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# The fixed point theorems of 1-set-contractive operators in Banach space

- Shuang Wang
^{1}Email author

**2011**:15

https://doi.org/10.1186/1687-1812-2011-15

© Wang; licensee Springer. 2011

**Received:**14 November 2010**Accepted:**19 July 2011**Published:**19 July 2011

## Abstract

In this paper, we obtain some new fixed point theorems and existence theorems of solutions for the equation *Ax* = *μx* using properties of strictly convex (concave) function and theories of topological degree. Our results and methods are different from the corresponding ones announced by many others.

**MSC: 47H09, 47H10**

## Keywords

- 1-Set-contractive operator
- Topological degree
- Convex function
- Concave function
- Fixed point theorems

## 1 Introduction

For convenience, we first recall the topological degree of 1-set-contractive fields due to Petryshyn [1].

*E*be a real Banach space,

*p*∈

*E*, Ω be a bounded open subset of

*E*. Suppose that is a 1-set-contractive operator such that

*I - W*)

*x ≠ p*, ∀

*x*∈ ∂

*D*, and so it is easy to see that deg(

*I*-

*W*,

*D*,

*p*) is well defined and independent of

*W*. Therefore, we are led to define the topological degree as follows:

Without loss of generality, we set *p* = *θ* in the above definition.

Let
be a 1-set-contractive operator. *A* is said to be a semi-closed 1-set-contractive operator, if *I* -*A* is closed operator (see [2]).

It should be noted that this class of operators, as special cases, includes completely continuous operators, strict set-contractive operators, condensing operators, semi-compact 1-set-contractive operators and others (see [2]).

Petryshyn [1] and Nussbaum [3] first introduced the topological degree of 1-set-contractive fields, studied its basic properties and obtained fixed point theorems of 1-set-contractive operators. Amann [4] and Nussbaum [5] have introduced the fixed point indices of *k*-set contractive operators (0 ≤ *k* < 1) and condensing operators to derive some fixed point theorems. As a complement, Li [2] has defined the fixed point index of 1-set-contractive operators and obtained some fixed point theorems of 1-set-contractive operators. Recently, Li [6] obtained some fixed point theorems for 1-set-contractive operators and existence theorems of solutions for the equation *Ax* = *μx*. Very recently, Xu [7] extended the results of Li [6] and obtained some fixed point theorems. In this paper, we continue to investigate boundary conditions, under which the topological degree of 1-set contractive fields, deg(*I - A*, Ω, *p*), is equal to unity or zero. Consequently, we obtain some new fixed point theorems and existence theorems of solutions for the equation *Ax* = *μx* using properties of strictly convex (concave) functions. Our results and methods are different from the corresponding ones announced by many others (e.g., Li [6], Xu [7]).

We need the following concepts and lemmas for the proof of our main results.

*θ*∉ (

*I*-

*A*)∂Ω, then, by the standard method, we can easily see that the topological degree has the basic properties as follows:

- (a)
(Normalization) deg(

*I*, Ω,*p*) = 1, when*p*∈ Ω; deg(*I*, Ω,*p*) = 0, when*p*∉ Ω; - (b)
(Solution property) If deg(

*I*-*A*, Ω,*θ*) ≠ 0, then*A*has at least one fixed point in Ω. - (c)
- (d)
and the measure of non-compactness γ(

*H*([0, 1] ×*Q*)) ≤ γ(*Q*) for every . Then deg(*I*-*H*_{ t }, Ω,*θ*) =*const*, for any*t*∈ [0, 1]. - (e)
Let

*B*be an open ball with center*θ*, a semi-closed 1-set-contractive operator and (*I*-*A*)*x*≠ 0 for all*x*∈ ∂*B*. Suppose that*A*is odd on ∂*B*(i.e.,*A*(-*x*) =*Ax*, for*x*∈ ∂*B*), then deg(*I*-*A*,*B*,*θ*) ≠ 0. - (f)
(Change of base) Let

*p*≠*θ*, then deg(*I*-*A*, Ω,*p*) = deg(*I*-*A*-*p*, Ω,*θ*).

**Lemma 1.1**. [7].

*Let E be a real Banach space*, Ω

*a bounded open subset of E and θ*∈ Ω.

*is a semi-closed 1-set-contractive operator and satisfies the Leray*-

*Schauder boundary condition*

*then* deg(*I* - *A*, Ω, *θ*) = 1 *and so A has a fixed point in* Ω.

then *φ* is called strictly concave function on *D*.

## 2 Main results

We are now in the position to apply the topological degree and properties of strictly convex (concave) function to derive some new fixed point theorems for semi-closed 1-set-contractive operators and existence theorems of solutions for the equation *Ax* = *μx* which generalize a great deal of well-known results and relevant recent ones.

**Theorem 2.1**.

*Let E*, Ω,

*A be the same as in Lemma 1.1. Moreover, if there exist strictly convex function φ*:

*R*

^{+}→

*R*

^{+}

*with φ*(0) = 0

*and real function*

*ϕ*:

*R*

^{+}→

*R with*

*ϕ*(

*t*) ≥ 1,

*for all t*> 1,

*such that*

*then* deg(*I* - *A*, Ω, *θ*) = 1 *if A has no fixed point on* ∂Ω, *and so A has at least one fixed point in*
.

*Proof*. If the operator *A* has a fixed point on ∂Ω, then *A* has at least one fixed point in
. Now suppose that *A* has no fixed points on ∂Ω. Next we shall prove that the condition (L-S) is satisfied.

Suppose this is not true. Then there exists *x*_{0} ∈ ∂Ω, *t*_{0} ≥ 1 such that *Ax*_{0} = *t*_{0}*x*_{0}, i.e.,
. It is easy to see that ||*Ax*_{0}|| ≠ 0 and *t*_{0} > 1.

which contradicts (4), and so the condition (L-S) is satisfied. Therefore, it follows from Lemma 1.1 that the conclusions of Theorem 2.1 hold. □

**Remark 2.2**. If there exist convex function *φ* : *R*^{+} → *R*^{+}, *φ*(0) = 0 and real function *ϕ* : *R*^{+} → *R*, *ϕ* (*t*) > 1, ∀*t* > 1 satisfied (1), the conclusions of Theorem 2.1 also hold.

**Theorem 2.3**.

*Let E*, Ω,

*A be the same as in Lemma 1.1. Moreover, if there exist strictly concave function φ*:

*R*

^{+}→

*R*

^{+}

*with φ*(0) = 0

*and real function*

*ϕ*:

*R*

^{+}→

*R*,

*ϕ*(

*t*) ≤ 1, ∀

*t*> 1,

*such that*

*then* deg(*I* - *A*, Ω, *θ*) = 1 *if A has no fixed point on* ∂Ω, *and so A has at least one fixed point in*
.

*Proof*. If the operator *A* has a fixed point on ∂Ω, then *A* has at least one fixed point in
. Now suppose that *A* has no fixed points on ∂Ω. Next we shall prove that the condition (L-S) is satisfied.

*x*

_{0}∈ ∂Ω,

*t*

_{0}≥ 1 such that

*Ax*

_{0}=

*t*

_{0}

*x*

_{0}, i.e., . It is easy to see that ||

*Ax*

_{0}|| ≠ 0 and

*t*

_{0}> 1. From (5), we have

which contradicts (8), and so the condition (L-S) is satisfied. Therefore, it follows from Lemma 1.1 that the conclusions of Theorem 2.3 hold. □

**Remark 2.4**. If there exist concave function *φ* : *R*^{+} → *R*^{+}, *φ* (0) = 0 and real function *ϕ* : *R*^{+} → *R*, *ϕ* (*t*) < 1, ∀*t* > 1 satisfied (5), the conclusions of Theorem 2.3 also hold.

**Corollary 2.5**.

*Let E*, Ω,

*A be the same as in Lemma 1.1. Moreover, if there exist α*∈ (-∞, 0) ∪ (1, +∞)

*and β*≥ 0

*such that*

*then* deg(*I* - *A*, Ω, *θ*) = 1 *if A has no fixed point on* ∂Ω, *and so A has at least one fixed point in*
.

*Proof*. Putting *φ*(*t*) = *t*^{
α
} , *ϕ*(*t*) = *t*^{
β
} , we have *φ* (*t*) is a strictly convex function with *φ* (0) = 0 and *ϕ*(*t*) ≥ 1, ∀*t* > 1. Therefore, from Theorem 2.1, the conclusions of Corollary 2.5 hold.. □

**Remark 2.6**. 1. Corollary 2.5 generalizes Theorem 2.2 of Xu [7] from *α* > 1 to *α* ∈ (-∞, 0) ∪ (1, +∞). Moreover, our methods are different from those in many recent works (e.g., Li [6], Xu [7]).

2. Putting *α* > 1, *β* = 0 in Corollary 2.5, we can obtain Theorem 5 of Li [6].

**Corollary 2.7**.

*Let E*, Ω,

*A be the same as in Lemma 1.1. Moreover, if there exist α*∈ (0, 1)

*and β*≤ 0

*such that*

*then* deg(*I* - *A*, Ω, *θ*) = 1 *if A has no fixed point on* ∂Ω, *and so A has at least one fixed point in*
.

*Proof*. Putting *φ*(*t*) = *t*^{
α
} , *ϕ*(*t*) = *t*^{
β
} , we have *φ*(*t*) is a strictly concave function with *φ* (0) = 0 and *ϕ*(*t*) ≤ 1, ∀*t* > 1. Therefore, from Theorem 2.3, the conclusions of Corollary 2.7 hold. □

**Remark 2.8**. Corollary 2.7 extends Theorem 8 of Li [6]. Putting *β* = 0 in Corollary 2.7, we can obtain Theorem 8 of Li [6].

**Theorem 2.9**.

*Let E*, Ω,

*A be the same as in Lemma 1.1. Moreover, if there exist strictly convex function φ*:

*R*

^{+}→

*R*

^{+}

*with φ*(0) = 0

*and real function*

*ϕ*:

*R*

^{+}→

*R with*

*ϕ*(

*t*) ≥ 1,

*for all t*> 1,

*such that*

*then* deg(*I* - *A*, Ω, *θ*) = 1 *if A has no fixed point on* ∂Ω, *and so A has at least one fixed point in*
.

*Proof*. If the operator *A* has a fixed point on ∂Ω, then *A* has at least one fixed point in
. Now suppose that *A* has no fixed points on ∂Ω. Next we shall prove that the condition (L-S) is satisfied.

*x*

_{0}∈ ∂Ω,

*t*

_{0}≥ 1 such that

*Ax*

_{0}=

*t*

_{0}

*x*

_{0}, i.e., . It is easy to see that ||

*Ax*

_{0}|| ≠ 0 and

*t*

_{0}> 1. By virtue of (9), we have

which contradicts (11), and so the condition (L-S) is satisfied. Therefore, it follows from Lemma 1.1 that the conclusions of Theorem 2.9 hold. □

**Remark 2.10**. If there exist convex function *φ* : *R*^{+} → *R*^{+}, *φ* (0) = 0 and real function *ϕ* : *R*^{+} → *R*, *ϕ*(*t*) > 1, ∀*t* > 1 satisfied (9), the conclusions of Theorem 2.9 also hold.

**Theorem 2.11**.

*Let E*, Ω,

*A be the same as in Lemma 1.1. Moreover, if there exist strictly concave function φ*:

*R*

^{+}→

*R*

^{+}

*with φ*(0) = 0

*and real function*

*ϕ*:

*R*

^{+}→

*R*,

*ϕ*(

*t*) ≤ 1, ∀

*t*> 1,

*such that*

*then* deg(*I* - *A*, Ω, *θ*) = 1 *if A has no fixed point on* ∂Ω, *and so A has at least one fixed point in*
.

*Proof*. If the operator *A* has a fixed point on ∂Ω, then *A* has at least one fixed point in
. Now suppose that *A* has no fixed points on ∂Ω. Next we shall prove that the condition (L-S) is satisfied.

*x*

_{0}∈ ∂Ω,

*t*

_{0}≥ 1 such that

*Ax*

_{0}=

*t*

_{0}

*x*

_{0}, i.e., . It is easy to see that ||

*Ax*

_{0}|| ≠ 0 and

*t*

_{0}> 1. By (12), we have

which contradicts (14), and so the condition (L-S) is satisfied. Therefore, it follows from Lemma 1.1 that the conclusions of Theorem 2.11 hold. □

**Remark 2.12**. If there exist convex function *φ* : *R*^{+} → *R*^{+}, *φ* (0) = 0 and real function *ϕ* : *R*^{+} → *R*, *ϕ* (*t*) > 1, ∀*t* > 1 satisfied (12), the conclusions of Theorem 2.11 also hold.

**Corollary 2.13**.

*Let E*, Ω,

*A be the same as in Lemma 1.1. Moreover, if there exist α*∈ (-∞, 0)∪(1, +∞)

*and β*≥ 0

*such that*

*then* deg(*I* - *A*, Ω, *θ*) = 1 *if A has no fixed point on* ∂Ω, *and so A has at least one fixed point in*
.

Taking *φ*(*t*) = *t*^{
α
} , *ϕ*(*t*) = *t*^{
β
} , we have *φ* (*t*) is a strictly convex function with *φ* (0) = 0 and *ϕ*(*t*) ≥ 1, ∀*t* > 1. Therefore, from Theorem 2.9, the conclusions of Corollary 2.13 hold. □

**Remark 2.14**. 1. Corollary 2.13 generalizes Theorem 2.4 of Xu [7] from *α* > 1 to *α* ∈ (-∞, 0) ∪ (1, +∞). Moreover, our methods are different from those in many recent works (e.g., Li [6], Xu [7]).

2. Putting *α* > 1, *β* = 0 in Corollary 2.13, we can obtain Theorem 5 of Li [6].

**Corollary 2.15**.

*Let E*, Ω,

*A be the same as in Lemma 1.1. Moreover, if there exist α*∈ (0, 1)

*and β*≤ 0

*such that*

*then* deg(*I* - *A*, Ω, *θ*) = 1 *if A has no fixed point on* ∂Ω, *and so A has at least one fixed point in*
.

Putting *φ*(*t*) = *t*^{
α
} , *ϕ*(*t*) = *t*^{
β
} , we have *φ* (*t*) is a strictly concave function with *φ* (0) = 0 and *ϕ*(*t*) ≤ 1, ∀*t* > 1. Therefore, from Theorem 2.11, the conclusions of Corollary 2.15 hold. □

**Remark 2.16**. Corollary 2.15 extends Theorem 8 of Li [6]. Putting *β* = 0 in Corollary 2.15, we can obtain Theorem 8 of Li [6].

**Theorem 2.17**.

*Let E*, Ω,

*A be the same as in Lemma 1.1. Moreover, if there exist α*∈ (-∞, 0)∪(1, +∞),

*β*≥ 0

*and μ*≥ 1

*such that*

*then the equation Ax* = *μx possesses a solution in*
.

It is easy to see that
*A* is a semi-closed 1-set-contractive operator. It follows from Corollary 2.5 that
, and so the equation *Ax* = *μx* possesses a solution in
.

**Remark 2.18**. Similarly, from Corollary 2.7, Corollary 2.13 or Corollary 2.15, we can obtain the equation *Ax* = *μx* possesses a solution in
.

## Declarations

### Acknowledgements

This study was supported by the Natural Science Foundation of Yancheng Teachers University under Grant(10YCKL022).

## Authors’ Affiliations

## References

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