1 Introduction

For convenience, we first recall the topological degree of 1-set-contractive 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 1-set-contractive operator such that

In addition, if there exists a k-set-contractive 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 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.

Suppose that is a semi-closed 1-set-contractive 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:

  1. (a)

    (Normalization) deg(I, Ω, p) = 1, when p ∈ Ω; deg(I, Ω, p) = 0, when p ∉ Ω;

  2. (b)

    (Solution property) If deg(I - A, Ω, θ) ≠ 0, then A has at least one fixed point in Ω.

  3. (c)

    (Additivity) For every pair of disjoint open subsets Ω1, Ω2 of Ω such that {x ∈ Ω |(I - A)x = 0} ⊂ Ω1 ∪ Ω2, we have

  4. (d)

    (Homotopy invariance) Let be a continuous operator such that

    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].

  5. (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.

  6. (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

(L-S)

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 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

(1)

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 x0 ∈ ∂Ω, t0 ≥ 1 such that Ax0 = t0x0, i.e., . It is easy to see that ||Ax0|| ≠ 0 and t0 > 1.

From (1), we have

which implies

(2)

By strict convexity of φ and φ(0) = 0, we obtain

(3)

It is easy to see from (2) and (3) that

(4)

Noting that t0 > 1 and ϕ(t) ≥ 1, for all t > 1, we have

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

(5)

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 x0 ∈ ∂Ω, t0 ≥ 1 such that Ax0 = t0x0, i.e., . It is easy to see that ||Ax0|| ≠ 0 and t0 > 1. From (5), we have

This implies that

(6)

By strict concavity of φ and φ (0) = 0, we obtain

(7)

It follows from (6) and (7) that

(8)

On the other hand, by t0 > 1 and ϕ(t) ≤ 1, ∀t > 1, 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) = tα , ϕ(t) = tβ , 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) = tα , ϕ(t) = tβ , 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

(9)

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 x0 ∈ ∂Ω, t0 ≥ 1 such that Ax0 = t0x0, i.e., . It is easy to see that ||Ax0|| ≠ 0 and t0 > 1. By virtue of (9), we have

which implies

(10)

By strict convexity of φ and φ (0) = 0, we obtain (3) holds. From (3) and (10), we have

(11)

Noting that t0 > 1 and ϕ(t) ≥ 1, for all t > 1, we have , and so

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

(12)

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 x0 ∈ ∂Ω, t0 ≥ 1 such that Ax0 = t0x0, i.e., . It is easy to see that ||Ax0|| ≠ 0 and t0 > 1. By (12), we have

which implies

(13)

By strict concavity of φ and φ (0) = 0, we have (7) holds. From (7) and (13), we obtain

(14)

On the other hand, by t0 > 1, we have . Therefore, it follows from ϕ(t) ≤ 1, ∀t > 1 that

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

(15)

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) = tα , ϕ(t) = tβ , 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

(16)

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) = tα , ϕ(t) = tβ , 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 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 .