Positive Fixed Point of Strict Set Contraction Operators on Ordered Banach Spaces and Applications

and Applied Analysis 3 wider variety of situations. Our results either improve or generalize the corresponding results due to 19–22 and many of others. As an application of our main results, we consider the existence of positive solutions for second-order differential equations with integral boundary conditions in an ordered Banach space. On the other hand, our conditions are weaker than those of 22 . The organization of this paper is as follows. We will introduce some lemmas and notations in the rest of this section. In Section 2, the main results will be stated and proved. In Section 3, as an application of our main results, the existence of positive solutions for a second-order boundary value problem with integral boundary conditions in ordered Banach spaces is considered. Finally, in Section 4, one example is also included to illustrate the main results. Basic facts about ordered Banach space E can be found in 1–4 . Here we just recall a few of them. The cone P in E induces a partial order on E, that is, x ≤ y if and only if y − x ∈ P . P is said to be normal if there exists a positive constant N such that θ ≤ x ≤ y implies ‖x‖ ≤ N‖y‖. Without loss of generality, suppose, in the present paper, the normal constant N 1. For a bounded set V in Banach space E, we denote α V the Kuratowski measure of noncompactness see 1–4 , for further understanding . The operator A : D → E D ⊂ E is said to be a k-set contraction ifA : D → E is continuous and bounded and there is a constant k ≥ 0 such that α A S ≤ kα S for any bounded S ⊂ D; a k-set contraction with k < 1 is called a strict set contraction. In the following, denote the Kuratowski’s measure of noncompactness by α · . For the application in the sequel, we first state the following definition and lemmas which can be found in 1 , and some notation. Definition 1.3. Let S be a bounded set of a real Banach space E. Let α S inf{δ > 0 : S be expressed as the union of a finite number of sets such that the diameter of each set does not exceed δ, that is, S ⋃m i 1 Si with diam Si ≤ δ, i 1, 2, . . . , m}. Clearly, 0 ≤ α S < ∞. α S is called the Kuratowski’s measure of noncompactness. Definition 1.4. Let P be a cone of a real Banach space E. If P ∗ {Ψ ∈ E∗ | Ψ x ≥ 0, ∀x ∈ P}, then P ∗ is a dual cone of cone P . Definition 1.5. Let P be a cone of real Banach space E. ρ : P → R is said to be a convex functional on P if ρ tx 1 − t y ≤ tρ x 1 − t ρ y for all x, y ∈ P and t ∈ 0, 1 . Definition 1.6. A subset X ⊂ E is said to be a retract of E if there exists a continuous mapping r : E → X satisfying r x x, x ∈ X. Lemma 1.7. Let D ⊂ E, D be a bounded set and f uniformly continuous and bounded from J × S into E; then α ( f J × S ) max t∈J α ( f t, S ) , ∀S ⊂ D. 1.1 2. Main Results Lemma 2.1 see 22 . Let P be a cone in a real Banach space E. If ρ : P → 0,∞ is a uniformly continuous convex functional with ρ θ 0 and ρ x > 0 for x / θ, then ∀r > 0, Dr {x ∈ P : ρ x ≥ r} is a retract of E. 4 Abstract and Applied Analysis Lemma 2.2. Let E be a real Banach space, ‖ · ‖ the norm in E, P a cone in E, andΩ {x ∈ E : ‖x‖ < R}, where R is a positive real number. Suppose that A : P ∩Ω → P is a k-set contraction with k < 1 and ρ : P → 0,∞ is a uniformly continuous convex functional with ρ θ 0, ρ x > 0, ∀x / θ and ρ x ≤ ‖x‖. If i infx∈P∩∂Ωρ x > 0, and there exists δ > 0 such that R/infx∈P∩∂Ωρ x ≤ 1 δ/k and ρ Ax ≥ k δ ρ x , ∀x ∈ P ∩ ∂Ω; ii Ax/ μx, μ ∈ 0, 1 , ∀x ∈ P ∩ ∂Ω, hold, then the fixed point index i A,P ∩Ω, P 0. Proof. Without loss of generality, we suppose k δ < 1 If k δ > 1, then let N k δ. The proof is the same as the following process . Let N 1/ k δ , then NA is a strict set contraction. Considering ht x tAx 1 − t NAx, t ∈ 0, 1 , x ∈ P ∩ ∂Ω. If there exists t0 ∈ 0, 1 , x0 ∈ P ∩ ∂Ω such that x0 t0Ax0 1− t0 NAx0, thenAx0 1/ t0 N 1− t0 x0, which contradicts with ii . Then by the homotopy invariance property of fixed point index, we have i NA,P ∩Ω, P i A,P ∩Ω, P . Let r infx∈P∩∂Ωρ x . Define Dr {x ∈ P | ρ x ≥ r}. It follows from θ /∈Dr that d infx∈Dr‖x‖ > 0. From the fact that ρ x ≤ ‖x‖, we have r ≤ d ≤ R. In fact, since P ∩ ∂ Ω ⊂ Dr, we have d ≤ R. On the other hand, for ∀x ∈ Dr, ρ x r, combining this with ρ x ≤ ‖x‖, we have r ≤ ‖x‖, ∀x ∈ Dr, then r ≤ infx∈Dr‖x‖ d. Let M R/r, then by i we obtain Mk/ k δ < 1 andMd > supx∈P∩Ω‖x‖, andMDr∩ P∩Ω ∅, whereMDr {Mx | x ∈ Dr}. Let H t, x 1 − t NAx tMNAx, ∀ t, x ∈ 0, 1 × P ∩Ω. Then, we have α H t, S ≤ 1 − t α NA S tα MNA S ≤ 1 − t k k δ α S t Mk k δ α S < Mk k δ α S , ∀S ⊂ P ∩Ω, 2.1 and then we obtain that H t, · : P ∩ Ω → P is the strict set contraction. In addition, it is obvious that H t, x is uniformly continuous about t for all x ∈ P ∩Ω. If there exists x1 ∈ P ∩ ∂Ω, t1 ∈ 0, 1 such that 1 − t1 NAx1 t1MNAx1 x1, then Ax1 N 1 − t1 t1M x1, which contradicts with ii . Thus by the homotopy invariance property of fixed point index, we have i MNA,P ∩Ω, P i NA,P ∩Ω, P . Since Dr is a retract of E by Lemma 2.1, there exists a retraction r : E → Dr satisfying r x x, x ∈ Dr. Let A1 NA, A1 r ◦ A1, then A1 is strict set contraction. From i and the definition of ρ, we have ρ A1x ρ NAx ≥ Nρ Ax ≥ ρ x ≥ r, ∀x ∈ P ∩ ∂Ω. 2.2 Therefore,A1 ∂Ω ⊂ Dr , that is,A1x A1x, ∀x ∈ P ∩∂Ω. Then i MA1, P ∩Ω, P i MA1, P ∩ Ω, P . If i A1, P ∩ Ω, P / 0, then i MA1, P ∩ Ω, P / 0, which implies that MA1 has a fixed point x∗ in P ∩Ω. Thus x∗ MA1x ∈ MDr. It is a paradox. The proof is complete. Abstract and Applied Analysis 5 Lemma 2.3 see 2 . Let P be a cone and Ω a bounded open set in E with θ ∈ Ω. Suppose that A : P ∩Ω → P is condensing andand Applied Analysis 5 Lemma 2.3 see 2 . Let P be a cone and Ω a bounded open set in E with θ ∈ Ω. Suppose that A : P ∩Ω → P is condensing and Ax/ μx, ∀x ∈ P ∩ ∂Ω, μ ≥ 1. 2.3 Then i A,P ∩Ω, P 1. Lemma 2.4. Let P be a cone and Ω a bounded open set in E. Suppose that A : P ∩ Ω → P is a k-set contraction with k < 1 and ρ : P → 0,∞ is a uniformly continuous convex functional with ρ θ 0 and ρ x > 0 for x / θ. If ρ Ax ≤ ρ x andAx/ x for x ∈ P∩∂Ω, then i A,P∩Ω, P 1. Proof. If there exist x1 ∈ P ∩ ∂Ω and μ1 ≥ 1 such that Ax1 μ1x1, then μ1 > 1. Therefore, ρ x1 ρ ( 1 μ1 Ax1 ) ≤ 1 μ1 ρ Ax1 ≤ 1 μ1 ρ x1 < ρ x1 . 2.4 It is a paradox. From Lemma 2.3, it follows that i A,P ∩Ω, P 1. The proof is complete. Theorem 2.5. LetΩ1 be a bounded open set in E such that θ ∈ Ω1, andΩ2 {x ∈ E | ‖x‖ < R} and Ω1 ⊂ Ω2. Suppose thatA : P ∩ Ω2 \Ω1 → P is a k-set contraction with k < 1 and ρ : P → 0,∞ is a uniformly continuous convex functional with ρ θ 0 and ρ x > 0, ∀x / θ and ρ x ≤ ‖x‖. If a ρ Ax ≤ ρ x , ∀x ∈ P ∩ ∂Ω1; b infx∈P∩∂Ω2ρ x > 0, and there exists δ > 0 such that R/infx∈P∩∂Ω2ρ x ≤ 1 δ/k and ρ Ax ≥ k δ ρ x , and Ax/ μx, μ ∈ 0, 1 , ∀x ∈ P ∩ ∂Ω2 hold, then A has at least one fixed point in P ∩ Ω2 \Ω1 . Proof. It is easy to obtain the results by Lemmas 2.2 and 2.4. So we omit it. Theorem 2.6. Let Ω1 {x ∈ E | ‖x‖ < R} and Ω2 a bounded open set in E such that Ω1 ⊂ Ω2. Suppose that A : P ∩ Ω2 \ Ω1 → P is a k-set contraction with k < 1, and ρ : P → 0,∞ is a uniformly continuous convex functional with ρ θ 0 and ρ x > 0, ∀x / θ and ρ x ≤ ‖x‖. If a infx∈P∩∂Ω1ρ x > 0, and there exists δ > 0 such that R/infx∈P∩∂Ω1ρ x ≤ 1 δ/k and ρ Ax ≥ k δ ρ x , and Ax/ μx, μ ∈ 0, 1 , ∀x ∈ P ∩ ∂Ω1; b ρ Ax ≤ ρ x , ∀x ∈ P ∩ ∂Ω2 are satisfied, then A has at least one fixed point in P ∩ Ω2 \Ω1 . Proof. It is easy to obtain the results by Lemmas 2.2 and 2.4. So we omit it. Remark 2.7. If we let k 0, then A is completely continuous. Comparing with Corollary 2.1 of 22 , our conditions are weaker. 6 Abstract and Applied Analysis Corollary 2.8. Let Ω1 be a bounded open set in E such that θ ∈ Ω1, Ω2 {x ∈ E | ‖x‖ < R}, and Ω1 ⊂ Ω2. Suppose thatA : P ∩ Ω2 \Ω1 → P is a k-set contraction with k < 1 and ρ : P → 0,∞ is a uniformly continuous convex functional with ρ θ 0 and ρ x > 0, ∀x / θ and ρ x ≤ ‖x‖. If a ρ Ax ≤ ρ x , ∀x ∈ P ∩ ∂Ω1; b infx∈P∩∂Ω2ρ x > 0 with R/infx∈P∩∂Ω2ρ x ≤ 1/k, ρ Ax ≥ ρ x and Ax/ μx, μ ∈ 0, 1 , ∀x ∈ P ∩ ∂Ω2 hold, then A has at least one fixed point in P ∩ Ω2 \Ω1 . Proof. It follows by taking k δ 1. Corollary 2.9. Let Ω1 {x ∈ E | ‖x‖ < R} and Ω2 a bounded open set in E such that Ω1 ⊂ Ω2. Suppose that A : P ∩ Ω2 \ Ω1 → P is a k-set contraction with k < 1 and ρ : P → 0,∞ is a uniformly continuous convex functional with ρ θ 0 and ρ x > 0, ∀x / θ and ρ x ≤ ‖x‖. If a infx∈P∩∂Ω1ρ x > 0 with R/infx∈P∩∂Ω1ρ x ≤ 1/k, ρ Ax ≥ ρ x , and Ax/ μx, μ ∈ 0, 1 , ∀x ∈ P ∩ ∂Ω1; b ρ Ax ≤ ρ x , ∀x ∈ P ∩ ∂Ω2 hold, then A has at least one fixed point in P ∩ Ω2 \Ω1 . Proof. It follows by taking k δ 1. 3. Applications Throughout the remainder of this paper, we apply the above results to a second-order differential equation in Banach spaces: x′′ f t, x θ, 0 < t < 1, 3.1 subject to the following integral boundary conditions: