}\langle l,y-z_{n}\rangle_{E}$
$\displaystyle\leq$ $\displaystyle\limsup_{n\to\infty}\left[\langle
B(u_{n},y),y-z_{n}\rangle_{E}+H^{0}(\delta_{2}u_{n},\gamma_{2}y;\gamma_{2}(y-z_{n}))+\theta(y)-\theta(z_{n})\right]$
$\displaystyle\leq$ $\displaystyle\limsup_{n\to\infty}\langle
B(u_{n},y),y-z_{n}\rangle_{E}+\limsup_{n\to\infty}H^{0}(\delta_{2}u_{n},\gamma_{2}y;\gamma_{2}(y-z_{n}))$
$\displaystyle+\theta(y)-\liminf_{n\to\infty}\theta(z_{n})$
$\displaystyle\leq$ $\displaystyle\langle
B(u,y),y-z\rangle_{E}+H^{0}(\delta_{2}u,\gamma_{2}y;\gamma_{2}(y-z))+\theta(y)-\theta(z).$

Hence

$\langle
A(w,x),x-v\rangle_{V}+J^{0}(\delta_{1}w,\gamma_{1}x;\gamma_{1}(x-v))+\psi(x)-\psi(v)\geq\langle
h,x-v\rangle_{V}$ (3.10)

for all $x\in C$, and

$\langle
B(u,y),y-z\rangle_{E}+H^{0}(\delta_{2}u,\gamma_{2}y;\gamma_{2}(y-z))+\theta(y)-\theta(z)\geq\langle
l,y-z_{n}\rangle_{E}$ (3.11)

for all $y\in D$. Let $t\in(0,1)$ and $r\in C$ be arbitrary. We insert
$x=x_{t}:=tr+(1-t)v$ into (3.10) and apply the positive homogeneity of
$v\mapsto J^{0}(w,u;v)$ and convexity of $\psi$ to get

$\displaystyle t\left[\langle
A(w,x_{t}),r-v\rangle_{V}+J^{0}(\delta_{1}w,\gamma_{1}x_{t};\gamma_{1}(r-v))+\psi(r)-\psi(v)\right]$
$\displaystyle\geq$ $\displaystyle t\langle
A(w,x_{t}),r-v\rangle_{V}+J^{0}(\delta_{1}w,\gamma_{1}x_{t};t\gamma_{1}(r-v))+\psi(x_{t})-\psi(v)$
$\displaystyle\geq$ $\displaystyle t\langle h,r-v\rangle_{V}.$

So, we obtain

$\langle
A(w,x_{t}),r-v\rangle_{V}+J^{0}(\delta_{1}w,\gamma_{1}x_{t};\gamma_{1}(r-v))+\psi(r)-\psi(v)\geq\langle
h,r-v\rangle_{V}.$

Passing to the upper limit as $t\downarrow 0$ in the inequality above and
using hypothesis $H(A)$(i) and upper semicontinuity of $(u,v)\mapsto
J^{0}(w,u;v)$, we have

$\displaystyle\langle
A(w,v),r-v\rangle_{V}+J^{0}(\delta_{1}w,\gamma_{1}v;\gamma_{1}(r-v))+\psi(r)-\psi(v)$
$\displaystyle\geq$ $\displaystyle\limsup_{t\downarrow 0}\left[\langle
A(w,x_{t}),r-v\rangle_{V}+J^{0}(\delta_{1}w,\gamma_{1}x_{t};\gamma_{1}(r-v))+\psi(r)-\psi(v)\right]$
$\displaystyle\geq$ $\displaystyle\langle h,r-v\rangle_{V}.$

Since $r\in C$ is arbitrary, so, we have that $v\in C$ is a solution of
problem (1.1) corresponding to $w\in D$. Similarly, we also can obtain that
$z\in D$ is a solution of problem (1.2) corresponding to $u\in C$. This
implies that $(v,z)\in\Gamma(u,w)$. Due to the weak closedness of
$\mathcal{M}$, we obtain that $(v,z)\in\mathcal{M}$, that is,
$(u,w)\in\Gamma^{-}(\mathcal{M})$. Therefore, we use Proposition 2.1 to
conclude that $\Gamma$ is weakly-weakly u.s.c..

We conclude that all conditions of the Tychonoff theorem, Theorem 1, have been
verified. Using this theorem, we deduce that $\Gamma$ has at least one fixed
point $(u^{*},w^{*})\in C\times D$ in $\mathcal{X}$. This implies that
$(u^{*},w^{*})\in C\times D$ is also a solution of Problem 1.1.

Step 5. The set ${\mathbb{S}}(h,l)$ is weakly compact.

From Step 1, we can see that the set ${\mathbb{S}}(h,l)$ is bounded. Because
of the reflexivity of $V\times E$, it is sufficient to show that
${\mathbb{S}}(h,l)$ is weakly closed. Let
$\\{(u_{n},w_{n})\\}\subset{\mathbb{S}}(h,l)$ be a sequence such that

$(u_{n},w_{n})\ \stackrel{{\scriptstyle w}}{{\longrightarrow}}\ (u,w)\mbox{ in
$V\times E$ as $n\to\infty$}$ (3.12)

for some $(u,w)\in V\times E$. In virtue of the definition of $\Gamma$, it
yields $(u_{n},w_{n})\in\Gamma(u_{n},w_{n})$. Keeping in mind that $\Gamma$ is
weakly-weakly u.s.c. and has nonempty, bounded, closed and convex values, it
follows from [10, Theorem 1.1.4] that $\Gamma$ is weakly-weakly closed. The
latter together with the convergence (3.12) implies that
$(u,v)\in\Gamma(u,v)$. From the definition of $\Gamma$, we infer that
$(u,v)\in{\mathbb{S}}(h,l)$, i.e., ${\mathbb{S}}(h,l)$ is weakly closed.
Consequently, we conclude that ${\mathbb{S}}(h,l)$ is weakly compact in
$V\times E$. This completes the proof. ∎

We formulate several corollaries of Theorem 2. To this end, we need the
following hypotheses.

${\underline{H(J^{\prime})}}$: $J\colon X\to\mathbb{R}$ is locally Lipschitz
continuous such that there exists a constant $c_{J}\geq 0$ such that

$\|\xi\|_{X^{*}}\leq c_{J}\left(1+\|u\|_{X}\right)$

for all $\xi\in\partial J(u)$ and $u\in X$.

${\underline{H(H^{\prime})}}$: $H\colon Y\to\mathbb{R}$ is locally Lipschitz
continuous such that there exists a constant $c_{H}\geq 0$ such that

$\|\eta\|_{Y^{*}}\leq c_{H}\left(1+\|w\|_{Y}\right)$

for all $\eta\in\partial H(w)$ and $w\in Y$.

${\underline{H(2^{\prime})}}$: $\gamma_{1}\colon V\to X$ and $\gamma_{2}\colon
E\to Y$ are bounded, linear and compact.

${\underline{H(A)}}$(ii)’: for each $w\in E$ the multivalued mapping $V\ni
u\mapsto A(w,u)+\gamma_{1}^{*}\partial J(\gamma_{1}u)\subset V^{*}$ is stable
$\psi$-pseudomonotone with respect to $\\{h\\}$.

${\underline{H(A)}}$(v)’: there exists a function
$r_{A}\colon\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}$ such that

$\langle A(w,u),u\rangle_{V}-J^{0}(\gamma_{1}u;-\gamma_{1}u)\geq
r_{A}(\|u\|_{V},\|w\|_{E})\|u\|_{V}\mbox{ for all $u\in V$ and $w\in E$},$

and

  * $\bullet$

for every nonempty and bounded set $O\subset\mathbb{R}_{+}$, we have
$r_{A}(t,s)\to+\infty$ as $t\to+\infty$ for all $s\in O$,

  * $\bullet$

for any constants $c_{1},c_{2}\geq 0$, it holds
$r_{A}(t,c_{1}t+c_{2})\to+\infty$ as $t\to+\infty$,

  * $\bullet$

for sequences $\\{s_{n}\\}\subset\mathbb{R}_{+}$ and $\\{t_{n}\\}\subset$ such
that

$s_{n}\to+\infty,\mbox{ $t_{n}\to+\infty$ and $\frac{t_{n}}{s_{n}}\to 0$ as
$n\to\infty$},$

we have

$r_{A}(s_{n},t_{n})\to+\infty\mbox{ as $n\to\infty$}.$

${\underline{H(B)}}$(ii)’: for each $u\in V$ the multivalued mapping $E\ni
w\mapsto B(u,w)+\gamma_{2}^{*}\partial H(\gamma_{2}w)\subset E^{*}$ is stable
$\theta$-pseudomonotone with respect to $\\{l\\}$.

${\underline{H(B)}}$(v)’: there exists a function
$r_{B}\colon\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}$ such that

$\langle B(u,w),w\rangle_{E}-H^{0}(\gamma_{2}w;-\gamma_{2}w)\geq
r_{B}(\|w\|_{E},\|u\|_{V})\|w\|_{E}\mbox{ for all $u\in V$ and $w\in E$},$

and

  * $\bullet$

for every nonempty and bounded set $O\subset\mathbb{R}_{+}$, we have
$r_{B}(t,s)\to+\infty$ as $t\to+\infty$ for all $s\in O$,

  * $\bullet$

for any constants $c_{1},c_{2}\geq 0$, it holds
$r_{B}(t,c_{1}t+c_{2})\to+\infty$ as $t\to+\infty$,

  * $\bullet$

for sequences $\\{s_{n}\\}\subset\mathbb{R}_{+}$ and $\\{t_{n}\\}\subset$ such
that

$s_{n}\to+\infty,\mbox{ $t_{n}\to+\infty$ and $\frac{t_{n}}{s_{n}}\to 0$ as
$n\to\infty$},$

we have

$r_{B}(s_{n},t_{n})\to+\infty\mbox{ as $n\to\infty$}.$

######  Corollary 3.1.

Suppose that $H(A)$(i), (ii)’, (iii), (iv)’, $H(B)$(i), (ii)’, (iii), (iv)’,
$H(0)$, $H(1)$, $H(2^{\prime})$, $H(J^{\prime})$, $H(H^{\prime})$, $H(\psi)$
and $H(\theta)$ hold. Then, the set of solutions to problem
$(\ref{prob1a})$–$(\ref{prob1b})$ is nonempty and weakly compact in $V\times
E$.

######  Corollary 3.2.

Suppose that $H(A)$, $H(B)$, $H(0)$, $H(1)$, $H(2)$, $H(J)$ and $H(H)$ are
fulfilled. Then, the set of solutions to problem
$(\ref{prob2a})$–$(\ref{prob2b})$ is nonempty and weakly compact in $V\times
E$.

${\underline{H(A)}}$(ii)”: for each $w\in E$, $V\ni u\mapsto A(w,u)\in V^{*}$
is stable $\psi$-pseudomonotone with respect to $\\{h\\}$.

${\underline{H(A)}}$(v)”: there exists a function
$r_{A}\colon\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}$ such that

$\langle A(w,u),u\rangle_{V}\geq r_{A}(\|u\|_{V},\|w\|_{E})\|u\|_{V}\mbox{ for
all $u\in V$ and $w\in E$},$

and

  * $\bullet$

for every nonempty and bounded set $O\subset\mathbb{R}_{+}$, we have
$r_{A}(t,s)\to+\infty$ as $t\to+\infty$ for all $s\in O$,

  * $\bullet$

for any constants $c_{1},c_{2}\geq 0$, it holds
$r_{A}(t,c_{1}t+c_{2})\to+\infty$ as $t\to+\infty$,

  * $\bullet$

for sequences $\\{s_{n}\\}\subset\mathbb{R}_{+}$ and $\\{t_{n}\\}\subset$ such
that

$s_{n}\to+\infty,\mbox{ $t_{n}\to+\infty$ and $\frac{t_{n}}{s_{n}}\to 0$ as
$n\to\infty$},$

we have

$r_{A}(s_{n},t_{n})\to+\infty\mbox{ as $n\to\infty$}.$

${\underline{H(B)}}$(ii)”: for each $u\in V$, $E\ni w\mapsto B(u,w)\in E^{*}$
is stable $\theta$-pseudomonotone with respect to $\\{l\\}$.

${\underline{H(B)}}$(v)”: there exists a function
$r_{B}\colon\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}$ such that

$\langle B(u,w),w\rangle_{E}\geq r_{B}(\|w\|_{E},\|u\|_{V})\|w\|_{E}\mbox{ for
all $u\in V$ and $w\in E$},$

and

  * $\bullet$

for every nonempty and bounded set $O\subset\mathbb{R}_{+}$, we have
$r_{B}(t,s)\to+\infty$ as $t\to+\infty$ for all $s\in O$,

  * $\bullet$

for any constants $c_{1},c_{2}\geq 0$, it holds
$r_{B}(t,c_{1}t+c_{2})\to+\infty$ as $t\to+\infty$,

  * $\bullet$

for sequences $\\{s_{n}\\}\subset\mathbb{R}_{+}$ and $\\{t_{n}\\}\subset$ such
that

$s_{n}\to+\infty,\mbox{ $t_{n}\to+\infty$ and $\frac{t_{n}}{s_{n}}\to 0$ as
$n\to\infty$},$

we have

$r_{B}(s_{n},t_{n})\to+\infty\mbox{ as $n\to\infty$}.$

######  Corollary 3.3.

Suppose that $H(A)$(i), (ii)”, (iii), (iv)”, $H(B)$(i), (ii)”, (iii), (iv)”,
$H(0)$, $H(1)$, $H(\psi)$ and $H(\theta)$ are satisfied. Then, the set of
solutions to problem $(\ref{eqn003})$–$(\ref{eqn004})$ is nonempty and weakly
compact in $V\times E$.

######  Remark 3.1.

Corollary 3.3 coincides with a result of Liu-Yang-Zeng-Zhao [11, Theorem 7].
In comparision with that result, in the present paper, we give a new proof
which is based on a multivalued version of the Tychonoff fixed point principle
in a Banach space along with the theory of nonsmoth analysis, the generalized
monotonicity arguments and the Minty approach. We conclude that our results
are much more general, improve the former one in several directions, and are
proved by using a new approach.

##  4\. Conclusions

In the present paper, a coupled system which consists of two nonlinear
variational-hemivariational inequalities with constraints in Banach spaces has
been investigated. A general existence result to the system was established by
using a multivalued version of the Tychonoff fixed point principle in a Banach
space together with the theory of nonsmoth analysis, generalized monotonicity
arguments and the Minty approach. Our result extends the recent ones obtained
in [11, Theorem 7].

There are plenty of problems arising in engineering applications which can be
formulated as a system of coupled variational-hemivariational inequalities.
With this motivation, in the future, we plan to utilize the theoretical
results established in this paper to study various real engineering problems.
Also, we will further develop the mathematical theory for systems of the
variational-hemivariational inequalities, to cover, for instance, stability
analysis, optimal control, sensitivity, and homogenization.

Acknowledgments

This project has received funding from the NNSF of China Grant Nos. 12001478
and 12101143, the European Union’s Horizon 2020 Research and Innovation
Programme under the Marie Sklodowska-Curie grant agreement No. 823731 CONMECH,
and the Startup Project of Doctor Scientific Research of Yulin Normal
University No. G2020ZK07. It is also supported by Natural Science Foundation
of Guangxi Grants Nos. 2021GXNSFFA196004, 2020GXNSFBA297137, GKAD21220144,
2018GXNSFAA281353, the Ministry of Science and Higher Education of Republic of
Poland under Grant No. 440328/PnH2/2019, and the National Science Centre of
Poland under Project No. 2021/41/B/ST1/01636.

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