International Journal of Analysis and Applications
ISSN 2291-8639
Volume 6, Number 2 (2014), 154-163
http://www.etamaths.com

INDEX FORMULAS FOR COUNTABLY ϕ−SET CONTRACTION

H. SALAHIFARD, S. M. VAEZPOUR∗

Abstract. In this paper, we study the index formulas for a class of bounded

linear operators, namely ϕ−set contractions, acting on a Banach space and we
discuss some application of this class of operators to the theory of bifurcation
points. In particular our results generalize and improve some recent results

mentioned in the literature.

1. Introduction and Preliminary

Finding necessary and sufficient conditions for the appearance of nontrivial so-
lutions arbitrary close to some points (called bifurcation points) of the trivial
branch, with assumption of existence of a known (trivial) branch of solutions for a
parametrized family of an equation, is one of the oldest problems of mathematics
which have created bifurcation theory. One of the most important role in bifurca-
tion theory is played by index formulas for suitable kind of operators. In recent
years, many authors have focused on set-contractive operators and obtained a lot
of valuable results (see [12, 7, 10]). Nussbaum (1969) [14] developed degree theory
for k-set contractive operators (0 ≤ k < 1), which was first introduced by Kura-
towski at 1930 [11], Stuart, Toland [18] and Amann (1976) [2] established the index
formula for k-set contractions and condensing operators. Kim (2008)[10] present-
ed an index formula for countably k-set contractive bounded linear operators in a
real Banach space, by using a degree theory for countably condensing operators.
In this paper, we continue to study set-contractive operators and investigate the
conditions under which the topological degrees can be defined for a larger class
of k-set contraction, namely ϕ−set-contractive operators. Moreover, we introduce
generalized ϕk−set-contractive operators. It should be noted that this class of oper-
ators, as special cases, includes linear bounded operators, nonexpansive operators,
completely continuous operators, k- set-contractive operators, condensing opera-
tors and 1−set contractive operators. Correspondingly, we can obtain some new
bifurcation theorems of these operators, which improve and extend many famous
theorems such as the Hetzer’s theorem, Nussbaum’s theorem, Kim’s theorem, etc.

Before proceeding to the main results of this paper, we must recall some nota-
tions, definitions and theorems we shall need.
A function γ : {B ⊂ X : B is bounded} → [0,∞) is said to be a measure of non-
compactness on a Banach space X, if it satisfies the following conditions:

2010 Mathematics Subject Classification. 47A13, 37G10.
Key words and phrases. Index formula; ϕ-set contractions; Fredholm operators; Bifurcation

points; Condensing map.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

154



INDEX FORMULA FOR ϕ−SET CONTRACTIONS 155

(1)(invariance under closure and convex hull): γ( ¯coB) = γ(B),
(2)(regularity): γ(B) = 0 if and only if B is relatively compact,
(3)(semi-additivity): γ(B1 ∪B2) = max{γ(B1),γ(B2)},
(4)(algebraic semi-additivity): γ(B1 + B2) ≤ γ(B1) + γ(B2)
(5)(semi-homogeneity): γ(αB) = |α|γ(B) for all α ∈ R, and
(6)(Lipschitzianity): |γ(B1)−γ(B2)| ≤ Lγρ(B1,B2), where ρ denotes the Hausdorff
semi-metric, that is,

ρ(B1,B2) = inf{� > 0 : B2 ⊂ B1 + �B̄(0, 1),B1 ⊂ B2 + �B̄(0, 1)}.
The most important examples of measures of noncompactness are the Kuratowski
measure of noncompactness (or set measure of noncompactness)

α(Ω) = inf{r > 0 : Xmay be covered by finitely many sets of diameter ≤ r},
and the Hausdorff measure of noncompactness (or ball measure of noncompactness)

β(Ω) = inf{r > 0 : there exists a finite r-net forΩ in X}.
A detailed account of theory and applications of measures of noncompactness may
be found in the monographs [1, 3]. Note that the Kuratowski measure of noncom-
pactness and the Hausdorff measure of noncompactness have the above properties;
see [1, 19].

A function ϕ : R+ → R+ is said to be a
• comparison function if ϕ(0) = 0 and ϕ(t) < t for each t > 0,
• semi-comparison function if ϕ(0) = 0 and ϕ(t) < kt for each t > 0 and some
k ≥ 0, Denote

Φk = {ϕ : R+ → R+,ϕ(t) < kt for t > 0, ϕ(0) = 0},
for k = 1, we denote Φ1 by Φ.
Let k ≥ 1. A continuous operator T : X → X (with respect to γ) is said to be
• countably k−set contractive [10]:
if γ(T(C)) ≤ kγ(C) for each countable bounded set C ⊆ X.
• countably k−set contraction [10]:
if γ(T(C)) ≤ kγ(C) for each countable bounded set C ⊆ X and 0 ≤ k < 1.
• k−set contraction [18]:
if γ(T(C)) ≤ kγ(C) for all bounded sets C ⊂ X.
and 1−set contraction, if k = 1 [6].
• countably condensing [10]:
if γ(T(C)) < γ(C) for each countable bounded set C ⊂ X with γ(C) > 0.
• countably ϕ−set contractive :
if γ(T(C)) ≤ ϕ(γ(C)) for some ϕ ∈ Φk and each countable bounded set C ⊆ X.
•countably ϕ−set contraction [5]:
if γ(T(C)) ≤ ϕ(γ(C)) for some ϕ ∈ Φ and each countable bounded set C ⊆ X.

Remark 1.1. Clearly, every k−set contractive mapping is a ϕ−set contractive
and every bounded linear operator is a k−set contractive mapping by property (6)
of measure of noncompactness γ, so every bounded linear operator is a ϕ−set con-
tractive.

Let T be a bounded linear operator of a Banach space X into a Banach space Y
. The null space and the range of T are denoted by N(T) and R(T), respectively.



156 SALAHIFARD AND VAEZPOUR

T is said to be a Fredholm operator if the dimension of N(T) is finite and the
codimension of R(T) is finite. If the dimension of N(T) is finite and R(T) is closed,
then T is said to be a semi-Fredholm operator. In this case, i(T) = dimN(T) −
codimR(T) is called the index of T .

2. Main Results

In this section, we present a necessary condition for the existence of bifurcation
points of

u = λGu,

where X is a Banach space, u ∈ X, λ ∈ R and G : X → X is a countably general-
ized ϕ−set contraction.
This result can be regarded as a generalization of the [9] and [8]. Moreover, we ex-
tend the index formula for countably ϕ−set contractive, comutable bounded linear
operators as follows:

Theorem 2.1. Let T : X → X be a countably ϕ−set contractive bounded linear
comutable operator on a real Banach space X. If 1 is not an eigenvalue of T , then
ind(T, 0) = (−1)ν, where ν is the sum of the multiplicities of the eigenvalues λ > 1
of T .

So the results from [10] can be obtained as consequences of our result.
In order to show the main theorem above, we shall need some notations, definitions
and lemmas to show that the decomposition is possible for countably generalized
ϕ−set contractive bounded linear operators.

Now let us state our main definition which determines an important class of
operators including linear bounded operators, nonexpansive operators, completely
continuous operators, k- set-contractive operators, condensing operators and 1−set
contractive operators.

Definition 2.2. A continuous operator T : Ω → X is said to be countably gener-
alized ϕ−set contractive if

γ(Tn(C)) ≤ ϕ(γ(C))
for some n ∈ N, ϕ ∈ Φk, and each countable bounded set C ⊆ X, and is said to be
countably generalized ϕ−set contraction if ϕ ∈ Φ.

The following provides two examples; one is an example of a ϕ−set contraction
which is not a k-set contraction for any 0 < k < 1, the other is an example of a
generalized ϕ−set contraction which is not a ϕ−set contraction.

Example 2.3. Let X = [0, 1] ∪{2, 3, 4, ...} For the metric, let

ρ(x,y) =



|x−y| ifx,y ∈ [0, 1],

x + y if one of x,y 6∈ [0, 1].
It is apparent that (X,ρ) is a complete metric space Define the mapping T : X → X
by

Tx =




x− 1
2
x2 ifx,y ∈ [0, 1],

x− 1 ifx ∈{2, 3, ...}.



INDEX FORMULA FOR ϕ−SET CONTRACTIONS 157

Then, for x,y ∈ [0, 1] with x?y = t > 0,

ρ(x,y) = (x−y)(1 −
1

2
(x + y)) ≤ t(l−

1

2
t)

and, if x ∈{2, 3, 4, ...} with x > y, then

ρ(x,y) = Tx + Ty < x− 1 + y = ρ(x,y) − 1.

Thus, if we define ψ(t) by

ψ(t) =




t− 1
2
t2 ift ∈ [0, 1],

t− 1 ift > 1.
then ψ ∈ Φ and T is a ϕ−set contraction because it is easy to prove that ev-
ery ϕ−contraction mapping is a ϕ−set contraction with respect to the Kuratowskii
measure of noncompactness.
However, as n →∞, ρ(Tn, 0)/ρ(n, 0) → 1 so there can be no 0 ≤ k < 1 for which
T is a k−set contraction.

Example 2.4. Let X = {1}∪{2n, 3n : n ∈ N} is equipped with the discrete metric
and T : X → X be defined by

T(x) =



{1} ifx = 1,

{3(2n + 1)} ifx = 2n,

{1} ifx = 3n and n is odd.

If α is the Kuratowski measure of noncompactness, then α(T2X) = 0. There-
fore, T is a generalized ϕ−set contraction. But if C = {2n : n ∈ N}, then
α(T(C)) = α(C) = 1 and so T could not be a ϕ−set contraction since ϕ(1) would
have to be one.

Lemma 2.5. Let X be a real or complex Banach space and T : X → X a bounded
linear operator which is countably generalized ϕ−set contraction. Let I denote the
identity operator in X. Then for any relatively compact set M ⊂ X and for any
bounded countable set C ⊂ X, M1 = {x ∈ C : x−Tx ∈ M} is relatively compact.

Proof. Let M be a relatively compact set in X and C a bounded countable set in
X and M1 = {x ∈ C : x − Tx ∈ M}. We will show that γ(M1) = 0. Suppose
that x ∈ M1, so that x = Tx + z for some z ∈ M. Substituting for x on the right,
x = T2x + Tz + z, and continuing in this way we find

x = Tnx + (Σn−1j=0 T
j)z.

If we write M2 = (Σ
n−1
j=0 T

j)(M), the set M2 is relatively compact because it is
the continuous image of a relatively compact set. Furthermore, the above equality
implies that M1 ⊂ Tn(M1) +M2, so that γ(M1) ≤ γ(TnM1). Since T is generalized
countably ϕ−set contraction we have γ(M1) ≤ ϕ(γ(M1)). Since ϕ ∈ Φ, we have
ϕ(γ(M1)) < γ(M1), which yields a contradiction. It follows that γ(M1) = 0 and
thus M1 is relatively compact.

�



158 SALAHIFARD AND VAEZPOUR

Proposition 2.6. Let X be a real or complex Banach space and T : X → X a
bounded linear operator which is countably generalized ϕ−set contraction, then I−T
is a semi-Fredholm operator.

Proof. We must show that, the null space of I − T is finite dimensional and the
range of I −T is closed in X. But it follows by previous Lemma and Proposition
2.1 in [10]. �

Remark 2.7. Let X be a real or complex Banach space and T : X → X a bounded
linear operator which is countably generalized ϕ−set contraction, then I −λT is a
semi-Fredholm operator for any λ ∈ [0, 1], since

γ((λT)n(C)) = λnγ(Tn(C)) ≤ λnϕ(γ(C)) ≤ ϕ(γ(C)),

so λT is a countably generalized ϕ−set contraction too and by last Proposition
I −λT is a semi-Fredholm operator.

Proposition 2.8. Let X be a real or complex Banach space and T : X → X a
bounded linear operator which is countably generalized ϕ−set contraction, then I−T
is a Fredholm operator of index zero.

Proof. By remark I −λT is a semi-Fredholm operator for any λ ∈ [0, 1]. Since the
index for semi-Fredholm operators is constant on its connected components, and
{I −λT : λ ∈ [0, 1]} is connected, we have

i(I −T) = i(I) = 0.

Therefore, I −T is a Fredholm operator of index zero. �

Proposition 2.9. Let X be a real or complex Banach space and T : X → X a
bounded linear operator, so by Remark 1.1 we have

(2.1) γ(Tn(C)) ≤ ϕ(γ(C)),

for some n ∈ N and some ϕ ∈ Φk.
Let µ ∈ R be such that µn < k−1 for the same n and k in 2.1. Then I −µT is a
Fredholm operator of index zero.

Proof. Since T is a linear operator and satisfying , we have

γ((µT)n(C)) = µnγ(Tn(C)) ≤ µnϕ(γ(C)) ≤ µnk(γ(C)) = ψ(γ(C)),

where ψ(t) = µnkt is belong to φk, thus µT is a countably generalized ϕ−set
contraction and therefore I −µT is a Fredholm operator of index zero. �

Let (X, ||.||) be a real Banach space and E = R × X a Banach space with the
norm

||(λ,u)|| = (|λ|2 + ||u||2)
1
2 for (λ,u) ∈ E.

Consider the following nonlinear equation:

(2.2) u = λGu, (λ,u) ∈ E.

Assume that the operator G : X → X satisfies the following conditions:
(H1) Gu = Lu + Hu for all u ∈ X.



INDEX FORMULA FOR ϕ−SET CONTRACTIONS 159

(H2) L : X → X is a bounded linear operator (so it satisfies 2.1).
(H3) H : X → X is a continuous operator such that

||Hu||
||u||

→ 0 as ||u||→ 0.

(H4) G : X → X is a generalized ϕ−set contraction.

A real number λ is called a characteristic value of L if there exists a nonzero
vector u in X such that u = λLu. We call the line {(λ, 0) : λ ∈ R} the set of trivial
solutions of 2.2. Let S denote the subset of E consisting of all nontrivial solutions
of 2.2. A point (µ, 0) is called a bifurcation point of 2.2 if, given any � > 0, there
exists an element (λ,u) ∈ S such that |λ−µ|2 + ||u||2 < �2.
Now we give a necessary condition for the existence of bifurcation points of the
above equation, for the case of countably generalized ϕ−set operators, which ex-
tends [10].

Theorem 2.10. Let X be a real Banach space and let G : X → X satisfy the
hypotheses (H1), (H2) and (H3). Suppose that µ ∈ R be such that µn ≤ k−1 and
that µ is not a characteristic value of L. Then (µ, 0) is not a bifurcation point of
2.2.

Proof. Suppose that µ is not a characteristic value of L. Then I −µL is injective.
From µn ≤ k−1 it follows by Proposition 2.8 that I−µL is Fredholm of index zero.
Hence codimR(I −µL) = dimN(I −µL) = 0 and so R(I −µL) = X. Since I −µL
is a bijective bounded linear operator on X, the bounded inverse theorem implies
that (I − µL)−1 is bounded. Assume on the contrary that (µ, 0) is a bifurcation
point of 2.2. Then there exist a sequence {un} in X \{0} and a sequence {µn} in
R such that un = µnGun, un 6= 0 and µn 6= µ as n →∞. Hence we have

||un|| ≤ ||(I −µL)−1||||(I −µL)un||
= ||(I −µL)−1||||µnHun + (µ−λ)Lun||
≤ ||(I −µL)−1||(|µn|||Hun|| + |µn −µ|||L||||un||).

Therefore

1 ≤ ||(I −µL)−1||(|µn|
||Hun||
||un||

+ |µn −µ|||L||).

Since the right-hand side of the last inequality tends to zero as n → ∞ by (H3),
this is a contradiction. We conclude that (µ, 0) is not a bifurcation point of 2.2. �

Definition 2.11. A bounded linear operator T : X → X on a complex Banach
space X is said to be commutable if there exists a finite-dimensional linear operator
F such that F commutes with T and I −T −F is a one-to-one operator of X onto
X. When we say an operataor is finite dimensional, we shall mean its range is
finite dimensional and when we say that a linear operator F commutes with T we
shall mean (i) the domain of F , D(F), contains the domain of T , (ii) F(x) ∈ D(T)
whenever x ∈ D(T), (iii) and TFx = FTx for x ∈ D(T2).

Browder defined the essential spectrum of a densely defined closed linear opera-
tor T on a Banach space, in symbols ess(T), to be the set of λ ∈ σ(T) such that
at least one of the following conditions holds:
(i) R(λI −T), the range of λI −T , is not closed.
(ii) λ is a limit point of σ(T).



160 SALAHIFARD AND VAEZPOUR

(iii) ∪∞ν=1N(λI − T)ν is infinite dimensional, where N(λI − T)ν denotes the null
space of (λI −T)ν.

If T is a densely defined closed linear operator on X, define re(T), the essential
spectral radius of T, by

re(T) := sup{|λ| : λ ∈ ess(T)}.
Note that ess(T) is the largest subset of the spectrum σ(T) which remains invariant
under perturbations of T by compact operators which commute with T, i.e.

ess(T) = {λ : λ ∈ σ(T + C)for every compact operator C such that
C(D(T)) ⊂ C, and TCx = CTx for x ∈ D(T2)}

Now, we may recall another notion of the essential spectrum, introduced by
Schechter [17], as follows:

If one takes the essential spectrum to be the largest subset of the spectrum which
remains invariant under arbitrary compact perturbations it yeilds to Schechter’s
definition. Let T be a closed linear operator on a Banach space X.

The Schechter essential spectrum of the operator T is defined by

σs(T) = ∩K∈K(X)σ(T + K),
where K(X) denote the set of all compact linear operators. It is clear that ess(T)
includes properly σs(T) and if we add to σs(T), all limit points of the spectrum,
then it will be equivalent to one given by Browder, i.e ess(T).

The following proposition gives a characterization of the Schechter essential spec-
trum by means of Fredholm operators:

Proposition 2.12. ([[16], Theorem 5.4, p. 180]). Let X be a Banach space and
T : X → X be a closed, densely defined linear operator. Then

λ 6∈ σs(T) if and only if λI −T is a Fredholm operator of index zero.

Lemma 2.13. (Nussbaum [13]). Let T be an operator on X and r > re(T). Then
there exists a finite dimensional operator F on X, which commutes with T , such
that σ(T + F) ⊂{λ ∈ C : |λ| ≤ r}.

Lemma 2.14. Let T : X → X be a bounded linear operator on a complex Banach
space X. If re(T) < 1 then T is comutable.

Proof. By Lemma 2.13 there exists a finite dimensional operator F on X, which
commutes with T , such that I −T −F is invertible operator of X onto X, so T is
commutable. �

Corollary 2.15. Let T : X → X be a densely defined closed linear operator on a
complex Banach space X which is countably generalized ϕ−set contraction. Then
re(T) < 1 if and only if 1 is not belong to limit points of spectrum of T .

Proof. Since T is a linear operator satisfying , for |t| ≤ 1, tT is generalized ϕ− set
contraction and therefore by 2.8, I − tT is a Fredholm operator of index zero, so
λI −T for {λ : |λ| ≥ 1} is a Fredholm operator of index zero.
Thus by Proposition 2.12, λ 6∈ σs(T) for λ ∈ C with |λ| ≥ 1, and since {λ ∈ C :
|λ| = 1} is not a limit point of spectrum of T, therefore

σe(T) ⊂{λ ∈ C : |λ| < 1}.



INDEX FORMULA FOR ϕ−SET CONTRACTIONS 161

Consequently re(T) < 1. �

Theorem 2.16. Let X be a real or complex Banach space and T : X → X a
bounded linear comutable operator which is countably generalized ϕ−set contraction.
Then there exist a finite-dimensional subspace N and a closed subspace E of finite
codimension such that X = N ⊕ E, N and E are both invariant under T , and
(I − tT)|E is a homeomorphism of E onto itself for each t ∈ [0, 1].

Proof. Since T is a countably generalized ϕ−set contraction, I −T is a Fredholm
operator of index zero. If C is a closed subspace of X such that I −T|C : C → C
is one-to-one, this implies that I −T |C is one-to-one and onto C.
Let ζ be the complexification of B and let T be the natural extension of T to ζ:

T(x + iy) = Tx + iTy for x,y ∈ C.
Since T is comutable, there exists a finite-dimensional complex linear operator F
such that I − T −F is a one-to-one operator of ζ onto ζ and F commutes with T.
Now applying Theorem 2.7 in [10], the desired result is obtained. �

Corollary 2.17. Let T : X → X be a comutable, countably generalized ϕ−set con-
tractive bounded linear operator. Then the sum of the multiplicities of the eigenval-
ues λ ≥ 1 of T is finite.
Proof. Let λ ≥ 1 be any eigenvalue of T . Suppose that x ∈ X is a nonzero vector
such that (λI −T)nx = 0 for some positive integer n. Let x = z + w, where z ∈ N
and w ∈ E, and X = N⊕E is the decomposition described in Theorem 2.16. Since
N and E are invariant under T , we have (λI−T)nz = −(λI−T)nw ∈ N∩E = {0},
and so (λI − T)nw = 0. Since (I − λ−1T)|E is one-to-one by Theorem 2.16, we
have w = 0, and therefore x = z ∈ N. Since N is finite dimensional, the conclusion
follows. �

Let Ω be a nonempty bounded open set in a Banach space X. If T : X → X is
a countably γ-condensing operator that has no fixed points on the boundary ∂Ω,
one may define the degree of I −T on Ω as an integer, denoted by deg(I −T, Ω, 0)
More details of this definition are given in [20].

The above degree has the following basic properties, see [18, Theorem 1.3] and
[18, Corollary 2.1].

Lemma 2.18. Let Ω be a nonempty bounded open set in a Banach space X and
T : Ω̄ → X a countably γ-condensing operator such that T has no fixed points on
∂Ω. Then the following statements hold:
(1) If deg(I −T, Ω, 0) = 0, then T has a fixed point in Ω.
(2) If 0 ∈ Ω, then deg(I, Ω, 0) = 1.
(3) (Homotopy invariance) If H : [0, 1] × Ω̄ → X is a countably γ-condensing
homotopy such that H(t,x) 6= x for all (t,x) ∈ [0, 1] ×∂Ω, then

deg(I −H(0, .), Ω, 0) = deg(I −H(1, .), Ω, 0).
Remark 2.19. If T is a generalized ϕ−set contraction, then Tn is a condensing
mapping, so we can use the above properties for it.

Definition 2.20. Let Ω be an open subset of a Banach space X and T : Ω → X a
ϕ−set contractive operator. If x0 is an isolated fixed point of T , then the index of
x0 for T is defined by

ind(T,x0) = deg(I −T,B(x0,r), 0);



162 SALAHIFARD AND VAEZPOUR

where B(x0,r) is an open ball in X centered at x0 with radius r so small that T
has no fixed points other than x0 in B(x0,r).

Now, it is time to mention our main theorem which shows that the index formula
holds for countably ϕ-set contractive bounded linear operators.

Proof. of 2.1:
Since 1 is not an eigenvalue of T , then 0 is the only fixed point of T and

ind(T,x0) = deg(I −T,B(x0,r), 0).

Let X = N ⊕E be the decomposition of X introduced in Theorem 2.16. Define an
operator S : X → X by S = T ◦P , where P denotes the projection onto N. Since
N is finite dimensional, we obtain that P is compact and so is S. Now consider a
continuous homotopy H : [0, 1] ×X → X defined by

H(t,x) = tSx + (1 − t)Tx for (t,x) ∈ [0, 1] ×X.

Then H is countably γ-condensing on [0, 1] × B̄(0, 1). In fact, for each countable
set C ⊂ B̄(0, 1) with γ(C) > 0 we have

γ(H([0, 1] ×C)) ≤ γ(co(S(C) ∪T(C)))
≤ max{γ(S(C)),γ(T(C))}
≤ ϕ(γ(C))
< γ(C)

because S is compact and T is countably ϕ−set contraction. We claim that
H(t,x) 6= x for all (t,x) ∈ [0, 1]×∂B(0, 1). Indeed, suppose that H(t0,x0) = x0 for
some (t0,x0) ∈ [0, 1] ×∂B(0, 1). Let x0 = z + w, where z ∈ N and w ∈ E. Then
z +w = t0Tz + (1−t0)Tz + (1−t0)Tw. By the invariance of N and E under T , we
have z = Tz and w = (1−t0)Tw. Since 1 is not an eigenvalue of T and T is ϕ−set
contraction, I−(1−t0)T|E is one-to-one by Theorem 2.7, we have z = 0 and w = 0
and hence x0 = 0, which contradicts the assumption that x0 ∈ ∂B(0, 1). Lemma
3.1 implies that deg(I −T,B(0, 1), 0) and deg(I −S,B(0, 1), 0) are equal. Since S
is compact, deg(I −S,B(0, 1), 0) is equal to the LeraySchauder degree. Using the
LeraySchauder formula for a compact linear operator (see e.g. [3, Theorem 8.10]),
we have

ind(T, 0) = deg(I −S,B(0, 1), 0) = (−1)ν,

where ν is the sum of the multiplicities of the eigenvalues λ > 1 of S. It remains
to show that Vn = Un for every positive integer n, where

Vn = {(λ,x) : λ > 1,x ∈ X,x 6= 0 and (I −T)nx = 0};
Un = {(λ,x) : λ > 1,x ∈ X,x 6= 0 and (I −S)nx = 0};

Suppose that (λ,x) ∈ Vn and put x = z + w, where z ∈ N and w ∈ E. Then
(λI−T)nz = −(λI−T)nw ∈ N∩E = {0}. Since (I−λ−1T)|E is one-to-one, we have
w = 0 and so Tx = T(Pz) = Sx. This shows that Vn ⊂ Un. Now let (λ,x) ∈ Un.
Since S = T on N and S = 0 on E, we have (λI−S)nz = −(λI−S)nw = −λnw = 0,
where x = z + w with z ∈ N and w ∈ E. Hence w = 0 and Sx = Tz = Tx. This
shows that Un ⊂ Vn. This completes the proof. �



INDEX FORMULA FOR ϕ−SET CONTRACTIONS 163

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Department of Mathematics and Computer Science, Amirkabir University of Tech-
nology, Tehran, Iran

∗Corresponding author