Journal of Convex Analysis Volume 4 (1997), No. 1, 149–164

Cone-Constrained Linear Equations in Banach Spaces^{1}

O. Hernandez-Lerma

Departamento de Matemáticas, CINVESTAV-IPN, A. Postal 14-740, México D.F. 07000, Mexico. e-mail: ohernand@math.cinvestav.mx

J. B. Lasserre

LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse Cédex, France. e-mail: lasserre@laas.fr

Received May 22, 1996

This paper presents necessary and sufficient conditions for the existence of solutions to cone-constrained linear equations in some function spaces. These conditions yield, in particular, the classical Fredholm alternative for compact operators. We use a formulation that under some conditions permits to apply the Generalized Farkas Theorem of Craven and Koliha. The Poissson Equation for stochastic (Markov) kernels, the Volterra and Fredholm equations for non-compact operators in L_{p} spaces, are among the particular cases of potential application.

Keywords: Linear equations in Banach spaces, Generalized Farkas Theorem, Fredholm alternative

1991 Mathematics Subject Classification: 28A33, 60B10, 60J05, 90C05, 90C48.

1. Introduction

In this paper we are concerned with the existence of solutions to the linear equations

Ax = b  (1.1)

and

Ax = b, x ∈ S,  (1.2)

where A : X → X is a linear operator, X is a Banach space, and S ⊆ X is a convex cone. Of course, the unconstrained equation (1.1) is a particular case of (1.2) with S := X. Other particular cases are the integral-type equations [7, 10, 11], the Poisson Equation [9] in L_{p} spaces and the Hilbert-Schmidt-type operators.

The Generalized Farkas Theorem of Craven and Koliha [3] permits to characterize existence of solutions to (1.1) or (1.2) provided the condition “A(S) is closed” is satisfied

1 This research was partially suported by the CNRS (France)- CONACYT (Mexico) Scientific Cooperation Program. Research also partially supported for the first author by a visiting professorship at Paul Sabatier University, Toulouse, France, and by CONACYT Grant 1332-E9206

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag

150 O.Hernandez-Lerma, J.B.Lasserre / Cone-constrained linear equations in Banach spaces

in some appropriate topology. However, this condition is rarely met in practice. For instance, consider the Poisson Equation (I − P)h = g for a Markov chain with associated stochastic kernel P, viewed as an operator on X := L_{p}(Ω, B, μ) (see [9]). The images (I − P)(X) or (I − P)(S) (where S is the positive cone in X) need not be closed. However, the case of compact operators P is a notable exception and we will show that the Fredholm alternative theorem (see e.g. [2, 10]) is an immediate consequence of the Generalized Farkas Theorem of Craven and Koliha [3]. In addition, we will also show that our conditions for existence of a solution to (1.1) also reduce to the standard Fredholm alternative for compact operators when X is a reflexive Banach space.

To circumvent the non-closedness of A(S), and in the same spirit but differently from [12], we formulate the problem in such a way that, by introducing some appropriate constraint, one may apply the Generalized Farkas Theorem to the modified system, which yields a new Generalized Farkas Theorem without this closure condition (see Theorem 2.4). We then apply this result to the particular case of L_{p}(Ω, B, μ) spaces.

The case X := L_{1}(Ω, B, μ) needs a special treatment and interestingly enough, although the proof is different, the resulting (necessary and sufficient) conditions of existence can be represented in a single theorem (Theorem 3.2) that covers all the spaces L_{p}(Ω, B, μ), 1 ≤ p ≤ ∞. Another slightly different Farkas-like theorem is also given for L_{1}(Ω, B, μ) when identified with a subspace of M(Ω), the space of bounded signed Borel measures on Ω.

The paper is organized as follows. In Section 2, after some preliminary results, we present the new Farkas Lemma without a closure condition. We also consider the special case of compact operators and show that our conditions then reduce to the Fredholm alternative. In Section 3, we consider cone-constrained linear equations in the spaces L_{p}(Ω, B, μ), and particularize the results obtained in Section 2 in a single theorem (Theorem 3.2 below) that covers all the cases. In Section 4, we present the proof of Theorem 3.2 with special attention to the case L_{1}. Finally, Section 5 is an appendix summarizing some basic results that we extensively use.

2. Notation, Definitions and Main Result

Let X be a separable Banach space with topological dual X^{∗}. The duality bracket between X and X^{∗} is denoted ⟨., .⟩. For a convex cone S in X we denote by S^{∗} its dual cone, i.e.

S^{∗} := {y ∈ X^{∗}| ⟨x, y⟩ ≥ 0 ∀x ∈ S},  (2.1)

and for a convex cone Ω in X^{∗} we define

Ω^{∗} := {x ∈ X^{∗∗}| ⟨x, y⟩ ≥ 0 ∀y ∈ Ω}  (2.2)

and

Ω^{+} := {x ∈ X | ⟨x, y⟩ ≥ 0 ∀y ∈ Ω}  (2.3)

Remark 2.1. Note that with the natural embedding of X into X^{∗∗}, Ω^{+} = Ω^{∗} ∩ X. In addition, if S is strongly closed, then (S^{∗})^{+} = S (see e.g. [3]). Moreover, (X ×R, X^{∗} ×R) is viewed as a dual pair with duality bracket ⟨(x, r), (y, ρ)⟩ := ⟨x, y⟩ + rρ.

2.1. A preliminary result

In the sequel we will use the following lemma:

Lemma 2.2. Let Γ ⊂ (X^{∗} × R) be the cone {(y, r) ∈ X^{∗} × R| ||y|| ≤ r}. Then Γ is a weak* closed convex cone, and Γ^{+} = Ω, where

Ω := {(x, z) ∈ X × R| ||x|| ≤ z}.  (2.4)

We also have

(Γ^{+})^{∗} = Γ, i.e. Ω^{∗} = Γ.  (2.5)

Proof. The fact that Γ is a convex cone is trivial. Now, to prove that it is weak* closed, from Theorem 5.2 (d), consider a sequence (y_{n}, r_{n}) in Γ such that (y_{n}, r_{n}) converges in the weak* topology to (a, b) in X^{∗} × R, i.e.,

y_{n} →^{w∗} a, r_{n} → b as n → ∞,  (2.6)

where →^{w∗} denotes the σ(X^{∗}, X) (weak*) topology in X^{∗}. We want to show that (a, b) ∈ Γ. From (2.6), b ≥ lim inf_{n} ||y_{n}|| ≥ ||a|| (see e.g. [2]) so that (a, b) ∈ Γ. Since Γ is weak* closed, then (Γ^{+})^{∗} = Γ (see e.g. Proposition 1 in [1]) which yields (2.5). We shall now prove that Γ^{+} = Ω.

1. Ω ⊆ Γ^{+}. It is obvious that

(x, z) ∈ Ω ⇒ ⟨(y, r), (x, z)⟩ = ⟨x, y⟩ + rz ≥ 0 ∀(y, r) ∈ Γ,

since (note that both r and z are nonnegative)

|⟨x, y⟩| ≤ ||x||.||y|| ≤ zr,

yields ⟨x, y⟩ + rz ≥ 0.

2. Γ^{+} ⊆ Ω.

(x, z) ∈ Γ^{+} ⇒ ⟨x, y⟩ + rz ≥ 0 for all (y, r) ∈ Γ. For any x ∈ X , ∃y(x) ∈ X^{∗} such that ||x|| = ⟨y(x), x⟩ and ||y(x)|| = 1 (see e.g. [2]). Thus, for any (x, z) ∈ Γ^{+}, (±y(x), 1) ∈ Γ so that

⟨x, ±y(x)⟩ + z.1 ≥ 0 ⇒ |⟨y(x), x⟩| (:= ||x||) ≤ z ⇒ (x, z) ∈ Ω,

which is the desired result.

Remark 2.3. Note that if X is a reflexive Banach space, then in Lemma 2.2, (Γ^{∗})^{∗} = Γ. Let H be a locally convex space with topological dual H′ equipped with the weak* topology σ(H′, H). If C and D are two convex cones in H with closures \overline{C} and \overline{D}, then (\overline{C} ∩ \overline{D})^{∗} = \overline{C^{∗} + D^{∗}} (see e.g. [1]).

2.2. A Farkas Lemma without a closedness condition

In this section, we consider the general linear system

Ax = b, x ∈ S^{∗}  (2.7)

where A : X^{∗} → Z is a linear mapping, X and Z are Banach spaces and S is a convex cone in X. W is another Banach space such that (Z, W) is a dual pair. Existence of solutions to (2.7) can be characterized via e.g. the Generalized Farkas Theorem of Craven and Koliha [3] (see Theorem 5.1 below). However, in addition to the usual continuity assumptions on the mapping A, a (restrictive) closure assumption is required on A(S). The purpose of this section is to present a new Farkas-like theorem without such a restrictive assumption.

Theorem 2.4. Let X be a separable Banach space with dual X^{∗}, equipped with the weak topology σ(X , X^{∗}) and the weak* topology σ(X^{∗}, X) respectively. Let (Z, W) be a dual pair of Banach spaces, Z (resp. W) being equipped with the weak topology σ(Z, W) (resp. σ(W, Z)). Let A : X^{∗} → Z be a weakly continuous linear mapping and S ⊂ X a strongly closed convex cone in X. Let A^{∗} : W → X be the adjoint mapping of A, and let Γ be the convex cone {(x, r) ∈ X^{∗} × R^{+}| ||x|| ≤ r}. Then,

(a) the following two propositions are equivalent for b ∈ Z:

(i) the system Ax = b has a solution x ∈ S^{∗} ⊂ X^{∗}.

(ii) [w ∈ W, z ∈ R and (A^{∗}w, z) ∈ (Γ ∩ (S^{∗} × R))^{+}] ⇒ ⟨b, w⟩ + Mz ≥ 0, for some

M > 0.

(b) If, in addition, X is reflexive, or if the cone Ω + (S × {0}) is strongly closed (where Ω := {(x, r) ∈ X × R^{+}| ||x|| ≤ r}), then the following two propositions are equivalent for b ∈ Z:

(i) the system Ax = b has a solution x ∈ S^{∗} ⊂ X^{∗}.

(ii) w ∈ W, s ∈ S ⇒ ⟨b, w⟩ + M||A^{∗}w − s|| ≥ 0, for some M > 0.

Proof. (a) The system {Ax = b, x ∈ S^{∗}} has a solution if and only if the system

Ax = b; r = M; ||x|| ≤ r; x ∈ S^{∗}  (2.8)

has a solution for some M > 0, or equivalently, if and only if

T (x, r) = (b, M), (x, r) ∈ Γ ∩ (S^{∗} × R) (with T (x, r) := (Ax, r))  (2.9)

has a solution for some M > 0. We first prove that T (Γ ∩ (S^{∗} × R)) is weakly-closed. Let (x_{α}, r_{α}) be a net (α ∈ D for some directed set (D, ≥)) in Γ ∩ (S^{∗} × R) such that T (x_{α}, r_{α}) converges weakly to some (a_{1}, a_{2}) in Z × R, i.e.,

Ax_{α} converges weakly to a_{1}, and r_{α} converges to a_{2}.  (2.10)

If a_{2} = 0, then obviously, since (x_{α}, r_{α}) ∈ Γ, ||x_{α}|| converges to zero so that Ax_{α} converges weakly to a_{1} = 0, and T (0, 0) = (0, 0). Now, if a_{2} ≠ 0 then, from (2.10) and the fact that (x_{α}, r_{α}) ∈ Γ there exists some α_{0} ∈ D such that for all α ≥ α_{0}, ||x_{α}|| ≤ 2a_{2}. Since X^{∗} is the dual of a separable Banach space, the set {x| ||x|| ≤ 2a_{2}} is weak* sequentially compact (see Theorem 5.2 (b),(c)). Thus, from the net {x_{α}}, one can extract a sequence {x_{α_{i}}} that converges to some x in the weak* topology σ(X^{∗}, X). Moreover, since A is weakly continuous, Ax_{α_{i}} converges weakly to Ax = a_{1}. In addition, since r_{α_{i}} converges to a_{2} and lim inf_{i} ||x_{α_{i}}|| ≥ ||x|| (see e.g. [2]), we get ||x|| ≤ a_{2} i.e. (x, a_{2}) ∈ Γ. Finally, noting that S^{∗} is weak* closed, then x ∈ S^{∗} so that (x, a_{2}) ∈ (Γ ∩ (S^{∗} × R^{+})). This combined with T (x, a_{2}) = (a_{1}, a_{2}) implies that T (Γ ∩ (S^{∗} × R)) is weakly closed. Since T ((Γ ∩ (S^{∗} × R)) is weakly-closed, we can appply the Generalized Farkas Theorem (Theorem 5.1 below) which states that the system {T (x, r) = (b, M), (x, r) ∈ Γ∩(S^{∗}×R)} has a solution if and only if

[(w, z) ∈ W × R, T^{∗}(w, z) ∈ (Γ ∩ (S^{∗} × R))^{+}] ⇒ ⟨b, w⟩ + Mz ≥ 0,  (2.11)

where T^{∗}(w, z) = (A^{∗}w, z), which yields part (a).

(b) To prove part (b), note that from Lemma 2.2, Γ is the dual cone of the strongly closed convex cone Ω := {(x, z) ∈ X × R| ||x|| ≤ z}. Note also that S^{∗} × R is the dual cone of the strongly closed convex cone S × {0}. Thus, as (0, 0) ∈ Ω ∩ (S × {0}),

Γ ∩ (S^{∗} × R) = (Ω + (S × {0}))^{∗}.  (2.12)

As X is reflexive, the cone Ω + (S × {0}) is strongly closed in X × R. Indeed, consider any sequence (x_{n}, s_{n}, r_{n}) with (x_{n}, r_{n}) ∈ Ω and s_{n} ∈ S, such that

x_{n} + s_{n} → a and r_{n} → r  (2.13)

for some (a, r) ∈ X × R and where the first convergence is in the strong topology of X. From (2.13) and the fact that ||x_{n}|| ≤ r_{n} we conclude that both x_{n} and s_{n} are uniformly bounded. From the weak* sequential compactness of the unit ball in X (since X is reflexive), there is a subsequence (x_{n_{i}}, s_{n_{i}}, r_{n_{i}}) such that x_{n_{i}} →^{w∗} x and s_{n_{i}} →^{w∗} s. Both Ω and S are weak* closed (as dual cones of Γ and S^{∗} respectively) so that (x, r) ∈ Ω and s ∈ S. Combining this and (2.13) yields (a, r) = (x, r) + (s, 0), i.e. (a, r) ∈ Ω + (S × {0}), which proves that Ω + (S × {0}) is strongly closed. Thus, by (2.12) and Remark 2.3,

(Γ ∩ (S^{∗} × R))^{+} = (Ω + (S × {0}))^{∗∗} = Ω + (S × {0}).  (2.14)

Hence, (2.11) reads

\begin{matrix}(w, z) ∈ W × R, (A^{∗}w, z) = (u + s, z), ||u|| ≤ z, s ∈ S \\ ⇒ ⟨b, w⟩ + Mz ≥ 0 \end{matrix}  (2.15)

for some M > 0, or, equivalently,

w ∈ W, s ∈ S ⇒ ⟨b, w⟩ + M||A^{∗}w − s|| ≥ 0,

since it suffices to check (2.15) for z := ||A^{∗}w − s||.

2.3. Compact operators

Let A := (I − P) where P : X^{∗} → X^{∗} is a compact operator, i.e. P maps the unit ball of X^{∗} into a relatively compact set in X^{∗}. We show that the Fredholm alternative (see e.g. [2]) is a particular case of the (Generalized Farkas) Theorem 5 of Craven and Koliha in [3]. We then show that our condition in Theorem 2.4 (b)(ii) also reduces to the Fredholm alternative when X is reflexive. Note that if P is a compact operator then range(A) is closed (see e.g. [2] Th. VI.6). In addition, A is also strongly continuous. Thus, one may apply Theorem 5.1 below, with X := X^{∗}, Y = X^{∗}, S := X^{∗}, A := (I − P) so that, since S^{∗} = {0}, we obtain

Corollary 2.5. Assume that P : X^{∗} → X^{∗} is compact. Let A := (I − P). Then

Ax = b has a solution x in X^{∗} iff [A^{∗}w = 0, w ∈ X] ⇒ ⟨b, w⟩ = 0,  (2.16)

which is the Fredholm alternative.

We now prove that our condition in Theorem 2.4 (b)(ii) also reduces to (2.16), assuming that X is reflexive.

Corollary 2.6. Assume that X is reflexive and let P and A be as in Corollary 2.5. Then the condition in Theorem 2.4 (b)(ii) reduces to (2.16).

Proof. As P : X^{∗} → X^{∗}, in Theorem 2.4 let (Z, W) be the dual pair (X^{∗}, X). With S := {0}, the condition (b)(ii) in Theorem 2.4 now reads

⟨b, w⟩ + M||A^{∗}w|| ≥ 0 ∀w ∈ X ,  (2.17)

for some M > 0, with A^{∗} = I − P^{∗}. Let V := N(A^{∗}) = {w ∈ X | A^{∗}w = 0}. As P is compact then so is P^{∗} (see Schauder Theorem in e.g. [2]) and thus V has finite dimension (see [2] p. 90). Therefore, it admits a topological supplement V^{c} such that V^{c} is closed, V ∩ V^{c} = {0} and X = V + V^{c}. Note that if w ∈ V, then −w ∈ V so that from (2.17) we must have ⟨b, w⟩ = 0. Hence it remains to show that

⟨b, w⟩ + M||A^{∗}w|| ≥ 0 ∀w ∈ V^{c}  (2.18)

is always satisfied for some M > 0, so that (2.17) reduces to the Fredholm alternative. Without loss of generality we may and will assume that ||w|| = 1 in (2.18). Let δ := inf{||A^{∗}w|| | ||w|| = 1, w ∈ V^{c}} and consider a minimizing sequence {w_{n}} in V^{c} such that ||w_{n}|| = 1 and ||A^{∗}w_{n}|| ↓ δ. We prove that δ > 0. By the weak* sequential compactness of the unit ball in X (recall that X is separable and reflexive, and see Theorem 5.2 (c)), ∃w and a subsequence {n_{i}} such that w_{n_{i}} →^{w∗} w and also A^{∗}w_{n_{i}} →^{w∗} A^{∗}w, where →^{w∗} denotes the (weak* or weak) σ(X , X^{∗}) convergence. As V^{c} is closed, it is also weakly-closed (i.e. σ(X , X^{∗})-closed). Thus w ∈ V^{c}. Let us now consider the two cases, w ≠ 0 and w = 0.

• If w ≠ 0 then A^{∗}w ≠ 0 and as A^{∗}w_{n_{i}} →^{w} A^{∗}w we have δ = lim inf_{i} ||A^{∗}w_{n_{i}}|| ≥ ||A^{∗}w|| > 0.

• Consider now the case where w = 0. Since P^{∗} is compact and ||w_{n_{i}}|| = 1 for all i, {P^{∗}w_{n_{i}}} is in a relatively compact set for the strong topology in X. Thus, for a subsequence again denoted {w_{n_{i}}}, P^{∗}w_{n_{i}} →^{s} q in X. Moreover, P^{∗}w_{n_{i}} →^{w∗} P^{∗}w = 0 and thus q = 0, which implies ||P^{∗}w_{n_{i}}|| ↓ 0. Now, from

||A^{∗}w_{n_{i}}|| ≥ ||w_{n_{i}}|| − ||P^{∗}w_{n_{i}}||

we conclude that ∃ϵ > 0 such that for i large enough

||A^{∗}w_{n_{i}}|| ≥ ||w_{n_{i}}|| − ϵ = 1 − ϵ

so that δ > 0.

Moreover, |⟨b, w_{n_{i}}⟩| ≤ ||b||.||w_{n_{i}}|| = ||b|| so that for M large enough, and w ∈ V^{c}

⟨b, w⟩ + M||A^{∗}w|| ≥ 0.

Hence, in Theorem 2.4, the condition (b)(ii)

⟨b, w⟩ + M||A^{∗}w|| ≥ 0 ∀w ∈ X

for some M > 0 reduces to

w ∈ X , A^{∗}w = 0 ⇒ ⟨b, w⟩ = 0,

and the proof is complete.

3. Linear systems in L_{p} spaces

General assumption. (X, B, μ) is a σ-finite complete measure space, with X a topological space, and B the completion (with respect to μ) of the σ-algebra of Borel subsets of X. In addition, for the particular case of L_{1}, we assume that X is a locally compact separable metric space.

For 1 ≤ p ≤ ∞, let q be the exponent conjugate to p, i.e. (1/p) + (1/q) = 1. We write L_{p} for L_{p}(X, B, μ), and L_{p}^{+} denotes the convex cone of nonnegative functions in L_{p}. Recall that L_{p} is a Banach space for every 1 ≤ p ≤ ∞, with topological dual L_{q} when 1 ≤ p < ∞, the corresponding “inner product” being

⟨u, v⟩ := ∫_{X} uvdμ, u ∈ L_{p}, v ∈ L_{q}.

In this section, we are concerned with the existence of solutions h ∈ L_{p} to the equation

(I − P)h = b,  (3.1)

and

(I − P)h = b, h ∈ S,  (3.2)

where P : L_{p} → L_{p} is a linear operator, b ∈ L_{p} a given function, and S a convex cone in L_{p}.

For instance, in solving equation (3.2) with S := L_{p}^{+}, one looks for nonnegative solutions h ∈ L_{p} to (3.1). The following examples show that (3.1), (3.2) include well known equations in analysis and probability.

3.1. Examples

P : L_{p} → L_{p} is a linear operator and there exists a measurable function K(x, y) on X ×X such that

P u(x) = ∫_{X} K(x, y)u(y)μ(dy) x ∈ X, ∀u ∈ L_{p}.

Among particular cases of the above type of linear operators, let us mention: Fredholm-type kernel. In this case, take for instance X := [a, b] a closed interval on the real line, and μ the Lebesgue measure. Then, define

P u(x) := λ ∫_{a}^{b} K(x, y)u(y)dy, x ∈ X  (3.3)

where λ is some fixed scalar. Volterra-type kernel. Again, take for instance X := [a, b] a closed interval on the real line, and μ the Lebesgue measure. Then, define

P u(x) := λ ∫_{a}^{x} K(x, y)u(y)dy, x ∈ X  (3.4)

The Poisson Equation. Let P be a stochastic kernel on (X, B), i.e. P (x, .) is a probability measure on X for every x ∈ X, and P (., B) is a measurable function on X for every B ∈ B. Let

P u(x) := ∫ P (x, dy)u(y),  x ∈ X,  (3.5)

and suppose that P (x, .) is absolutely continuous with respect to μ, with density K(x, .), i.e.

P u(x) := ∫_{X} K(x, y)u(y)μ(dy),  x ∈ X.  (3.6)

3.2. Existence of solutions in L_{p}

With P as in (3.1), (3.2), we suppose that for some given p ∈ [1, ∞]:

Assumption 3.1.

(a) P maps L_{p} into itself.

(b) The adjoint P^{∗} of P maps L_{q} into itself.

(c) In addition, if p = 1, P^{∗} maps C_{0}(X) into itself, where C_{0}(X) is the separable Banach space of real-valued continuous functions on X that vanish at infinity, endowed with the sup-norm (see e.g. [4] or [13]).

We now state the following main result:

Theorem 3.2. Suppose that Assumption 3.1 holds for a given p ∈ [1, ∞]. Then:

(a) the equation (3.1) has a solution in L_{p} if and only if

⟨b, w⟩ + M||(I − P^{∗})w||_{q} ≥ 0 ∀w ∈ L_{q},  (3.7)

for some M > 0.

(b) The equation (3.1) has a solution in L_{p}^{+} if and only if

⟨b, w⟩ + M|| min[0, (I − P^{∗})w]||_{q} ≥ 0 ∀w ∈ L_{q},  (3.8)

for some M > 0.

The proof of Theorem 3.2 requires different arguments depending on whether p = 1 or 1 < p ≤ ∞. The proof is given in the next section.

4. Proof of Theorem 3.2

4.1. The case 1 < p ≤ ∞

Suppose that p ∈ (1, ∞] is fixed and b is a given function in L_{p}, and we wish to find a solution h in L_{p} (case (a)) or a nonnegative solution h in L_{p} to (3.1) (case (b)). Then, Theorem 2.4 with the identification

X^{∗} := L_{p}; X := L_{q}; Z := L_{p}; W := L_{q}; A := (I − P)

and S^{∗} := X^{∗} (case (a)) yields Theorem 3.2 (a) in the case 1 < p ≤ ∞. Similarly, to obtain part (b) for 1 < p < ∞, let S^{∗} be the positive cone in L_{p} with dual cone S = the positive cone in L_{q}, and recall that the spaces L_{p} are reflexive when 1 < p < ∞. For the case p = ∞, although L_{∞} is not reflexive, the cone Ω + (S × {0}) in Theorem 2.4, is strongly closed when X := L_{1}. Indeed, let (f_{n}, g_{n}, r_{n}) be a sequence in L_{1} such that

||f_{n}||_{1} ≤ r_{n}, r_{n} → r, g_{n} ≥ 0 and lim_{n} ||f_{n} + g_{n} − u||_{1} = 0.

Then, using the standard notation u^{+} := max[u, 0] and u^{−} := max[−u, 0], we wish to prove that u = u^{+} − u^{−} is in the cone Ω + (S × {0}), for which (as u^{+} ≥ 0) it is sufficient to show that ||u^{−}||_{1} ≤ r. To prove this, let {m} be a subsequence of {n} such that f_{m} + g_{m} converges to u μ-a.e., so that, in particular,

(f_{m} + g_{m})^{−} → u^{−} μ -a.e..

This, in turn (as g_{m} ≥ 0 implies (f_{m} + g_{m})^{−} ≤ f_{m}^{−}), yields

u^{−} ≤ lim inf f_{m}^{−},

and we get ||u^{−}||_{1} ≤ r since, by Fatou’s Lemma,

||u^{−}||_{1} ≤ lim inf ||f_{m}^{−}||_{1} ≤ lim inf ||f_{m}||_{1} ≤ r.

This proves that Ω + (S × {0}) is strongly closed and, therefore, Theorem 2.4 (b) is valid. To see that (3.8) in Theorem 3.2 is equivalent to Theorem 2.4 (b)(ii), note that if S is the positive cone in L_{q}, then 2.4 (b)(ii) with b ∈ L_{p} is true if and only if

⟨b, w⟩ + M|| min[0, A^{∗}w]||_{q} ≥ 0 ∀w ∈ L_{q},

since for any s ∈ S, ||A^{∗}w − s||_{q} ≥ || min[0, A^{∗}w]||_{q} and thus it suffices to check the condition for s := max[0, A^{∗}w].

4.2. The case p = 1

We now consider the special case of L_{1} where (X, B, μ) is a σ-finite complete measure space, X is a locally compact separable metric space, and B is the completion (with respect to μ) of the σ-algebra of Borel subsets of X. As L_{1} is not the dual of L_{∞}, we cannot use the weak* topology as we extensively did in the proof of Theorem 2.4. Suppose that b is a given function in L_{1}, and we wish to find a nonnegative solution h in L_{1} to (3.1). Then, (3.1) has a solution in L_{1}^{+} if and only if the following system

(I − P)h = b, ⟨h, 1⟩ ≤ M, h ∈ L_{1}^{+}  (4.1)

has a solution for some M > 0, or equivalently, if and only if the system

(I − P)h = b, ⟨h, 1⟩ + r = M,  (4.2)

has a solution (h, r) in L_{1}^{+} × R^{+} for some M > 0. The dual pair (L_{1} × R, L_{∞} × R) is endowed with the inner product

⟨(h, r), (u, ρ)⟩ := ⟨h, u⟩ + rρ

where ⟨h, u⟩ := ∫ hudμ for h ∈ L_{1} and u ∈ L_{∞}. Thus we now consider the linear operator A_{1} : L_{1} × R → L_{1} × R and its adjoint A_{1}^{∗} : L_{∞} × R → L_{∞} × R given by

A_{1}(h, r) := ((I − P)h, ⟨h, 1⟩ + r),  (4.3)
A_{1}^{∗}(u, ρ) := ((I − P^{∗})u + ρ, ρ).  (4.4)

Note that, by Assumption 3.1 (b), A_{1} is weakly continuous and, on the other hand, (4.1) is equivalent to

A_{1}(h, r) = (b, M) has a solution (h, r) in L_{1}^{+} × R^{+}  (4.5)

for some M ≥ 0. Similarly, if we wish to find solutions h = h^{+} − h^{−} in L_{1}, we consider the operators

A_{1} : (L_{1})^{2} × R → L_{1} × R, and A_{1}^{∗} : L_{∞} × R → (L_{∞})^{2} × R

given by

A_{1}(h_{1}, h_{2}, r) := ((I − P)(h_{1} − h_{2}), ⟨h_{1} + h_{2}, 1⟩ + r),  (4.6)

A_{1}^{∗}(u, ρ) := ((I − P^{∗})u + ρ, ρ − (I − P^{∗})u, ρ).  (4.7)

Again, A_{1} is weakly continuous, and (3.1) has a solution in L_{1} if and only if

A_{1}(h_{1}, h_{2}, r) = (b, M) has a solution (h_{1}, h_{2}, r) ∈ (L_{1}^{+})^{2} × R^{+}  (4.8)

for some M ≥ 0. Thus Lemma 4.2 below and Theorem 5.1 yield the following proposition ≡ Theorem 3.2 for p = 1.

Proposition 4.1. Suppose that b ∈ L_{1} and Assumption 3.1 holds. Then:

(a) The equation (3.1) has a solution in L_{1} if and only if

[u ∈ L_{∞}, ρ ∈ R^{+}, and − ρ ≤ (I − P^{∗})u ≤ ρ] ⇒ ⟨b, u⟩ + Mρ ≥ 0

for some M ≥ 0, or, equivalently, if and only if

⟨b, u⟩ + M||(I − P^{∗})u||_{∞} ≥ 0 ∀u ∈ L_{∞}

for some M ≥ 0.

(b) The equation (3.1) has a solution in L_{1}^{+} if and only if

[u ∈ L_{∞}, ρ ∈ R^{+}, and (I − P^{∗})u ≥ −ρ] ⇒ ⟨b, u⟩ + Mρ ≥ 0

for some M ≥ 0, or, equivalently, if and only if

⟨b, u⟩ + M|| min[0, (I − P^{∗})u||_{∞} ≥ 0 ∀u ∈ L_{∞}

for some M ≥ 0.

Lemma 4.2.

(a) With A_{1} as in (4.3), A_{1}(L_{1}^{+} × R^{+}) is weakly closed.

(b) With A_{1} as in (4.6), A_{1}((L_{1}^{+})^{2} × R^{+}) is weakly closed.

Proof. The proof of Lemma 4.2 requires in particular Lemma 5.3 (a) in the appendix, which is an extension of the Vitali-Hahn-Saks theorem.

Remark 4.3. We use below the following notation: M(X) denotes the Banach space of finite signed measures on (X, B), endowed with the total variation norm. By the Riesz theorem (see e.g. [13] p. 130) M(X) is the dual of the separable Banach space C_{0}(X) in Assumption 3.1 (c).

Proof of Lemma 4.2 (b). We first give the proof of part (b), and then show that it also contains the proof of (a). Let us write the convex cone (L_{1}^{+})^{2}×R^{+} as S_{1}, and for some directed set (D, ≤), let {(v_{α}, w_{α}, r_{α}), α ∈ D} be a net in S_{1} such that A_{1}(v_{α}, w_{α}, r_{α}), with A_{1} as in (4.6), converges to (a, b) ∈ L_{1} × R in the weak topology σ(L_{1} × R, L_{∞} × R); that is, for all (u, ρ) in L_{∞} × R:

⟨(I − P)(v_{α} − w_{α}), u⟩ + (⟨v_{α} + w_{α}, 1⟩ + r_{α})ρ → ⟨a, u⟩ + bρ.  (4.9)

We wish to show that (a, b) is in A_{1}(S_{1}), i.e. there is (h_{1}, h_{2}, r) in S_{1} with

(I − P)(h_{1} − h_{2}) = a, and ⟨h_{1} + h_{2}, 1⟩ + r = b.  (4.10)

Now, in (4.9) take ρ = 0, and then ρ = 1, u = 0 to get

⟨(I − P)(v_{α} − w_{α}), u⟩ → ⟨a, u⟩ ∀u ∈ L_{∞},  (4.11)

and

⟨v_{α} + w_{α}, 1⟩ + r_{α} → b  (4.12)

respectively. If b = 0, then we are done because in such a case r_{α}, ⟨v_{α}, 1⟩ and ⟨w_{α}, 1⟩ → 0 and we may take h_{1} = h_{2} = 0 and r = 0 in (4.10) since a has to be 0. Let us now consider the case b > 0. By (4.10), there is α_{0} ∈ D such that

0 ≤ ||v_{α}||_{1} + ||w_{α}||_{1} + r_{α} ≤ 2b ∀α ≥ α_{0},  (4.13)

where we have used that ⟨v_{α}, 1⟩ := ∫ v_{α}dμ = ||v_{α}||_{1} and similarly for w_{α}. For every α ≥ α_{0} consider the (nonnegative) measures φ_{α}, ψ_{α} defined as

φ_{α}(B) := ∫_{B} v_{α}dμ, and ψ_{α}(B) := ∫_{B} w_{α}dμ, B ∈ B,  (4.14)

which, by (4.13), are uniformly bounded by 2b. Therefore (see Remark 4.3), by Theorem 5.2 (b),(c), there is a sequence {α_{i}} in D, such that {φ_{α_{i}}} and {ψ_{α_{i}}} converge in the weak* topology σ(M(X), C_{0}(X)) to measures φ and ψ respectively, i.e., ∀u ∈ C_{0}(X):

⟨φ_{α_{i}}, u⟩ → ⟨φ, u⟩ and ⟨ψ_{α_{i}}, u⟩ → ⟨ψ, u⟩.  (4.15)

From (4.14)–(4.15) and Lemma 5.3 (together with the Radon-Nikodym Theorem and the fact that φ_{α} and ψ_{α} are uniformly bounded, finite measures) there exist functions h_{1} and h_{2} in L_{1}^{+} such that

φ(B) = ∫_{B} h_{1}dμ and ψ(B) = ∫_{B} h_{2}dμ ∀B ∈ B.  (4.16)

Moreover, (4.15)–(4.16) yield ∀u ∈ C_{0}(X):

⟨v_{α_{i}}, u⟩ → ⟨h_{1}, u⟩  (4.17)

since

⟨v_{α_{i}}, u⟩ = ∫ uv_{α_{i}}dμ = ⟨φ_{α_{i}}, u⟩ → ⟨φ, u⟩ = ⟨h_{1}, u⟩.

Similarly,

⟨w_{α_{i}}, u⟩ → ⟨h_{2}, u⟩ ∀u ∈ C_{0}(X).  (4.18)

In addition (as p = 1), Assumption 3.1 (c) yields, ∀u ∈ C_{0}(X):

⟨P v_{α_{i}}, u⟩ → ⟨P h_{1}, u⟩  (4.19)

since

⟨P v_{α_{i}}, u⟩ = ⟨v_{α_{i}}, P^{∗}u⟩ = ⟨φ_{α_{i}}, P^{∗}u⟩ → ⟨φ, P^{∗}u⟩ = ⟨h_{1}, P^{∗}u⟩ = ⟨P h_{1}, u⟩.

Similarly,

⟨P w_{α_{i}}, u⟩ → ⟨P h_{2}, u⟩ ∀u ∈ C_{0}(X).  (4.20)

Thus combining (4.18)–(4.20) and (4.11)–(4.12) we see that h_{1}, h_{2} and the nonnegative number r := b − ⟨h_{1} + h_{2}, 1⟩ satisfy (4.10). As h_{1}, h_{2} are in L_{1}^{+} this completes the proof of part (b). In fact, the latter also yields part (a), taking w_{α} = h_{2} = 0 in (4.9)–(4.20) - - i.e. “deleting” w_{α} and h_{2} (in which case note that (4.6) reduces to (4.3)).

4.3. The case p = 1: another Farkas-like lemma

In this section we provide another Farkas-like theorem for linear systems in L_{1}. We now identify L_{1} with the linear subspace N of finite signed measures in M(X) which are absolutely continuous with respect to μ, and we shall use again Remark 4.3. Note that by Theorem 5.2 (d) and Lemma 5.3, N is weak* closed in M(X). Moreover, with p = 1, consider the case where P has a kernel K(x, y) on X × X. Let P (B|x) := ∫_{B} K(x, y)μ(dy), B ∈ B, and assume that P ν(B) := ∫ P (B|x)ν(dx) is finite for all B ∈ B, ν ∈ M(X). Then, P may be viewed as a linear operator on M(X) and (3.1) (with p = 1) is equivalent to

(I − P)φ = ν_{b}, φ ∈ N,  (4.21)

with ν_{b} ∈ M(X) and ν_{b}(B) := ∫_{B} bdμ ∀B ∈ B; moreover, if we look for a nonnegative solution, (3.2) is equivalent to

(I − P)φ = ν_{b}, φ ∈ Δ ∩ N,  (4.22)

where now Δ is the positive cone in M(X). The orthogonal complement of N, i.e. N^{⊥} := {f ∈ C_{0}(X)| ⟨f, φ⟩ = 0 ∀φ ∈ N}, is (weakly) σ(C_{0}(X), M(X))-closed and thus strongly closed. In addition, (N^{⊥})^{⊥} := {φ ∈ M(X)| ⟨f, φ⟩ = 0 ∀f ∈ N^{⊥}} coincides with the (weak*) σ(M(X), C_{0}(X))-closure of N (see [2] p. 24) and therefore (N^{⊥})^{⊥} = N since N is weak* closed. Then, we can apply Theorem 2.4 with

X := C_{0}(X); X^{∗} := M(X); Z := M(X); W := C_{0}(X); A := (I − P)

(X being equipped with the sup norm ||.||) and S^{∗} := N = (N^{⊥})^{⊥} = (N^{⊥})^{∗} in the case of equation (3.1) or S^{∗} := Δ ∩ N = Δ ∩ (N^{⊥})^{∗} in the case of (3.2), which yields

Theorem 4.4. Suppose that b ∈ L_{1} and Assumption 3.1 holds. Then:

(a) The equation (3.1) has a solution in L_{1} if and only if

u, w ∈ C_{0}(X), w ∈ N^{⊥} ⇒ ⟨b, u⟩ + M||(I − P)^{∗}u − w|| ≥ 0

for some M > 0.

(b) The equation (3.2) has a nonnegative solution in L_{1} if and only if

u, w, h ∈ C_{0}(X), w ≥ 0, h ∈ N^{⊥} ⇒ ⟨b, u⟩ + M||(I − P)^{∗}u − w − h|| ≥ 0

for some M > 0, or equivalently, if and only if

u, h ∈ C_{0}(X), h ∈ N^{⊥} ⇒ ⟨b, u⟩ + M|| min[0, (I − P)^{∗}u] − h|| ≥ 0

for some M > 0.

Proof. Because of Assumption 3.1, the hypotheses of Theorem 2.4 (a) are satisfied. In the case of (3.2),

S^{∗} = (Δ ∩ N) = (G + N^{⊥})^{∗}

with G := {f ∈ C_{0}(X), f ≥ 0}. As X is not reflexive, it remains to show that Ω+(S×{0}) is strongly closed in C_{0}(X). We first consider the case S = N^{⊥}. Let (f_{n}, g_{n}, r_{n}) be a sequence in C_{0}(X) × N^{⊥} × R^{+} such that

||f_{n}|| ≤ r_{n}; g_{n} ∈ N^{⊥}; r_{n} → r and lim_{n} ||f_{n} + g_{n} − f|| = 0,

where ||.|| denotes the sup norm in C_{0}(X). C_{0}(X) with the sup norm is complete so that f ∈ C_{0}(X). Let B_{1} := {x ∈ X| f(x) > r} and B_{2} := {x ∈ X| f(x) < −r}. Assume that μ(B_{1}) > 0. Then as strong convergence implies weak convergence, and g_{n} ∈ N^{⊥}, we have

∫ (f_{n} + g_{n})dφ = ∫ f_{n}dφ → ∫ fdφ, ∀φ ∈ N.

In particular, take a nonnegative measure φ in N with φ(B_{1}) = 1 and φ(B_{1}^{c}) = 0. We would have ∫ f_{n}dφ → ∫ fdφ = r+δ for some δ > 0. On the other hand, as ||f_{n}|| ≤ r_{n} ∀n, for n sufficiently large, ||f_{n}|| ≤ r + δ/2 so that | ∫ f_{n}dφ| ≤ r + δ/2 < r + δ a contradiction. Therefore, we must have μ(B_{1}) = 0 and similarly μ(B_{2}) = 0. In addition, {x ∈ X| |f(x)| ≥ r} is compact as f ∈ C_{0}(X). Consider the functions f_{1}(x) := f(x) if |f(x)| ≤ r and sign(f(x))r otherwise, f_{2}(x) := f(x) − r if f(x) ≥ r, f(x) + r if f(x) ≤ −r and 0 otherwise. Both are in C_{0}(X). In addition, f_{2} is in N^{⊥}, and f = f_{1} + f_{2}. It then suffices to note that ||f_{1}|| ≤ r. For the case where S = G+N^{⊥} consider a sequence (f_{n}, h_{n}, g_{n}, r_{n}) in C_{0}(X)×G×N^{⊥}×R^{+} such that

||f_{n}|| ≤ r_{n}; g_{n} ∈ N^{⊥}; r_{n} → r and lim_{n} ||f_{n} + h_{n} + g_{n} − f|| = 0,

f_{n} = f_{n}^{+}+f_{n}^{−} with ||f_{n}^{−}|| ≤ r_{n}. Rewrite f_{n}+h_{n} as w_{n}^{+}+w_{n}^{−} so that as h_{n} ≥ 0, ||w_{n}^{−}|| ≤ r_{n}. Again consider the set B_{2} as above. μ(B_{2}) = 0 for the same reasons. Indeed, with φ such that φ(B_{2}) = 1 and φ(B_{2}^{c}) = 0

∫ (f_{n} + h_{n} + g_{n})dφ = ∫ (w_{n}^{+} + w_{n}^{−})dφ → ∫ fdφ = −r − δ

for some δ > 0. But ∫ (w_{n}^{+}+w_{n}^{−})dφ ≥ ∫ w_{n}^{−}dφ ≥ −r−δ/2 for n sufficiently large. Consider the functions f_{1}(x) := f(x) if −r ≤ f ≤ 0, −r if f ≤ −r and 0 if f ≥ 0; f_{2}(x) := f(x) + r if f ≤ −r and 0 otherwise; f_{3}(x) := max[0, f(x)]. Again f_{i} ∈ C_{0}(X), ∀i; f = f_{1} +f_{2} +f_{3}, f_{2} ∈ N^{⊥}, f_{3} ≥ 0, and ||f_{1}|| ≤ r, which proves that Ω + (S × {0}) is closed in C_{0}(X).

Note that with this Farkas-like theorem, one uses functions in C_{0}(X) and the sup-norm rather than functions in L_{∞} with ||.||_{∞} as in Theorem 3.2.

5. Appendix

For ease of reference we collect in this appendix some results used in the paper, including Theorem 5.1 below that is a special case of the Generalized Farkas Theorem of Craven and Koliha ([3], Theor. 2). If X is Banach space with topological dual X^{∗}, the weak topology on X is denoted σ(X , X^{∗}) and the weak* topology on X^{∗} is denoted σ(X^{∗}, X). U denotes the closed unit sphere in X^{∗}, i.e. U := {f ∈ X^{∗}| ||f|| ≤ 1}. If S is a convex cone in X, its dual cone is

S^{∗} := {f ∈ X^{∗}| ⟨f, x⟩ ≥ 0 ∀x ∈ S}.

Theorem 5.1. (cf. [3] Theor. 2). Let X and Y be Banach spaces with topological duals X^{∗} and Y^{∗} respectively. Let S be a convex cone in X, and let A : X → Y be a weakly continuous linear map with adjoint A^{∗} : Y^{∗} → X^{∗}. If A(S) is weakly closed, then the following are equivalent conditions on b ∈ Y:

(a) The equation Ax = b has a solution x in S.

(b) A^{∗}y^{∗} ∈ S^{∗} ⇒ ⟨b, y^{∗}⟩ ≥ 0.

Theorem 5.2. Let X be a Banach space with topological dual X^{∗}.

(a) If x_{n} converges to x in the weak topology σ(X , X^{∗}), then ||x_{n}|| is bounded and lim inf ||x_{n}|| ≥ ||x||.

(b) The unit sphere U in X^{∗} is compact in the weak* topology.

(c) If X is separable, then the weak* topology of U is metrizable.

(d) If X is separable, then a convex subset K of X^{∗} is closed in the weak* topology if and

only if

(x_{n}^{∗} ∈ K and ⟨x, x_{n}^{∗}⟩ → ⟨x, x^{∗}⟩ ∀x ∈ X) ⇒ x^{∗} ∈ K.

Theorem 5.2 (b) is the so-called Alaoglu (or Banach-Alaoglu-Bourbaki) theorem. For a proof of Theorem 5.2 see e.g. [2] or [6].

Lemma 5.3. Let (X, B, μ) be as in Section 3. Let {φ_{n}} and φ be σ-finite measures on (X, B) such that

⟨φ_{n}, u⟩ → ⟨φ, u⟩ ∀u ∈ C_{0}(X),  (5.1)

where ⟨φ, u⟩ := ∫ udφ. Suppose, in addition, that every φ_{n} is absolutely continuous (a.c.) with respect to μ. Then

(a) φ is a.c. with respect to μ. Moreover (by the Radon-Nikodym theorem), let u_{n} and u be nonnegative measurable functions such that

φ_{n}(B) = ∫_{B} u_{n}dμ, and φ(B) = ∫_{B} udμ ∀B ∈ B.

164 O.Hernandez-Lerma, J.B.Lasserre / Cone-constrained linear equations in Banach spaces

(b) If (for a given 1 ≤ p ≤ ∞) u_{n} ∈ L_{p} ∀n, and lim inf_{n} ||u_{n}||_{p} ≤ M for some constant M, then u is in L_{p}.

For a proof of Lemma 5.3 see [8].

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