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On the noncompact component of solutions for nonlinear inclusions
Mathematical and Computer Modelling 45 (2007) 795–800 www.elsevier.com/locate/mcm
On the noncompact component of solutions for nonlinear inclusions In-Sook Kim 1 Department of Mathematics and Institute of Basic Science, Sungkyunkwan University, Suwon 440-746, Republic of Korea Received 19 May 2006; received in revised form 18 July 2006; accepted 28 July 2006
Abstract We give a bifurcation theorem for boundedly 0-epi multi-valued maps and then show the existence of a noncompact component of solutions for nonlinear inclusions. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Solutions; Noncompact component; Inclusion; Boundedly 0-epi maps, degree theory
1. Introduction Furi and Pera [4] showed the existence of an unbounded connected set of solutions for nonlinear equations in Banach spaces, where the proof is mainly based on the theory of 0-epi maps introduced in [3]. For a survey, we refer to [5]. V¨ath [9] used a variant of this result in a more general setting to obtain a global bifurcation theorem of Rabinowitz type [8] as follows: under certain conditions, a parameterized equation F(u, λ) = 0 has an unbounded connected set of solutions (u, λ) which contains a point of the form (u, 0). The aim of this paper is to prove the existence of a noncompact connected set of solutions for nonlinear inclusions of the form J (x) + G(x) ∈ M(x, λ) where X is a Banach space, Y is a normed space, Λ is a finite-dimensional space, J : X → Y is a linear homeomorphism, G : X → Y is a compact continuous map, and M : X × Λ ( Y is a compact upper semicontinuous multi-valued map with nonempty closed convex values. To this end, we introduce the notion of boundedly 0-epi multivalued maps and give a bifurcation theorem for such maps, where the method is to use a result on the noncompactness of solution branch stated in [10]. Moreover, we apply Borsuk’s theorem to find conditions sufficient for a multi-valued map to be boundedly 0-epi, where the degree theory of Ma [7] is considered. Consequently, the above main result is obtained under these observations.
E-mail address: [email protected]. 1 Current address: Mathematisches Institut, Ludwig-Maximilians Universit¨at M¨unchen, Theresienstr. 39, D-80333 M¨unchen, Germany. c 2006 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2006.07.022
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2. Boundedly 0-epi maps Let X and Y be two normed spaces and Ω a nonempty subset of X . We denote by k(Y ) the collection of all nonempty closed subsets of Y and by kc(Y ) the collection of all nonempty closed convex subsets of Y , respectively. We first introduce a notion of 0-epi multi-valued maps. A concept of F-epi single-valued maps has been developed in [1], where each ϕ ∈ F is a multi-valued map. Definition 2.1. A multi-valued map F : Ω ( Y is said to be 0-epi on Ω provided that it is upper semicontinuous on Ω and 0 6∈ F(x) for all x ∈ ∂Ω and for every upper semicontinuous map ϕ : Ω → kc(Y ) such that the range ϕ(Ω ) has a compact closure and ϕ|∂ Ω = 0, the coincidence problem (F, ϕ) has a solution in Ω . This means that there exists an x ∈ Ω such that F(x) ∩ ϕ(x) 6= ∅, or denoted by CoinΩ (F, ϕ) := {x ∈ Ω : F(x) ∩ ϕ(x) 6= ∅} 6= ∅. We begin with two basic properties of 0-epi multi-valued maps. Proposition 2.2 (Restriction). Let Ω0 ⊆ Ω . If F : Ω ( Y is 0-epi on Ω and 0 6∈ F(x) outside the interior of Ω0 , then F is 0-epi on Ω0 . Proof. The proof is analogous to that of Proposition 2.7.
Proposition 2.3 (Homotopy Invariance). Let F : Ω → k(Y ) be a 0-epi map on Ω and H : [0, 1] × Ω → kc(Y ) an upper semicontinuous map such that the range of H has a compact closure and H (0, ·) = 0. For each t ∈ [0, 1], let G t : Ω → k(Y ) be defined by G t (x) := F(x) − H (t, x) for x ∈ Ω . If 0 6∈ G t (x) for all t ∈ [0, 1] and all x ∈ ∂Ω , then each G t is 0-epi on Ω . Proof. It suffices to prove that G 1 is 0-epi on Ω . Let ϕ : Ω → kc(Y ) be an upper semicontinuous map such that the range ϕ(Ω ) has a compact closure and ϕ|∂ Ω = 0. Let S := {x ∈ Ω : G t (x) ∩ ϕ(x) 6= ∅ for some t ∈ [0, 1]}. Then S is nonempty and closed. Indeed, let (xn ) be a sequence in S and x ∈ Ω such that xn → x as n → ∞. For each n, there are tn ∈ [0, 1] and z n , wn ∈ Y such that z n ∈ ϕ(xn ), wn ∈ H (tn , xn ) and z n + wn ∈ F(xn ). Since [0, 1] is compact and the ranges of ϕ and H have a compact closure, respectively, we may assume without loss of generality that tn → t, z n → z, and wn → w as n → ∞ for some t ∈ [0, 1] and some z, w ∈ Y . Since the maps ϕ, H , and F have a closed graph, respectively, we obtain z ∈ ϕ(x), w ∈ H (t, x), and z + w ∈ F(x) and hence z ∈ ϕ(x) ∩ G t (x) and therefore x ∈ S. We have shown that S is closed in X . Since assumptions on G t and ϕ imply that S ∩ ∂Ω = ∅, there is a continuous function α : Ω → [0, 1] such that α = 0 on ∂Ω and α = 1 on S. Consider a map ψ : Ω → kc(Y ) given by ψ(x) := H (α(x), x) + ϕ(x) for x ∈ Ω . Then ψ is upper semicontinuous on Ω and its range has a compact closure, and ψ|∂ Ω = 0. Since F is 0-epi on Ω , by Definition 2.1, there exists an x ∈ Ω such that F(x) ∩ ψ(x) 6= ∅, that is, G α(x) (x) ∩ ϕ(x) 6= ∅. This implies that x ∈ S and so α(x) = 1. We conclude that G 1 is 0-epi on Ω . In the proof of Theorem 2.5, we make use of the following classical lemma from [6]. Lemma 2.4. Let A and B be two disjoint closed subsets of a compact Hausdorff space X . Then either there is a component of X which intersects A and B, or X divides into two disjoint closed sets which contain A and B, respectively. We give a bifurcation result for 0-epi multi-valued maps. In the single-valued case, a topological bifurcation result in a more general setting is studied in [9, Theorem 9]. Theorem 2.5. Let X, Y1 , and Y2 be three normed spaces and Ω a nonempty subset of X . Suppose that F = (F1 , F2 ) : Ω → k(Y1 × Y2 ) is 0-epi on Ω and X i := {x ∈ Ω : 0 ∈ Fi (x)} is a subset of Ω for i = 1, 2 such that the following conditions are satisfied: (1) X 1 is compact. (2) There exists a point of Y2 that does not belong to F2 (X 1 ). Then X 1 contains a component which intersects the sets X 2 and ∂Ω .
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Proof. Assume that X 1 has no component which intersects X 2 and ∂Ω . Notice that the set X 1 ∩ X 2 is closed in X because F has a closed graph. By Lemma 2.4, the compact Hausdorff space X 1 divides into two disjoint closed sets (in X 1 and thus in X ) which contain X 1 ∩ X 2 and X 1 ∩ ∂Ω , respectively. Hence there exist two disjoint open sets Ω1 and Ω2 in X such that X 1 ∩ X 2 ⊆ Ω1 , X 1 ∩ ∂Ω ⊆ Ω2 , and X 1 ⊆ Ω1 ∪ Ω2 . Put Ω0 := Ω1 ∩ Ω . Since ∂Ω0 ⊆ ∂Ω1 ∪ ∂Ω and since the sets Ω1 and Ω2 are open and disjoint, we have X 1 ∩ ∂Ω0 ⊆ [(X 1 ∩ ∂Ω1 ) ∪ (X 1 ∩ ∂Ω )] ∩ Ω 1 ⊆ Ω2 ∩ Ω 1 = ∅. Choose a point y2 ∈ Y2 such that y2 6∈ F2 (X 1 ). Consider a homotopy H : [0, 1] × Ω 0 → Y1 × Y2 defined by H (t, x) := (0, t y2 ) for (t, x) ∈ [0, 1] × Ω 0 . Then H is a continuous single-valued map and its range is compact. For each t ∈ [0, 1], let G t : Ω 0 → k(Y1 × Y2 ) be given by G t (x) = F(x) − H (t, x) = {(z 1 , z 2 ) ∈ Y1 × Y2 : z 1 ∈ F1 (x), z 2 + t y2 ∈ F2 (x)}. For all t ∈ [0, 1] and all x ∈ ∂Ω0 , it follows from x 6∈ X 1 that (0, 0) 6∈ G t (x). Since {x ∈ Ω : 0 ∈ F(x)} = X 1 ∩ X 2 ⊆ Ω0 , Proposition 2.2 implies that F is 0-epi on Ω0 . By Proposition 2.3, G 1 is 0-epi on Ω0 . Since CoinΩ0 (G 1 , 0) 6= ∅, there exists an x ∈ Ω0 such that 0 ∈ F1 (x) (and thus x ∈ X 1 ) and y2 ∈ F2 (x), which contradicts the choice of y2 . This completes the proof. To achieve bifurcation for nonlinear inclusions, another notion is necessary. Definition 2.6. An upper semicontinuous multi-valued map ϕ : Ω ( Y is said to be compact if the image ϕ(B) has a compact closure for any bounded set B ⊆ Ω . A multi-valued map F : Ω ( Y is said to be boundedly 0-epi on Ω provided that it is upper semicontinuous on Ω and 0 6∈ F(x) for all x ∈ ∂Ω and for every compact upper semicontinuous map ϕ : Ω → kc(Y ) such that ϕ(x) = {0} for all x ∈ ∂Ω and for all x outside some bounded set, the coincidence problem (F, ϕ) has a solution in Ω . The class of boundedly 0-epi multi-valued maps has the usual properties as follows: Proposition 2.7 (Restriction). Let Ω0 ⊆ Ω . If F : Ω ( Y is boundedly 0-epi on Ω and 0 6∈ F(x) outside the interior of Ω0 , then F is boundedly 0-epi on Ω0 . In particular, if Ω0 is bounded, then F is 0-epi on Ω0 . Proof. Let ϕ : Ω 0 → all x ∈ Ω 0 \ B, where ϕ(x) ϕ(x) ˜ := {0}
kc(Y ) be a compact upper semicontinuous map such that ϕ(x) = {0} for all x ∈ ∂Ω0 and for B ⊆ X is some bounded set. Define a map ϕ˜ : Ω → kc(Y ) by for x ∈ Ω 0 for x 6∈ Ω0 .
Then ϕ˜ is upper semicontinuous and compact, and ϕ(x) ˜ = {0} for all x ∈ ∂Ω and for all x ∈ Ω \ B. Since F is ˜ 0 ) 6= ∅. From 0 6∈ F(x) and boundedly 0-epi on Ω , by Definition 2.6, there exists an x0 ∈ Ω such that F(x0 ) ∩ ϕ(x ϕ(x) ˜ = {0} for all x ∈ Ω \ Ω0 it follows that x0 belongs to Ω0 and so F(x0 ) ∩ ϕ(x0 ) 6= ∅. Thus, F is boundedly 0-epi on Ω0 . Proposition 2.8 (Homotopy Invariance). Let F : Ω → k(Y ) be a boundedly 0-epi map on Ω and H : [0, 1] × Ω → kc(Y ) an upper semicontinuous map such that H (0, ·) = 0 and H ([0, 1] × B) has a compact closure for any bounded set B ⊆ Ω . For each t ∈ [0, 1], let G t : Ω → k(Y ) be defined by G t (x) := F(x) − H (t, x) for x ∈ Ω . If 0 6∈ G t (x) for all t ∈ [0, 1], all x ∈ ∂Ω and all x outside some bounded set which does not depend on t, then each G t is boundedly 0-epi on Ω . Proof. Let ϕ : Ω → kc(Y ) be a compact upper semicontinuous map such that ϕ(x) = {0} for all x ∈ ∂Ω and for all x ∈ Ω \ B, where B is some bounded set in X . We may assume that B is open in X and 0 6∈ G t (x) for all t ∈ [0, 1] and all x ∈ ∂Ω ∪ (Ω \ B), by enlarging B if necessary. In particular, 0 6∈ G t (x) for all t ∈ [0, 1] and all x ∈ ∂(B ∩ Ω ). Since 0 6∈ F(x) outside the interior of B ∩ Ω , the restriction F| B∩Ω is 0-epi on B ∩ Ω , by Proposition 2.7. Since H ([0, 1] × (B ∩ Ω )) has a compact closure, Proposition 2.3 implies that each G t | B∩Ω is 0-epi on B ∩ Ω . Since ϕ(B ∩ Ω ) has a compact closure and ϕ|∂(B∩Ω ) = 0, there is an x ∈ B ∩ Ω such that G t (x) ∩ ϕ(x) 6= ∅. Therefore, each G t is boundedly 0-epi on Ω .
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3. Main results The following crucial criterion is a key tool in finding a noncompact solution branch; see [10, Theorem 2.2]. Lemma 3.1. Let X be a regular space, A ⊆ X compact, and S ⊆ X such that S ∩ A is closed. Suppose that S is locally compact and that there exists an open (possibly empty) set U0 ⊇ (S \ S) ∩ A such that U 0 ∩ S is compact. Then, for each open set U ⊇ A ∪ U 0 , the following statements are equivalent: (1) For each open set O ⊇ A with O ⊆ U and compact O ∩ S, there is an x ∈ ∂ O in S. (2) There is a connected closed (in S) subset C ⊆ S ∩ U which intersects A and is either noncompact or intersects ∂U . (3) There is a connected set C ⊆ (S ∩ U ) \ A such that C ∩ S intersects A and is either noncompact or intersects ∂U . We can prove the following bifurcation result for boundedly 0-epi multi-valued maps. The single-valued case is considered in [9, Theorem 18], where the method of approach is different from ours. Theorem 3.2. Let X , Y1 and Y2 be three normed spaces and Ω a nonempty subset of X . Suppose that F = (F1 , F2 ) : Ω → k(Y1 × Y2 ) is boundedly 0-epi on Ω and X i := {x ∈ Ω : 0 ∈ Fi (x)} is a subset of Ω for i = 1, 2 such that the following conditions are satisfied: (1) X 1 is locally compact. (2) X 1 ∩ X 2 is compact. (3) For each bounded set B ⊆ Ω , there exists a point of Y2 that does not belong to F2 (X 1 ∩ B). Then there is a connected set C ⊆ X 1 which intersects X 2 and is either noncompact or intersects ∂Ω . Proof. Let A = X 1 ∩ X 2 and S = X 1 , then we may take U0 = ∅, where U0 is given as in Lemma 3.1. We may assume that Ω is an open set in X . Let O be any open set in X such that A ⊆ O ⊆ O ⊆ Ω , and O ∩ S is compact. Let K := O ∩ S. Since K is compact and S is locally compact, there is a bounded open set Ω0 in X such that K ⊆ Ω0 ⊆ Ω 0 ⊆ Ω and Ω 0 ∩ S is compact. Since 0 6∈ F(x) outside Ω0 , the map F is 0-epi on Ω0 , by Proposition 2.7. Note that there exists a point y ∈ Y2 that does not belong to F2 (S ∩ Ω 0 ), by assumption (3). Applying Theorem 2.5 with F |Ω 0 , the compact set S ∩ Ω 0 has a component which intersects X 2 and ∂Ω0 ; hence there is an x ∈ ∂ O in S. By Lemma 3.1, there is a connected set C ⊆ (S ∩ Ω ) which intersects A and is either noncompact or intersects ∂Ω .
Based on Borsuk’s theorem, we observe when a multi-valued map F is boundedly 0-epi. Proposition 3.3. Let X be a Banach space and Y a normed space. Let J : X → Y be a linear homeomorphism and G : X → Y a compact continuous map, and F := J + G a map such that F(x) 6= 0 for all x outside some bounded set. If the map ψ := J −1 ◦ (−G) : X → X is odd, then F is boundedly 0-epi on X . Proof. Let ϕ : X → kc(Y ) be a compact upper semicontinuous map such that there is a bounded set B ⊆ X with ϕ(x) = {0} for all x ∈ X \ B. We may assume that B is an open ball with center at 0 and F(x) 6= 0 for all x ∈ X \ B, by enlarging B if necessary. Then the coincidence set Coin X (F, ϕ) = {x ∈ X : F(x) ∈ ϕ(x)} is contained in B. Let ψ1 : B → kc(X ) be a map given by ψ1 (x) := J −1 ◦ (ϕ − G)(x). Then the sets ψ1 (B) and ψ(B) are relatively compact because ϕ and G are compact on X and J −1 is continuous on Y . Consider a map H : [0, 1] × B → kc(Y ) defined by H (t, x) := J −1 ◦ (tϕ − G)(x)
for (t, x) ∈ [0, 1] × B.
Then the range of H has a compact closure and x 6∈ H (t, x) for all (t, x) ∈ [0, 1] × ∂ B, because ϕ(x) = {0} and F(x) 6= 0 for all x ∈ ∂ B. The homotopy invariance of Ma’s degree [7] for compact multi-valued maps implies that Deg (I − ψ1 , B, 0) = Deg (I − ψ, B, 0), where I denotes the identity map of X . Since the latter has odd degree by Borsuk’s theorem, see e.g. [2, Theorem 8.3], we obtain that ψ1 has a fixed point in B. Since the fixed point of ψ1 belongs to the set Coin X (F, ϕ), we conclude that F is boundedly 0-epi on X .
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Now we give the following global bifurcation result in a more concrete situation. Theorem 3.4. Let X be a Banach space, Y a normed space, and Λ a finite-dimensional space. Let J : X → Y be a linear homeomorphism and G : X → Y a compact continuous map such that the composition J −1 ◦ (−G) is odd. Let M : X × Λ → kc(Y ) be a compact upper semicontinuous map. Suppose that the set [ S := {x ∈ X : J (x) + G(x) ∈ t M(x, 0)} t∈[0,1]
is bounded. Then the set X 1 := {(x, λ) ∈ X × Λ : J (x) + G(x) ∈ M(x, λ)} has a noncompact component C which intersects X × {0}. Proof. For t ∈ [0, 1], let Φt : X × Λ → k(Y × Λ) be defined by Φt (x, λ) := (J (x) + G(x) − t M(x, λ), λ)
for (x, λ) ∈ X × Λ.
We will show that any Φt is boundedly 0-epi on X × Λ. Consider the homotopy H : [0, 1] × X × Λ → kc(Y × Λ) given by H (t, (x, λ)) := (t M(x, λ), 0). It is easily verified that H is compact and has a closed graph and so is upper semicontinuous because M is compact and has a closed graph. Note that Φt = Φ0 − H (t, ·) and the set [ {(x, λ) ∈ X × Λ : (0, 0) ∈ Φt (x, λ)} = S × {0} t∈[0,1]
is bounded. The map Φ0 is written as the sum of the linear homeomorphism J0 (x, λ) := (J (x), λ) and the compact continuous map G 0 (x, λ) := (G(x), 0). Since J0−1 ◦(−G 0 ) is odd, it follows from Proposition 3.3 that Φ0 is boundedly 0-epi on X × Λ. Hence Proposition 2.8 implies that each Φt is boundedly 0-epi on X × Λ. Now we apply Theorem 3.2 with Y1 := Y, Y2 := Λ, F1 (x, λ) := J (x) + G(x) − M(x, λ), and F2 (x, λ) := λ. Note that X 1 = {(x, λ) ∈ X × Λ : 0 ∈ F1 (x, λ)} X 2 = {(x, λ) ∈ X × Λ : F2 (x, λ) = 0} = X × {0}, and so X 1 ∩ X 2 ⊆ S × {0}. Since M is compact and has a closed graph and J and G are continuous, the set X 1 is closed in X × Λ. Since X 2 is closed in X × Λ, the set X 1 ∩ X 2 is closed in X × Λ and bounded. The intersection of X 1 with any closed ball B is compact. In fact, denoting the projection of B ∩ X 1 to X and Λ by B X and BΛ , respectively, we have J (B X ) ⊆ M(B ∩ X 1 ) − G(B X ). Since G and M are compact and J −1 is continuous, the set J (B X ) is relatively compact and hence B X is contained in the compact set J −1 (J (B X )). Since B ∩ X 1 ⊆ B X × BΛ and Λ has a finite dimension, we conclude that the closed set B ∩ X 1 is compact because it is contained in the Cartesian product of two relatively compact sets. From this fact, it follows that X 1 is locally compact. Since the bounded set X 1 ∩ X 2 is contained in some closed ball B and X 1 ∩ B is compact, the closed set X 1 ∩ X 2 is compact. Moreover, for any bounded set B ⊆ X × Λ, the set Y2 \ F2 (X 1 ∩ B) = Λ \ {λ : (x, λ) ∈ X 1 ∩ B} is not empty. Since all the conditions in Theorem 3.2 are satisfied, there is a noncompact connected set C ⊆ X 1 which intersects X 2 . This completes the proof. As an immediate consequence of Theorem 3.4, we have the single-valued case in [9, Corollary 22]. Corollary 3.5. Let X, Y, Λ, J, G be as in Theorem 3.4. Let M : X × Λ → Y be a compact continuous single-valued map. Suppose that the set [ S= {x ∈ X : J (x) + G(x) = t M(x, 0)} t∈[0,1]
is bounded. Then the set X 1 = {(x, λ) ∈ X × Λ : J (x) + G(x) = M(x, λ)} has a noncompact component C which intersects X × {0}.
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Acknowledgment I would like to thank Professor Heinrich Steinlein for staying at the Ludwig-Maximilians Universit¨at M¨unchen on my sabbatical leave during the period 7 July 2005 – 30 July 2006. References [1] J. Appell, M. V¨ath, A. Vignoli, F -epi maps, Topol. Methods Nonlinear Anal. 18 (2001) 373–393. [2] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. [3] M. Furi, M. Martelli, A. Vignoli, On the solvability of nonlinear operator equations in normed spaces, Ann. Mat. Pura Appl. 124 (1980) 321–343. [4] M. Furi, M.P. Pera, On the existence of an unbounded connected set of solutions for nonlinear equations in Banach spaces, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 67 (1979) 31–38. [5] D.H. Hyers, G. Isac, T.M. Rassias, Topics in Nonlinear Analysis and Applications, World Scientific, Singapore, 1997. [6] K. Kuratowski, Topology, vol. II, Academic Press, New York, London, 1968. [7] T.-W. Ma, Topological degrees of set-valued compact fields in locally convex spaces, Diss. Math. 92 (1972). [8] P.H. Rabinowitz, A note on a nonlinear elliptic equation, Indiana Univ. Math. J. 22 (1) (1972) 43–49. [9] M. V¨ath, Global bifurcation of the p-Laplacian and related operators, J. Differential Equations 213 (2005) 389–409. [10] M. V¨ath, Global solution branches and a topological implicit function theorem, Ann. Mat. Pura Appl. (2006) (in press).
On the noncompact component of solutions for nonlinear inclusions In-Sook Kim 1 Department of Mathematics and Institute of Basic Science, Sungkyunkwan University, Suwon 440-746, Republic of Korea Received 19 May 2006; received in revised form 18 July 2006; accepted 28 July 2006
Abstract We give a bifurcation theorem for boundedly 0-epi multi-valued maps and then show the existence of a noncompact component of solutions for nonlinear inclusions. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Solutions; Noncompact component; Inclusion; Boundedly 0-epi maps, degree theory
1. Introduction Furi and Pera [4] showed the existence of an unbounded connected set of solutions for nonlinear equations in Banach spaces, where the proof is mainly based on the theory of 0-epi maps introduced in [3]. For a survey, we refer to [5]. V¨ath [9] used a variant of this result in a more general setting to obtain a global bifurcation theorem of Rabinowitz type [8] as follows: under certain conditions, a parameterized equation F(u, λ) = 0 has an unbounded connected set of solutions (u, λ) which contains a point of the form (u, 0). The aim of this paper is to prove the existence of a noncompact connected set of solutions for nonlinear inclusions of the form J (x) + G(x) ∈ M(x, λ) where X is a Banach space, Y is a normed space, Λ is a finite-dimensional space, J : X → Y is a linear homeomorphism, G : X → Y is a compact continuous map, and M : X × Λ ( Y is a compact upper semicontinuous multi-valued map with nonempty closed convex values. To this end, we introduce the notion of boundedly 0-epi multivalued maps and give a bifurcation theorem for such maps, where the method is to use a result on the noncompactness of solution branch stated in [10]. Moreover, we apply Borsuk’s theorem to find conditions sufficient for a multi-valued map to be boundedly 0-epi, where the degree theory of Ma [7] is considered. Consequently, the above main result is obtained under these observations.
E-mail address: [email protected]. 1 Current address: Mathematisches Institut, Ludwig-Maximilians Universit¨at M¨unchen, Theresienstr. 39, D-80333 M¨unchen, Germany. c 2006 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2006.07.022
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2. Boundedly 0-epi maps Let X and Y be two normed spaces and Ω a nonempty subset of X . We denote by k(Y ) the collection of all nonempty closed subsets of Y and by kc(Y ) the collection of all nonempty closed convex subsets of Y , respectively. We first introduce a notion of 0-epi multi-valued maps. A concept of F-epi single-valued maps has been developed in [1], where each ϕ ∈ F is a multi-valued map. Definition 2.1. A multi-valued map F : Ω ( Y is said to be 0-epi on Ω provided that it is upper semicontinuous on Ω and 0 6∈ F(x) for all x ∈ ∂Ω and for every upper semicontinuous map ϕ : Ω → kc(Y ) such that the range ϕ(Ω ) has a compact closure and ϕ|∂ Ω = 0, the coincidence problem (F, ϕ) has a solution in Ω . This means that there exists an x ∈ Ω such that F(x) ∩ ϕ(x) 6= ∅, or denoted by CoinΩ (F, ϕ) := {x ∈ Ω : F(x) ∩ ϕ(x) 6= ∅} 6= ∅. We begin with two basic properties of 0-epi multi-valued maps. Proposition 2.2 (Restriction). Let Ω0 ⊆ Ω . If F : Ω ( Y is 0-epi on Ω and 0 6∈ F(x) outside the interior of Ω0 , then F is 0-epi on Ω0 . Proof. The proof is analogous to that of Proposition 2.7.
Proposition 2.3 (Homotopy Invariance). Let F : Ω → k(Y ) be a 0-epi map on Ω and H : [0, 1] × Ω → kc(Y ) an upper semicontinuous map such that the range of H has a compact closure and H (0, ·) = 0. For each t ∈ [0, 1], let G t : Ω → k(Y ) be defined by G t (x) := F(x) − H (t, x) for x ∈ Ω . If 0 6∈ G t (x) for all t ∈ [0, 1] and all x ∈ ∂Ω , then each G t is 0-epi on Ω . Proof. It suffices to prove that G 1 is 0-epi on Ω . Let ϕ : Ω → kc(Y ) be an upper semicontinuous map such that the range ϕ(Ω ) has a compact closure and ϕ|∂ Ω = 0. Let S := {x ∈ Ω : G t (x) ∩ ϕ(x) 6= ∅ for some t ∈ [0, 1]}. Then S is nonempty and closed. Indeed, let (xn ) be a sequence in S and x ∈ Ω such that xn → x as n → ∞. For each n, there are tn ∈ [0, 1] and z n , wn ∈ Y such that z n ∈ ϕ(xn ), wn ∈ H (tn , xn ) and z n + wn ∈ F(xn ). Since [0, 1] is compact and the ranges of ϕ and H have a compact closure, respectively, we may assume without loss of generality that tn → t, z n → z, and wn → w as n → ∞ for some t ∈ [0, 1] and some z, w ∈ Y . Since the maps ϕ, H , and F have a closed graph, respectively, we obtain z ∈ ϕ(x), w ∈ H (t, x), and z + w ∈ F(x) and hence z ∈ ϕ(x) ∩ G t (x) and therefore x ∈ S. We have shown that S is closed in X . Since assumptions on G t and ϕ imply that S ∩ ∂Ω = ∅, there is a continuous function α : Ω → [0, 1] such that α = 0 on ∂Ω and α = 1 on S. Consider a map ψ : Ω → kc(Y ) given by ψ(x) := H (α(x), x) + ϕ(x) for x ∈ Ω . Then ψ is upper semicontinuous on Ω and its range has a compact closure, and ψ|∂ Ω = 0. Since F is 0-epi on Ω , by Definition 2.1, there exists an x ∈ Ω such that F(x) ∩ ψ(x) 6= ∅, that is, G α(x) (x) ∩ ϕ(x) 6= ∅. This implies that x ∈ S and so α(x) = 1. We conclude that G 1 is 0-epi on Ω . In the proof of Theorem 2.5, we make use of the following classical lemma from [6]. Lemma 2.4. Let A and B be two disjoint closed subsets of a compact Hausdorff space X . Then either there is a component of X which intersects A and B, or X divides into two disjoint closed sets which contain A and B, respectively. We give a bifurcation result for 0-epi multi-valued maps. In the single-valued case, a topological bifurcation result in a more general setting is studied in [9, Theorem 9]. Theorem 2.5. Let X, Y1 , and Y2 be three normed spaces and Ω a nonempty subset of X . Suppose that F = (F1 , F2 ) : Ω → k(Y1 × Y2 ) is 0-epi on Ω and X i := {x ∈ Ω : 0 ∈ Fi (x)} is a subset of Ω for i = 1, 2 such that the following conditions are satisfied: (1) X 1 is compact. (2) There exists a point of Y2 that does not belong to F2 (X 1 ). Then X 1 contains a component which intersects the sets X 2 and ∂Ω .
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Proof. Assume that X 1 has no component which intersects X 2 and ∂Ω . Notice that the set X 1 ∩ X 2 is closed in X because F has a closed graph. By Lemma 2.4, the compact Hausdorff space X 1 divides into two disjoint closed sets (in X 1 and thus in X ) which contain X 1 ∩ X 2 and X 1 ∩ ∂Ω , respectively. Hence there exist two disjoint open sets Ω1 and Ω2 in X such that X 1 ∩ X 2 ⊆ Ω1 , X 1 ∩ ∂Ω ⊆ Ω2 , and X 1 ⊆ Ω1 ∪ Ω2 . Put Ω0 := Ω1 ∩ Ω . Since ∂Ω0 ⊆ ∂Ω1 ∪ ∂Ω and since the sets Ω1 and Ω2 are open and disjoint, we have X 1 ∩ ∂Ω0 ⊆ [(X 1 ∩ ∂Ω1 ) ∪ (X 1 ∩ ∂Ω )] ∩ Ω 1 ⊆ Ω2 ∩ Ω 1 = ∅. Choose a point y2 ∈ Y2 such that y2 6∈ F2 (X 1 ). Consider a homotopy H : [0, 1] × Ω 0 → Y1 × Y2 defined by H (t, x) := (0, t y2 ) for (t, x) ∈ [0, 1] × Ω 0 . Then H is a continuous single-valued map and its range is compact. For each t ∈ [0, 1], let G t : Ω 0 → k(Y1 × Y2 ) be given by G t (x) = F(x) − H (t, x) = {(z 1 , z 2 ) ∈ Y1 × Y2 : z 1 ∈ F1 (x), z 2 + t y2 ∈ F2 (x)}. For all t ∈ [0, 1] and all x ∈ ∂Ω0 , it follows from x 6∈ X 1 that (0, 0) 6∈ G t (x). Since {x ∈ Ω : 0 ∈ F(x)} = X 1 ∩ X 2 ⊆ Ω0 , Proposition 2.2 implies that F is 0-epi on Ω0 . By Proposition 2.3, G 1 is 0-epi on Ω0 . Since CoinΩ0 (G 1 , 0) 6= ∅, there exists an x ∈ Ω0 such that 0 ∈ F1 (x) (and thus x ∈ X 1 ) and y2 ∈ F2 (x), which contradicts the choice of y2 . This completes the proof. To achieve bifurcation for nonlinear inclusions, another notion is necessary. Definition 2.6. An upper semicontinuous multi-valued map ϕ : Ω ( Y is said to be compact if the image ϕ(B) has a compact closure for any bounded set B ⊆ Ω . A multi-valued map F : Ω ( Y is said to be boundedly 0-epi on Ω provided that it is upper semicontinuous on Ω and 0 6∈ F(x) for all x ∈ ∂Ω and for every compact upper semicontinuous map ϕ : Ω → kc(Y ) such that ϕ(x) = {0} for all x ∈ ∂Ω and for all x outside some bounded set, the coincidence problem (F, ϕ) has a solution in Ω . The class of boundedly 0-epi multi-valued maps has the usual properties as follows: Proposition 2.7 (Restriction). Let Ω0 ⊆ Ω . If F : Ω ( Y is boundedly 0-epi on Ω and 0 6∈ F(x) outside the interior of Ω0 , then F is boundedly 0-epi on Ω0 . In particular, if Ω0 is bounded, then F is 0-epi on Ω0 . Proof. Let ϕ : Ω 0 → all x ∈ Ω 0 \ B, where ϕ(x) ϕ(x) ˜ := {0}
kc(Y ) be a compact upper semicontinuous map such that ϕ(x) = {0} for all x ∈ ∂Ω0 and for B ⊆ X is some bounded set. Define a map ϕ˜ : Ω → kc(Y ) by for x ∈ Ω 0 for x 6∈ Ω0 .
Then ϕ˜ is upper semicontinuous and compact, and ϕ(x) ˜ = {0} for all x ∈ ∂Ω and for all x ∈ Ω \ B. Since F is ˜ 0 ) 6= ∅. From 0 6∈ F(x) and boundedly 0-epi on Ω , by Definition 2.6, there exists an x0 ∈ Ω such that F(x0 ) ∩ ϕ(x ϕ(x) ˜ = {0} for all x ∈ Ω \ Ω0 it follows that x0 belongs to Ω0 and so F(x0 ) ∩ ϕ(x0 ) 6= ∅. Thus, F is boundedly 0-epi on Ω0 . Proposition 2.8 (Homotopy Invariance). Let F : Ω → k(Y ) be a boundedly 0-epi map on Ω and H : [0, 1] × Ω → kc(Y ) an upper semicontinuous map such that H (0, ·) = 0 and H ([0, 1] × B) has a compact closure for any bounded set B ⊆ Ω . For each t ∈ [0, 1], let G t : Ω → k(Y ) be defined by G t (x) := F(x) − H (t, x) for x ∈ Ω . If 0 6∈ G t (x) for all t ∈ [0, 1], all x ∈ ∂Ω and all x outside some bounded set which does not depend on t, then each G t is boundedly 0-epi on Ω . Proof. Let ϕ : Ω → kc(Y ) be a compact upper semicontinuous map such that ϕ(x) = {0} for all x ∈ ∂Ω and for all x ∈ Ω \ B, where B is some bounded set in X . We may assume that B is open in X and 0 6∈ G t (x) for all t ∈ [0, 1] and all x ∈ ∂Ω ∪ (Ω \ B), by enlarging B if necessary. In particular, 0 6∈ G t (x) for all t ∈ [0, 1] and all x ∈ ∂(B ∩ Ω ). Since 0 6∈ F(x) outside the interior of B ∩ Ω , the restriction F| B∩Ω is 0-epi on B ∩ Ω , by Proposition 2.7. Since H ([0, 1] × (B ∩ Ω )) has a compact closure, Proposition 2.3 implies that each G t | B∩Ω is 0-epi on B ∩ Ω . Since ϕ(B ∩ Ω ) has a compact closure and ϕ|∂(B∩Ω ) = 0, there is an x ∈ B ∩ Ω such that G t (x) ∩ ϕ(x) 6= ∅. Therefore, each G t is boundedly 0-epi on Ω .
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3. Main results The following crucial criterion is a key tool in finding a noncompact solution branch; see [10, Theorem 2.2]. Lemma 3.1. Let X be a regular space, A ⊆ X compact, and S ⊆ X such that S ∩ A is closed. Suppose that S is locally compact and that there exists an open (possibly empty) set U0 ⊇ (S \ S) ∩ A such that U 0 ∩ S is compact. Then, for each open set U ⊇ A ∪ U 0 , the following statements are equivalent: (1) For each open set O ⊇ A with O ⊆ U and compact O ∩ S, there is an x ∈ ∂ O in S. (2) There is a connected closed (in S) subset C ⊆ S ∩ U which intersects A and is either noncompact or intersects ∂U . (3) There is a connected set C ⊆ (S ∩ U ) \ A such that C ∩ S intersects A and is either noncompact or intersects ∂U . We can prove the following bifurcation result for boundedly 0-epi multi-valued maps. The single-valued case is considered in [9, Theorem 18], where the method of approach is different from ours. Theorem 3.2. Let X , Y1 and Y2 be three normed spaces and Ω a nonempty subset of X . Suppose that F = (F1 , F2 ) : Ω → k(Y1 × Y2 ) is boundedly 0-epi on Ω and X i := {x ∈ Ω : 0 ∈ Fi (x)} is a subset of Ω for i = 1, 2 such that the following conditions are satisfied: (1) X 1 is locally compact. (2) X 1 ∩ X 2 is compact. (3) For each bounded set B ⊆ Ω , there exists a point of Y2 that does not belong to F2 (X 1 ∩ B). Then there is a connected set C ⊆ X 1 which intersects X 2 and is either noncompact or intersects ∂Ω . Proof. Let A = X 1 ∩ X 2 and S = X 1 , then we may take U0 = ∅, where U0 is given as in Lemma 3.1. We may assume that Ω is an open set in X . Let O be any open set in X such that A ⊆ O ⊆ O ⊆ Ω , and O ∩ S is compact. Let K := O ∩ S. Since K is compact and S is locally compact, there is a bounded open set Ω0 in X such that K ⊆ Ω0 ⊆ Ω 0 ⊆ Ω and Ω 0 ∩ S is compact. Since 0 6∈ F(x) outside Ω0 , the map F is 0-epi on Ω0 , by Proposition 2.7. Note that there exists a point y ∈ Y2 that does not belong to F2 (S ∩ Ω 0 ), by assumption (3). Applying Theorem 2.5 with F |Ω 0 , the compact set S ∩ Ω 0 has a component which intersects X 2 and ∂Ω0 ; hence there is an x ∈ ∂ O in S. By Lemma 3.1, there is a connected set C ⊆ (S ∩ Ω ) which intersects A and is either noncompact or intersects ∂Ω .
Based on Borsuk’s theorem, we observe when a multi-valued map F is boundedly 0-epi. Proposition 3.3. Let X be a Banach space and Y a normed space. Let J : X → Y be a linear homeomorphism and G : X → Y a compact continuous map, and F := J + G a map such that F(x) 6= 0 for all x outside some bounded set. If the map ψ := J −1 ◦ (−G) : X → X is odd, then F is boundedly 0-epi on X . Proof. Let ϕ : X → kc(Y ) be a compact upper semicontinuous map such that there is a bounded set B ⊆ X with ϕ(x) = {0} for all x ∈ X \ B. We may assume that B is an open ball with center at 0 and F(x) 6= 0 for all x ∈ X \ B, by enlarging B if necessary. Then the coincidence set Coin X (F, ϕ) = {x ∈ X : F(x) ∈ ϕ(x)} is contained in B. Let ψ1 : B → kc(X ) be a map given by ψ1 (x) := J −1 ◦ (ϕ − G)(x). Then the sets ψ1 (B) and ψ(B) are relatively compact because ϕ and G are compact on X and J −1 is continuous on Y . Consider a map H : [0, 1] × B → kc(Y ) defined by H (t, x) := J −1 ◦ (tϕ − G)(x)
for (t, x) ∈ [0, 1] × B.
Then the range of H has a compact closure and x 6∈ H (t, x) for all (t, x) ∈ [0, 1] × ∂ B, because ϕ(x) = {0} and F(x) 6= 0 for all x ∈ ∂ B. The homotopy invariance of Ma’s degree [7] for compact multi-valued maps implies that Deg (I − ψ1 , B, 0) = Deg (I − ψ, B, 0), where I denotes the identity map of X . Since the latter has odd degree by Borsuk’s theorem, see e.g. [2, Theorem 8.3], we obtain that ψ1 has a fixed point in B. Since the fixed point of ψ1 belongs to the set Coin X (F, ϕ), we conclude that F is boundedly 0-epi on X .
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Now we give the following global bifurcation result in a more concrete situation. Theorem 3.4. Let X be a Banach space, Y a normed space, and Λ a finite-dimensional space. Let J : X → Y be a linear homeomorphism and G : X → Y a compact continuous map such that the composition J −1 ◦ (−G) is odd. Let M : X × Λ → kc(Y ) be a compact upper semicontinuous map. Suppose that the set [ S := {x ∈ X : J (x) + G(x) ∈ t M(x, 0)} t∈[0,1]
is bounded. Then the set X 1 := {(x, λ) ∈ X × Λ : J (x) + G(x) ∈ M(x, λ)} has a noncompact component C which intersects X × {0}. Proof. For t ∈ [0, 1], let Φt : X × Λ → k(Y × Λ) be defined by Φt (x, λ) := (J (x) + G(x) − t M(x, λ), λ)
for (x, λ) ∈ X × Λ.
We will show that any Φt is boundedly 0-epi on X × Λ. Consider the homotopy H : [0, 1] × X × Λ → kc(Y × Λ) given by H (t, (x, λ)) := (t M(x, λ), 0). It is easily verified that H is compact and has a closed graph and so is upper semicontinuous because M is compact and has a closed graph. Note that Φt = Φ0 − H (t, ·) and the set [ {(x, λ) ∈ X × Λ : (0, 0) ∈ Φt (x, λ)} = S × {0} t∈[0,1]
is bounded. The map Φ0 is written as the sum of the linear homeomorphism J0 (x, λ) := (J (x), λ) and the compact continuous map G 0 (x, λ) := (G(x), 0). Since J0−1 ◦(−G 0 ) is odd, it follows from Proposition 3.3 that Φ0 is boundedly 0-epi on X × Λ. Hence Proposition 2.8 implies that each Φt is boundedly 0-epi on X × Λ. Now we apply Theorem 3.2 with Y1 := Y, Y2 := Λ, F1 (x, λ) := J (x) + G(x) − M(x, λ), and F2 (x, λ) := λ. Note that X 1 = {(x, λ) ∈ X × Λ : 0 ∈ F1 (x, λ)} X 2 = {(x, λ) ∈ X × Λ : F2 (x, λ) = 0} = X × {0}, and so X 1 ∩ X 2 ⊆ S × {0}. Since M is compact and has a closed graph and J and G are continuous, the set X 1 is closed in X × Λ. Since X 2 is closed in X × Λ, the set X 1 ∩ X 2 is closed in X × Λ and bounded. The intersection of X 1 with any closed ball B is compact. In fact, denoting the projection of B ∩ X 1 to X and Λ by B X and BΛ , respectively, we have J (B X ) ⊆ M(B ∩ X 1 ) − G(B X ). Since G and M are compact and J −1 is continuous, the set J (B X ) is relatively compact and hence B X is contained in the compact set J −1 (J (B X )). Since B ∩ X 1 ⊆ B X × BΛ and Λ has a finite dimension, we conclude that the closed set B ∩ X 1 is compact because it is contained in the Cartesian product of two relatively compact sets. From this fact, it follows that X 1 is locally compact. Since the bounded set X 1 ∩ X 2 is contained in some closed ball B and X 1 ∩ B is compact, the closed set X 1 ∩ X 2 is compact. Moreover, for any bounded set B ⊆ X × Λ, the set Y2 \ F2 (X 1 ∩ B) = Λ \ {λ : (x, λ) ∈ X 1 ∩ B} is not empty. Since all the conditions in Theorem 3.2 are satisfied, there is a noncompact connected set C ⊆ X 1 which intersects X 2 . This completes the proof. As an immediate consequence of Theorem 3.4, we have the single-valued case in [9, Corollary 22]. Corollary 3.5. Let X, Y, Λ, J, G be as in Theorem 3.4. Let M : X × Λ → Y be a compact continuous single-valued map. Suppose that the set [ S= {x ∈ X : J (x) + G(x) = t M(x, 0)} t∈[0,1]
is bounded. Then the set X 1 = {(x, λ) ∈ X × Λ : J (x) + G(x) = M(x, λ)} has a noncompact component C which intersects X × {0}.
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Acknowledgment I would like to thank Professor Heinrich Steinlein for staying at the Ludwig-Maximilians Universit¨at M¨unchen on my sabbatical leave during the period 7 July 2005 – 30 July 2006. References [1] J. Appell, M. V¨ath, A. Vignoli, F -epi maps, Topol. Methods Nonlinear Anal. 18 (2001) 373–393. [2] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. [3] M. Furi, M. Martelli, A. Vignoli, On the solvability of nonlinear operator equations in normed spaces, Ann. Mat. Pura Appl. 124 (1980) 321–343. [4] M. Furi, M.P. Pera, On the existence of an unbounded connected set of solutions for nonlinear equations in Banach spaces, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 67 (1979) 31–38. [5] D.H. Hyers, G. Isac, T.M. Rassias, Topics in Nonlinear Analysis and Applications, World Scientific, Singapore, 1997. [6] K. Kuratowski, Topology, vol. II, Academic Press, New York, London, 1968. [7] T.-W. Ma, Topological degrees of set-valued compact fields in locally convex spaces, Diss. Math. 92 (1972). [8] P.H. Rabinowitz, A note on a nonlinear elliptic equation, Indiana Univ. Math. J. 22 (1) (1972) 43–49. [9] M. V¨ath, Global bifurcation of the p-Laplacian and related operators, J. Differential Equations 213 (2005) 389–409. [10] M. V¨ath, Global solution branches and a topological implicit function theorem, Ann. Mat. Pura Appl. (2006) (in press).