Nonlinear Spectrum and Fixed Point Index for a Class of Decomposable Operators

: We study a class of nonlinear operators that can be written as the composition of a linear operator and a nonlinear map. We obtain results on ﬁxed point index based on parameters that are related to the deﬁnitions of nonlinear spectra. As a particular case, existence of positive solutions for a second-order differential equation with separated boundary conditions is proved. The result also provides a spectral interval for the corresponding Hammerstein integral operator.


Introduction
Nonlinear spectral theory has been shown to have applications in the study of existence of solutions for operator equations, particularly in integral equations [1,2]. On the other hand, fixed point index is well known as a popular technique to prove existence and multiplicity of positive solutions for Boundary Value Problems (BVPs). For example, a common method in studying differential equations with various boundary conditions is to convert the problem to an integral equation using the Green's function, then apply a fixed point theorem. Usually, the integral equation can be written as composition of a bounded linear operator and a nonlinear map.
In this paper, we are interested in operators in the form LF : P → P ⊂ E, where L is a linear operator, F is a nonlinear map, and P is an order cone of the Banach space E. We obtain results on fixed point index of the nonlinear operator LF based on parameters that are related to the nonlinear spectra. We also extend the continuation principle for stably-solvable maps to the operator LF on a cone. The stably-solvable property is a key concept in the definition of nonlinear spectra [3,4]. As a particular case, we prove existence of positive solutions for a second-order differential equation with separated boundary conditions [5] and thus obtain a spectral interval for the Hammerstein integral operator.
Let E, F be Banach spaces and f : E → F be a continuous nonlinear map. The Furi-Martelli-Vignoli-spectrum (fmv-spectrum) [3,4] is defined by two parameters d( f ), ω( f ) and the stably-solvable property. Later, the Feng-spectrum [1,6] was introduced with the parameters ω( f ), ν( f ) and m( f ). It is shown that the Feng-spectrum (σ F ( f )) contains all eigenvalues of the operator f . We briefly review definitions of the related parameters. Let α(Ω) denote the Measure of Noncompactness of Ω ⊂ E [1]. Then, where | f | is called the quasinorm of f .

Definition 1.
The nonlinear map f : E → F is stably-solvable if and only if given any compact map h : E → F with |h| = 0, the equation has a solution in E.
Next, an order cone of Banach space introduces a partial order for the space so that positive solutions can be studied.

Definition 2.
Let E be a Banach space, P is a subset of E. P is called an order cone iff: (i) P is closed, nonempty, and P = {0}; (ii) a, b ∈ R, a, b ≥ 0, x, y ∈ P ⇒ ax + by ∈ P; (iii) x ∈ P and −x ∈ P ⇒ x = 0.
Let P be an order cone of the Banach space E. For r > 0, denote P r = {u ∈ P, u < r}, and ∂P r = {u ∈ P, u = r}.
The following two lemmas on fixed point index [7] have been applied to prove existence of solutions for boundary value problems [8] and many other applications [7,9]. Lemma 1. Let N : P → P be a completely continuous mapping. If Nu = µu, for all u ∈ ∂P r , and all µ ≥ 1, then the fixed point index i(N, P r , P) = 1. Lemma 2. let N : P → P be a completely continuous mapping and satisfy Nu = u for u ∈ ∂P r . If Nu ≥ u , for u ∈ ∂P r , then the fixed point index i(N, P r , P) = 0.

Stably-Solvable Maps and Fixed Point Index
Let E be a Banach space and P ⊂ E be an order cone. We consider the linear homeomorphism L : E → E. It is known that [4,6] Let F : P → P be a nonlinear map. We use the following notations, The stably-solvable maps on a cone P ⊂ E are defined below.
Definition 3. The nonlinear map F : P → P is stably-solvable on the cone P if and only if given any compact map h : P → P with |h| P = 0, the equation has a solution x ∈ P.
The following theorem is an extension of the continuation principle for stably-solvable maps to the class of decomposable operators LF : P → P. Theorem 1. If F : P → P is stably-solvable on the cone P and L : P → P is bijective.
(2) Assume that h : If F(S) is bounded, then the equation has a solution x ∈ P.
Proof. (1) If h : P → P is a compact operator with |h| P = 0. Then, L −1 h : P → P is compact and |L −1 h| P ≤ L |h| P = 0. Therefore, the equation has a solution x ∈ P. Thus LF(x) = h(x) has a solution. By definition, LF is stably-solvable on P.
(2) Consider the operator L −1 h : As F is stably-solvable on P, S = {x ∈ P : Our next result is on the fixed point index of the nonlinear operator LF based on the parameters such as |F| P and d(F) P that are related to the definition of the fmv-spectrum [4].
Theorem 2. Assume that L : E → E is a linear homeomorphism and F : P → P is a nonlinear map such that the composition LF : P → P is completely continuous.
We prove that under condition (1), We only prove (1) and (4). (2) and (3) can be proved following the similar ideas. (1) . This contradicts the condition On the other hand, if condition (4) holds, assume there exists x n ∈ O 2 such that x n → 0 as n → ∞. We have Thus, This contradicts the assumption d(F) 0 d(L) > 1. Next, if O 1 is bounded, we can select R large enough such that LFx = µx, for all x ∈ ∂P R , and all µ ≥ 1.
On the other hand, if O 1 is bounded below, we can select r small enough such that LFx = µx, for all x ∈ ∂P r , and all µ ≥ 1.
Again by Lemma 1, we have i(LF, P r , P) = 1.
By Lemma 2, we have i(LF, P R , P) = 0 Similarly, if O 2 is bounded below, we can select r small enough such that LFx ≥ x , for all x ∈ ∂P r .
By Lemma 2, we have i(LF, P r , P) = 0 The proof is complete.
Theorem 2 can be used to prove existence of positive solutions for nonlinear operator equations involving a parameter. Then, the operator equation λLF(x) = x has a positive solution x ∈ P for 1 ≤ λ < d(L −1 ) By Theorem 2 (4), there exists r > 0 small enough such that i(λLF, P r , P) = 0. On the other side, if λ < . By Theorem 2 (1), there exists R > 0 large enough such that i(λLF, P R , P) = 1. Therefore, there exists a fixed point λLF( As the Feng-spectrum contains all eigenvalues and it is closed [6], the following result on spectral interval follows from Theorem 3.

Corollary 1. Under the conditions of Theorem 3, the nonlinear operator LF has the spectral interval
[ L |F| P , 1] ⊂ σ F (LF).
In the following, we prove existence of positive solutions of BVP (3)-(5) using Lemmas 1 and 2 and obtain a spectral interval for the corresponding Hammerstein integral operator that can be written as the composition of a linear operator L and a nonlinear map F.
Notice that existence of a solution for (3)-(5) is equivalent to the existence of a fixed point for the following Hammerstein operator [5]: where the Green's function Let C[0, 1] denote a Banach space of continuous functions with the norm We use the cone P with parameter 0 < c 0 < 1: Then, N(λ, u) = λ(LF)(u). Note that the linear operator L is not a homeomorphism on the space C[0, 1]. However, we will show that L : P → P and is injective on P. Following Lemma 2.1 of [5], we know that the Green's function G satisfies the strong positivity condition [9]: For ∀u ∈ P, (10) ensures that Therefore, N(λ, P) ⊂ P. We first prove a property of the linear operator L that is related to the so-called u 0positive linear operator on a cone [16], that later was generalized to u 0 -positive linear operator relative to a pair of cones [9,17]. The following lemma shows that L actually satisfies stronger conditions than the requirements of u 0 -positive linear operators. Lemma 3. Let L be defined by (9). Then L : P → P is completely continuous and satisfies k 1 u(1) ≤ Lu ≤ k 2 u(1), for any u ∈ P, (12) for some k 1 , k 2 > 0.
Proof. For ∀u ∈ P, by property (10), we have So L(P) ⊂ P. Moreover, Thus Applying the Ascoli-Arzela theorem, we can prove that L is completely continuous.
Next, property (12) ensures that c 0 k 1 u ≤ Lu ≤ k 2 u , for any u ∈ P.
For u ∈ P, if L(u) = 0, then u = 0. Therefore, L is injective on P. The spectral radius of L, r(L) > 0 [9]. We now prove existence of a positive solution for problem (3)-(5) which implies a spectral interval for the operator LF. The proof follows similar ideas as that of [8].
On the other hand, select M large enough such that λMc 0 1 0 G(1, s)ds > 1.
As d( f ) = ∞, there exists M 1 > 0, such that f (t,x) x > M for x > M 1 . We take M 1 > max{c 0 , 2δ} and let R = M 1 c 0 . For u ∈ ∂P R , we have