1 Introduction
The concept of well-posedness, which was firstly introduced by Tykhonov in [
1] for a minimization problem and thus was called Tykhonov well-posedness, has been studied widely in recent years for optimization problems, variational inequality problems, hemivariational inequality problems, fixed point problems, saddle point problems, equilibrium problems, and their related problems because of their important applications in physics, mechanics, engineering, economics, management science, etc. (see, for example, [
2‐
13]). Tykhonov well-posedness for an optimization problem is defined by requiring the existence and uniqueness of its solution and the convergence to the unique solution of its approximating sequences. There are a great many kinds of generalizations for the concept of well-posedness, such as Levitin-Polyak well-posedness, parametric well-posedness, and
α-well-posedness, to optimization problems, variational inequality problems, and their related problems (see, for example, [
14‐
21]).
Due to the close relationship between optimization problems and variational inequality problems, the concept of well-posedness for optimization problems is generalized to variational inequalities and their related problems. The earliest research work of well-posedness for variational inequalities should at least date back to 1980s when Lucchetti and Patrone [
22,
23] firstly introduced the concept of well-posedness for a variational inequality and proved some important results. After that, Lignola and Morgan [
20], Fang and Hu [
24], Huang and Yao [
25] have made significant contributions to the study of well-posedness for variational inequalities. As an important generalization of variation inequality, hemivariational inequality has drawn much attention of mathematical researchers due to its abundant applications in mechanics and engineering. With the tools of nonsmooth analysis and nonlinear analysis, many kinds of hemivariational inequalities have been studied since 1980s [
7,
26‐
30]. Also, many kinds of concepts of well-posedness hemivariational inequalities have been studied since Goeleven and Mentagui [
31] firstly introduced the concept of well-posedness to a hemivariational inequality in 1995. For more research work on the well-posedness for variational inequalities and hemivariational inequalities, we refer the readers to [
14,
20,
32‐
35].
Split variational inequality, which was introduced by Censor et al. in [
36], can be regarded as a generalization of variational inequality and includes as a special case, the split feasibility problem, which is an important model for a wide range of practical problems arising from signal recovery, image processing, and tensity-modulated radiation therapy treatment planning (see, for example, [
37‐
41]). Thus, the concepts of well-posedness and Levitin-Polyak well-posedness for various split variational inequalities were studied by Hu and Fang recently [
42]. Obviously, split hemivariational inequality could be regarded as a generalization of split variational inequality. It could arise in a system of hemivariational inequalities for modeling some frictional contact problems in mechanics, where two hemivariational inequalities are linked by a linear constraint. Also, when nonconvex and nonsmooth functionals are involved, the model for the above mentioned practical problems, such as signal recovery and image processing, turns to split hemivariational inequality rather than split variational inequality. However, as far as we know, there are few research works studying well-posedness for split hemivariational inequalities.
Inspired by recent research works on the well-posedness for split variational inequalities and hemivariational inequalities, in this paper, we focus on studying metric characterization of well-posedness for a class of split hemivariational inequalities specified as follows:
Find
\((u_{1},u_{2})\in V_{1}\times V_{2}\) such that
$$ \textstyle\begin{cases} u_{2}=Tu_{1}, \\ \langle A_{1}u_{1}-f_{1},v_{1}-u_{1}\rangle_{V_{1}^{*}\times V_{1}}+J_{1}^{\circ}(u_{1};v_{1}-u_{1})\geq0, \quad \forall v_{1}\in V_{1}, \\ \langle A_{2}u_{2}-f_{2},v_{2}-u_{2}\rangle_{V_{2}^{*}\times V_{2}}+J_{2}^{\circ}(u_{2};v_{2}-u_{2})\geq0, \quad \forall v_{2}\in V_{2}, \end{cases} $$
(SHI)
where, for
\(i=1,2\),
\(\langle\cdot,\cdot\rangle_{V_{i^{*}}\times V_{i}}\) denotes the duality paring between Banach space
\(V_{i}\) and its dual space
\(V_{i}^{*}\),
\(A_{i}:V_{i}\to V_{i}^{*}\) is an operator from
\(V_{i}\) to
\(V_{i}^{*} \),
\(f_{i}\) is a given point in
\(V_{i}^{*}\),
\(J_{i}:V_{i}\to\mathbb{R}\) is a locally Lipschitz functional on
\(V_{i}\) with
\(J_{i}^{\circ}(u_{i};v_{i}-u_{i})\) being its generalized directional derivative at
\(u_{i}\) in direction of
\(v_{i}-u_{i}\), which will be defined in the next section, and
\(T:V_{1}\to V_{2}\) is a continuous mapping from
\(V_{1}\) to
\(V_{2}\). After defining the concept of well-posedness for the split hemivariational inequality (
SHI), we present some metric characterizations for its well-posedness under very mild assumptions.
The remainder of the paper is organized as follows. In Sect.
2, we recall some crucial definitions and results. Under very mild assumptions on involved operators, Sect.
3 presents several results on the metric characterizations of well-posedness for the split hemivariational inequality (
SHI). At last, some concluding remarks are provided in Sect.
4.
3 Well-posedness and metric characterizations
In this section, we aim to extend the well-posedness to the split hemivariational inequality (
SHI). We first give the definition of well-posedness for the split hemivariational inequality (
SHI), and then we prove its metric characterizations for the well-posedness by using two useful sets defined.
In order to establish the metric characterizations for well-posedness of the split hemivariational inequality (
SHI), we first define two sets on
\(V_{1}\times V_{2}\) as follows: for
\(\varepsilon>0\),
$$\begin{aligned}& \Omega(\epsilon)=\left \{ (u_{1},u_{2})\in V_{1}\times V_{2} :\vphantom{\textstyle\begin{array}{l} \|u_{2}-Tu_{1}\|_{V_{2}}\leq\epsilon, \mbox{and } \forall v_{1}\in V_{1},v_{2}\in V_{2}, \\ \langle A_{1}u_{1}-f_{1},v_{1}-u_{1}\rangle_{V_{1}^{\ast}\times V_{1}}+ J_{1}^{\circ}(u_{1};v_{1}-u_{1})\geq-\varepsilon\|v_{1}-u_{1}\| _{V_{1}}, \\ \langle A_{2}u_{2}-f_{2},v_{2}-u_{2}\rangle_{V_{2}^{\ast}\times V_{2}}+ J_{2}^{\circ}(u_{2};v_{2}-u_{2})\geq-\varepsilon\|v_{2}-u_{2}\|_{V_{2}} \end{array}\displaystyle }\right. \\& \hphantom{\Omega(\epsilon)={}}{} \left.\textstyle\begin{array}{l} \|u_{2}-Tu_{1}\|_{V_{2}}\leq\epsilon, \mbox{and } \forall v_{1}\in V_{1},v_{2}\in V_{2}, \\ \langle A_{1}u_{1}-f_{1},v_{1}-u_{1}\rangle_{V_{1}^{\ast}\times V_{1}}+ J_{1}^{\circ}(u_{1};v_{1}-u_{1})\geq-\varepsilon\|v_{1}-u_{1}\| _{V_{1}}, \\ \langle A_{2}u_{2}-f_{2},v_{2}-u_{2}\rangle_{V_{2}^{\ast}\times V_{2}}+ J_{2}^{\circ}(u_{2};v_{2}-u_{2})\geq-\varepsilon\|v_{2}-u_{2}\|_{V_{2}} \end{array}\displaystyle \right \}, \\& \Psi(\epsilon)=\left \{ (u_{1},u_{2})\in V_{1}\times V_{2} :\vphantom{\textstyle\begin{array}{l} \|u_{2}-Tu_{1}\|_{V_{2}}\leq\epsilon, \mbox{and } \forall v_{1}\in V_{1},v_{2}\in V_{2}, \\ \langle A_{1}v_{1}-f_{1},v_{1}-u_{1}\rangle_{V_{1}^{\ast}\times V_{1}}+ J_{1}^{\circ}(u_{1};v_{1}-u_{1})\geq-\varepsilon\|v_{1}-u_{1}\| _{V_{1}}, \\ \langle A_{2}v_{2}-f_{2},v_{2}-u_{2}\rangle_{V_{2}^{\ast}\times V_{2}}+ J_{2}^{\circ}(u_{2};v_{2}-u_{2})\geq-\varepsilon\|v_{2}-u_{2}\|_{V_{2}} \end{array}\displaystyle }\right. \\& \hphantom{\Psi(\epsilon)={}}\left. \textstyle\begin{array}{l} \|u_{2}-Tu_{1}\|_{V_{2}}\leq\epsilon, \mbox{and } \forall v_{1}\in V_{1},v_{2}\in V_{2}, \\ \langle A_{1}v_{1}-f_{1},v_{1}-u_{1}\rangle_{V_{1}^{\ast}\times V_{1}}+ J_{1}^{\circ}(u_{1};v_{1}-u_{1})\geq-\varepsilon\|v_{1}-u_{1}\| _{V_{1}}, \\ \langle A_{2}v_{2}-f_{2},v_{2}-u_{2}\rangle_{V_{2}^{\ast}\times V_{2}}+ J_{2}^{\circ}(u_{2};v_{2}-u_{2})\geq-\varepsilon\|v_{2}-u_{2}\|_{V_{2}} \end{array}\displaystyle \right \}. \end{aligned}$$
With the definition of two sets \(\Omega(\epsilon)\) and \(\Psi (\epsilon)\), we can get the following properties.
With Lemmas
3.1 and
3.2, it is easy to get the following corollary on the closedness of
\(\Omega(\epsilon)\) for any
\(\epsilon>0\), which is crucial to the metric characterizations for well-posedness of the split hemivariational inequality (
SHI).
Now, with properties of the set
\(\Omega(\epsilon)\) given above, we are in a position to prove metric characterizations for the split hemivariational inequality (
SHI)by using similar methods for studying well-posedness of variational inequalities and hemivariational inequalities in research works [
17,
25,
46,
47].
The following is a concrete example to illustrate the metric characterization of well-posedness for a hemivariational inequality.
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