Introduction
Nonlinear boundary value problems involving the generalized p-Laplacian operator arise from many physical phenomena, such as reaction-diffusion problems, petroleum extraction, flow through porous media and non-Newtonian fluids, just to name a few. Hence, the study of such problems and their generalizations have attracted numerous attention in recent years. For example, based on Calvert and Gupta’s [1] result on perturbations of the ranges of m-accretive mappings (stated as Theorem 1.1 in Section 1.2), Wei and Agarwal [2] have studied the following nonlinear elliptic boundary value problem involving the generalized p-Laplacian:
$$ \left \{ \textstyle\begin{array}{l} -\operatorname{div} [(C(x)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u ]+\varepsilon|u|^{q-2}u+g(x,u(x))= f(x),\quad \mbox{a.e. in } \Omega, \\ - \langle\vartheta,(C(x)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u \rangle\in\beta_{x}(u(x)),\quad \mbox{a.e. on } \Gamma, \end{array}\displaystyle \right . $$
(1.1)
where \(0\leq C(x)\in L^{p}(\Omega)\), \(\beta_{x}\) is the subdifferential of a proper, convex, and lower-semi-continuous function, ε is a non-negative constant and ϑ denotes the exterior normal derivative of Γ. It is shown that (1.1) has solutions in \(L^{s}(\Omega)\) under some conditions, where \(\frac{2N}{N+1} < p\leq s <+\infty\), \(1\leq q<+\infty\) if \(p\geq N\), and \(1\leq q \leq \frac{Np}{N-p}\) if \(p< N\), for \(N \geq1\).
Recently, the work on the generalized p-Laplacian operator problem (1.1) is extended to the so-called p-Laplacian-like problem
$$ \left \{ \textstyle\begin{array}{l} -\operatorname{div} [(C(x)+|\nabla u|^{2})^{\frac{s}{2}}|\nabla u|^{m-1}\nabla u ] + \varepsilon|u|^{q-2}u + g(x,u(x)) = f(x),\quad \mbox{in } \Omega, \\ - \langle\vartheta,(C(x)+|\nabla u|^{2})^{\frac {s}{2}}|\nabla u|^{m-1}\nabla u \rangle \in\beta_{x}(u), \quad \mbox{on } \Gamma. \end{array}\displaystyle \right . $$
(1.2)
Using Theorem 1.1 again, it is shown in [3] that (1.2) has solutions in \(L^{p}(\Omega)\) under some conditions, where \(\frac{2N}{N+1} < p <+\infty\), \(1\leq q<+\infty\) if \(p\geq N\), and \(1\leq q \leq\frac{Np}{N-p}\) if \(p< N\), for \(N \geq1\).
Since one system, expressed by one equation, interacts with another system in reality, the study of nonlinear systems with \((p,q)\)-Laplacian is also an important topic. In the non-Newtonian theory, the quantity \((p,q)\) is a characteristic of the medium. Media with \((p,q)> (2,2)\) are called dilatant fluids, those with \((p,q) < (2,2)\) are called pseudodoplastics, and if \((p,q) = (2,2)\), they are called Newtonian fluids. The studies on the p-Laplacian boundary value problems have been extended to cases of nonlinear Neumann elliptic systems with \((p,q)\)-Laplacian. For example, in [4] the following system with Neumann boundaries has been discussed:
$$ \left \{ \textstyle\begin{array}{l} -\Delta_{p}u + \varepsilon_{1} |u|^{p-2}u+g(x,u(x),v(x)) = f_{1}(x),\quad \mbox{a.e. in } \Omega, \\ -\Delta_{q}v+ \varepsilon_{2} |v|^{q-2}v + g(x,v(x),u(x)) = f_{2}(x),\quad \mbox{a.e. in } \Omega, \\ - \langle\vartheta, |\nabla u|^{p-2}\nabla u \rangle \in \beta_{x}(u(x)) ,\quad \mbox{a.e. on } \Gamma, \\ - \langle\vartheta, |\nabla v|^{q-2}\nabla v \rangle\in \beta_{x}(v(x)),\quad \mbox{a.e. on } \Gamma. \end{array}\displaystyle \right . $$
(1.3)
Inspired by Theorem 1.1 again, a sufficient condition on the existence of a solution in \(L^{p}(\Omega)\times L^{q}(\Omega)\) is presented in [4].
On the other hand, based on Brezis’ result [5] (stated as Theorem 1.2 in Section 1.2), Wei et al. [6] have studied the following nonlinear Dirichlet elliptic system in \(W^{1,p}(\Omega)\times W^{1,q}(\Omega)\):
$$ \left \{ \textstyle\begin{array}{l} -\Delta_{p}u+\varepsilon_{1}|u|^{p-2}u-\Delta_{q} v+\varepsilon_{2}|v|^{q-2}v =f_{1}(x)+f_{2}(x),\quad \mbox{a.e. in } \Omega, \\ \gamma_{1} u = g_{1}(x) , \qquad \gamma_{2} v = g_{2}(x),\quad \mbox{a.e. on } \Gamma, \end{array}\displaystyle \right . $$
(1.4)
and then extend (1.4) to the following two cases with generalized \((p,q)\)-Laplacian:
$$ \left \{ \textstyle\begin{array}{l} -\operatorname{div}(\alpha_{1}(\operatorname{grad} u))+\varepsilon_{1}|u|^{p-2}u-\operatorname{div}(\alpha_{2}(\operatorname{grad} v))+\varepsilon_{2}|v|^{q-2}v \\ \quad =f_{1}(x)+f_{2}(x), \quad \mbox{a.e. in } \Omega, \\ \gamma_{1} u = g_{1}(x) ,\qquad \gamma_{2} v = g_{2}(x), \quad \mbox{a.e. on } \Gamma \end{array}\displaystyle \right . $$
(1.5)
and
$$ \left \{ \textstyle\begin{array}{l} -\operatorname{div} [(C_{1}(x)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u ]+\varepsilon_{1}|u|^{p-2}u -\operatorname{div} [(C_{2}(x)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v ]+ \varepsilon_{2}|v|^{q-2}v \\ \quad =f_{1}(x)+f_{2}(x),\quad \mbox{a.e. in } \Omega, \\ \gamma_{1} u = g_{1}(x) ,\qquad \gamma_{2} v = g_{2}(x),\quad \mbox{a.e. on } \Gamma. \end{array}\displaystyle \right . $$
(1.6)
Integro-differential equation is also a much-studied topic in applied mathematics. Most of the existing techniques used to discuss the existence and uniqueness of the solution to integro-differential equation involves the finite element method. In [7], a new method based on a result of Zeidler [8] (stated as Theorem 1.3 in Section 1.2) is employed to tackle the following nonlinear integro-differential equation involving the generalized p-Laplacian operator with mixed boundary conditions:
$$ \left \{ \textstyle\begin{array}{l} \frac{\partial u}{\partial t}-\operatorname{div} [(C(x,t)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u ] +\varepsilon|u|^{q-2}u + a \frac{\partial}{\partial t}\int_{\Omega}u \,dx \\ \quad = f(x,t),\quad (x,t)\in\Omega\times(0,T), \\ - \langle\vartheta, (C(x,t)+|\nabla u|^{2})^{\frac {p-2}{2}}\nabla u \rangle\in\beta_{x}(u) ,\quad (x,t)\in\Gamma\times (0,T), \\ u(x,0) = u(x,T), \quad x \in \Omega. \end{array}\displaystyle \right . $$
(1.7)
It is proved that (1.7) has a unique solution in \(L^{p}(0 , T; W^{1,p}(\Omega))\), where \(1 < q \leq p < +\infty\).
Inspired by the work on (1.7), the following nonlinear integro-differential system involving the generalized \((p,q)\)-Laplacian is investigated in [9]:
$$ \left \{ \textstyle\begin{array}{l} \frac{\partial u(x,t)}{\partial t} -\operatorname{div} [(C_{1}(x,t)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u ] + \varepsilon_{1} |u|^{r-2}u + g_{1}(x,u, \nabla u)+ a_{1} \frac{\partial}{\partial t} \int_{\Omega}u\,dx \\ \quad = f_{1}(x,t),\quad (x,t) \in\Omega\times(0,T), \\ \frac{\partial v(x,t)}{\partial t} - \operatorname{div} [(C_{2}(x,t)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v ] + \varepsilon_{2} |v|^{s-2}v + g_{2}(x,v,\nabla v)+ a_{2} \frac{\partial}{\partial t} \int_{\Omega}v\,dx \\ \quad =f_{2}(x,t),\quad (x,t) \in\Omega\times(0,T), \\ - \langle\vartheta,(C_{1}(x,t)+|\nabla u|^{2})^{\frac {p-2}{2}}\nabla u \rangle\in\beta_{x}(u),\quad (x,t) \in\Gamma\times(0,T), \\ - \langle\vartheta,(C_{2}(x,t)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v \rangle\in \beta_{x}(v),\quad (x,t) \in\Gamma\times(0,T), \\ u(x,0) = u(x, T), \qquad v(x,0) = v(x,T),\quad x \in\Omega, \end{array}\displaystyle \right . $$
(1.8)
where \(\nabla u = (\frac{\partial u}{\partial x_{1}}, \frac {\partial u}{\partial x_{2}}, \ldots, \frac{\partial u}{\partial x_{N}} )\) and \(x = (x_{1}, x_{2}, \ldots, x_{N}) \in\Omega\). Based on a result of [10] (stated as Theorem 1.4 in Section 1.2), the existence of the unique non-trivial solution of (1.8) in \(L^{p}(0 , T;W^{1,p}(\Omega))\times L^{q}(0 , T; W^{1,q}(\Omega))\) is presented, where \(N\geq1\), \(\frac{2N}{N+1} < r \leq\min\{p,p'\} < +\infty\), and \(\frac{2N}{N+1} < s \leq\min\{q,q'\} < +\infty\). (Here, \(\frac{1}{p}+\frac{1}{p'}=1\), \(\frac{1}{q}+\frac{1}{q'}=1\).)
Parabolic equations are equally important as elliptic equations and integro-differential equations. The generalized \((p,q)\)-Laplacian parabolic equation with mixed boundaries has been extensively studied in [11],
$$ \left \{ \textstyle\begin{array}{l} \frac{\partial u}{\partial t}-\operatorname{div} [(C(x,t)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u ] + \varepsilon|u|^{p-2}u = f(x,t),\quad (x,t) \in\Omega\times(0,T), \\ - \langle\vartheta,(C(x,t)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u \rangle\in\beta(u) - h(x,t),\quad (x,t) \in\Gamma\times (0,T), \\ u(x,0) = u(x,T),\quad x \in\Omega. \end{array}\displaystyle \right . $$
(1.9)
It is shown that (1.9) has a unique solution in \(L^{p}(0,T; W^{1,p}(\Omega))\) where \(p \geq2\). The discussion of (1.9) in [11] is mainly based on Theorem 1.2 and a result of Reich [12] (stated as Theorem 1.5 in Section 1.2).
From the above research, we notice that it is not easy to check the assumptions presented in Theorems 1.1-1.5. As such we are motivated to extend the previous work to new problems and also to simplify the proof of the result. Indeed, motivated by the systems (1.4)-(1.6), (1.8), and (1.9), in this paper we shall employ a result of Zeidler [8] (stated as Theorem 1.6 in Section 1.2) as the main tool to obtain sufficient conditions for the existence and uniqueness of solutions for three nonlinear systems - the first is a nonlinear elliptic system involving the generalized \((p,q)\)-Laplacian with Neumann boundaries, the second is a nonlinear parabolic system involving the generalized \((p,q)\)-Laplacian with mixed boundaries, and the third is a nonlinear integro-differential system involving the generalized \((p,q)\)-Laplacian with mixed boundaries. The three systems considered are as follows:
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} -\operatorname{div} [(C_{1}(x)+|\nabla u|^{2})^{\frac {p-2}{2}}\nabla u ] + \varepsilon_{1} |u|^{r-2}u - \operatorname{div} [(C_{2}(x)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v ] \\ \qquad {}+ \varepsilon_{2} |v|^{s-2}v + g_{1}(x, u , \nabla u)+ g_{2}(x,v,\nabla v) \\ \quad = f_{1}(x)+ f_{2}(x),\quad x \in\Omega, \\ - \langle\vartheta,(C_{1}(x)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u \rangle- \langle\vartheta,(C_{2}(x)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v \rangle \\ \quad \in \beta_{x}(u)+\beta_{x}(v),\quad x \in\Gamma; \end{array}\displaystyle \right . \end{aligned}$$
(1.10)
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} \frac{\partial u(x,t)}{\partial t} -\operatorname{div} [(C_{1}(x,t)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u ] + \varepsilon_{1} |u|^{r-2}u + g_{1}(x,u, \nabla u)+\frac{\partial v(x,t)}{\partial t} \\ \qquad {} - \operatorname{div} [(C_{2}(x,t)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v ] + \varepsilon_{2} |v|^{s-2}v + g_{2}(x,v,\nabla v) \\ \quad = f_{1}(x,t) +f_{2}(x,t),\quad (x,t) \in\Omega\times(0,T), \\ - \langle\vartheta,(C_{1}(x,t)+|\nabla u|^{2})^{\frac {p-2}{2}}\nabla u \rangle- \langle\vartheta,(C_{2}(x,t)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v \rangle \\ \quad \in\beta_{x}(u)+ \beta_{x}(v),\quad (x,t) \in\Gamma\times (0,T), \\ u(x,0) = u(x, T), \qquad v(x,0) = v(x,T), \quad x \in\Omega; \end{array}\displaystyle \right . \end{aligned}$$
(1.11)
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} \frac{\partial u(x,t)}{\partial t} -\operatorname{div} [(C_{1}(x,t)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u ] + \varepsilon_{1} |u|^{r-2}u+ g_{1}(x,u, \nabla u) \\ \qquad {}+a_{1}\frac{\partial}{\partial t }\int_{\Omega}u \,dx+ \frac{\partial v(x,t)}{\partial t} - \operatorname{div} [(C_{2}(x,t)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v ] \\ \qquad {} + \varepsilon_{2} |v|^{s-2}v + g_{2}(x,v,\nabla v)+a_{2}\frac{\partial}{\partial t }\int_{\Omega}v \,dx \\ \quad = f_{1}(x,t) +f_{2}(x,t), \quad (x,t) \in\Omega\times(0,T), \\ - \langle\vartheta,(C_{1}(x,t)+|\nabla u|^{2})^{\frac {p-2}{2}}\nabla u \rangle- \langle\vartheta,(C_{2}(x,t)+|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v \rangle \\ \quad \in \beta_{x}(u)+\beta_{x}(v),\quad (x,t) \in \Gamma\times(0,T), \\ u(x,0) = u(x, T), \qquad v(x,0) = v(x,T), \quad x \in\Omega. \end{array}\displaystyle \right . \end{aligned}$$
(1.12)
The investigation of systems (1.10)-(1.12) will be presented in Sections 2-4, respectively, and more details of these systems will be introduced in these sections. Finally, in Section 5 we shall present some examples of (1.10)-(1.12).
Preliminaries
Let X be a real Banach space with its dual \(X^{*}\) being strictly convex. We shall use \((\cdot,\cdot)\) to denote the generalized duality pairing between X and \(X^{*}\). For any subset G of X, we denote by intG its interior and G̅ its closure, respectively. For two subsets \(G_{1}\) and \(G_{2}\) in X, if \(\overline{G}_{1}= \overline{G}_{2}\) and \(\operatorname{int} G_{1} = \operatorname{int} G_{2}\), then we say that \(G_{1}\) is almost equal to \(G_{2}\), denoted by \(G_{1} \simeq G_{2}\). We use ‘\(w\mbox{-}\!\lim\)’ to denote the weak convergence. A mapping \(T:D(T)= X\rightarrow X^{*}\) is said to be hemi-continuous on X [13] if \(w\mbox{-}\!\lim_{t \rightarrow 0}T(x+ty) = Tx\), for any \(x,y \in X\). A mapping \(T:D(T)= X\rightarrow X^{*}\) is said to be demi-continuous on X [13] if \(w\mbox{-}\!\lim_{n \rightarrow\infty}Tx_{n} = Tx\), for any sequence \(\{x_{n}\}\) strongly converges to x in X.
Let \(J_{r}\) denote the duality mapping from X into \(2^{X^{*}}\), which is defined by
$$J_{r}(x)=\bigl\{ f\in X^{*} : (x, f) = \|x\|^{r}, \|f\|=\|x\|^{r-1}\bigr\} ,\quad x \in X, $$
where \(r > 1\) is a constant. If \(r \equiv2\), then we use J to denote \(J_{2}\), which is called the normalized duality mapping. It is well known that, in general, \(J_{r}(x) = \|x\|^{r-2}J(x)\), for all \(x \neq0\). Since \(X^{*}\) is strictly convex, J is a single-valued mapping [1, 14].
A multi-valued mapping \(B:X\rightarrow2^{X^{*}}\) is said to be monotone [14] if \((u_{1}-u_{2},w_{1}-w_{2})\ge0\), for any \(u_{i}\in D(B)\) and \(w_{i}\in Bu_{i}\), \(i=1,2\). The monotone operator B is said to be maximal monotone if \(R(J+rB) = X^{*}\), for any \(r> 0 \). The mapping \(B:X\rightarrow2^{X^{*}}\) is said to be strictly monotone [14] if \((u_{1}-u_{2},w_{1}-w_{2})= 0\), for \(w_{i}\in Bu_{i}\), \(i=1,2\), implies \(u_{1} = u_{2}\). The mapping B is said to be coercive [13, 14] if \(\lim_{n\rightarrow +\infty}{(x_{n},x^{*}_{n})}/{\|x_{n}\|}=+\infty\) for all \(x_{n}\in D(B)\), \(x^{*}_{n}\in Bx_{n}\) such that \(\lim_{n\rightarrow +\infty}\|x_{n}\|= +\infty\).
Let \(B : X \rightarrow2^{X^{*}}\) be a maximal monotone operator such that \(0 \in B0\), then the equation \(J(u_{t} - u) + t Bu_{t} \ni 0\) has a unique solution \(u_{t} \in D(B)\) for every \(u \in X\) and \(t > 0\). The resolvent
\(J_{t}^{B}\) and the Yosida approximation
\(B_{t}\) of B are defined by \(J_{t}^{B} u = u_{t}\) and \(B_{t} u = -\frac{1}{t}J(u_{t}-u)\) for all \(u \in X\) and \(t > 0\) [14].
For \(k \in(-\infty, +\infty)\), a multi-valued mapping \(\widetilde{A}: D(\widetilde{A})\subset X \rightarrow2^{X}\) is said to be k-accretive [10] if
$$ \bigl(v_{1}-v_{2},J(u_{1}-u_{2}) \bigr)\geq k \|u_{1}-u_{2}\|^{2}, $$
(1.13)
for any \(u_{i} \in D(\widetilde{A})\) and \(v_{i}\in\widetilde{A}u_{i}\), \(i=1,2\). For \(k > 0 \) in inequality (1.13), we say that à is strongly accretive while for \(k = 0\), à is simply called accretive. An accretive mapping à is said to be m-accretive if \(R(I+\lambda\widetilde{A})=X\) for some \(\lambda>0\). We say that a mapping \(\widetilde{A}:X\rightarrow2^{X}\) is boundedly-inversely-compact [1] if, for any pair of bounded subsets G and \(G'\) of X, the subset \(G\cap \widetilde{A}^{-1}(G')\) is relatively compact in X.
Let C be a closed convex subset of X and let \(A: C \rightarrow 2^{X^{*}}\)be a multi-valued mapping. Then A is said to be a pseudo-monotone operator [14] provided that
-
(i)
for each \(x \in C\), the image Ax is a non-empty closed and convex subset of \(X^{*}\);
-
(ii)
if \(\{x_{n}\}\) is a sequence in C converging weakly to \(x \in C \) and if \(f_{n} \in Ax_{n}\) is such that \(\limsup_{n\rightarrow \infty}(x_{n} - x,f_{n}) \leq0\), then to each element \(y \in C\), there corresponds an \(f(y) \in Ax \) with the property that \((x - y, f(y))\leq\liminf_{n \rightarrow\infty} (x_{n} - x, f_{n})\);
-
(iii)
for each finite-dimensional subspace F of X, the operator A is continuous from \(C \cap F\) to \(X^{*}\) in the weak topology.
A function Φ is called a proper convex function on X [14] if Φ is defined from X to \((-\infty, +\infty]\), not identically +∞, such that \(\Phi((1-\lambda)x+\lambda y)\leq(1-\lambda)\Phi(x)+\lambda\Phi(y)\), whenever \(x , y \in X\) and \(0 \leq\lambda\leq1\).
A function \(\Phi: X \rightarrow(-\infty, +\infty]\) is said to be lower-semi-continuous on X [14] if \(\liminf_{y \rightarrow x}\Phi(y)\geq\Phi(x)\), for any \(x \in X\).
Given a proper convex function Φ on X and a point \(x \in X\), we denote by \(\partial\Phi(x)\) the set of all \(x^{*} \in X^{*}\) such that \(\Phi(x)\leq\Phi(y)+ (x - y, x^{*})\), for any \(y \in X\). Such element \(x^{*}\) is called the subgradient of Φ at x, and \(\partial\Phi(x)\) is called the subdifferential of Φ at x [14].
For easy reference of the reader, Theorems 1.1-1.5 mentioned in Section 1.1 are stated as follows.
Theorem 1.1
[1]
Let
X
be a real Banach space with a strictly convex dual space
\(X^{*}\). Let
\(J_{r}: X\rightarrow X^{*}\)
be a duality mapping on
X
and there exists a function
\(\eta: X \rightarrow[0,+\infty)\)
such that for all
\(u, v \in X\),
$$ \|J_{r}u - J_{r}v\| \leq\eta(u-v). $$
(1.14)
Let
\(A, C_{1}:X\rightarrow2^{X}\)
be accretive mappings such that
-
(i)
either both
A
and
\(C_{1}\)
satisfy the following condition (1.15), or
\(D(A)\subset D(C_{1})\)
and
\(C_{1}\)
satisfies the condition (1.15):
$$ \left \{ \textstyle\begin{array}{l} \textit{for }u\in D(A)\textit{ and }v \in Au, \textit{there exists a constant }C(a,f) \textit{ such that} \\ (v - f, J_{r}(u - a)) \geq C(a,f); \end{array}\displaystyle \right . $$
(1.15)
-
(ii)
\(A+C_{1}\)
is
m-accretive and boundedly-inversely-compact.
Let
\(C_{2}:X\rightarrow X\)
be a bounded continuous mapping such that, for any
\(y\in X\), there is a constant
\(C(y)\)
satisfying
\((C_{2}(u+y),J_{r}u)\ge-C(y)\)
for any
\(u\in X\). Then the following results hold:
-
(a)
\(\overline{[R(A)+R(C_{1})]}\subset\overline {R(A+C_{1}+C_{2})}\);
-
(b)
\(\operatorname{int}[R(A)+R(C_{1})]\subset \operatorname{int} R(A+C_{1}+C_{2})\).
Theorem 1.2
[5]
Let
\(T: X \rightarrow X^{*}\)
be a bounded and pseudo-monotone operator, and
K
be a closed and convex subset of
X. Suppose that Φ is a lower-semi-continuous and convex function defined on
K, which is not always +∞, such that
\(\Phi(v) \in(-\infty, +\infty]\), for any
\(v \in K\). Suppose there exists
\(v_{0} \in K\)
such that
\(\Phi(v_{0})<+\infty\), and
$$\frac{(v - v_{0}, Tv) + \Phi(v)}{\|v\|} \rightarrow\infty, $$
as
\(\|v\|\rightarrow\infty\), \(v \in K\). Then there exists
\(u \in K\)
such that
\((u - v,Tu) \leq\Phi(v)- \Phi(u)\), for all
\(v \in K\).
Theorem 1.3
[8]
Let
X
be a real reflexive Banach space with
\(X^{*}\)
being its dual space. Let
C
be a non-empty closed convex subset of
X. Assume that
-
(i)
the mapping
\(A: C \rightarrow2^{X^{*}} \)
is a maximal monotone operator;
-
(ii)
the mapping
\(B: C \rightarrow X^{*}\)
is pseudo-monotone, bounded, and demi-continuous;
-
(iii)
if the subset
C
is unbounded, then the operator
B
is
A-coercive with respect to the fixed element
\(b \in X^{*}\), i.e., there exist an element
\(u_{0} \in C \cap D(A)\)
and a number
\(r>0\)
such that
$$ (u - u_{0}, Bu)> (u - u_{0}, b), $$
(1.16)
for all
\(u \in C\)
with
\(\|u\| > r\).
Then the equation
\(b \in Au + Bu\)
has a solution.
Theorem 1.4
[10]
Let
X
be a smooth Banach space, \(A : D(A) \subset X \rightarrow2^{X}\)
be an
m-accretive mapping, and
\(S: D(S)\subset X \rightarrow X\)
be continuous and strongly accretive with
\(\overline{D(A)}\subset D(S)\). Then, for any
\(z \in X\), the equation
\(z \in Sx + \lambda Ax\)
has a unique solution
\(x_{\lambda}\), for any
\(\lambda> 0\).
Theorem 1.5
[12]
Let
X
be a real reflexive Banach space with both
X
and
\(X^{*}\)
being strictly convex. Let
\(J : X \rightarrow X^{*}\)
be the normalized duality mapping on
X. Let
A
and
B
be two maximal monotone operators in
X. If there exist
\(0 \leq k < 1\)
and
\(C_{1}, C_{2} > 0\)
such that
$$ \bigl(a, J^{-1}(B_{t} v)\bigr) \geq- k \|B_{t} v \|^{2}-C_{1} \|B_{t} v\| - C_{2}, $$
(1.17)
for any
\(v \in D(A)\), \(a \in Av\)
and
\(t > 0\) (\(B_{t}\)
is the Yosida approximation of
B), then
\(R(A) + R(B) \simeq R(A+B)\).
The following results will be needed in subsequent discussion.
Lemma 1.1
[14]
If
A
and
B
are maximal monotone operators in
X
such that
\((\operatorname{int} D(A))\cap D(B) \neq\emptyset\), then
\(A+B\)
is maximal monotone.
Lemma 1.2
[14]
If
\(\Phi: X \rightarrow R\)
is proper, convex, and lower-semi-continuous, then
∂Φ is maximal monotone.
Lemma 1.3
[14]
If
\(B: X \rightarrow2^{X^{*}}\)
is everywhere defined, monotone, and hemi-continuous, then
B
is maximal monotone.
Theorem 1.6
[8]
Assume that
X
is a real reflexive Banach space and the following conditions hold:
-
(H1)
The linear operator
\(L: D(L) \subseteq X \rightarrow X^{*}\)
is maximal monotone in
X.
-
(H2)
The operator
\(A: X \rightarrow2^{X^{*}}\)
is monotone.
-
(H3)
The functional
\(\varphi: X \rightarrow(-\infty, +\infty]\)
is convex, lower-semi-continuous, and
\(\varphi\neq +\infty\).
-
(H4)
One of the following conditions is satisfied:
-
(H4.1)
\(A : X \rightarrow X^{*}\)
is single-valued and hemi-continuous;
-
(H4.2)
A
is maximal monotone and
\(\operatorname{int} D(A) \cap D(\partial \varphi)\neq\emptyset\);
-
(H4.3)
A
is maximal monotone and
\(D(A) \cap \operatorname{int} D(\partial \varphi)\neq\emptyset\).
-
(H5)
The sum
\(L+A+\partial\varphi: X \rightarrow2^{X^{*}}\)
is coercive with respect to 0, i.e., there exist
\(r > 0\)
and
\(u_{0} \in D(L) \cap D(A) \cap D(\partial\varphi)\)
such that
$$\bigl(u - u_{0}, u^{*}\bigr) > 0, $$
for all
\((u,u^{*}) \in L + A +\partial\varphi\)
with
\(\|u\|> r\).
-
(H6)
\(D(L) \cap D(A+\partial\varphi)\neq\emptyset\).
Then the equation
$$0 \in Lu + Au + \partial\varphi(u), \quad u \in X, $$
has a solution.
Definition 1.1
For \(1 < p < +\infty\) and \(1 < q < +\infty\), we use Y to denote the product of two spaces \(W^{1,p}(\Omega)\) and \(W^{1,q}(\Omega)\), i.e., \(Y = W^{1,p}(\Omega)\times W^{1,q}(\Omega)= \{(u,v): u \in W^{1,p}(\Omega), v \in W^{1,q}(\Omega)\}\). The dual space of Y will be denoted by \(Y^{*}\). Also, Y will be endowed with the norm
$$\bigl\Vert (u,v)\bigr\Vert _{Y} = \sqrt{\|u \|_{1,p,\Omega}^{2}+\|v\|_{1,q,\Omega}^{2}},\quad (u,v) \in Y, $$
where \(\|\cdot\|_{1,p,\Omega}\) and \(\|\cdot\|_{1,q,\Omega}\) denote the norm in \(W^{1,p}(\Omega)\) and \(W^{1,q}(\Omega)\), respectively.
Definition 1.2
[15]
For \(1 < p < +\infty\), let \(L^{p}(0,T ; X)\) denote the space of all X-valued strongly measurable functions \(x(t)\) defined a.e. on \((0, T)\) such that \(\|x(t)\|^{p}_{X}\) is Lebesgue integrable over \((0,T)\). It is well known that \(L^{p}(0,T; X)\) is a Banach space with the norm defined by \(\|x\|_{L^{p}(0,T ; X)} = (\int_{0}^{T} \|x(t)\|_{X} ^{p}\,dt )^{\frac{1}{p}}\). If X is reflexive, then \(L^{p}(0,T ; X)\) is reflexive, and its dual space coincides with \(L^{p'}(0,T ; X^{*})\), where \(\frac{1}{p}+ \frac{1}{p'}= 1\). Moreover, \(L^{p}(0,T ; X)\) is reflexive in the case when X is reflexive, and \(L^{p}(0,T ; X)\) is strictly (uniformly) convex in the case when X is strictly (uniformly) convex.
Definition 1.3
For \(1 < p < +\infty\) and \(1 < q < +\infty\), we use Z to denote the product of two spaces \(L^{p}(0,T; W^{1,p}(\Omega))\) and \(L^{q}(0,T; W^{1,q}(\Omega))\), i.e., \(Z = L^{p}(0,T; W^{1,p}(\Omega))\times L^{q}(0,T; W^{1,q}(\Omega)) = \{(u,v): u \in L^{p}(0,T; W^{1,p}(\Omega)), v \in L^{q}(0,T; W^{1,q}(\Omega))\}\). The dual space of Z is denoted by \(Z^{*}\). Also, Z will be endowed with the norm
$$\bigl\Vert (u,v)\bigr\Vert _{Z} = \sqrt{\|u \|_{L^{p}(0,T; W^{1,p}(\Omega))}^{2}+\|v\|_{L^{q}(0,T; W^{1,q}(\Omega))}^{2}},\quad (u,v) \in Z. $$