Jrl Syst Sci & Complexity (2009) 22: 542–554
A NONLINEAR KREIN RUTMAN THEOREM∗
K. C. CHANG
Received: 24 June 2009
c
2009
Springer Science + Business Media, LLC
Abstract A nonlinear version of Krein Rutman Theorem is established. This paper presents a
unified proof of the Krein Rutman Theorem for linear operators and for nonlinear operators, and of
the Perron-Frobenius theorem for nonnegative matrices and for nonnegative tensors.
Key words Eigenvector, increasing map, p-Laplace operator, positive operator, spectral radius.
1 Introduction
Krein Rutman Theorem is a fundamental theorem in positive linear compact operator theory. It has been widely applied to Partial Differential Equations, Dynamical systems, Markov
Process, Fixed Point Theory, and Functional Analysis.
The Krein-Rutman Theorem (K-R, in short) reads as:
Theorem 1.1 Let T be a linear, positive and compact operator with r(T ) > 0. Then r(T )
is an eigenvalue with eigenvector x0 ∈ Ṗ . If further int(P ) 6= ∅, and T is strongly positive, then
1. r(T ) > 0 is an eigenvalue, with eigen-vector x0 ∈ int(P ), i.e., T x0 = r(T )x0 .
2. r(T ) is algebraically simple.
3. If λ 6= r(T ) is an eigenvalue (real or complex) of T , then |λ| < r(T ).
See, for instance, [1–3]. The counterpart of this theorem in finite dimensional case is Perron
Frobenius Theorem (P-F, in short) for nonnegative matrices.
Theorem 1.2[4] Let A be a nonnegative square matrix, then r(A), the spectral radius of A,
is an eigenvalue with a nonnegative vector x0 6= 0 such that
Ax0 = r(A)x0 .
(1)
If further, A is irreducible, then
1. r(A) > 0 is an eigenvalue.
2. There exists a nonnegative vector x0 > 0, i.e., all components of x0 are positive, such
that Ax0 = r(A)x0 .
3. (Uniqueness) If λ is an eigenvalue with a nonnegative eigenvector, then λ = r(A).
4. The eigenvalue r(A) is algebraically simple.
5. If λ is an eigenvalue of A, then |λ| ≤ r(A).
Although (P-F) is for finite dimensional nonnegative operators, while (K-R) is for infinite
dimensional, the previous one is not a special case of the later. Indeed, the positive spectral
K. C. CHANG
Laboratory of Mathematics and Applied Mathematics, School of Mathematical Sciences, Peking University,
Beijing 100871, China. Email: [email protected].
∗ In memory of Prof. Guan Zhao Zhi
A NONLINEAR KREIN RUTMAN THEOREM
543
radius assumption is not needed in (P-F), and also the irreduciblity is weaker than the strong
positiveness for nonnegative square matrices.
The nonlinear version of the theorem has been extended to positive eigenvalue problem for
increasing, positively 1-homogeneous, compact, continuous mappings by R. Nussbaum[5−6] , J.
Mallet Paret and R. Nussbaum[7] , and R. Mahadevan[8] .
Let X be a real Banach space with total order cone P , i.e., X = P − P and P ∩ −P = {θ}.
The cone induces an ordering ≤:
∀x, y ∈ X, x ≤ y ⇔ y − x ∈ P.
Let Ṗ = P \{θ}, and denote x < y if y − x ∈ Ṗ . Let P ∗ be the dual cone of P, i.e.,
P ∗ = {x∗ ∈ X ∗ | hx∗ , xi ≥ 0, ∀x ∈ P }.
A map T : X → X is called (strictly) increasing of x ≤ y ⇒ T x ≤ T y (x < y ⇒ T x <
T y, respectively). If the P has non-empty interior: intP 6= ∅ and if T : Ṗ → int(P ), then T is
called strongly positive.
It is called positively 1-homogeneous, if T (tx) = tT x, ∀ t > 0, ∀ x ∈ X.
A pair (λ, x) ∈ R1 + ×Ṗ is called a positive eigen-pair if
T x = λx.
The following nonlinear Krein Rutman Theorem has been established in [8].
Theorem 1.3 Let T : X → X be an increasing, positively 1-homogeneous, compact,
continuous mapping for which there exists a nonzero u ∈ P and M > 0 such that
M T u ≥ u.
(2)
Then T has a positive eigen-pair (λ0 , x0 ).
Furthermore, if T is strongly positive and strictly increasing, then
1. If λ is an eigenvalue with nonnegative eigenvector, then λ = λ0 .
2. λ0 is geometrically simple.
3. If λ is a real eigenvalue of T , then |λ| ≤ λ0 .
We note: The spectral radius r(T ) does not appear in Mahadevan’s Theorem. In this
sense, the above nonlinear version of the Krein-Rutman theorem is an existence result of a
positive eigen-pair, but no qualitative information on what the positive eigenvalue is. However,
the notion of spectral radius for linear operators has been extended to increasing, positively
1-homogeneous, continuous mapping T in [7].
The purpose of this paper is threefold:
1. To define two numbers r∗ (T ) and r∗ (T ) related to T . They are used to bound all positive
eigenvalues of T .
2. To define two kinds of positiveness condition, which are stronger than the positive cone
preserving condition, but weaker than the strong positiveness condition. One of which is an
extension of the irreducibility for matrices, and the other is related to a notion for linear
operators due to Krasnosel’ski[9] and Webb[10] .
3. To provide a unified proof for linear and nonlinear, also for finite dimension and infinite
dimension.
Namely, we have the following
Theorem 1.4 Assume that T : X → X, is a increasing, positively 1-homogeneous, compact
mapping. If one of the following cases holds:
544
K. C. CHANG
1. dimX < ∞,
2. r∗ (T ) > 0.
3. r∗ (T ) > 0,
Then there exists a positive eigen-pair: (λ0 , x0 ), such that
1. λ0 ≥ 0 in case (1),
2. λ0 ≥ r∗ (T ) in case (2),
3. λ0 = r∗ (T ) in case (3).
i) If furthermore, intP 6= ∅, and T is semi-strongly positive, then λ0 = r∗ (T ) = r∗ (T ) is the
unique positive eigenvalue with positive eigenvector.
Moreover, the eigenvalue with respect to λ0 is geometrically simple. And λ0 is the largest among
all real eigenvalues of T .
ii) If further, T is e-positive for some e ∈ Ṗ , then λ0 = r∗ (T ) = r∗ (T ) is the unique positive
eigenvalue with positive eigenvector.
Moreover, the eigenvector with respect to λ0 is unique up to a multiplicative constant.
The paper is organized as follow: In Section 2, the two spectral radii r∗ (T ) and r∗ (T ) are
defined, and some of their basic properties are studied. The existence result is proved in Section
3. In Section 4, the notion of strong positiveness of T is extended in two directions: semi-strong
positiveness and e-positiveness. Under which we show that r∗ (T ) = r∗ (T ). In the last section,
few applications, including the positive eigenvalue problem for p-Laplace systems and that for
nonnegative tensors are studied.
2 Generalized Spectral Radii r ∗(T ) and r∗(T )
In this section, we introduce two numbers r∗ (T ) and r∗ (T ) related to an increasing, positively
1-homogeneous, compact, continuous mapping T . They are used as an replacement of the
spectral radius r(T ) for a linear, positive, compact operator T .
Definition 2.1 ∀ x ∈ Ṗ , we define
P ∗ (x) = {x∗ ∈ P ∗ | hx∗ , xi > 0}
hx∗ , T xi
µ∗ (x) = ∗ inf∗
,
x ∈P (x) hx∗ , xi
hx∗ , T xi
µ∗ (x) = sup
,
∗
x∗ ∈P ∗ (x) hx , xi
and
r∗ (T ) = sup µ∗ (x),
x∈Ṗ
r∗ (T ) = inf µ∗ (x).
x∈Ṗ
The following properties hold:
1) ∀ x ∈ Ṗ , P ∗ (x) is dense in P˙∗ .
In fact, ∀ x∗ ∈ P˙∗ \P ∗ (x), we have hx∗ , xi = 0. According to the Extension Theorem for
Positive Functionals (Hahn-Banach) (see Theorem 6 in [1], and Proposition 19.3(a) in [2]).
∃x∗0 ∈ P˙∗ with kx∗0 k = 1, such that hx∗0 , xi > 0. Then x∗ + ǫx∗0 ∈ P ∗ (x), ∀ ǫ > 0. Obviously,
x∗ + ǫx∗0 → x∗ as ǫ → 0.
2) If intP 6= ∅, then ∀ x ∈ intP, P ∗ (x) = Ṗ ∗ (see [2], Proposition 19.3(b), p.222).
3) If ∃c > 0 and ∃x ∈ Ṗ such that cx ≥ T x (cx ≤ T x), then
r∗ (T ) ≤ µ∗ (x) ≤ c, (r∗ (T ) ≥ µ∗ (x) ≥ c, respectively).
545
A NONLINEAR KREIN RUTMAN THEOREM
4) A pair (λ, x) is a positive eigen-pair, if and only if (λ, x) ∈ R1 +×Ṗ and µ∗ (x) = λ = µ∗ (x).
And then, r∗ (T ) ≤ λ ≤ r∗ (T ), if λ ≥ 0 is an eigenvalue of T with a positive eigenvector.
5) The function x 7→ µ∗ (x) (x 7→ µ∗ (x)) is l.s.c (u.s.c., respectively), and positively 0homogeneous.
In the following, we present few examples of the functions µ∗ and µ∗ :
Example 2.2 Let T be the linear operator on R2 defined by the matrix:
20
01
Then
µ∗ (x, y) =
(
µ∗ (x, y) =
(
2,
1
x > 0,
x = 0.
1,
2,
y > 0,
y = 0.
And then, r∗ (T ) = 1, r∗ (T ) = 2.
Example 2.3 If T ∈ L(X, X) is positive and compact with r(T ) > 0, then r∗ (T ) ≥ r(T ) ≥
r∗ (T ).
This is a direct consequence of Krein-Rutman Theorem and (4). However, we would like to
give a different proof without using Krein-Rutman Theorem.
Proof Let λ0 = r(T ).
1. ∀ ε > 0 small, λ0 + ε ∈ ρ(T ), the resultant set of T . Since
((λ0 + ε)I − T )−1 =
X
n≥0
Tn
(λ0 + ε)n+1
is positive, ∀ u ∈ Ṗ , x = ((λ0 + ε)I − T )−1 u ∈ Ṗ .
It follows,
(λ0 + ε)x = T x + u ≥ T x.
Applying (3), r∗ (T ) ≤ λ0 + ε. Since ε > 0 is arbitrarily small, we have r∗ (T ) ≤ λ0 .
2. Now, we prove: r∗ (T ) ≥ λ0 . It is sufficient to show that ∀ c < λ0 , ∃ xc ∈ Ṗ , such that
T xc ≥ cxc .
(3)
Suppose that (2.1) is not true, then
x 6= tλ−1
0 T x,
∀ (x, t) ∈ S+ × [0, 1].
where S+ = P ∩ ∂B1 (θ). Since T is compact, there exists δ > 0, such that
tλ−1
0 T xk ≥ δ. One chooses u ∈ Ṗ with kuk <
max{kT xk | kxk ≤ 1}, ∀ (x, t) ∈ S+ × [0, 1],
λ0
4 δ.
inf
(x,t)∈S+ ×[0,1]
kx −
λ20
Now, ∀ ε ∈ (0, 4M δ), where M =
kx − t(λ0 − ε)−1 (T x − u)k > 0.
(4)
546
K. C. CHANG
According to the homotopy invariance of the Leray-Schauder degree, it follows
iP ((I − (λ0 − ε)−1 (T − u), P ∩ B1 (θ)) = iP (I, P ∩ B1 (θ)) = 1,
where iP is the Leray Schauder index with respect to the cone P .
Then there exists xε ∈ P ∩ B1 (θ) satisfying
(λ0 − ε)xε = T xε − u.
Therefore, T xε ≥ (λ0 − ε)xε . This is a contradiction. Therefore (2.1) is true. Again by (3),
λ0 ≤ r∗ (T ).
3 Existence Theorem
Theorem 3.1 Assume that T : P → P is an increasing, positively 1-homogeneous, compact,
continuous mapping. If one of the following cases holds:
1. dimX < ∞,
2. r∗ (T ) > 0.
3. r∗ (T ) > 0,
Then there exists a positive eigen-pair: (λ0 , x0 ), such that λ0 ≥ 0 in case (1), λ0 ≥ r∗ (T )
in case (2), and λ0 = r∗ (T ) in case (3).
Proof 1. We choose u ∈ Ṗ and ε > 0 arbitrarily, and then define a mapping:
1
fε (x, µ) = µT (x + εu) : P × R+
→ P.
Since fε (x, 0) = θ, according to Leray Schauder Theorem, see Rabinowitz[11] , the solution set
of the equation
x = fε (x, µ) = µT (x + εu)
(5)
1
has a nontrivial connected unbounded component Cε+ = {(xε , µ) ∈ P × R+
| fε (xε , µ) = xε }
passing through (θ, 0).
From the equation, we have
xε (µ) = µT (xε (µ) + εu) ≥ µεT u,
provided T being increasing and positively 1-homogeneous. According to the Extension Theorem of Positive Functionals (Hahn-Banach), there exists x∗ ∈ Ṗ ∗ , with kx∗ k = 1, and
c = hx∗ , T ui > 0. Then we have
kxε (µ)k ≥ hx∗ , xε (µ)i ≥ hx∗ , µεT ui ≥ µεc.
(6)
Moreover, T is compact, there exists M > 0 such that
kT xk ≤ M kxk.
(7)
Fixing any ε > 0, we have shown that the component Cε+ is unbounded, i.e., either µ or kxε (µ)k
is unbounded. Provided by (3.2) there exists µε such that kxε k = 1, where xε := xε (µε ).
It follows from (3.3),
1 = kxε k = µε kT (xε (µ) + εu)k ≤ µε M kxε + εuk ≤ µε M (1 + εkuk),
we conclude:
A NONLINEAR KREIN RUTMAN THEOREM
547
µε ≥ (M (1 + εkuk))−1 ,
i.e., µε is bounded from 0.
In case (1), dimX < ∞. There exists a subsequence: εn → 0, such that xn → x0 , with
kx0 k = 1, where xn = xεn .
Let µn = µεn and λn = µ1n . Then µn is either bounded or unbounded, after a subsequence,
we have µn → µ0 ≥ M −1 or µn → ∞. These imply λn → λ0 with λ0 ≥ 0 and T x0 = λ0 x0 .
In case (2), from the equation (3.1), we have
xε = xε (µε ) = µε T (xε (µε ) + εµ) ≥ µε T xε .
Applying Proposition (3), we have r∗ (T ) ≤ λε := (µε )−1 .
By the assumption, r∗ (T ) > 0, µε is bounded. Since T is compact, we obtain subsequences
{xn } and {µn }, where xn = xεn , and µn = µεn such that T xn → y0 , µn → µ0 , for some
y0 ∈ X, µ0 ∈ [M −1 , r∗ (T1)−1 ].
From the equation:
xn = µn T (xn + εn u) → µ0 y0 ,
and let x0 = µ0 y0 , it follows T x0 = λ0 x0 , and xn → x0 . Therefore, kx0 k = 1, and λ0 =
(µ0 )−1 ∈ [r∗ (T ), M ].
In case (3), as we have seen in the proof of Example 2.3, the assumption r∗ (T ) > 0 implies
that ∀ε ∈ (0, r∗ (T )), ∃ yε ∈ Ṗ , such that (r∗ (T ) − ε)yε ≤ T yε . By a standard argument we can
show that:
x ≥ (µ(r∗ (T ) − ε))n εu, ∀ n = 1, 2, · · · ,
whenever (x, µ) ∈ C + ε, and then µ ≤ (r∗ (T ) − ε)−1 , see Rabinowitz[11] , see also Mahadevan[8] .
By the same argument used in previous paragraph to the remaining part, we can prove
that there exist (x0 , µ0 ), with kx0 k = 1, x0 ∈ P and λ0 ≥ r∗ (T ) satisfying: T x0 = λ0 x0 . Since
now λ0 is an positive eigenvalue with positive eigenvector, applying (4) in section 2, we have
λ0 ≤ r∗ (T ). Thus λ0 = r∗ (T ).
Remark 3.2 Part (1) contains the weak form of Perron Frobenius Theorem as a special
case. Part (3) of Theorem 3.1 has been essentially obtained in [5,8].
Remark 3.3 Combining Example 2.3, Part (3) of Theorem 3.1 provides a topological proof
for the first part of (K-R) Theorem.
In fact, if r(T ) > 0, according to Example 2.3, then we have r ∗ (T ) ≥ r(T ) > 0, applying the
case (3), we conclude the existence of a positive eigen-pair (λ0 , x0 ) satisfying λ0 = r∗(T ) ≥ r(T ).
On the other hand, r(T ) is the spectral radius of T , we must have λ0 ≤ r(T ). This proves that
the spectral radius is an eigenvalue with positive eigenvector if it is positive. Moreover, in this
case, r∗ (T ) = r(T ).
4 Extensions of the Strong Positiveness
In the following we study the relation between r∗ (T ) and r∗ (T ).
Definition 4.1 ∀ u ∈ Ṗ , v ∈
/ P, let us define
1
| u + tv ∈ P }.
δu (v) = sup{t ∈ R+
By definition, we have
548
K. C. CHANG
1. u + tv ∈ P, ∀ t ∈ [0, δu (v)],
2. u + tv ∈
/ P, ∀ t > δu (v).
The proof is trivial; see, for instance Zeidler (p.292, lemma 7.3.1)[3] .
First, we assume int(P ) 6= ∅.
Lemma 4.2 Assume int(P ) 6= ∅, and u ∈ int(P ), then ∀v ∈
/ P, then δu (v) > 0.
Lemma 4.3 Assume int(P ) 6= ∅, and that T : P → P is an increasing, positively 11
1
homogeneous mapping. If there exist (λ, x) ∈ R+
\{0} × int(P ) and (µ, y) ∈ R+
× Ṗ , such that
T x ≤ λx and T y ≥ µy, then µ ≤ λ.
Proof We apply Lemma 4.2 to the pair (x, −y), and obtain
1. x − ty ∈ P, ∀ t ∈ [0, δx (−y)],
2. x − ty ∈
/ P, ∀ t > δx (−y).
From (1), we have
λx ≥ T x ≥ δx (−y)T y ≥ µδx (−y)y.
From (2), it follows µ ≤ λ.
Now, let us define
◦r∗ (T ) =
inf
µ∗ (x),
sup
µ∗ (x).
x∈int(P )
and
◦r∗ (T ) =
x∈int(P )
1
By definition, if (λ, x) ∈ R+
× int(P ) is a positive eigen-pair, then ◦r∗ (T ) ≤ λ ≤ ◦r∗ (T ).
Conversely, we have
Theorem 4.4 Assume int(P ) 6= ∅. If T : P → P is an increasing, positively 1-homogeneous,
compact, continuous mapping. Then
r∗ (T ) ≤ ◦r∗ (T ).
Proof
means
1. By definition, ∀ ε > 0, ∃ yε ∈ Ṗ such that µ∗ (yε ) ≤ r∗ (T ) < µ ∗ (yε ) + ε. This
r∗ (T ) − ε < µ∗ (yε ) ≤
hx∗ , T yε i
,
hx∗ , yε i
∀ x∗ ∈ P ∗ (yε ).
It follows
hx∗ , T yε − (r∗ (T ) − ε)yε i ≥ 0, ∀ x∗ ∈ P ∗ (yε ).
According to property (1), the inequality holds for all x∗ ∈ P ∗ . Again by the Extension Theorem
of Positive Functionals (Hahn- Banach)[1] [Theorem 6. p.570], see also [2] [Proposition 19.3(a),
p.222], we obtain T yε ≥ (r∗ (T ) − ε)yε .
2. On the other hand, by the same reason, we have xε ∈ int(P ) such that (◦r∗ (T ) + ε)xε ≥
T xε .
3. Now, we apply lemma 4.3, and conclude: r∗ (T ) − ε ≤ ◦r∗ (T ) + ε. Since ε > 0 is arbitrary,
the theorem is proved.
Recall that a mapping T : P → P is strongly positive, if T : Ṗ → int(P ).
Definition 4.5 Assume int(P ) 6= ∅. T is called semi-strongly positive, if there exists
x∗ ∈ P ∗ such that
hx∗ , T xi > 0 = hx∗ , xi, ∀ x ∈ Ṗ \int(P ).
According to the Extension Theorem of Positive Functionals, a strongly positive mapping
is semi-strongly positive.
A NONLINEAR KREIN RUTMAN THEOREM
549
An easy example for a semi-strongly positive mapping, but not strongly positive is the
following matrix


0001
1 0 0 0


0 1 0 0 .
0010
More generally, we have
Lemma 4.6 Let T = (tij ) be an n × n nonnegative matrix. Then as a nonnegative linear
operator, T is semi-strong positiveness if and only if the matrix is irreducible.
Proof In fact, x = (x1 , x2 , · · · , xn ) ∈ Ṗ \int(P ), if and only if there exists a unique proper
subset I of {1, 2, · · · , n} such that xi = 0, ∀ i ∈ I and xj > 0, ∀ j ∈
/ I. Let XI = {x ∈ Rn | xi =
0, ∀ i ∈ I, xj 6= 0, ∀ j ∈
/ I}, this means x ∈ XI ∩ int(P ).
If T is reducible, i.e., there exists a proper invariant subspace X ⊂ Rn , then X ∩ P is
invariant under T . We choose x ∈ Ṗ \int(P ), with XI = X. ∀ x∗ = (y1 , y2 , · · · , yn ) ∈ P ∗ such
that hx∗ , xi = 0, we have yj = 0, ∀ j ∈
/ I, this implies hx∗ , T xi = 0. Then T is not semi-strong
positive.
Conversely, if T is not semi-strong positive, i.e., ∃ x ∈ Ṗ \int(P ), such that ∀ x∗ ∈ P ∗ , hx∗ , xi
= 0 implies hx∗ , T xi = 0, then T x ∈ XI , where I is the proper subset related to x, i.e.,
X
tij xj = 0, ∀ i ∈ I.
j ∈I
/
It implies tij = 0, ∀ (i, j) ∈ I × I ′ , where I ′ is the complement set of I. Then T is reducible.
Remark 4.7 Let T be a nonnegative matrix. From the above proof, it is easily seen the
following equivalences:
1. T is irreducible.
2. ∀ x ∈ XI ∩ int(P ) ⇒ T x ∈
/ XI .
3. T has no positive eigen-pair (λ0 , x0 ) such that x0 ∈ Ṗ \int(P ).
We leave the proof to the reader.
Theorem 4.8 Let T be a semi-strongly positive, positively 1-homogeneous, strictly increasing, compact, continuous mapping with either r∗ (T ) > 0 or r∗ (T ) > 0. Then λ0 = r∗ (T ) =
r∗ (T ) is the unique positive eigenvalue with positive eigenvector.
Moreover, the eigenvalue λ0 is geometrically simple, and the eigenvector x0 ∈ int(P ). Also,
λ0 is the largest among all real eigenvalues of T.
Proof 1. We claim µ∗ (x) = +∞, ∀ x ∈ Ṗ \int(P ). In fact, by definition, ∀ x ∈ Ṗ \int(P ), ∃ x∗ ∈
P ∗ such that
hx∗ , T xi > 0 = hx∗ , xi.
∀ y ∗ ∈ P ∗ (x), we define x∗n = x∗ + n1 y ∗ ∈ P ∗ (x). It satisfies
hx∗n , T xi ≥ hx∗ , T xi,
and
hx∗n , xi =
Therefore,
µ∗ (x) ≥
1 ∗
hy , xi → 0.
n
hx∗n , T xi
→ +∞.
hx∗n , xi
550
K. C. CHANG
2. Accordingly to Theorem 4.4,
r∗ (T ) = inf µ∗ (x) =
x∈Ṗ
int
x∈int(P )
µ∗ (x) = ◦r∗ (T ) ≥ r∗ (T ).
3. From Theorem 3.1, the first conclusion λ0 = r∗ (T ) = r∗ (T ) follows. Moreover, from the
definitions of the semi-strong positiveness and the positive eigen-pair, the associate eigenvector
x0 ∈ int(P ),
4. Now, we prove the geometric simplicity of λ0 . Suppose that there exists x1 ∈ X satisfying
T x1 = λ0 x1 , but x1 6= x0 . First we assume x1 ∈
/ P . According to lemma 4.2, the number
δx0 (x1 ) > 0. Since x0 +δx0 (x1 )x1 ∈ P, and T is strictly increasing and positively 1-homogeneous,
if x0 6= −δx0 (x1 )x1 , then T x0 > T (−δx0 (x1 )x1 ). Therefore, x0 > −δx0 (x1 )x1 . This contradicts
with the definition of δx0 (x1 ). We proved: x0 and x1 are co-linear.
Next, we assume x1 ∈ Ṗ , by the same manner, again, we can show x0 and x1 are co-linear.
5. Finally, let (λ, x) be an eigen-pair, we want to show |λ| ≤ λ0 . Since x0 ∈ int(P ), we have
δx0 (±x) ∈ P, i.e., x0 ≥ ∓δx0 (±x)x. Since T is increasing and positively 1-homogeneous, which
implies λ0 x0 = T x0 ≥ ∓δx0 (±x)T x = ∓δx0 (±x)λx, or
x0 ± δx0 (±x)
λ
x ∈ P.
λ0
Therefore, |λ| ≤ λ0 .
Remark 4.9 Since algebraic simplicity, which implies geometric simplicity, is only defined
for linear operators, for nonlinear mappings, we are only concerned with geometric simplicity.
In this sense, Theorem 4.8 includes Theorems 1.1, 1.2, and 1.3 as special cases except (3) in
Theorem 1.1 and (5) in Theorem 1.2.
Now, we avoid the assumption int(P ) 6= ∅, but make assumptions on the mapping T .
Definition 4.10 Let e ∈ Ṗ , a mapping T : P → P is called e-positive, if ∀ x ∈ Ṗ there
exist c(x), d(x) > 0, such that
c(x)e ≤ T x ≤ d(x)e.
Obviously, strongly positive mapping is e-positive, ∀ e ∈ int(P ). There are many examples of
e-positive mappings, but not strongly positive. In particular, on those Banach function spaces,
e.g., C0 (Ω ), Lp (Ω ), 1 ≤ p ≤ ∞, etc. the positive cones have no interior, The e-positiveness
plays the role of strong positiveness.
Lemma 4.11 If T is e-positive, then ∀ x, y ∈ Ṗ , δT x (−T y) > 0.
Proof It follows from the fact
(c(x) − td(y))e ≤ T x − tT y.
Lemma 4.12 Let T : P → P be an increasing, positively 1-homogeneous, e-positive map1
ping. If (λ, x), (µ, y) ∈ R+
× Ṗ satisfy T x ≤ λx, and T y ≥ µy, then µ ≤ λ.
Proof We follow the procedure in the proof of lemma 4.3. Since now δT x (−T y) > 0, we
have
0 ≤ T x − δT x (−T y)T y ≤ λx − δT x (−T y)µy,
i.e., x − δT x (−T y) λµ y ∈ P. Since T is increasing, T x ≥ δT x (−T y) λµ T y. By the definition, we
obtain µ ≤ λ.
Parallel to Theorem 4.8, we have
Theorem 4.13 Let T : P → P be an e-positive, increasing, positively 1-homogeneous,
compact, and continuous mapping with either r∗ (T ) > 0 or r∗ (T ) > 0. Then there exists
A NONLINEAR KREIN RUTMAN THEOREM
551
unique eigenvalue λ0 = r∗ (T ) = r∗ (T ) with positive eigenvector x0 ∈ Ṗ . If further, T is strictly
increasing, then x0 is the unique positive eigenvector up to a multiplicative constant.
Proof We follow the procedure in the proof of Theorem 4.8 , ∀ε > 0 we find xε , yε ∈ Ṗ ,
such that
T xε ≤ (r∗ (T ) + ε)xε , T yε ≥ (r∗ (T ) − ε)yε .
According to lemma 4.12 , r∗ (T ) ≤ r∗ (T ).
1
Applying Theorem 3.1, there is an positive eigen-pair (λ0 , x0 ) ∈ R+
× Ṗ . Therefore r∗ (T ) ≤
∗
λ0 ≤ r∗ (T ) ≤ r (T ), and then they are all equal.
Next, we turn to prove the uniqueness of positive eigenvectors. Assume x, y ∈ Ṗ satisfying
T x = λ0 x, and T y = λ0 y. According to lemma 4.11, δx (−y) = δT x (−T y) > 0, and then
x − δx (−y)y ∈ P . If x 6= δx (−y)y, since T is strictly increasing and 1-homogeneous, then
T x 6= δx (−y)T y. This contradicts with the definition of δx (−y)y.
As a byproduct, we obtain the following minimax characterization of the positive eigenvalue
of T , which is an extension of a result due to Collatz[12] .
Corollary 4.14 Let T be an increasing, positively 1-homogeneous, compact, and continuous
mapping with either r∗ (T ) > 0 or r ∗ (T ) > 0. If T is either semi-strongly positive or e-positive
for some e ∈ Ṗ , then we have the equality:
hx∗ , T xi
hx∗ , T xi
∗
=
r
(T
)
=
r
(T
)
=
sup
inf
.
∗
∗
∗
∗
∗
x∈Ṗ x∗ ∈P ∗ (x) hx , xi
x∈Ṗ x ∈P (x) hx , xi
inf
sup
5 Applications
We present few applications of the above theorems to some nonlinear problems.
5.1 p-Laplace Systems
Let Ω ⊂ Rn be a bounded domain with a smooth, connected boundary ∂Ω . Let ∆p =
div(|∇u|p−2 ∇u), 1 < p < ∞, be the p-Laplace operator. It is known that ∀ f ∈ Lq (Ω ) with
q > pn/(p − 1), the following equation:
−∆p u = f
possesses a unique solution u ∈ W01,p (Ω ) satisfying u ∈ C 1,α for some α ∈ (0, 1), see for instance,
di Benedetto[13] , Tolksdorf[14] . If f ≥ 0, then u ≥ 0. Moreover, if u does not vanish identically
on Ω , it is positive everywhere in Ω , and ∂u
∂ν > 0 on ∂Ω , where ν is the interior normal, see
[15]
Vazques .
Now, we study the positive eigenvalue problem for the p-Laplace system: Let M = (mij ) be
an N × N nonnegative irreducible matrix, and let u = (u1 , u2 , · · · , uN )T be a vector function
in W01,p (Ω , RN ). We consider the positive eigen-pair (λ, u) of the system:
−∆p u = λM(|u1 |p−2 u1 , |u2 |p−2 u2 , · · · , |uN |p−2 uN )T .
Define X = C 1,α (Ω , RN )∩C0 (Ω , RN ). Let us define a strictly increasing, positively 1-homogeneous
mapping A : X → X as follow:
u 7→ (−∆p • IN ×N )−1 v,
where v = V (x) = M(|u1 |p−2 u1 , |u2 |p−2 u2 , · · · , |uN |p−2 uN )T .
Obviously, A is compact and continuous. If all entries of M are positive, then A is strongly
positive. However, we only assume M being irreducible, let us verify the semi strong positiveness of A.
552
K. C. CHANG
Let P be positive cone of X, ∀ u ∈ P , let w = W = (w1 (x), w2 (x), · · · , wN (x))T = Au, and
p−1
p−1
y = (y1 , y2 , · · · , yN ) = (ku1 kp−1
p−1 , ku2 kp−1 , · · · , kuN kp−1 ).
According to the Strong Maximum Principle, ∀ i
wi ≡ 0 ⇔ vi (x) = 0, a.e., ⇔ (My T )i = 0.
Let I = {i ∈ [1, 2, · · · , N ] | yi = 0}.
Now, for u ∈ Ṗ \int(P ), if I = ∅, then by the Strong Maximum Principle, V ∈ int(P ).
Obviously, there exist e∗ ∈ X ∗ satisfying he∗ , wi > 0 = he∗ , ui.
Otherwise, I 6= ∅. Since M is irreducible, following from remark 4.7, ∃ i0 ∈ I, such that
(My T )i0 > 0. Again, due to the Strong Maximum Principle, wi0 (x) > 0, a.e. Define e∗ =
(z1 (x), z2 (x), · · · , zN (x)), where
zi (x) =
Then,
he∗ , ui =
XZ
i∈I
and
∗
1, if i ∈ I,
0, if i ∈
/ I.
∗
ui (x)dx = 0,
Ω
he , Aui = he , wi ≥
Z
wi0 (x)dx > 0.
Ω
The semi-strong positiveness of A has been proved.
Now, we verify the condition: r∗ (A) > 0. It is sufficient to show that there exist u ∈ Ṗ , δ > 0,
such that Au ≥ δu.
In fact, let ϕ ∈ D(Ω ), with ϕ ≥ 0, but not identical to 0. One chooses u = (ϕ, ϕ, · · · , ϕ)T .
PN
From the irreducibility of M, we have j=1 mij > 0, ∀ i. It follows, w = Au ∈ int(P ).
Applying Theorem 1.4, we conclude the existence of a unique positive eigen-pair (λ0 , u0 ) ∈
1
R+
× W01,p (Ω , RN ) satisfying:
−∆p u = λM(|u1 |p−2 u1 , |u2 |p−2 u2 , · · · , |uN |p−2 uN )T ,
with ui (x) > 0, ∀ x ∈ Ω , ∀ i = 1, 2, · · · N.
Remark 5.1 The positive eigenvalue problem for the p-Laplace operator has been studied
by many authors, see [16], etc. via variational methods and Mahadevan[8] via his nonlinear
version of Krein Rutman Theorem. However, the system studied above can not be obtained by
all of their methods.
5.2 Eigenvalues for tensors
An m-order n-dimensional tensor C is a set of nm real entries
C = (ci1 i2 ···im ),
ci1 i2 ···im ∈ R,
1 ≤ i1 , i2 , · · · , im ≤ n.
C is called nonnegative (or respectively positive) if ci1 ···im ≥ 0 (or respectively ci1 ···im > 0).
To an n-vector x = (x1 , x2 , · · · , xn ), real or complex, we define an n-vector:
m−1
Cx
:=
n
X
i2 ,··· ,im =1
cii2 ···im xi2 · · · xim
1≤i≤n
.
A NONLINEAR KREIN RUTMAN THEOREM
553
Suppose Cxm−1 6= 0, a pair (λ, x) ∈ C × (Cn \ {0}) is called an eigenvalue and an eigenvector,
if they satisfy
Cxm−1 = λx[m−1] ,
, xm−1
, · · · , xm−1
). When m is even, and C is symmetric, this was
where x[m−1] = (xm−1
n
2
1
introduced by Qi[17] ; when m is odd, Lim[18] used (xm−1
sgnx1 , xm−1
sgnx2 , · · · , xm−1
sgnxn ) on
n
1
2
the right-hand side instead, and the notion has been generalized in Chang Pearson Zhang[19] .
Let us define a mapping T from P , the positive cone of Rn into itself:
1
1
1 u 7→ T u = (Cxm−1 )1m−1 , (Cxm−1 )2m−1 , · · · , (Cxm−1 )nm−1 .
Obviously, T is strictly increasing, positively 1-homogeneous, compact, and continuous. If
C is positive, then T is strongly positive. Similar to the matrices (m = 2), we introduce the
notion of reducibility for tensors[19] .
Definition 5.2 (Reducibility) A tensor C = (ci1 i2 ···im ) of order m dimension n is called
reducible, if there exists a nonempty proper index subset I ⊂ {1, 2, · · · , n} such that
ci1 i2 ···im = 0,
∀i1 ∈ I,
∀i2 , i3 · · · , im ∈
/ I.
If C is not reducible, then we call C irreducible.
Again, similar to Lemma 4.6, we can show: If C is irreducible, then T is semi-strongly
positive[19] .
Then Theorem 1.4 is applied, we have
Theorem 5.3 If A is a nonnegative tensor of order m dimension n, then there exist λ0 ≥ 0
and a nonnegative vector x0 6= 0 such that
[m−1]
Axm−1
= λ0 x0
0
.
(8)
Theorem 5.4 If A is an irreducible nonnegative tensor of order m dimension n, then the
pair (λ0 , x0 ) in equation (5.2) satisfy:
1. λ0 > 0 is an eigenvalue.
2. x0 > 0, i.e., all components of x0 are positive.
3. If λ is an eigenvalue with nonnegative eigenvector, then λ = λ0 . Moreover, the nonnegative eigenvector is unique up to a multiplicative constant.
Remark 5.5 Both Theorems 5.3 and 5.4 were obtained in [19]
References
[1]
[2]
[3]
[4]
Z. Z. Guan, Lectures on Functional Analysis (in Chinese), High Education Press, 1958.
K. Deimling, Nonlinear Functional Analysis, Spronger-Verlag, 1985.
E. Zeidler, Nonlinear Functional Analysis and its Applications, Springer-Verlag, 1986.
A. Berman and R. Plemmom, Nonnegative Matrices in the Mathematical Sciences, Acad. Press,
1979.
[5] R. D. Nussbaum, Eigenvectors of Nonlinear positive operators and the linear Krein-Rutman theorem, Lect. Notes in Math, 1981, 866: 303–330.
[6] R. D. Nussbaum, Eigenvectors of order preserving linear operators, J. London Math. Soc., 1996,
(2): 480–496.
[7] Marllet-Paret and R. D. Nussbaum, Eigenvalues for a class of homogeneous cone maps arising from
max-plus operators, Discrete Continuous Dynamical Systems, 2002, 8(3): 519–562.
554
K. C. CHANG
[8] R. Mahadeva, A note on a non-linear Krein-Rutman theorem, Nonlinear Analysis, TMA, 2007,
67(11): 3084–3090.
[9] M. A. Krasnosel’ski, Positive Solutions of Operator Equations, Nordhoff, Groningen, 1964.
[10] J. R. L. Webb, Remarks on u0 positive operators, J. of Fixed Point Theory and Applications, 2009,
5(1): 37–46.
[11] P. Rabinowitz, Theorie du degré topologique et applications á des problémes aux limittes non
linéaires, Laboratoire analyse et numerique, Univ. Paris VI, 1975.
[12] L. Collatz, Einschliessungssatz fur die charakteristischen Zahlen von Matrizen Math. Z, 1946, 48:
221–226.
[13] E. di Benedetto, C 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear
Analysis, TMA, 1983, 7: 827–850.
[14] P. Tolksdorf, Regularity for a more general class of quasiliner elliptic equations, J. Diff. Equ., 1984,
51: 126–150.
[15] J. L. Vazquez, A strong Maximum Principle for some quasilinear elliptic equations, Applied Math.
and Optimizations, 1984, 12: 191–202.
[16] A. Anane, Simplicité et isolation de la première valeur du p-lalplacien avec poids, C. R. Acad. Sci.
Paris, 1987, 305(16): 725–728.
[17] L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Computation, 2005, 40: 1302–1324.
[18] L. H. Lim, Singular values and eigenvalues of tensors, A variational approach, in Proc. 1st IEEE
International Workshop on Computational Advances of Multi-Tensor Adaptive Processing, Dec.
13–15, 2005, 129–132.
[19] K. C. Chang, K. Pearson, and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 2008, 6: 507–520.