Structure of solutions set of nonlinear eigenvalue problems

J. Math. Anal. Appl. 435 (2016) 1410–1425

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Structure of solutions set of nonlinear eigenvalue problems ✩ Sun Jingxian, Xu Xian ∗ Department of Mathematics, Jiangsu Normal University, Xuzhou, Jiangsu, 221116, PR China

a r t i c l e

i n f o

Article history: Received 16 September 2013 Available online 12 November 2015 Submitted by P.J. McKenna Keywords: Bifurcation theories Leray–Schauder degree Connected component

a b s t r a c t In this paper, by using the global bifurcation theory we obtain some results for structure of the solution set of some nonlinear equations with parameters. We show a result concerning the existence of a connected component, which either has a loop, or is unbounded both from left and right hand side. Especially, in this paper we also give some sufficient conditions for a bounded connected component of solution set being a loop. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Bifurcation phenomena arise in many fields of mathematical physics. The studying of their nature is of practical as well as theoretical importance. There have been many results concerning the local or global bifurcation theories by using the topological degree theories, general set point theories and the linearization considerations; See [1–3,5–10,12–18,20,21]. In particular, some authors obtained the results concerning the existence of the connected component which has a loop. Now let us recall some such kind of global bifurcations results; See [2,7,9,15]. The author of [9] considered the problem of global bifurcation of nontrivial solutions of x = λLx + H(x, λ), where L is linear and compact on a Banach space E, H : E × R → E is completely continuous, H(0, λ) = 0 on R and x−1 H(x, λ) → 0 as x → 0 locally uniformly in λ. The author of [9] proved global bifurcation of two semi-global continua Cμ± of non-trivial solutions bifurcating from (0, μ), where μ is a characteristic value of L. It is shown by the author that if μ has geometric multiplicity 1 and odd algebraic multiplicity, then Cμ+ and Cμ− are both unbounded, or Cμ+ ∩Cμ− \{(0, μ)} = ∅. Moreover, if Cμ+ and Cμ− are both bounded, Cμ+ ∩ Cμ− contains (0, α), where α = μ, that is, C + and C − have a loop structure. ✩ This paper is supported by NSFC11501260, Natural Science Foundation of Jiangsu Education Committee (09KJB110008) and Qing Lan Project. * Corresponding author. E-mail address: [email protected] (X. Xu).

http://dx.doi.org/10.1016/j.jmaa.2015.11.024 0022-247X/© 2015 Elsevier Inc. All rights reserved.

J. Sun, X. Xu / J. Math. Anal. Appl. 435 (2016) 1410–1425

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The authors of [15] studied bifurcations of positive solutions from the trivial branch of one parameter families of compact vector fields on ordered Banach spaces. They proved the existence of a connected component of positive solutions emanating from a singular point λ0 of the linearization which is either unbounded or goes back to the trivial branch, provided: (i) the kernel of the linearization of the family at λ0 is one dimensional and generated by a vector in the positive cone, (ii) the Leray–Schauder degree changes at λ0 , (iii) the derivative of the compact part sends positive elements into the interior of the cone. Obviously, as the connected component emanating from a singular point λ0 goes back to the trivial branch, then the connected component may form a loop structure. Now an interesting question is: whether there is a nonlinear operator equation with parameters which has a bifurcation diagram that is an upright counterpart of those of the main theorems in [2,7,9,15]. The purpose of this paper is to study the structure of solution set of some nonlinear operator equations. We shall show a result for the existence of a connected component C, which either has a loop, or is unbounded both from left and right hands. Roughly speaking, we may think the bifurcation diagrams of our main results – Theorems 2.1 and 2.2 – are an upright counterpart of those of the main theorems in [2,7,9,15]. In this paper we shall give some sufficient conditions for a bounded connected component of solutions set being a loop; See Corollaries 2.1–2.3 and Theorem 3.1 below. These results can be applied to elliptic boundary value problems, ordinary differential boundary value problems etc. to give the existence results for loops emitting from non-trivial solutions as well as trivial solutions. From our points of view, although we can obtain the existence results for loops by using the main results of [2,7,9,15], however, those results of [2,7,9,15] would be more suitable for finding loops emitting from trivial solution. This seems to be a difference between our main results and those of [2,7,9,15]. 2. The main results First let us introduce some symbols we will use in the sequel of this paper. Let M be a metric space, U ⊂ M . In the sequel we will use U M to denote the closure of U in M , ∂M U to denote the boundary of U in M . Let E be a real Banach space. For simplicity we will use U to denote the closure of U in the Banach space E, and ∂U the boundary of U in the Banach space E. In the sequel of this paper we always assume that A : E → E is a completely continuous operator, L = {(λ, x) ∈ R × E : x = λAx, x = 0}R×E , and S = {(λ, x) ∈ R × E : x = λAx}. From [20] we have the following Definitions 2.1 and 2.2. Definition 2.1. Let C be a connected component of L, (λ∗ , x∗ ) ∈ C. Then (λ∗ , x∗ ) ∈ C is called a regular point of C, if there exists r0 > 0 small enough such that ({λ∗ } × B(0, r0 )) ∩ L = {(λ∗ , x∗ )}, where B(0, r0 ) = {x ∈ E : x < r0 }. Assume that (λ∗ , x∗ ) is a regular point on a connected component C of L. For each 0 < r < r0 , let O+ ((λ∗ , x∗ ), r) = {(λ, x) : (λ, x) ∈ R × E, (|λ − λ∗ |2 + x − x∗ 2 ) 2 < r, λ > λ∗ }, 1

O− ((λ∗ , x∗ ), r) = {(λ, x) : (λ, x) ∈ R × E, (|λ − λ∗ |2 + x − x∗ 2 ) 2 < r, λ < λ∗ }. 1

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J. Sun, X. Xu / J. Math. Anal. Appl. 435 (2016) 1410–1425

Fig. 1. An illustration of the proof of Theorem 2.1.

  Let Cr+ be the connected component of L ∩ R × E\O− ((λ∗ , x∗ ), r) containing (λ∗ , x∗ ), and Cr− be the connected component of L ∩ (R × E\O+ (x∗ , r)) containing (λ∗ , x∗ ). Let C+ =



Cr+ R×E ,

0


C− =

Cr− R×E .

0
Definition 2.2. Let C + and C − be defined as above. Then we call C + the right hand connected continuum of L passing through (λ∗ , x∗ ), and C − the left hand connected continuum of L passing through (λ∗ , x∗ ). Remark 2.1. Obviously, C + and C − are connected and closed, and (λ∗ , x∗ ) ∈ C + ∩ C − . It should be pointed out that, from Definition 2.2, C + may satisfy C + ∩ ((−∞, λ∗ ] × E)\{(λ∗ , x∗ )} = ∅, and C − may satisfy C − ∩ ([λ∗ , +∞) × E)\{(λ∗ , x∗ )} = ∅; See Fig. 1. From [22] we have the following Lemma 2.1. Lemma 2.1. Let X be a compact metric space. Assume that A and B are two disjoint closed subsets of X. Then either there exist a connected component of X meeting both A and B or X = ΩA ∪ ΩB , where ΩA , ΩB are disjoint compact subsets of X containing A and B, respectively. Let U be an open and bounded subset of the metric space [a, b] × E. We set U (λ) = {x ∈ E : (λ, x) ∈ U }, whose boundary is denoted by ∂U (λ). Consider a map h(λ, x) = x − k(λ, x), such that k(λ, ·) is completely continuous and 0 ∈ / h(∂U ). Such a map h will also be called an admissible homotopy on U . If h is an admissible homotopy, for every λ ∈ [a, b] and every x ∈ ∂U (λ), one has that hλ (x) := h(λ, x) = 0 and it makes sense to evaluate deg(hλ , U (λ), 0). From [1] we have the following Lemma 2.2. Lemma 2.2. If h is an admissible homotopy on U ⊂ [a, b] × E, then deg(hλ , U (λ), 0) is constant for all λ ∈ [a, b]. Theorem 2.1. Let C be a connected component of L. Assume that C ∩ (R × {0}) = ∅, (λ∗ , x∗ ) is a regular point of C and ind(I − λ∗ A, x∗ ) = 0. Then one of the following conclusions holds: (1) Both C + and C − are unbounded; (2) There exists (λ∗ , x∗∗ ) with x∗∗ = x∗ such that (λ∗ , x∗∗ ) ∈ C + ∩ C − .

(2.1)

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Proof. The process to show Theorem 2.1 can be illustrated by Fig. 1. Assume that the conclusion (2) doesn’t hold. Now we shall show that the conclusion (1) holds. Assume on the contrary that (1) doesn’t hold. Then, at least one of C + and C − is bounded. Assume without loss of generality that C + is bounded. Let D = C + ∩ (({λ∗ } × E)\B((λ∗ , x∗ ), r0 )). Obviously, D is a closed subset of {λ∗ } × E. We have two cases: Case 1. D = ∅. For each p ∈ D, let E − (p) be the connected component of L ∩ ((−∞, λ∗ ] × E) passing through (λ∗ , x∗ ). Obviously, E − (p) is bounded because C + is bounded by our assumption, and (λ∗ , x∗ ) ∈ / E − (p) for every p ∈ D. From our assumption that the conclusion (2) doesn’t hold, we easily see that there exists a bounded neighborhood U1− (p) of E − (p) in the metric space (−∞, λ∗ ] × E such that U1− (p)(−∞,λ

∗

]×E

∩ C − = ∅,

(λ∗ , x∗ ) ∈ / U1− (p)(−∞,λ

∗

]×E

,

and U1− (p)(−∞,λ

∗

]×E

∩ ((−∞, λ∗ ] × {0}) = ∅.

Assume that ∂(−∞,λ∗ ]×E U1− (p) ∩ L = ∅. ∗

Then U1− (p)(−∞,λ ]×E ∩ L is a compact set. From the maximal connectedness of E − (p), we see that there exists no connected subset of L meeting both ∂(−∞,λ∗ ]×E U1− (p) ∩ L and E − (p). By Lemma 2.1, there ∗

are disjoint compact subsets Ω1 and Ω2 of the metric space U1− (p)(−∞,λ ]×E ∩ L containing E − (p) and ∂(−∞,λ∗ ]×E U1− (p) ∩L, respectively. Let U2− (p) be a δ30 -neighborhood of Ω1 in the metric space (−∞, λ∗ ] ×E, where δ0 = ρ(Ω1 , Ω2 ), ρ(Ω1 , Ω2 ) denotes the distance of Ω1 and Ω2 . Thus we can define  −

U (p) =

U1− (p),

if ∂(−∞,λ∗ ]×E U1− (p) ∩ L = ∅;

U1− (p) ∩ U2− (p), if ∂(−∞,λ∗ ]×E U1− (p) ∩ L = ∅.

Obviously, U − (p) is a bounded neighborhood of E − (p) in the metric space (−∞, λ∗ ] × E such that ∂(−∞,λ∗ ]×E U − (p) ∩ L = ∅,

(2.2)

and so ∂(−∞,λ∗ ]×E U − (p) ∩ S = ∅. Assume without loss of generality that U − (p)(−∞,λ

∗

]×E

∩ ({λ∗ } × E) ⊂ ({λ∗ } × E)\({λ∗ } × B((λ∗ , x∗ ),

r0 {λ∗ }×E )) . 2

(2.3)

It is easy to see that {U − (p) ∩ ({λ∗ } × E) : p ∈ D} is an open cover of the set D, and D is a compact subset of {λ∗ } × E. Then there exist finitely many subsets, say U − (p1 ) ∩ ({λ∗ } × E), U − (p2 ) ∩ ({λ∗ } × E), · · · , U − (pm ) ∩ ({λ∗ } × E) such that D⊂

m   −  U (pi ) ∩ ({λ∗ } × E) . i=1

Let U − =

m  i=1

U − (pi ). Then, by (2.2) we have

J. Sun, X. Xu / J. Math. Anal. Appl. 435 (2016) 1410–1425

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∂(−∞,λ∗ ]×E U − ∩ L ⊂

m  

 ∂(−∞,λ∗ ]×E U − (pi ) ∩ L = ∅.

i=1

So, ∂(−∞,λ∗ ]×E U − ∩ L = ∅.

(2.4)

Moreover, by (2.3) we have U − (−∞,λ

∗

]×E

∩ ({λ∗ } × E) ⊂ ({λ∗ } × E)\({λ∗ } × B((λ∗ , x∗ ),

r0 {λ∗ }×E )) , 2

(2.5)

and (C + \{(λ∗ , x∗ )}) ∩ ((−∞, λ∗ ] × E) ⊂ U − . Take λ < λ∗ such that U − (λ ) = ∅. Then, by Lemma 2.2, we have deg(I − λ∗ A, U − (λ∗ ), 0) = deg(I − λ A, U − (λ ), 0) = 0.

(2.6)

Similarly, we can construct a bounded open neighborhood U + of C + ∩ ([λ∗ , +∞) × E) in the metric space ∗ [λ∗ , +∞) × E such that U + [λ ,+∞)×E ∩ ([λ∗ , +∞) × {0}) = ∅ and ∂[λ∗ ,+∞)×E U + ∩ L = ∅,

(2.7)

and so ∂[λ∗ ,+∞)×E U + ∩ S = ∅. There is a minor difference between the construction U − and U + . In the process of the construction of U + , we need to consider E + ((λ∗ , x∗ )) which is the connected component of L ∩ ([λ∗ , +∞) × E) passing through (λ∗ , x∗ ). Thus, (λ∗ , x∗ ) ∈ U + . Because (λ∗ , x∗ ) is the unique solution of x = λ∗ Ax in {λ∗ } × B((λ∗ , x∗ ), r0 ), we may assume that U + ∩ ({λ∗ } × B((λ∗ , x∗ ), r0 )) ⊂ {λ∗ } × B((λ∗ , x∗ ),

r0 ). 2

(2.8)

Let r0 )), 2   r0 ∗ U2+ (λ∗ ) = U + (λ∗ ) ∩ {λ∗ } × E\{λ∗ } × B((λ∗ , x∗ ), ){λ }×E . 2 U1+ (λ∗ ) = U + (λ∗ ) ∩ ({λ∗ } × B((λ∗ , x∗ ),

Then, U + (λ∗ ) = U1+ (λ∗ ) ∪ U2+ (λ∗ ), and U1+ (λ∗ ) ∩ U2+ (λ∗ ) = ∅. Take λ > λ∗ large enough such that U + (λ ) = ∅. Also by Lemma 2.2, we have deg(I − λ∗ A, U + (λ∗ ), 0) = deg(I − λ A, U + (λ ), 0) = 0.

(2.9)

By (2.6), (2.8) and the excision property of Leray–Schauder degree, we have deg(I − λ∗ A, U2+ (λ∗ ), 0) = deg(I − λ∗ A, U − (λ∗ ) ∩ U2+ (λ∗ ), 0) = deg(I − λ∗ A, U − (λ∗ ), 0) = 0.

(2.10)

By (2.9), (2.10) we have deg(I − λ∗ A, {λ∗ } × B((λ∗ , x∗ ), r0 ), 0) = deg(I − λ∗ A, U1+ (λ∗ ), 0) = − deg(I − λ∗ A, U2+ (λ∗ ), 0) = 0, which is a contradiction of (2.1).

(2.11)

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Case 2. D = ∅. In this case we can easily construct an open neighborhood U + of C + in the metric space [λ∗ , +∞) × E such that (2.7) holds, U + (λ∗ ) ⊂ {λ∗ } × B((λ∗ , x∗ ), r0 ),

U + ∩ ([λ∗ , +∞) × {0}) = ∅.

Take λ > λ∗ large enough such that U + (λ ) = ∅. Then, (2.9) holds, and so deg(I − λ∗ A, {λ∗ } × B((λ∗ , x∗ ), r0 ), 0) = 0. Now we also get a contradiction of (2.1). The proof is complete. Corollary 2.1. Let C be a bounded connected component of L. Assume that C ∩ (R × {0}) = ∅, and C has a regular point (λ∗ , x∗ ) such that (2.1) holds. Then there exists (λ∗ , x∗∗ ) with x∗∗ = x∗ such that (λ∗ , x∗∗ ) ∈ C + ∩ C − , that is, C is a loop. Corollary 2.2. Assume that there exists λ∗ ∈ R such that there are only finitely many connected component meeting {λ∗ } × E and every such connected component is bounded, and meets {λ∗ } × E at a unique point, but meets R × {0} nowhere. Moreover, ind(I − λ∗ A, ∞) − ind(I − λ∗ A, 0) = 0. Then L has at least one loop. Next we consider the case of C ∩(R × {0}) = ∅. For this purpose we first recall some results for computing Leray–Schauder degree; See Lemmas 2.3–2.5 below. The main elements of the proofs of Lemmas 2.3–2.5 can be found in P.H. Rabinowitz [18], E.N. Dancer [7] and J.A. Izé [12]. Lemmas 2.3–2.5 were originally written in Chinese in [20]. As we do not know an exact reference for our setting, we give the proof of Lemma 2.3 in the Appendix of the paper for the convenience of the readers. Lemma 2.3. Assume that A : E → E is completely continuous, A(0) = 0, A is Fréchet differentiable at 0. Let U be a bounded open subset of [λ , λ ] × E, and ∂[λ ,λ ]×E U ∩ L = ∅. Then the following conclusions hold: 

(1) If (λ , 0) ∈ / U [λ ,λ



]×E



, (λ , 0) ∈ / U [λ ,λ



]×E

, then

deg(I − λ A, U (λ ), 0) = deg(I − λ A, U (λ ), 0) + 2J; (2) If (λ , 0) ∈ U , (λ , 0) ∈ U , then + deg(I − λ A, U (λ ), 0) = deg(I − λ A, U (λ ), 0) + 2J − γ(μ− 1 ) + γ(μn ); 

(3) If (λ , 0) ∈ U , (λ , 0) ∈ / U [λ ,λ



]×E

, then

deg(I − λ A, U (λ ), 0) = deg(I − λ A, U (λ ), 0) + 2J − γ(μ− 1 );

J. Sun, X. Xu / J. Math. Anal. Appl. 435 (2016) 1410–1425

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(4) If (λ , 0) ∈ / U [λ ,λ



]×E

, (λ , 0) ∈ U , then

deg(I − λ A, U (λ ), 0) = deg(I − λ A, U (λ), 0) + 2J + γ(μ+ n ), where μ1 , μ2 , · · · , μn are characteristic values of A (0) with odd algebraic multiplicity and satisfying (μi , 0) ∈ U (1  i  n), λ < μ1 < μ2 < · · · < μn < λ , γ(μ− i ) = lim ind(I − λA, 0), λ→μ− i

γ(μ+ i ) = lim+ ind(I − λA, 0), λ→μi

J=

n 

i = 1, 2, · · · , n,

γ(μ− i ).

i=1

From Lemma 2.3 we have the following Lemmas 2.4 and 2.5. Lemma 2.4. Assume that A : E → E is completely continuous, A(0) = 0, A is Fréchet differentiable at 0, U is a bounded open subset of [λ , +∞) × E, and ∂[λ ,+∞)×E U ∩ L = ∅. Then the following conclusions hold: 

(1) If (λ , 0) ∈ / U [λ ,+∞)×E , then deg(I − λ A, U (λ ), 0) ≡ 0 (mod 2); (2) If (λ , 0) ∈ U , then deg(I − λ A, U (λ ), 0) ≡ 1 (mod 2). 



Proof. Take λ large enough such that U [λ ,+∞)×E ∩ ({λ } × E) = ∅, then (λ , 0) ∈ / U [λ ,+∞)×E . It follows from the conclusions (1) and (3) of Lemma 2.3 that the conclusions hold. The proof is complete. Similarly, we have the following Lemma 2.5. Lemma 2.5. Assume that A : E → E is completely continuous, A(0) = 0, A is Fréchet differentiable at 0, U is a bounded open subset of (−∞, λ ] × E, and ∂(−∞,λ ]×E U ∩ L = ∅. Then the following conclusions hold: 

(1) If (λ , 0) ∈ / U (−∞,λ ]×E , then deg(I − λ A, U (λ ), 0) ≡ 0 (mod 2);  (2) If (λ , 0) ∈ U , then deg(I − λ A, U (λ ), 0) ≡ 1 (mod 2). The following Theorem 2.2 comes from [20]. Theorem 2.2. Let C be a connected component of L with C ∩ (R × {0}) = ∅. Assume that A(0) = 0, A is Fréchet differentiable at 0, (λ∗ , x∗ ) is a regular point of C, x∗ = 0 and ind(I − λ∗ A, x∗ ) ≡ 1 (mod 2). Then one of the following conclusions holds: (1) Both C + and C − are unbounded; (2) There exists (λ∗ , x∗∗ ) with x∗∗ = x∗ such that (λ∗ , x∗∗ ) ∈ C + ∩ C − .

(2.12)

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Proof. As in Theorem 2.1, arguing indirectly, we suppose that the conclusion (2) doesn’t hold and C + is bounded, and then derive a contradiction. Let D be defined as in Theorem 2.1. Then we also have the following three cases. Case 1. D = ∅ and (λ∗ , 0) ∈ / D. By the method of Theorem 2.1 we construct a bounded open neighborhood U − of C + ∩ ((−∞, λ∗ ] × E) in the metric space (−∞, λ∗ ] × E such that (2.4) and (2.5) hold. Moreover, we may assume that (λ∗ , 0) ∈ / (−∞,λ∗ ]×E − U . Then we have by Lemma 2.5 deg(I − λ∗ A, U − (λ∗ ), 0) ≡ 0 (mod 2).

(2.13)

Let us construct a bounded open neighborhood U + of C + ∩([λ∗ , +∞) ×E) in the metric space [λ∗ , +∞) ×E ∗ such that (2.7) and (2.8) hold. Also, we may assume that (λ∗ , 0) ∈ / U + [λ ,+∞)×E . Then we have by Lemma 2.4 deg(I − λ∗ A, U + (λ∗ ), 0) ≡ 0 (mod 2).

(2.14)

Let U1+ , U2+ be defined as in Theorem 2.1. Then we have deg(I − λ∗ A, U2+ (λ∗ ), 0) = deg(I − λ∗ A, U − (λ∗ ) ∩ U2+ (λ∗ ), 0) = deg(I − λ∗ A, U − (λ∗ ), 0) ≡ 0 (mod 2).

(2.15)

Therefore, deg(I − λ∗ A, {λ∗ } × B((λ∗ , x∗ ), r0 ), 0) = deg(I − λ∗ A, U1+ (λ∗ ), 0) = − deg(I − λ∗ A, U2+ (λ∗ ), 0) ≡ 0 (mod 2), which is a contradiction of (2.12). Case 2. D = ∅ and (λ∗ , 0) ∈ D. By the method of Theorem 2.1 we construct U − and U + . Because (λ∗ , 0) ∈ D, we have (λ∗ , 0) ∈ U − and (λ∗ , 0) ∈ U + . By Lemmas 2.4 and 2.5 we have deg(I − λ∗ A, U − (λ∗ ), 0) ≡ 1 (mod 2)

(2.16)

deg(I − λ∗ A, U + (λ∗ ), 0) ≡ 1 (mod 2).

(2.17)

and

From (2.16) and (2.17), as in the proof of case 1, we can easily get a contradiction. Case 3. D = ∅. In this case we can easily construct a bounded open neighborhood U + of C + in the metric space ∗ [λ , +∞) × E such that (2.7) holds, U + (λ∗ ) ⊂ {λ∗ } × B((λ∗ , x∗ ), r0 ), and (λ∗ , 0) ∈ / U + [λ

∗

,+∞)×E

. By Lemma 2.4 we have

deg(I − λ∗ A, {λ∗ } × B((λ∗ , x∗ ), r0 ), 0) = deg(I − λ∗ A, U + (λ∗ ), 0) ≡ 0 (mod 2), which is a contradiction of (2.12). The proof is complete.

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Corollary 2.3. Let C be a bounded connected component of L with C ∩ (R × {0}) = ∅. Assume that A(0) = 0, A is Fréchet differentiable at 0, (λ∗ , x∗ ) is a regular point of C and (2.12) holds. Then there exists (λ∗ , x∗∗ ) with x∗∗ = x∗ such that (λ∗ , x∗∗ ) ∈ C + ∩ C − , that is, C is a loop. Next let us consider the case A(0) = 0. We have the following result. Theorem 2.3. Assume A(0) = 0. Let C be the connected component of L ∩ [0, ∞) emitting from (0, 0), (λ∗ , x∗ ) ∈ C be a regular point, λ∗ > 0 and ind(I − λ∗ A, x∗ ) ∈ / {1, −1, 0}.

(2.18)

Let C + and C − be the right and left hand connected continua emitting from (λ∗ , x∗ ), C1+ be the connected continuum of C + ∩ [0, ∞), and C1− be the connected continuum of C − ∩ [0, ∞), respectively. Then both C1+ and C1− are unbounded. Proof. By using Lemma 2.2 we can easily prove that C ∩[0, ∞) is unbounded. Because C ∩[0, ∞) = C1+ ∪C1− , ¯ x ¯ x ¯ x then (0, 0) ∈ C1+ or (0, 0) ∈ C1− . If there exists (λ, ¯) ∈ C ∩ [0, ∞), (λ, ¯) = (λ∗ , x∗ ) such that (λ, ¯) ∈ + − − + C1 ∩ C1 , then C1 = C1 = C ∩ [0, ∞), and so the conclusion holds. Thus, assume without loss of generality that C1+ ∩ C1− = {(λ∗ , x∗ )}. We have the following two cases: (1) (0, 0) ∈ C1+ ; (2) (0, 0) ∈ C1− . We only show the case (1). In a similar may we can show the case (2). Assume C1+ is bounded. By the method of Theorem 2.1 we can construct a bounded open set U − in the metric space [0, λ∗ ] × E such that (C1+ \{(λ∗ , x∗ )}) ∩ ([0, λ∗ ] × E) ⊂ U − , ∂[0,λ∗ ]×E U − ∩ S = ∅, (0, 0) ∈ U − and U − ∩ ({λ∗ } × B((λ∗ , x∗ ), r0 )) = ∅. It is easy to see that deg(I, U − (0), 0) = 1.

(2.19)

It follows from Lemma 2.2 and (2.19) that deg(I − λ∗ A, U − (λ∗ ), 0) = 1.

(2.20)

Construct a bounded open subset U + in the metric space [λ∗ , +∞) × E such that ∂[λ∗ ,+∞)×E U + ∩ S = ∅,

C1+ ∩ ([λ∗ , +∞) × E) ⊂ U + .

By Lemma 2.2 we have deg(I − λ∗ A, U + (λ∗ ), 0) = 0.

(2.21)

Let U1+ , U2+ be defined as in the proof of Theorem 2.2. Then, from (2.20) and (2.21) we have ind(I − λ∗ A, x∗ ) = deg(I − λ∗ A, U1+ (λ∗ ), 0) = − deg(I − λ∗ A, U2+ (λ∗ ) ∩ U − (λ∗ ), 0) = − deg(I − λ∗ A, U − (λ∗ ), 0) = −1, which is a contradiction of (2.18). Thus, C1+ is unbounded. In a similar may as in the proof of Theorem 2.1 we can show C1− is unbounded since C1− ∩ (R × {0}) = ∅ and ind(I − λ∗ A, x∗ ) = 0. The proof is complete.

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Obviously, the main results can be extended to the case when the nonlinear operator has the form of A(λ, x). For purpose of application to some elliptic boundary value problems, we now briefly give some results in this case. Let A(·, ·) : R × E → E be a completely continuous operator and C is a connected component of L1 , where L1 = {(λ, x) ∈ R × E : x = A(λ, x), x = 0}R×E . Define the regular point on C in a similar way as in Definition 2.1. Then we have the following result. Theorem 2.4. Let C be a connected component of L1 with C ∩ (R × {0}) = ∅. Assume that A(λ, 0) = 0, A(λ, ·) is Fréchet differentiable at 0 for every λ ∈ R, (λ∗ , x∗ ) is a regular point of C, x∗ = 0 and ind(I − A(λ∗ , ·), x∗ ) ≡ 1 (mod 2). Then one of the following conclusions holds: (1) Both C + and C − are unbounded; (2) There exists (λ∗ , x∗∗ ) with x∗∗ = x∗ such that (λ∗ , x∗∗ ) ∈ C + ∩ C − . Next we give an application of the main results. Consider the following semilinear elliptic boundary value problem 

−Δu = λm(x)u + b(x)uγ ,

for x ∈ Ω;

∂u(x) ∂n

on ∂Ω,

= 0,

(2.22)

¯ → R are where Ω is a bounded region with smooth boundary in RN , λ is a real parameter, m and b : Ω + smooth functions which change sign in Ω, m(x) changes sign on Ωb , and γ is a positive integer such that + N +2 1<γ< N −2 , where Ωb = {x ∈ Ω : b(x) > 0}. The author of [2] investigated the local and global nature of the bifurcation diagrams which can occur for (2.22). It was shown that closed loops of positive and negative solutions occur naturally for such problems and properties of these loops are investigated. Now we will prove  the existence of loops by using Theorem 2.4 in the case of Ω m(x)dx < 0. As proved in [2] the following eigenvalue problem 

−Δu = λm(x)u, for x ∈ Ω; ∂u(x) ∂n

= 0,

on ∂Ω

(2.23)

has two principle eigenvalues λ1 = 0 and λ1 (m) > 0. Let φ1 be the corresponding eigenfunction of (2.23) corresponding to λ1 (m). Let X = {u ∈ C 2+α (Ω) :

∂u = 0 on ∂Ω}. ∂n

Define the operator F : R × X → C α (Ω) by F (λ, u) = Δu + λm(x)u + b(x)uγ . By the method of [2] and Crandall–Rabinowitz theorem [5] on bifurcation from a simple eigenvalue, we see that bifurcation occurs at λ1 (m) and there is a curve of bifurcating solutions given by u(s) = s(φ1 + ψ(s));

λ(s) = λ1 (m) + μ(s)

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 for s ∈ J0 = [−ε, ε], where ψ : R → {u ∈ X : Ω muφ1 = 0} and μ : R → R are smooth functions such that ψ(0) = 0 and μ(0) = 0. By Corollary 1.13 and Theorem 1.16 in [6], there exist open intervals J1 , J2 with λ1 (m) ∈ J1 and 0 ∈ J2 ⊂ J0 and continuously differentiable functions ζ : J1 → R, η : J2 → R, h : J1 → X, w : J2 → X such that Fu (λ, 0)h(λ) = ζ(λ)h(λ)

for λ ∈ J1 ,

Fu (λ(s), u(s))w(s) = η(s)w(s)

for s ∈ J2 .

Moreover, ζ(λ1 (m)) = η(0) = 0, h(λ1 (m)) = φ1 = w(0), η(s) and −sλ (s)ζ  (λ1 (m)) have the same zeros. From [2] and L’Hospital principle we have  bφγ+1 μ(s) μ (s) Ω 1 2 . = = − s→0 (γ − 1)sγ−2 sγ−1 mφ1 Ω lim

(2.24)

 Note that ζ  (λ1 (m)) = 0. Assume that Ω b = 0. By (2.24) we easily see that there exists s0 = 0 small enough such that −s0 λ (s0 )ζ  (λ1 (m)) = 0, and so η(s0 ) = 0. This means (λ(s0 ), u(s0 )) is a non-degenerate point. From standard elliptic theory, the boundary value problem 

−Δu = v, in Ω; ∂u ∂n

= 0;

on ∂Ω

has a unique solution u = Kv for each v ∈ C α (Ω), and the operator K : C α (Ω) → C 2+α (Ω) can be extended to a linear completely continuous operator from E =: C(Ω) to E. Define the operator G(·, ·) : R×E → C α (Ω) by G(λ, u) = λm(x)u + b(x)uγ and the operator A(λ, u) = KG(λ, u) : E → E. Then A(·, ·) : R × E → E is a completely continuous operator. Since 1 is not an eigenvalue of Au (λ(s0 ), u(s0 )), we see that (λ(s0 ), u(s0 )) is a regular point on the connected component of L2 = {(λ, u) : (λ, u) is a solution of (2.22), u = 0} passing through (λ(s0 ), u(s0 )), and ind(I − A(λ(s0 ), ·), u(s0 )) = ±1. Using the method of [11] one can easily obtain a priori bounds for u on the connected component of L2 passing through (λ(s0 ), u(s0 )). On the other hand, by using the properties on eigenvalues of elliptic boundary value problems one can easily get a priori bounds in λ; See [2]. So, by using Theorem 2.4 we see that the connected component C of L2 passing through (λ(s0 ), u(s0 )) is a loop. Obviously, there are positive solutions, negative solutions as well as zero solution of (2.22) on C. 3. Some remarks on the main results Remark 3.1. Theorem 2.1 has a clear geometric meaning. It implies that the left hand connected continuum and right hand connected continuum either tend to infinity when alternative 1 occurs, or have a loop when alternative 2 occurs.

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Remark 3.2. If x∗ = 0, and the condition (2.12) is replaced with ind(I − λ∗ A, 0) ≡ 0 (mod 2), then all conclusions of Theorem 2.2 also hold. Remark 3.3. If the condition (2.12) in Theorem 2.2 is replaced with (2.1), then for C + at least one of the following alternatives 1–3 holds: Alternative 1: C + is unbounded; Alternative 2: There exists (λ∗ , x∗∗ ) with x∗∗ = x∗ , such that (λ∗ , x∗∗ ) ∈ C + ∩ C − ; Alternative 3: There exists a characteristic value μ+ of A (0) with odd algebraic multiplicity such that (μ+ , 0) ∈ C + . The similar conclusion also holds for C − . Remark 3.4. In the studying of global bifurcation theories, many authors studied the existence of connected component of solutions set of nonlinear operator equations which have turning point. These turning points may provide us with very important information about the connected components; See [8,13,17]. To show the existence of turning points, peoples have employed different methods. For examples, E.N. Dancer in [8] by computing the Morse index obtained the existence of a connected component which has infinitely many turning points; Zhaoli Liu in [13] by using the time-maps method studied a differential boundary value problem with concave and convex nonlinearity and obtained the existence of connected component of solutions set which has exactly one turning point; Tiancheng Ouyang and Junping Shi in [17] by using the Turning point theorem of Crandall–Rabinowitz obtained some results concerning the existence of solution branches which may have turning points. Let C be a connected component of L. By Lemma 2.2 one can easily get a necessary condition for a regular point (λ∗ , x∗ ) ∈ C being a turning point is ind(I − λ∗ A, x∗ ) = 0.

(3.1)

Obviously, (3.1) also holds for those points (λ∗ , x∗ ) on a connected component C which are regular points satisfying either −∞ < λ∗ = sup{λ| there exits x ∈ E, such that (λ, x) ∈ C} < +∞, or −∞ < λ∗ = inf{λ| there exits x ∈ E, such that (λ, x) ∈ C} < +∞. Furthermore, if A (x∗ ) exists, then, by the famous topological degree formula of Leray–Schauder, λ∗ must be a characteristic value of A (x∗ ), that is, h = λ∗ A (x∗ )h for some h ∈ E\{0}. Note the Implicit Function Theorem no longer holds in these points. Actually, E.N. Dancer in [8] showed the existence of turning points on a connected component by the method of finding those points that the Implicit Function Theorem no longer hold. From these observations and Theorem 2.2, we have the following conclusion. Theorem 3.1. Let C be a bounded component of L, A (x) exist for every (λ, x) ∈ C and there exist (λ∗ , x∗ ) ∈ C such that λ∗ is not a characteristic value of A (x∗ ), x∗ = 0. Then C must be a loop and there exist (λ , x ), (λ , x ) ∈ C such that λ < λ∗ < λ , and

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h1 = λ A (x )h1 ,

h2 = λ A (λ )h2

for some h1 , h2 ∈ E\{0}. Remark 3.5. The formula of computing Leray–Schauder degree in Lemma 2.3 has many applications to the studying of bifurcation theories. For example, by using Lemma 2.3 we can easily prove the famous global bifurcation theorem of Rabinowitz (see [18]). Moreover, we can use Lemma 2.3 to analyze the minimal cardinal of the λ-slices of a semi-bounded components of L. Here, by a semi-bounded component C of L, we mean C is a bounded connected component of ([λ0 , +∞) ×E) ∩L or ((−∞, λ0 ] ×E) ∩L for some λ0 ∈ R. Now we assume that C is a bounded connected component of ([λ0 , +∞) × E) ∩ L such that C(λ0 ) ∩ (R × {0}) = ∅, and I − λ0 A has finite different isolated zeros u1 , u2 , · · · , un , such that ind(I − λ0 A, uj ) ∈ {−1, 0, 1},

for 1 ≤ j ≤ n.

By the method of Theorem 2.2 we can easily construct a bounded open neighborhood U of C in [λ0 , +∞) ×E ¯ [λ0 ,+∞)×E and U not contain any points (λs , 0), where λs is a such that ∂[λ0 ,+∞)×E U ∩ L = ∅, (λ0 , 0) ∈ /U characteristic value of A (0) and (λs , 0) ∈ / C. Take λ > λ0 large enough such that U (λ ) = ∅. Then by the conclusion (1) of Lemma 2.3 we have deg(I − λ0 A, U (λ0 ), 0) = 2J, where J =

n i=1

 γ(μ− i ), μ1 , μ2 , · · · , μn are characteristic values of A (0) with odd algebraic multiplicity and

satisfying (μi , 0) ∈ C (1  i  n), λ0 < μ1 < μ2 < · · · < μn < +λ , γ(μ− i ) is defined as Lemma 2.3. Now by using the properties of Leray–Schauder degree we have Card C(λ0 ) ≥ |2J|. Thus we have obtained a result very similar to Theorem 5.3 in [16]. In [16] the minimal cardinal of λ-slices of semi-bounded component of the solution set of the nonlinear operator equation A(λ, u) = 0,

(λ, u) ∈ R × E

had been investigated, where A(λ, u) := L(λ)u + M (λ, u) for each (λ, u) ∈ R × E, Au (λ, 0) = L(λ) for each λ ∈ R. It is easy to see the spectrum Σ of the family L(λ) may be not discrete. Here, different from Theorem 5.3 in [16] we do not use the requirement λ0 ∈ / Σ and only require that (λ0 , 0) ∈ / C. Remark 3.6. There have been some related works to our main results. For example, Theorem 4.1 in [23], Theorem 3.5.3 in [4] are similar to results here. The readers also can find some other references with related results in [14,19]. Obviously, we can substitute the regular point by an isolated connected component of L ∩ ({λ∗ } × E). Furthermore, we may extend the main results of this paper by substituting the regular point by a domain on the metric space. For these purposes, we will give some further results in a new paper to discuss these problems. Acknowledgments The authors thank an anonymous referee for a careful reading and helpful suggestions, and they also thank Prof. Du Yihong for reading the manuscript and giving some helpful comments.

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Fig. 2. An illustration of the proof of Lemma 2.3.

Appendix A. The proof of Lemma 2.3 Now we only show the conclusion (1) of Lemma 2.3. In a similar way we can show the conclusions (2)–(4). Assume that μ ¯1 , · · · , μ ¯m are all of the bifurcation points of A satisfying (¯ μi , 0) ∈ U (1  i  m), and λ < μ ¯1 < μ ¯2 < · · · < μ ¯m < λ . For each μ ¯i (i = 1, 2, · · · , m), take δi > 0 small enough such that for each i = 1, 2, · · · , m, [¯ μi − δi , μ ¯ i + δi ] ¯i , has no characteristic values of A (0) which is different to μ [¯ μi − δi , μ ¯i + δi ] ⊂ [λ , λ ],

[¯ μi − δ i , μ ¯i + δi ] × {0} ⊂ U,

and [¯ μi − δi , μ ¯i + δi ] ∩ [¯ μj − δj , μ ¯j + δj ] = ∅,

i, j = 1, 2, · · · , m, i = j.

The proof can be illustrated by Fig. 2, where we only consider the case m = 1. Obviously, S ∩ ∂[¯μi −δi ,¯μi +δi ]×E U = ∅. From Lemma 2.2 we have deg(I − (¯ μi − δi )A, U (¯ μi − δi ), 0) = deg(I − (¯ μi + δi )A, U (¯ μi + δi ), 0). Let Δ∗ = [λ , λ ]\

m 



[¯ μi − δi , μ ¯i + δi ]. Obviously, (Δ∗ × {0}) ∩ (U [λ ,λ

i=1 [λ ,λ ]×E



ρ(Δ∗ × {0}, U ∩ L) > 0, where ρ(Δ∗ × {0}, U [λ ,λ   U [λ ,λ ]×E ∩ L. On the other hand,



]×E



]×E

∩ L) = ∅. Therefore, ρ1 =

∩ L) denotes the distance of Δ∗ × {0} and

m

  ρ2 = ρ ∂[λ ,λ ]×E U, ([¯ μi − δ i , μ ¯i + δi ] × {0}) > 0. i=1

Let r0 =

1 2

min{ρ1 , ρ2 }, B(0, r0 ) = {x ∈ E|x < r0 },

(A.1)

¯ r0 )). U ∗ = U \([λ , λ ] × B(0,

Then U ∗ is a bounded open subset of [λ , λ ] × E, and   S ∩ ∂[λ ,λ ]×E U ∗ ∩ (Δ∗ × E) = ∅.

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By Lemma 2.2 we have deg(I − (¯ μi + δi )A, U ∗ (¯ μi + δi ), 0) = deg(I − (¯ μi+1 − δi+1 )A, U ∗ (¯ μi+1 − δi+1 ), 0),

i = 1, 2, · · · , m − 1;

deg(I − λ A, U ∗ (λ ), 0) = deg(I − (¯ μ1 − δ1 )A, U ∗ (¯ μ1 − δ1 ), 0), ∗



∗



deg(I − (¯ μm + δm )A, U (¯ μm + δm ), 0) = deg(I − λ A, U (λ ), 0).

(A.2) (A.3) (A.4)

Obviously, we have for each i = 1, 2, · · · , m, {¯ μi + δi } × B(0, r0 ) ⊂ U (¯ μi + δi ),

{¯ μi − δi } × B(0, r0 ) ⊂ U (¯ μi − δi ).

Note that deg(I − (¯ μi ± δi )A, B(0, r0 ), 0) = γ(¯ μ± i ). Then we have deg(I − (¯ μi ± δi )A, U (¯ μi ± δi ), 0) = deg(I − (¯ μi ± δi )A, U ∗ (¯ μi ± δi ), 0) + γ(¯ μ± i ). 

Since (λ , 0) ∈ / U [λ ,λ



]×E



, (λ , 0) ∈ / U [λ ,λ



]×E

, we have

deg(I − λ A, U (λ ), 0) = deg(I − λ A, U ∗ (λ ), 0), 

(A.5)





∗



deg(I − λ A, U (λ ), 0) = deg(I − λ A, U (λ ), 0).

(A.6) (A.7)

It follows from (A.1)–(A.7) that deg(I − λ A, U (λ ), 0) = deg(I − λ A, U (λ ), 0) +

m  (γ(¯ μ− μ+ i )). i ) − γ(¯

(A.8)

i=1

Note γ(¯ μ− μ+ ¯i is a characteristic value of A (0) with odd algebraic multiplicity, and i ) when μ i ) = −γ(¯ − + γ(¯ μi ) = γ(¯ μi ) when μ ¯i is a characteristic value of A (0) with even algebraic multiplicity. Then we can easily see that the conclusion (1) holds. The proof is complete. References [1] A. Ambrosett, A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge University Press, Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sáo Paulo, 2006. [2] K.J. Brown, Local and global bifurcation results for a semilinear boundary value problem, J. Differential Equations 239 (2) (2007) 296–310. [3] Santiago Cano-Casanova, Julián López-Gómez, Marcela Molina-Meyer, Bounded components of positive solutions of nonlinear abstract equations, Ukr. Mat. Visn. 2 (1) (2005) 38–51; translation in: Ukr. Math. Bull. 2 (1) (2005) 39–52. [4] K.C. Chang, Methods in Nonlinear Analysis, Springer-Verlag, Berlin, Heidelberg, 2005. [5] Michael G. Crandall, Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971) 321–340. [6] Michael G. Crandall, Paul H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Ration. Mech. Anal. 52 (1973) 161–180. [7] E.N. Dancer, On the structure of solutions of nonlinear eigenvalue problems, Indiana Univ. Math. J. 33 (1974) 1069–1076. [8] E.N. Dancer, Infinitely many turning points for some supercritical problems, Ann. Mat. Pura Appl. (4) 178 (2000) 225–233. [9] E.N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. Lond. Math. Soc. 34 (5) (2002) 533–538. [10] E.N. Dancer, Yi Hong Du, Positive solutions for a three-species competition system with diffusion. I. General existence results, Nonlinear Anal. 24 (3) (1995) 337–357. [11] Yihong Du, Shujie Li, Nonlinear Liouville theorems and a priori estimates for indefinite superlinear elliptic equations, Adv. Differential Equations 10 (8) (2005) 841–860. [12] J.A. Izé, Bifurcation Theory for Fredholm Operators, Mem. Amer. Math. Soc., vol. 174, Amer. Math. Soc., Providence, RI, 1976. [13] Zhaoli Liu, Exact number of solutions of a class of two-point boundary value problems involving concave and convex nonlinearities, Nonlinear Anal. 46 (2) (2001) 181–197.

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