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- Open Access
Constrained extremal solutions of multi-valued linear inclusions in Banach spaces
- Zi Wang^{1, 2},
- Boying Wu^{1}Email author and
- Yuwen Wang^{2}
https://doi.org/10.1186/s13660-016-1008-1
© Wang et al. 2016
- Received: 5 August 2015
- Accepted: 4 February 2016
- Published: 12 May 2016
Abstract
Let X and Y be Banach spaces, 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. \(S=g+N\). A central problem in our general setting is to determine, for a given \(y\in Y\), a vector \(w\in S\cap D ( L )\) such that, for some \(z\in L ( w )\), \(\| z-y\| =\operatorname{dist} ( y, L ( S\cap D ( L ) ) )\), such a vector w is called the constrained extremal solution of multi-valued linear inclusions \(y\in L ( x )\) in Banach spaces. We establish three equivalent characterizations of 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 L, and the dual mapping of the spaces. As follows from the main results in this paper, 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. The setting in this paper includes large classes of constrained extremal problems and optimal control problems subject to generalized boundary conditions.
Keywords
- Banach space
- linear manifold
- constrained extremal solution
- metric generalized inverse
- algebraic operator part
MSC
- 47A05
1 Introduction
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].
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
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.
Definition 2.1
[2]
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.
Definition 2.2
- (i)
\(u\in D ( A ) \cap S\);
- (ii)there exists \(z\in A|_{S} ( u ) \) such thatwhere \(\operatorname{dist} ( y,R ( A|_{S} ) ) =\inf_{z\in R ( A|_{S} ) }\| y-z\| \).$$ \| y-z\| =\operatorname{dist} \bigl( y,R ( A|_{S} ) \bigr), $$
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.
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.
Proposition 2.3
[8]
- (i)
\(\pi_{G} ( x ) =x\) if \(x\in G\);
- (ii)
\(\pi_{G}^{2} ( x ) =\pi_{G} ( x ) \) for any \(x\in X\);
- (iii)
\(\pi_{G} ( \lambda x ) =\lambda\pi _{G} ( x ) \) for any \(x\in X\) and \(\lambda\in R^{1}\);
- (iv)
\(\pi_{G} ( x+y ) =\pi_{G} ( x ) +y\) for any \(x\in X\), \(y\in G\).
Proof
See Theorem 4.1 in [8]. □
We also use the dual mapping of Banach space, let us recall it.
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.
Lemma 2.4
[10] (Generalized orthogonal decomposition theorem)
Remark 2.5
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.
Definition 2.6
[5]
Remark 2.7
If both X and Y are Hilbert spaces, the metric generalized inverse \(A^{\#}\) of A is just the orthogonal generalized inverse (see [1–3, 7]).
Remark 2.8
If X and Y are Banach spaces, \(T:X\rightarrow Y\) is a linear operator, and \(N ( T )\), \(R ( T )\) are Chebyshev subspaces in X and Y, respectively, then \(T^{\#}\) is just the Moore-Penrose metric generalized inverse of T, denoted by \(T^{M}\).
For convenience we list them as the following propositions the results in [5].
Proposition 2.9
[5]
- (i)
\(u\in D ( A ) \) is the extremal solution of linear inclusion \(y\in A ( x )\);
- (ii)
\(u\in D ( A ) \) and \(\pi_{R ( A ) } ( y ) \in A ( u )\);
- (iii)
\(u\in D ( A ) \) and \(y\in A ( u ) \dotplus F_{Y}^{-1} ( R ( A ) ^{\bot} )\).
Proof
See Theorem 5.1 in [5]. □
Proposition 2.10
[5]
- (i)Let K be any algebraic operator part of \(A^{-1}\), then the cosetis the set of all extremal solutions of \(y\in A ( x ) \).$$ K \bigl( \pi_{R ( A ) } ( y ) \bigr) +N ( A ) $$
- (ii)Let \(A^{\#}\) be the metric generalized inverse of A, then the cosetis the set of all extremal solutions of \(y\in A ( x ) \).$$ A^{\#} ( y ) +N ( A ) $$
- (iii)
\(u=A^{\#} ( y ) \) is the unique least extremal solution of \(y\in A ( x ) \).
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.
Theorem 3.1
- (1)
\(w\in D ( L ) \cap S\) is a constrained extremal solution of the linear inclusion \(y\in L ( x ) \) with respect to S;
- (2)
\(k:=g-w\in D ( L ) \cap N\) is an extremal solution of the linear inclusion \(L_{S,P} ( g ) -y\in A ( x )\);
- (3)\(w\in D ( L ) \cap S\) and$$ L_{S,P} ( w ) -y\in L ( \theta ) +F_{Y}^{-1} \bigl( R ( A ) ^{\bot} \bigr); $$(3.1)
- (4)\(g\in D ( L ) \) such that$$ L_{S,P} ( g ) -y\in R ( A ) \dotplus F_{Y}^{-1} \bigl( R ( A ) ^{\bot} \bigr). $$(3.2)
Proof
Theorem 3.2
Proof
(ii) For any \(y\in Y\), we see that \(L_{S,P} ( g ) -y\in R ( A ) \dotplus F_{Y}^{-1} ( R ( A ) ^{\bot } )\), hence \(\Omega_{y}\neq\emptyset\) by (i).
Corollary 3.3
- (i)
\(w \in D ( L ) \cap S\) is a constrained extremal solution of the linear operator equation \(L(x)=y\) with respect to S;
- (ii)\(k:=A^{\#} ( z ) -w \in D ( L ) \cap N ( A ) \) is an extremal solution of the linear operator equation$$ L \bigl(A^{\#} ( z ) \bigr)-y=T ( x ); $$
- (iii)\(w\in D ( L ) \cap S\) and$$ L ( w ) -y\in F_{Y}^{-1} \bigl( R ( T ) ^{\bot } \bigr). $$
In particular, if \(R ( T ) \) is closed in Y, then \(L ( x ) =y\) always has a constrained extremal solution with respect to S.
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.
Corollary 3.4
[7]
- (i)
w is a restricted least-squares solution (LSS) of the linear inclusion \(y\in L ( x ) \) with respect to S.
- (ii)\(k:=g-w\) is an LSS of$$ L_{S,P} ( g ) -h\in M ( x ). $$
- (iii)\(w\in S\cap D ( L ) \) andwhere \(M^{\ast}:= \{ ( x,y ) : ( -y,x ) \in M^{\bot } \} \) is the adjoint subspace of the linear manifold \(M\subset H_{1}\times H_{2}\), and \(M^{\bot}\) is the orthogonal complement of M in Hilbert space \(H_{1}\times H_{2}\).$$ L_{S,P} ( g ) -h\in L ( \theta ) +N \bigl( M^{\ast } \bigr), $$
- (iv)\(g\in D ( L ) \) such thatIn particular, if \(R ( M )\) is closed, then a restricted LSS exists for each \(h\in H_{2}\).$$ L_{S,P} ( g ) -h\in R ( M ) \dotplus N \bigl( M^{\ast } \bigr). $$
Proof
In Theorem 3.1, take \(X=H_{1}\) and \(Y=H_{2}\), \(A=M=L|_{N}\), since \(H_{1}\), \(H_{2}\), and \(H_{1}\times H_{2}\) are Hilbert spaces, \(F_{Y}=I\) the identity operator of \(H_{2}\), and \(N ( M^{\ast} ) =R ( M ) ^{\bot}=R ( A ) ^{\bot}\) (see [13]). (I) in Corollary 3.4 follows from Theorem 3.1, and (II) in Corollary 3.4 follows from Theorem 3.2. □
Remark 3.5
In [7], the authors gave an application of Theorem 3.1, i.e. Corollary 3.4, to concrete cases of singular optimal control problems involving ordinary differential equations with general boundary conditions where both the control space and the state space are Hilbert space \(L_{2}^{m}=L_{2}([a,b],\mathbb{C}^{m})\) and \(L_{2}^{n}=L_{2}([a,b],\mathbb{C}^{n})\), but for the same problem with the control space \(L_{p}^{m}=L_{p}([a,b],\mathbb{C}^{m})\) and the state space \(L_{p}^{n}=L_{p}([a,b],\mathbb{C}^{n})\) (\(1< p<\infty\)), we cannot apply Theorem 3.1 in [7], while we can apply Theorem 3.1, and Theorem 3.2, in this paper.
Remark 3.6
Declarations
Acknowledgements
This research was supported by the National Natural Science Foundation of China (No. 11471091).
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Authors’ Affiliations
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