1 Introduction
For convenience, we first recall some related notations. Throughout this paper,
X and
Y denote Banach spaces and
A denotes a linear manifold in the product space
\(X\times Y\). We may view
A as a multi-valued linear operator from
X to
Y by taking
\(A ( x ) = \{ y: \{ x, y \} \in A \}\). The domain, range, and null space of
A are defined, respectively, by
$$\begin{aligned}& D ( A ) = \bigl\{ x\in X: \{ x,y \} \mbox{ for some }y\in Y \bigr\} ; \\& R ( A ) = \bigl\{ y\in Y: \{ x,y \} \mbox{ for some }x\in X \bigr\} ; \\& N ( A ) = \bigl\{ x\in D ( A ) : \{ x,\theta \} \in A \bigr\} . \end{aligned}$$
It is well known that the quadratic control problem subject to a certain class of boundary conditions can be equivalently formulated as the problem of finding a least-squares solutions (or extremal solutions) of an appropriate linear operator equations in Hilbert spaces (or Banach spaces). When the generalized quadratic cost function and the generalized boundary conditions are involved, the problem can be reformulated as a constrained least-squares solution (or extremal solution) of multi-valued linear operators
\(y\in A ( x )\) between Hilbert spaces (or Banach spaces)
X and
Y (see [
1]). If
X and
Y are Hilbert spaces, the orthogonal operator parts, the orthogonal generalized inverse of a linear manifold
A in
\(X\times Y\) and the least-squares solutions or the constrained least-squares solutions of multi-valued linear operators
\(y\in A ( x )\) were investigated by Lee and Nashed [
1‐
3]. If
X and
Y are Banach spaces, Lee and Nashed [
4] also introduced a concept of a generalized inverse
\(A^{\#}\) for the linear manifold
A in
\(X\times Y\) by means of algebraic projection and topological projection. In order to give the characterization of the set of all extremal solutions or least-extremal solutions of a linear inclusion
\(y\in A (x )\) in Banach space, in 2005, Wang and Liu [
5] introduced the concept of the metric generalized inverse
\(A^{\#}\) by means of the metric projection, which is nonlinear in general. In 2012, Wang
et al. [
6] also gave the criteria for the metric generalized inverse of multi-valued linear operators in Banach space.
Let
L be a linear manifold in
\(X\times Y\), or, equivalently, the graph of a multi-valued linear operator from
X to
Y and let
S be a prescribed hyperplane in
X,
i.e.
\(S=g+N\), we denote
\(A:=L|_{N}\). The problem in our general setting is to determine, for a given
\(y\in Y\), a vector
\(\omega \in S\cap D ( A )\) such that, for some
\(z\in A|_{S} ( \omega )\),
\(\| z-y\| =\operatorname{dist} ( y,R ( A|_{S} ) )\), such a vector
w is called the constrained extremal solution of multi-valued linear inclusions
\(y\in A ( x )\) in Banach space. The main purpose of this paper is to investigate the constrained extremal solution problem in Banach spaces in an abstract general setting. We first establish three equivalent characterizations of a constrained extremal solution of linear inclusions in Banach spaces by means of the algebraic operator parts, the metric generalized inverse of multi-valued linear operator, and the dual mapping of the spaces. It follows from the main results in this paper that we may get the constrained extremal solution of multi-valued linear inclusions, by using the extremal solution of some interrelated multi-valued linear inclusions in the same spaces, which are well investigated by using the algebraic operator parts, the metric generalized inverse of multi-valued linear operator in [
5] and [
6]. The setting in this paper includes large classes of constrained extremal problems and optimal control problems subject to generalized boundary conditions [
7].
In this paper,
X,
Y, and
Z denote Banach spaces. The following are standard notations (see [
1‐
3,
7]), but for convenience, we recall them again. For
\(A, B\subset X\times Y\),
\(C\subset Z\times X\),
$$\begin{aligned}& AC = \bigl\{ \{ z,y \} : \{ z,x \} \in C, \{x,y \} \in A \bigr\} ; \\& \alpha A = \bigl\{ \{ x,\alpha y \} : \{ x,y \}\in A \bigr\} ,\quad \alpha\in R^{1}; \\& A\dotplus B = \{ a+b:a\in A,b\in B \} ; \\& A+B = \bigl\{ \{ x,y+z \} : \{ x,y \}, \{ x,z \} \bigr\} . \end{aligned}$$
The main mathematics tools in this investigation are the algebraic operator part and metric generalized inverse of linear manifold in Banach spaces, we recall and describe them in Section
2 (see [
5] and [
6]). The other mathematics method is the generalized orthogonal decomposition theorem in Banach space, which is given by one of the authors in another paper (see Lemma
2.4 in Section
2).
2 Preliminaries and basic notions
Let
X and
Y be Banach spaces, and
A be a linear manifold (or linear subspace) in the product space
\(X\times Y\). The inverse relation
\(A^{-1}\), which is the graph inverse of
A, always exists and is given by
$$ A^{-1}= \bigl\{ \{ y,x \} : \{ x,y \} \in A \bigr\} . $$
If
S is a set in
X, then
$$ A ( S ) = \bigl\{ y: \{ x,y \} \in A,\mbox{ for some }x\in S \bigr\} . $$
It is possible that
\(A ( S )\) is empty. The restriction of
A to
S will be denoted by
$$ A|_{S}:= \bigl\{ \{ x,y \} : \{ x,y \} \in A\mbox{ and } x \in S \bigr\} . $$
If
\(T:X\rightarrow Y\) is a single-valued operator from
X to
Y with domain
\(D ( T ) \), then the gragh of
T, denote
\(Gr ( T )\), defined by
$$ Gr ( T ) = \bigl\{ \bigl( x,T ( x ) \bigr) :x\in D ( T ) \bigr\} . $$
For a multi-valued linear operator A from X into Y, we may introduce a single-valued operator from \(D ( A )\) into Y, denoted \(A_{S,P}\), which is defined as follows.
In this case, for any \(x\in D ( A )\), we may have \(A ( x ) =A_{S,P} ( x ) +A ( \theta )\) and express the variational set \(A ( x )\) as a variable \(A_{S,P} ( x )\) plus the fixed set \(A ( \theta )\). Since \(A ( \theta ) \) is a fixed subspace of Y, then there forever is an algebraic operator part of A.
Next, we introduce the concept of a constrained extremal solution of multi-valued linear inclusions in Banach spaces.
If
\(S=X\), the constrained extremal solution of the linear inclusion
\(y\in A ( x )\) with respect to
S is just the extremal solution or the extremal solution, which was defined in [
5].
Now we recall some notions and results in [
5], which were used in this paper on many occasions.
A subset
G in a Banach space
X is said to be proximal if every element
\(x\in X\) has at least one element of best approximation in
G,
i.e.
$$ \mathcal{P}_{G} ( x ) = \bigl\{ x_{0}\in G: \| x-x_{0}\| =\inf_{y\in G}\|x-y\| \bigr\} \neq \emptyset. $$
G is said to be a semi-Chebyshev set, if every element
\(x\in X\) has at most one element of best approximation in
G,
i.e.
\(x\in X\),
\(x_{1}, x_{2}\in \mathcal{P}_{G} ( x )\) implies
\(x_{1}=x_{2}\).
G is said a Chebyshev set if it is simultaneously a proximal and a semi-Chebyshev set (see [
8]). When
G is a Chebyshev set, we denote
\(\mathcal{P}_{G} ( x ) = \{ \pi_{G} ( x ) \} \), where
\(\pi_{G}\) is called the metric projector from
X onto
G.
It is well known that if
X is a reflexive Banach space and
\(G\subset X \) is a convex closed set, then
G is a proximal set, while if
X is a strictly convex Banach space and
G is a convex closed set, then
G is a semi-Chebyshev set (see [
8]).
We may use some properties of the metric projector, now we recall them.
We also use the dual mapping of Banach space, let us recall it.
Let
X be a Banach space, the set-valued mapping
\(F_{X}:X\rightarrow 2^{X^{\ast}}\), defined by
$$ F_{X} ( x ) = \bigl\{ x^{\ast}\in X^{\ast}: \bigl\langle x^{\ast },x \bigr\rangle = \bigl\| x^{\ast} \bigr\| ^{2}= \| x\| ^{2} \bigr\} $$
for
\(x\in X\), is called the dual mapping of
X, where
\(\langle x^{\ast }, x\rangle\) denotes the value of the functional
\(x^{\ast}\in X^{\ast }\) on
\(x\in X\) (see [
9]).
In Banach space, there is no the concept of the orthogonal property just as in Hilbert space. By using the dual mapping of the Banach space X and the Chebyshev property of subspace G, we can extend the Riesz orthogonal decomposition theorem from Hilbert space into Banach space.
Next, we recall the concept of the metric generalized inverse \(A^{\#}\), which is a single-valued operator of a multi-valued linear operator A from X into Y. By means of the metric generalized inverse \(A^{\#}\), we can express the constrained extremal solution of multi-valued linear inclusions \(y\in A ( x )\) in Banach space.
For convenience we list them as the following propositions the results in [
5].
3 Main theorems
In this section, we consider the constrained extremal problems for a linear inclusions restricted to a hyperplane in Banach space. By using Proposition
2.3, we establish several equivalent characterizations of the constrained extremal solution of linear inclusions in Banach spaces by means of the algebraic operator parts, the metric generalized inverse of multi-valued linear operator, and the dual mapping of the spaces. It follows from these results that we may get the constrained extremal solution of multi-valued linear inclusions by using the extremal solution of some interrelated multi-valued linear inclusions in the same space, which are well investigated in [
5] and [
6]. These characterizations involve algebraic operator parts, the metric generalized inverse, and the dual mapping of the spaces.
The main results, (i)-(iii) in Theorem 3.1 and (i)-(ii) in Theorem 3.2 in [
7], will be especial cases of Theorem
3.1 and Theorem
3.2. We express them as the following corollary.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
ZW and BW conceived and designed the study. ZW wrote and edited the manuscript. BW and YW examined all the steps of the proofs in this research and gave some advice. All authors read and approved the final manuscript.