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The fixed point theorems of 1setcontractive operators in Banach space
Fixed Point Theory and Applications volume 2011, Article number: 15 (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
1 Introduction
For convenience, we first recall the topological degree of 1setcontractive fields due to Petryshyn [1].
Let E be a real Banach space, p ∈ E, Ω be a bounded open subset of E. Suppose that is a 1setcontractive operator such that
In addition, if there exists a ksetcontractive operator such that
then (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 1setcontractive operator. A is said to be a semiclosed 1setcontractive 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 setcontractive operators, condensing operators, semicompact 1setcontractive operators and others (see [2]).
Petryshyn [1] and Nussbaum [3] first introduced the topological degree of 1setcontractive fields, studied its basic properties and obtained fixed point theorems of 1setcontractive operators. Amann [4] and Nussbaum [5] have introduced the fixed point indices of kset 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 1setcontractive operators and obtained some fixed point theorems of 1setcontractive operators. Recently, Li [6] obtained some fixed point theorems for 1setcontractive 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 1set 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.
Suppose that is a semiclosed 1setcontractive operator and 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)
(Additivity) For every pair of disjoint open subsets Ω_{1}, Ω_{2} of Ω such that {x ∈ Ω (I  A)x = 0} ⊂ Ω_{1} ∪ Ω_{2}, we have

(d)
(Homotopy invariance) Let be a continuous operator such that
and the measure of noncompactness 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 semiclosed 1setcontractive 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 semiclosed 1setcontractive operator and satisfies the LeraySchauder boundary condition
then deg(I  A, Ω, θ) = 1 and so A has a fixed point in Ω.
Definition 1.2. Let D be a nonempty subset of R. If φ : D → R is a real function such that
then φ is called strictly convex function on D. If φ : D → R is a real function such that
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 semiclosed 1setcontractive operators and existence theorems of solutions for the equation Ax = μx which generalize a great deal of wellknown 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 (LS) 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.
From (1), we have
which implies
By strict convexity of φ and φ(0) = 0, we obtain
It is easy to see from (2) and (3) that
Noting that t_{0} > 1 and ϕ(t) ≥ 1, for all t > 1, we have
which contradicts (4), and so the condition (LS) 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 (LS) 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. From (5), we have
This implies that
By strict concavity of φ and φ (0) = 0, we obtain
It follows from (6) and (7) that
On the other hand, by t_{0} > 1 and ϕ(t) ≤ 1, ∀t > 1, we have
which contradicts (8), and so the condition (LS) 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 (LS) 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. By virtue of (9), we have
which implies
By strict convexity of φ and φ (0) = 0, we obtain (3) holds. From (3) and (10), we have
Noting that t_{0} > 1 and ϕ(t) ≥ 1, for all t > 1, we have , and so
which contradicts (11), and so the condition (LS) 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 (LS) 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. By (12), we have
which implies
By strict concavity of φ and φ (0) = 0, we have (7) holds. From (7) and (13), we obtain
On the other hand, by t_{0} > 1, we have . Therefore, it follows from ϕ(t) ≤ 1, ∀t > 1 that
which contradicts (14), and so the condition (LS) 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.
Proof. From (15), we have
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.
Proof. From (16), we have
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.
Proof. Without loss of generality, suppose that A has no fixed point on ∂Ω. From (17), we have
which implies
It is easy to see that A is a semiclosed 1setcontractive 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 .
References
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Acknowledgements
This study was supported by the Natural Science Foundation of Yancheng Teachers University under Grant(10YCKL022).
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Wang, S. The fixed point theorems of 1setcontractive operators in Banach space. Fixed Point Theory Appl 2011, 15 (2011). https://doi.org/10.1186/16871812201115
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DOI: https://doi.org/10.1186/16871812201115
Keywords
 1Setcontractive operator
 Topological degree
 Convex function
 Concave function
 Fixed point theorems