Bifurcation points and asymptotic bifurcation points of nonlinear operators in M-PN spaces

Nonlinear Analysis 71 (2009) 4960–4966

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Bifurcation points and asymptotic bifurcation points of nonlinear operators in M-PN spaces Qiuying Li ∗ , Chuanxi Zhu, Sanhua Wang School of Science, Nanchang University, Nanchang 330031, PR China

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Article history: Received 29 September 2007 Accepted 23 March 2009 MSC: 60A99 47H99 34K18

abstract In the paper, the new concepts of bifurcation points and asymptotic bifurcation points of the compact continuous operator T are introduced in M-PN spaces. Some sufficient conditions for the existence of bifurcation points and asymptotic bifurcation points are obtained, and some theorems on the existence of intrinsic values are obtained. Meanwhile some theorems are generalized. © 2009 Elsevier Ltd. All rights reserved.

Keywords: M-PN space Bifurcation point Asymptotic bifurcation point Intrinsic value Topological degree

1. Introduction Deciding the bifurcation point of a nonlinear mapping is one of the basic problems of the bifurcation theory. But up to now most existence theorems about the bifurcation point are obtained in Banach space [1–4]. In this paper, the new concepts of bifurcation points and asymptotic bifurcation points are given in the more general space—M-PN space, and some sufficient conditions for the existence of bifurcation points and asymptotic bifurcation points are studied. Let us recall some concepts of the theory of probabilistic metric spaces. Let R be the set of all real numbers, R+ be the set of all nonnegative numbers, ∆ be the t-norm. A mapping f : R → R+ is called a distribution function if it is a nondecreasing function and has left continuation satisfying the following conditions: supt ∈R f (t ) = 1, inft ∈R f (t ) = 0. D is the set of all distribution functions. A Menger probabilistic metric linear normed space (Menger PN space, or M-PN space) is a triple (E , F , ∆) where E is a real space and F : E → D is a mapping (we shall denote the distribution function F (x) by fx ) satisfying the following conditions:

(PN (PN (PN (PN

− 1) fx (0) = 0; − 2) fx (t ) = 1 for all t > 0 if and only if x = θ ; − 3) for all real number α 6= 0, fαx (t ) = fx (t / |α|) ; − 4) if fx (t1 ) = 1, fy (t2 ) = 1, then fx+y (t1 + t2 ) = 1;
(PN − 5) fx+y (t1 + t2 ) ≥ ∆ fx (t1 ) , fy (t2 ) for all x, y ∈ E and t1 , t2 ∈ R+ .

∗

Corresponding author. E-mail addresses: [email protected] (Q. Li), [email protected] (C. Zhu).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.03.068

Q. Li et al. / Nonlinear Analysis 71 (2009) 4960–4966

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Definition 1 ([5]). Let (E , F , ∆) be a M-PN space. W is a subset of E with θ ∈ W . Let T : W → E be a continuous compact operator satisfying T θ = θ . x0 ∈ W and x0 6= θ , λ ∈ R is called the intrinsic value of T in W and x0 is the intrinsic element corresponding with λ, if fTx0 (s) = fλx0 (s) for any s > 0. Definition 2. Let (E , F , ∆) be a M-PN space, Ω is an open subset of E. Suppose that T : Ω → E be a continuous compact operator satisfying T θ = θ . µ0 ∈ R is called the bifurcation point of T , if for every ε > 0, there exist µ ∈ R and x ∈ E such that fx (s) = fµTx (s), ∀s > 0, |µ − µ0 | < ε, kxk < ε . Definition 3. Let (E , F , ∆) be a M-PN space. Suppose that T : Ω → E be a continuous compact operator satisfying T θ = θ . λ0 ∈ R is called the asymptotic bifurcation point of T , if for every ε > 0, there exists λ ∈ R and x ∈ E such that fx (s) = fλTx (s), ∀s > 0, |λ − λ0 | < ε, kxk > 1/ε. For some references on related work, see [8–14]. 2. Main results To complete the proofs of main results, we need the following lemmas. Lemma 1 ([6]). If (E , F ) is a Menger PM-space and ∆ is continuous, then the probabilistic distribution function, F , is a lower semi-continuous function of points, i.e. for every fixed x, if qn → q and pn → p, then lim inf Fpn ,qn (x) = Fp,q (x).

n→∞

Lemma 2 ([6]). Let Ω be an open subset of an infinite dimension M-PN space (E , F , ∆) with ∆ (t , t ) ≥ t for all t ∈ [0, 1]. Suppose that T : Ω → E is a continuous compact mapping satisfying the following conditions: (i) θ 6∈ T (∂ Ω ); (ii) Tx = 6 µx for all µ ∈ [0, 1] and x ∈ ∂ Ω . Then Deg (I − T , Ω , θ) = 0. Theorem 1. Let (E , F , ∆) be an infinite dimension M-PN space, where ∆ is continuous. Let T : E → E be a continuous compact mapping satisfying T θ = θ . Suppose that there exist r0 ∈ R+ \ {0} and α, β ∈ (0, 1), α < β such that fα x (s) ≥ fTx (s) ≥ fβ x (s),

for all s > 0, and kxk < r0 .

(1)

Then T has a bifurcation point in [1/β, 1/α ], but has no bifurcation points in (0, 1/β) ∪ (1/α, +∞). Proof. First, we prove that T has no bifurcation points in (0, 1/β) ∪ (1/α, +∞). Suppose 0 < λ < 1/β , kxk < r0 such that fx (s) = fλTx (s) for all s > 0. Then it follows that fx (s) = fλTx (s) = fTx (s/λ) ≥ fβ x (s/λ) = fx (s/βλ)

for all s > 0.

By virtue of the character of the distribution function, we have s/βλ ≤ s. Since s > 0, it follows that 1/βλ ≤ 1, i.e.

λ ≥ 1/β, which contradicts with 0 < λ < 1/β . Therefore, T has no bifurcation points in (0, 1/β). As the discussion above, T has no bifurcation points in (1/α, +∞). At last, we will prove that T has a bifurcation point in [1/β, 1/α ]. For any n ∈ N, n > max {1/β, 1/r0 }, let Hn (t , x) = x − t where B θ ,

1 n






1

β

−

1 n





Tx for all x ∈ B θ ,

1 n



and t ∈ [0, 1] ,


= x ∈ E : kxk ≤ 1n . Then Hn (t , x) 6= θ for all x ∈ ∂ B θ , 1n and t∈ [0, 1].

Suppose that it is not the case, then there exist n0 > max {1/β, 1/r0 }, x0 ∈ ∂ B θ , Hn0 (t0 , x0 ) = θ .

1 n0

and t0 ∈ [0, 1] such that

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It follows that x0 − t0



1

x0 = t0

1

−

β



1

−

β



n0

1 n0



Tx0 = θ , i.e.

Tx0 .

It is obvious that t0 6= 0. So we have fx0 (s) = f  1

(s) for all s > 0,   which implies that t0 β1 − n1 is a bifurcation point of T . But t0 β1 − 0 bifurcation points in (0, 1/β). 

t0 β − n1 Tx0 0



1 n0



∈ (0, 1/β), which contradicts that T has no

By the homotopy invariance and the normality of the topological degree in [7], we obtain that



 I−

Deg

1

β

−

1



n



T, B θ,

1



n

,θ



    1 = Deg I , B θ , , θ = 1.

(2)

n

On the other hand, for the natural number n mentioned above, let Ω = B θ , (a) θ 6∈ T0 (∂ Ω ). Suppose that it is not the case, then there exist yn ∈ T0 (∂ Ω ) such that

1 n

, T0 = α1 +




1 n




T , we are going to prove:

yn → θ . Thus there exist {xn } ⊂ ∂ Ω such that yn = T0 xn → θ , i.e.



1

α

1

+



n

Txn → θ ,

which implies that Txn → θ . By Lemma 1, it follows that lim inf fTxn (s) = fθ (s) = 1

for any s > 0.

n→∞

Then we have fTxn (s) ≥ 1 for any s > 0 as n → ∞. By (1) we have fα xn (s) ≥ 1, which implies that α xn = θ , i.e. α = θ or xn = θ , this contradicts that α > 0 and xn ∈ ∂ Ω . (b) T0 x 6= µx for all µ ∈ [0, 1]
and x ∈ ∂ Ω . Suppose T0 x = µx, i.e. α1 + 1n Tx = µx. Then 1

Tx =

1

α

+

µx =

1 n

nα n+α

µx ,

for x ∈ ∂ Ω .

It is obvious that µ 6= 0 (otherwise Tx = θ , by (1), we have fα x (s) ≥ 1 for all s > 0, which implies α x = θ , i.e. α = 0 or x = θ , this contradicts that α > 0 and x ∈ ∂ Ω ). Further we have fTx (s) = f

nα n+α µx

(s) = fαx



n+α nµ

 s

for any s > 0.

By (1), it follows that

 fα x

n+α nµ

 s

≤ fαx (s) for any s > 0.

Since fα x ∈ D, we have

(n + α)s ≤ s. nµ Since s > 0, we get nn+α ≤ 1, which implies µ ≥ n+α > 1. µ n Notice that dim E = +∞. By Lemma 2, we have



 I−

Deg

1

α

+

1



n

T , Ω, θ



= Deg (I − T0 , Ω , θ) = 0.

By (2) and (3), it follows that

 Deg

 I−

1

β

−

1 n



T , Ω, θ



    1 1 6= Deg I − + T , Ω, θ . α n

(3)

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By the of the topological degree in [7], for any natural number n > max {1/β, 1/r0 }, there exist  homotopy invariance 

λn ∈

1

β

− 1n , α1 +

and xn ∈ ∂ Ω such that

1 n

xn = λn Txn . Therefore there exist λnk ⊂ {λn } and λ∗ ∈





λnk → λ∗ ,

xn k → θ

h

1

β

i , α1 such that

and xnk = λnk Txnk ,

Thus fxnk (s) = fλ∗ Txnk (s)

for any s > 0,

which implies that λ is the bifurcation point of T in [1/β, 1/α ]. Then we finish the proof of Theorem 1.  ∗

Remark 1. Theorem 1 is a generalization of Theorem 1 in [1]. Theorem 2. Let (E , F , ∆) be an infinite dimension M-PN space, ∆ is a continuous t-norm. Ω is an open subset of E. Let T : Ω → E is a continuous compact mapping with T θ = θ . Suppose that there exist λ0 ∈ R+ \ {0}, xn ∈ Ω and xn → θ as n → ∞ such that lim fkTxn k−λ0 kxn k (s) = H (s)

n→∞

for all s ∈ R,

Then (i) 1/λ0 is a bifurcation point of T ; (ii) For any µ ∈ R \ {0} and ε ∈ (0, |µ| λ0 ), there exists t ∈ (0, 1) such that ε/µt is the intrinsic value of T . Proof. For any xn ∈ Ω such that kxn k → θ as n → ∞. By assumptions it follows that there exists n0 ∈ N such that fkTxn k−λ0 kxn k (s) > 1 − λ

for any s > 0, λ > 0 and n > n0 .

Let λ → 0, then for any n > n0 , we have fkTxn k−λ0 kxn k (s) = 1

for any s > 0,

which implies that kTxn k = λ0 kxn k for any n > n0 , i.e.

kTxk = λ0 > 0. x∈Ω ,kxk→0 kxk

(4)

lim

For λ0 /2 > 0, there exists r0 > 0 such that for any r ∈ (0, r0 ],

λ0 −

kTxk kTxk < λ0 /2, ≤ λ0 − kx k kx k

x ∈ Ω and kxk ≤ r ,

which implies that

kTxk λ0 ≥ kxk 2

for all x ∈ Ω and kxk ≤ r .

As Ω is open and θ ∈ Ω , we can take r0 > 0 such that for any r ∈ (0, r0 ], B (θ , r ) ⊂ Ω . It is obvious that λ2 T : B (θ , r ) → E 0 is a continuous compact mapping, θ ∈ B (θ , r ), λ2 T θ = θ 6∈ ∂ B (θ , r ) and 0



2

Tx ≥ kxk

λ0

for any x ∈ B (θ , r ) , r ∈ (0, r0 ] .

(5)

Now we are going to prove that there exist tr ∈ (0, 1) and xr ∈ ∂ B (θ , r ) such that Txr =

λ0 2

xr

for any r ∈ (0, r0 ] .

By (5) we have: (a) θ 6∈ λ2 T (∂ B (θ , r )). 0 Suppose λ2 T has no fixed points on ∂ B (θ , r ) otherwise the assertion (6) holds, then 0

(6)

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(b) λ2 Tx 6= tx for all t ∈ [0, 1] and x ∈ ∂ B (θ , r ). 0 In fact, suppose that there exist t0 ∈ [0, 1] and x0 ∈ ∂ B (θ , r ) such that λ2 Tx0 = t0 x0 . It is obvious that t0 6= 1 because we 0 assumed that λ2 T has no fixed points on ∂ B (θ , r ), and it follows that 0



2

Tx0 = t0 kx0 k < kx0 k ,

λ0 which contradicts with (5). Notice that dim E = +∞. By Lemma 2, we have



2

T , B (θ , r ) , θ



= 0. (7) λ0 Let H (t , x) = x − t λ2 Tx for t ∈ [0, 1] and x ∈ B (θ , r ), then H : [0, 1] × B (θ , r ) → E is a continuous compact mapping. 0 Suppose for any t ∈ [0, 1] and x ∈ ∂ B (θ , r ), H (t , x) 6= θ , then by the homotopy invariance of the topological degree in [7], I−

Deg

we have



2

I−

Deg

λ0

T , B (θ , r ) , θ



= Deg (I − H (1, ·) , B (θ , r ) , θ) = Deg (I − H (0, ·) , B (θ , r ) , θ) = Deg (I , B (θ , r ) , θ) = 1,

which contradicts (7). So there exist tr ∈ [0, 1] and xr ∈ ∂ B (θ , r ) such that H (tr , xr ) = θ , i.e. xr = tr

2

λ0

Txr .

It is obvious that tr 6= 0 and tr 6= 1, therefore Txr =

λ0 2tr

xr ,

tr ∈ (0, 1) .

Since all the discussions above for any r ∈ (0, r0 ] satisfying Txr =

(8) λ0 2tr

xr we have

kxr k = r → 0 (as r → 0), and

λ0 2tr

=

kTxr k → λ0 (r → 0). kxr k

(9)

 λ0 1 0 r Thus 2t → (as r → 0). By (9), for any ε > 0, there exists r > 0 such that − λ < ε for any r ∈ 0, r 0 . Let 0 λ0 λ0 2tr 
rε ∈ 0, min r 0 , r0 , ε .Then by the assertion (6), there exist trε ∈ (0, 1) and xrε ∈ ∂ B (x, rε ) such that λ0

λ0 − λ0 < ε, xrε = Txrε , 0 < xrε = rε < ε, 2tr

2trε

i.e. xrε =

2trε

λ0

Txrε ,

0 < xrε < ε.



Thus fxrε (s) = f 2trε

λ0 Txrε

(s) for any s > 0,

which implies that 1/λ0 is a bifurcation point of T , i.e. the conclusion (i) holds. For any µ ∈ R \ {0} and ε ∈ (0, |µ| λ0 ), |µ| λ0 − ε > 0, by (4) it follows that

kµTxk → |µ| λ0 (as kxk → 0). kxk Then there exists rµ,ε > 0 such that

|µ| λ0 −

 kµTxk kµTxk ≤ |µ| λ0 − < |µ| λ0 − ε for any r ∈ 0, rµ,ε and kxk ≤ r , kx k kx k

Q. Li et al. / Nonlinear Analysis 71 (2009) 4960–4966

4965

hence kµTxk > ε kxk, kxk ≤ r, i.e.

µ

Tx ≥ kxk , ε

r ∈ 0, rµ,ε

kx k ≤ r




.

(10)

As we get the assertion (6) from (5), we can get the following by (10): there exist t ∈ (0, 1) and x ∈ ∂ B x, rµ,ε such that




Tx =

ε x, µt

kxk = rµ,ε .

Further we have fTx (s) = f µεt x (s)

for any s > 0,

which implies that the conclusion (ii) holds. Then we finish the proof of Theorem 2.  Remark 2. Theorem 2 is a generalization of Theorem 1 in [2]. Theorem 3. Let (E , F , ∆) be an infinite dimension M-PN space, where ∆ is continuous. Let T : E → E be a continuous compact mapping satisfying T θ = θ . Suppose that there exist λ0 > 0, xn ∈ E and kxn k → ∞ as n → ∞ such that lim fkTxn k−λ0 kxn k (s) = H (s)

n→∞

for all s ∈ R,

Then (i) 1/λ0 is an asymptotic bifurcation point of T ; (ii) For any µ ∈ R \ {0} and ε ∈ (0, |µ| λ0 ), there exists t ∈ (0, 1) such that ε/µt is an intrinsic value of T . Proof. For any xn ∈ E, kxn k → ∞ as n → ∞, by the assumption it follows that there exists n0 ∈ N such that fkTxn k−λ0 kxn k (s) > 1 − λ for any s > 0, λ > 0 and n > n0 . Let λ → 0, then for any n > n0 , we have fkTxn k−λ0 kxn k (s) = 1 for any s > 0, which implies that kTxn k = λ0 kxn k for any n > n0 , i.e. lim

x∈E ,kxk→∞

kTxk = λ0 > 0. kx k

(11)

For λ0 /2 > 0, there exists r0 > 0 such that for any r ≥ r0 ,

λ0 −

kTxk kTxk < λ0 /2, ≤ λ0 − kx k kx k

kxk ≥ r0 and x ∈ E ,

which implies that

kTxk λ0 ≥ kxk 2

for kxk ≥ r0 and x ∈ E ,

i.e.



2

Tx ≥ kxk

λ

for any x ∈ ∂ B (θ , r ) .

(12)

0

As we get the assertion (6) we can get the following from (12): there exist tr ∈ (0, 1) and xr ∈ ∂ B (θ , r ) such that Txr =

λ0 2tr

xr

(r ≥ r0 ).

Since

kxr k = r → ∞,

λ0 2trε

=

kTxr k → λ0 as r → ∞. kx r k

Then for any ε > 0, there exists r1 > 0 such that

λ0 2t − λ0 < ε as r > r1 . r  Let rε > max r0 , r1 , 1ε . Then λ0 λ0 − λ0 < ε, Txrε = xrε , 2trε

2trη



xr = rε > 1 . ε ε

(13)

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Q. Li et al. / Nonlinear Analysis 71 (2009) 4960–4966 2t

Thus λrε Txrε = xrε . By (13), we have 0 2trε

λ0

→

1

λ0

as r → ∞.

Hence fxrε (s) = f 2trε

λ0 Txrε

(s) for all s > 0,

which implies that 1/λ0 is an asymptotic bifurcation point of T , i.e. the conclusion (i) holds. The proof the conclusion (ii) is as the same as that of the conclusion in the Theorem 2 and so omitted. Then we finish the proof of Theorem 3.  Remark 3. Theorem 3 is a generalization of Theorem 2 in [2].

Acknowledgment Supported by the National Natural Science Foundation of China (No.10461007 and No. 10761007), the Natural Science Foundation of Jiangxi Province (No. 0411043) and the School Foundation (No. Z03363). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

Z. Yang, Bifurcation points of nonlinear operators, J. Eng. Math. 17 (2) (2000) 109–112 (in Chinese). C. Zhao, Ambiguous point on the completely continuous operator, J. Math. Tech. 15 (1999) 17–20 (in Chinese). C. Zhao, Ambiguous point on the condensing mapping and applications, Coll. Math. 19 (3) (2003) 73–76 (in Chinese). Z. Yang, J. Sun, Asymptotic bifurcation points of nonlinear operators, J. Systems Sci. Math. Sci. 20 (1) (2000) 47–54 (in Chinese). C. Zhu, T. Zhu, Some problems of probabilistic analysis in the Z-P-S spaces, J. Systems Sci. Math. Sci. 26 (1) (2006) 1–4 (in Chinese). S. Chang, Y. Cho, S. Kang, Probabilistic Metric Spaces and Nonlinear Operator Theory, Sichuan University Press, Chengdu, 1994. S. Chang, Y. Chen, Topological degree theory and fixed point theorems in probabilistic metric spaces, Appl. Math. Mech. 10 (6) (1989) 495–505. C. Zhu, Some new fixed point theorems in probabilistic metric spaces, Appl. Math. Mech. 16 (2) (1995) 179–185. C. Zhu, Some theorems in the X-M-PN space, Appl. Math. Mech. 21 (2) (2000) 181–184. C. Zhu, Some problems in the Z-C-X Space, Appl. Math. Mech. 23 (8) (2002) 942–947. C. Zhu, Several nonlinear operator problems in the Menger PN space, Nonlinear Anal. 65 (2006) 1281–1284. C. Zhu, Generalizations of Krasnoselskii’s theorem and Petryshyn’s theorem, Appl. Math. Lett. 19 (2006) 628–632. Q. Li, C. Zhu, Solution of nonlinear operator equation in Z-P-S spaces, J. Systems Sci. Math. Sci. 28 (4) (2008) 400–405 (in Chinese). C. Zhu, A class of random operator equations in the Hilbert space, Acta Math. Sin. 47 (4) (2004) 641–646.