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Two-parameter nonresonance condition for the existence of fourth-order boundary value problems
J. Math. Anal. Appl. 308 (2005) 121–128 www.elsevier.com/locate/jmaa
Two-parameter nonresonance condition for the existence of fourth-order boundary value problems ✩ Yongxiang Li Department of Mathematics, Northwest Normal University, Lanzhou 730070, People’s Republic of China Received 12 August 2004 Available online 23 February 2005 Submitted by D. O’Regan
Abstract In this paper, we discuss the existence of the fourth-order boundary value problem u(4) = f (t, u, u ), 0 < t < 1, u(0) = u(1) = u (0) = u (1) = 0, where f : [0, 1] × R × R → R is continuous, and partly solve the Del Pino and Manasevich’s conjecture on the nonresonance condition involving the two-parameter linear eigenvalue problem. We also present a two-parameter nonresonance condition described by circle. 2004 Elsevier Inc. All rights reserved. Keywords: Existence; Nonresonance condition; Two-parameter linear eigenvalue problem; Equivalent norm
1. Introduction and main results In this paper we deal with the existence of the fourth-order ordinary differential equation boundary value problem (BVP) u(4) (t) = f (t, u(t), u (t)), 0 < t < 1, (1) u(0) = u(1) = u (0) = u (1) = 0, ✩
Research supported by NNSF of China (10271095) and the NSF of Gansu Province (ZS031-A25-003-Z). E-mail address: [email protected].
0022-247X/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2004.11.021
122
Y. Li / J. Math. Anal. Appl. 308 (2005) 121–128
where f : [0, 1] × R × R → R is continuous. This problem models deformations of an elastic beam in equilibrium state, whose two ends are simply supported. Owing to its importance in physics, the existence of this problem has been studied by many authors, see [1–12]. In [1], Aftabizadeh showed the existence of a solution to PBV (1) under the restriction that f is a bounded function. In [2, Theorem 1], Yang extended Aftabizadeh’s result and showed the existence for BVP (1) under the growth condition of the form f (t, u, v) a|u| + b|v| + c, (2) where a, b and c are positive constants such that πa4 + πb2 < 1. In [6], Del Pino and Manasevich further extended the result of Yang, and obtained the following existence theorem: Theorem A. Assume that the pair (α, β) satisfies β α + = 1, (kπ)4 (kπ)2
∀k ∈ N,
(3)
and that there are positive constants a, b and c such that a max k∈N
1 k2π 2 + b max < 1, |k 4 π 4 − α − k 2 π 2 β| k∈N |k 4 π 4 − α − k 2 π 2 β|
and f satisfies the growth condition f (t, u, v) − (αu − βv) a|u| + b|v| + c,
∀t ∈ [0, 1], u, v ∈ R.
(4)
(5)
Then the BVP (1) possesses at least one solution. Obviously, the result of Yang follows from Theorem A by just setting (α, β) = (0, 0). The condition (3)–(5) concerns a nonresonance condition involving the two-parameter linear eigenvalue problem (LEVP) u(4) (t) + βu (t) − αu(t) = 0, (6) u(0) = u(1) = u (0) = u (1) = 0. In [6] it has been proved that (α, β) is an eigenvalue pair of LEVP (6) if and only if β α + (kπ) 2 = 1 for some k ∈ N. Hence, for k ∈ N the straight line (kπ)4 α β Lk = (α, β) + = 1 (kπ)4 (kπ)2 is called an eigenline of LEVP (6). The condition (3)–(4) trivially implies that a + bk 2 π 2 < 1, |k 4 π 4 − α − k 2 π 2 β|
∀k ∈ N.
It is easily to prove that condition (7) is equivalent to the fact that the rectangle R(α, β; a, b) = [α − a, α + a] × [β − b, β + b]
(7)
Y. Li / J. Math. Anal. Appl. 308 (2005) 121–128
123
does not intersect any of the eigenline Lk of LEVP (6). In [6], Del Pino and Manasevich conjecture that Theorem A is also valid if (4) is replaced by (7). But they could not prove the conjecture. Now we affirmatively answer this conjecture. But in this case, the condition (5) should be modified as f (t, u, v) − (αu − βv) a 2 u2 + b2 v 2 + c, ∀t ∈ [0, 1], u, v ∈ R. (8) Our result is as follows. Theorem 1. Assume that pair (α, β) satisfies (3) and that there are positive constants a, b and c such that (7) and (8) hold. Then the BVP (1) has at least one solution. Condition (7)–(8) is a directly extension of singe parameter nonresonance condition to two-parameter, and we call it two-parameter nonresonance condition √ described by rectangle. In the case that the partial derivative fu and fv exist, if r = u2 + v 2 large enough, the pair fu (t, u, v), −fv (t, u, v) lies on a certain ellipse, says (x − α)2 (y − β)2 E(α, β; a, b) = (x, y) + 1 , a2 b2 and corresponding close rectangle R(α, β; a, b) does not intersect any of the eigenline Lk of LEVP (6), by theorem of differential mean value, we easily see that (3), (7) and (8) hold. Hence, by Theorem 1 we have Corollary 1. Assume that the partial derivative fu and fv exist in [0, 1] × R × R. If there exists an ellipse E(α, β; a, b) such that fu (t, u, v), −fv (t, u, v) ∈ E(α, β; a, b), ∀r = u2 + v 2 R0 , for a positive real number R0 large enough, and corresponding close rectangle R(α, β; a, b) satisfies R(α, β; a, b) ∩ Lk = ∅,
∀k ∈ N,
the BVP (1) has at least one solution. In the nonresonance condition of Theorem 1, the rectangle R(α, β; a, b) can be replaced by the circle ¯ B(α, β; r) = (x, y) | (x − α)2 + (y − β)2 r 2 . In other words, we have the following result. ¯ Theorem 2. Assume that there exist a circle B(α, β; r) and a positive constant c such that ¯ B(α, β; r) ∩ Lk = ∅,
∀k ∈ N,
(9)
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Y. Li / J. Math. Anal. Appl. 308 (2005) 121–128
and f satisfies the growth condition f (t, u, v) − (αu − βv) r u2 + v 2 + c,
∀t ∈ [0, 1], u, v ∈ R.
(10)
Then the BVP (1) has at least one solution. Correspondingly, we call condition (9)–(10) the two-parameter nonresonance condition described by circle, which is another extension of singe parameter nonresonance condition. Similarly to Corollary 1, we have Corollary 2. Assume that f has derivative fu and fv in [0, 1] × R × R. If there exists a ¯ circle B(α, β; r), which does not intersect any of the eigenline Lk of LEVP (6), such that ¯ fu (t, u, v), −fv (t, u, v) ∈ B(α, β; r), ∀r = u2 + v 2 R0 , (11) for a positive real number R0 large enough, then the BVP (1) has at least one solution. Theorem 2 and Corollary 2 are the improvement of Theorem 1 and Corollary 1 in the case of that a = b = r, respectively. In Theorem 2 and Corollary 2, the radius of circle ¯ B(α, β; r) can be decided by 0 < r < dist (α, β), L , (12)
∞ where L = k=1 Lk is the set of all eigenvalue pairs of LEVP (6). We note that L is a closed set in R2 , and hence, if (3) holds, dist((α, β), L) > 0.
2. Proof of the main results Let pair (α, β) not be an eigenvalue pair of LEVP (6), i.e., (α, β) ∈ / L. Given h ∈ L2 (I ), where I denotes the unit interval [0, 1], we consider the linear boundary value problem (LBVP) u(4) (t) + βu (t) − αu(t) = h(t), t ∈ I, (13) u(0) = u(1) = u (0) = u (1) = 0. For each h ∈ L2 (I ), by Fredholm alternative, the LBVP (13) has a unique solution u ∈ H 4 (I ). If h ∈ C(I ), the solution u ∈ C 4 (I ). We define an operator T by T h = u,
h ∈ L2 (I ).
(14)
: L2 (I ) → H 4 (I )
Then T is a bounded linear operator. By compactness of the embedding H 4 (I ) → H 2 (I ), T : L2 (I ) → H 2 (I ) is a compact linear operator. Since {sin kπt | k ∈ N} is a complete orthogonal system of L2 (I ), every h ∈ L2 (I ) can be expressed by the Fourier series expansion h(t) =
∞ k=1
hk sin kπt,
(15)
Y. Li / J. Math. Anal. Appl. 308 (2005) 121–128
where hk = 2
h 22 =
1
125
h(s) sin kπs ds, k = 1, 2, . . . , and the Parseval equality
0
∞
|hk |2
(16)
k=1
holds, where u 2 is the norm in L2 (I ). Now by the uniqueness of Fourier series expansion, the solution of LBVP (13), u = T h has the Fourier series expansions u(t) =
∞ k=1
hk sin kπt, k4π 4 − α − k2π 2β
(17)
and u can be expressed by the Fourier series
u (t) = −
∞ k=1
k 2 π 2 hk sin kπt. k4π 4 − α − k2π 2β
(18)
Moreover, by Parseval equality, we have that
u 22 =
∞ k=1
|hk |2 |k 4 π 4 − α − k 2 π 2 β|2
(19)
and
u 22 =
∞ k=1
|k 4 π 4
k 4 π 4 |hk |2 . − α − k 2 π 2 β|2
(20)
Let a, b > 0. We choose an equivalent norm in Sobolev space H 2 (I ) by
u a,b = a 2 u 22 + b2 u 22 and denote the Banach space of H 2 (I ) reendowed norm · a,b by Ea,b . Lemma 1. Let (α, β) ∈ / L. Then the solution operator of LBVP (6), T : L2 (I ) → Ea,b is a compact linear operator and its norm satisfies
T B(L2 (I ),Ea,b ) max k∈N
a + bk 2 π 2 . |k 4 π 4 − α − k 2 π 2 β|
(21)
Proof. We need to prove that (21) holds only. For every h ∈ L2 (I ), let u = T h. From (19), (20) and (16), we have that
T h 2a,b = u 2a,b = a 2 u 22 + b2 u 22 =
∞ (a 2 + b2 k 4 π 4 )|hk |2 |k 4 π 4 − α − k 2 π 2 β|2 k=1 ∞ k=1
(a + bk 2 π 2 )2 |hk |2 |k 4 π 4 − α − k 2 π 2 β|2
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Y. Li / J. Math. Anal. Appl. 308 (2005) 121–128
2 ∞ a + bk 2 π 2 max 4 4 |hk |2 k∈N |k π − α − k 2 π 2 β| k=1 2 2 2 a + bk π = max 4 4
h 22 . k∈N |k π − α − k 2 π 2 β|
This implies that (21) holds. Proof of Theorem 1. We define mapping F : Ea,b → L2 (I ) by F (u)(t) = f t, u(t), u (t) − αu(t) + βu (t), ∀u ∈ Ea,b .
(22)
From (8) it follows that F : Ea,b → L2 (I ) is continuous and it satisfies F (u) u a,b + c, ∀u ∈ Ea,b . 2
(23)
Therefore the mapping defined by Q = T ◦ F : Ea,b → Ea,b
(24)
is a completely continuous mapping. By the definition of T , the solution of BVP (1) is equivalent to fixed point of Q. From (7) and (21) it follows that T B(L2 (I ),Ea,b ) < 1. We choose R
c · T B(L2 (I ),Ea,b ) 1 − T B(L2 (I ),Ea,b )
(25)
¯ ¯ and let B(θ, R) = {u ∈ Ea,b | u a,b R}. Then for ∀u ∈ B(θ, R), from (23) and (25) we have Q(u) = T F (u) T 2 B(L (I ),Ea,b ) · F (u) 2 a,b a,b T B(L2 (I ),Ea,b ) u a,b + c T B(L2 (I ),Ea,b ) (R + c) R, ¯ ¯ ¯ namely, Q(u) ∈ B(θ, R). Therefore, Q(B(θ, R)) ⊂ B(θ, R). By Schauder fixed theorem, ¯ Q has a fixed point in B(θ, R), which is a solution of BVP (1). The proof of Theorem 1 is completed. 2 Let r > 0. Now we choose another equivalent norm in Sobolev space H 2 (I ) by
u r = r u 22 + u 22 and denote the Banach space of H 2 (I ) reendowed norm · r by Er . Similarly to Lemma 1, we have Lemma 2. Let (α, β) ∈ / L. Then T : L2 (I ) → Er is a compact linear operator and its norm satisfies r . (26)
T B(L2 (I ),Er ) dist((α, β), L)
Y. Li / J. Math. Anal. Appl. 308 (2005) 121–128
127
Proof. For all h ∈ L2 (I ), letting u = T h, from (19), (20) and (16) we have that
T h 2r = u 2r = r 2 u 22 + u 22 ∞
(1 + k 4 π 4 )|hk |2 |k 4 π 4 − α − k 2 π 2 β|2 k=1 √ 2 ∞ 1 + k4π 4 2 r max 4 4 |hk |2 k∈N |k π − α − k 2 π 2 β| k=1 √ 2 4 4 1+k π = r 2 max 4 4
h 22 . k∈N |k π − α − k 2 π 2 β| = r2
From this it follows that
√
1 + k4π 4 k∈N |k 4 π 4 − α − k 2 π 2 β| 1 = r · max k∈N dist((α, β), Lk ) r = . dist((α, β), L)
T B(L2 (I ),Er ) r · max
The proof of Lemma 2 is completed.
2
¯ Proof of Theorem 2. From (9) we see that B(α, β; r) ∩ L = ∅. This implies that dist((α, β), L) > r. Thus, from (26) we obtain that T B(L2 (I ),Er ) < 1. Let F be the mapping defined by (22). Then from (10) it follows that F : Er → L2 (I ) is continuous and it satisfies F (u) u r + c, ∀u ∈ Er . (27) 2
Thus Q = T ◦ F : Er → Er is a completely continuous mapping. Using (27) and the same argument in the proof of Theorem 1, we easily see that Q has a fixed point, which is a solution of BVP (1). The proof of Theorem 2 is completed. 2
References [1] A.R. Aftabizadeh, Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl. 116 (1986) 415–426. [2] Y. Yang, Fourth-order two-point boundary value problems, Proc. Amer. Math. Soc. 104 (1988) 175–180. [3] C.P. Gupta, Existence and uniqueness theorems for a bending of an elastic beam equation, Appl. Anal. 26 (1988) 289–304. [4] C.P. Gupta, Existence and uniqueness results for the bending of an elastic beam equation at resonance, J. Math. Anal. Appl. 135 (1988) 208–225. [5] R. Agarwal, On fourth-order boundary value problems arising in beam analysis, Differential Integral Equations 2 (1989) 91–110. [6] M.A. Del Pino, R.F. Manasevich, Existence for a fourth-order boundary value problem under a twoparameter nonresonance condition, Proc. Amer. Math. Soc. 112 (1991) 81–86.
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[7] C. De Coster, C. Fabry, F. Munyamarere, Nonresonance conditions for fourth-order nonlinear boundary value problems, Internat. J. Math. Math. Sci. 17 (1994) 725–740. [8] R. Ma, H. Wang, On the existence of positive solutions of fourth-order ordinary differential equations, Appl. Anal. 59 (1995) 225–231. [9] R. Ma, J. Zhang, S. Fu, The method of lower and upper solutions for fourth-order two-point boundary value problems, J. Math. Anal. Appl. 215 (1997) 415–422. [10] Z. Bai, The method of lower and upper solutions for a bending of an elastic beam equation, J. Math. Anal. Appl. 248 (2000) 195–202. [11] Y. Li, Positive solutions of fourth-order boundary value problems with two parameters, J. Math. Anal. Appl. 281 (2003) 477–484. [12] Y. Li, Existence and method of lower and upper solutions for fourth-order nonlinear boundary value problems, Acta Math. Sci. Ser. A 23 (2003) 245–252 (in Chinese).
Two-parameter nonresonance condition for the existence of fourth-order boundary value problems ✩ Yongxiang Li Department of Mathematics, Northwest Normal University, Lanzhou 730070, People’s Republic of China Received 12 August 2004 Available online 23 February 2005 Submitted by D. O’Regan
Abstract In this paper, we discuss the existence of the fourth-order boundary value problem u(4) = f (t, u, u ), 0 < t < 1, u(0) = u(1) = u (0) = u (1) = 0, where f : [0, 1] × R × R → R is continuous, and partly solve the Del Pino and Manasevich’s conjecture on the nonresonance condition involving the two-parameter linear eigenvalue problem. We also present a two-parameter nonresonance condition described by circle. 2004 Elsevier Inc. All rights reserved. Keywords: Existence; Nonresonance condition; Two-parameter linear eigenvalue problem; Equivalent norm
1. Introduction and main results In this paper we deal with the existence of the fourth-order ordinary differential equation boundary value problem (BVP) u(4) (t) = f (t, u(t), u (t)), 0 < t < 1, (1) u(0) = u(1) = u (0) = u (1) = 0, ✩
Research supported by NNSF of China (10271095) and the NSF of Gansu Province (ZS031-A25-003-Z). E-mail address: [email protected].
0022-247X/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2004.11.021
122
Y. Li / J. Math. Anal. Appl. 308 (2005) 121–128
where f : [0, 1] × R × R → R is continuous. This problem models deformations of an elastic beam in equilibrium state, whose two ends are simply supported. Owing to its importance in physics, the existence of this problem has been studied by many authors, see [1–12]. In [1], Aftabizadeh showed the existence of a solution to PBV (1) under the restriction that f is a bounded function. In [2, Theorem 1], Yang extended Aftabizadeh’s result and showed the existence for BVP (1) under the growth condition of the form f (t, u, v) a|u| + b|v| + c, (2) where a, b and c are positive constants such that πa4 + πb2 < 1. In [6], Del Pino and Manasevich further extended the result of Yang, and obtained the following existence theorem: Theorem A. Assume that the pair (α, β) satisfies β α + = 1, (kπ)4 (kπ)2
∀k ∈ N,
(3)
and that there are positive constants a, b and c such that a max k∈N
1 k2π 2 + b max < 1, |k 4 π 4 − α − k 2 π 2 β| k∈N |k 4 π 4 − α − k 2 π 2 β|
and f satisfies the growth condition f (t, u, v) − (αu − βv) a|u| + b|v| + c,
∀t ∈ [0, 1], u, v ∈ R.
(4)
(5)
Then the BVP (1) possesses at least one solution. Obviously, the result of Yang follows from Theorem A by just setting (α, β) = (0, 0). The condition (3)–(5) concerns a nonresonance condition involving the two-parameter linear eigenvalue problem (LEVP) u(4) (t) + βu (t) − αu(t) = 0, (6) u(0) = u(1) = u (0) = u (1) = 0. In [6] it has been proved that (α, β) is an eigenvalue pair of LEVP (6) if and only if β α + (kπ) 2 = 1 for some k ∈ N. Hence, for k ∈ N the straight line (kπ)4 α β Lk = (α, β) + = 1 (kπ)4 (kπ)2 is called an eigenline of LEVP (6). The condition (3)–(4) trivially implies that a + bk 2 π 2 < 1, |k 4 π 4 − α − k 2 π 2 β|
∀k ∈ N.
It is easily to prove that condition (7) is equivalent to the fact that the rectangle R(α, β; a, b) = [α − a, α + a] × [β − b, β + b]
(7)
Y. Li / J. Math. Anal. Appl. 308 (2005) 121–128
123
does not intersect any of the eigenline Lk of LEVP (6). In [6], Del Pino and Manasevich conjecture that Theorem A is also valid if (4) is replaced by (7). But they could not prove the conjecture. Now we affirmatively answer this conjecture. But in this case, the condition (5) should be modified as f (t, u, v) − (αu − βv) a 2 u2 + b2 v 2 + c, ∀t ∈ [0, 1], u, v ∈ R. (8) Our result is as follows. Theorem 1. Assume that pair (α, β) satisfies (3) and that there are positive constants a, b and c such that (7) and (8) hold. Then the BVP (1) has at least one solution. Condition (7)–(8) is a directly extension of singe parameter nonresonance condition to two-parameter, and we call it two-parameter nonresonance condition √ described by rectangle. In the case that the partial derivative fu and fv exist, if r = u2 + v 2 large enough, the pair fu (t, u, v), −fv (t, u, v) lies on a certain ellipse, says (x − α)2 (y − β)2 E(α, β; a, b) = (x, y) + 1 , a2 b2 and corresponding close rectangle R(α, β; a, b) does not intersect any of the eigenline Lk of LEVP (6), by theorem of differential mean value, we easily see that (3), (7) and (8) hold. Hence, by Theorem 1 we have Corollary 1. Assume that the partial derivative fu and fv exist in [0, 1] × R × R. If there exists an ellipse E(α, β; a, b) such that fu (t, u, v), −fv (t, u, v) ∈ E(α, β; a, b), ∀r = u2 + v 2 R0 , for a positive real number R0 large enough, and corresponding close rectangle R(α, β; a, b) satisfies R(α, β; a, b) ∩ Lk = ∅,
∀k ∈ N,
the BVP (1) has at least one solution. In the nonresonance condition of Theorem 1, the rectangle R(α, β; a, b) can be replaced by the circle ¯ B(α, β; r) = (x, y) | (x − α)2 + (y − β)2 r 2 . In other words, we have the following result. ¯ Theorem 2. Assume that there exist a circle B(α, β; r) and a positive constant c such that ¯ B(α, β; r) ∩ Lk = ∅,
∀k ∈ N,
(9)
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Y. Li / J. Math. Anal. Appl. 308 (2005) 121–128
and f satisfies the growth condition f (t, u, v) − (αu − βv) r u2 + v 2 + c,
∀t ∈ [0, 1], u, v ∈ R.
(10)
Then the BVP (1) has at least one solution. Correspondingly, we call condition (9)–(10) the two-parameter nonresonance condition described by circle, which is another extension of singe parameter nonresonance condition. Similarly to Corollary 1, we have Corollary 2. Assume that f has derivative fu and fv in [0, 1] × R × R. If there exists a ¯ circle B(α, β; r), which does not intersect any of the eigenline Lk of LEVP (6), such that ¯ fu (t, u, v), −fv (t, u, v) ∈ B(α, β; r), ∀r = u2 + v 2 R0 , (11) for a positive real number R0 large enough, then the BVP (1) has at least one solution. Theorem 2 and Corollary 2 are the improvement of Theorem 1 and Corollary 1 in the case of that a = b = r, respectively. In Theorem 2 and Corollary 2, the radius of circle ¯ B(α, β; r) can be decided by 0 < r < dist (α, β), L , (12)
∞ where L = k=1 Lk is the set of all eigenvalue pairs of LEVP (6). We note that L is a closed set in R2 , and hence, if (3) holds, dist((α, β), L) > 0.
2. Proof of the main results Let pair (α, β) not be an eigenvalue pair of LEVP (6), i.e., (α, β) ∈ / L. Given h ∈ L2 (I ), where I denotes the unit interval [0, 1], we consider the linear boundary value problem (LBVP) u(4) (t) + βu (t) − αu(t) = h(t), t ∈ I, (13) u(0) = u(1) = u (0) = u (1) = 0. For each h ∈ L2 (I ), by Fredholm alternative, the LBVP (13) has a unique solution u ∈ H 4 (I ). If h ∈ C(I ), the solution u ∈ C 4 (I ). We define an operator T by T h = u,
h ∈ L2 (I ).
(14)
: L2 (I ) → H 4 (I )
Then T is a bounded linear operator. By compactness of the embedding H 4 (I ) → H 2 (I ), T : L2 (I ) → H 2 (I ) is a compact linear operator. Since {sin kπt | k ∈ N} is a complete orthogonal system of L2 (I ), every h ∈ L2 (I ) can be expressed by the Fourier series expansion h(t) =
∞ k=1
hk sin kπt,
(15)
Y. Li / J. Math. Anal. Appl. 308 (2005) 121–128
where hk = 2
h 22 =
1
125
h(s) sin kπs ds, k = 1, 2, . . . , and the Parseval equality
0
∞
|hk |2
(16)
k=1
holds, where u 2 is the norm in L2 (I ). Now by the uniqueness of Fourier series expansion, the solution of LBVP (13), u = T h has the Fourier series expansions u(t) =
∞ k=1
hk sin kπt, k4π 4 − α − k2π 2β
(17)
and u can be expressed by the Fourier series
u (t) = −
∞ k=1
k 2 π 2 hk sin kπt. k4π 4 − α − k2π 2β
(18)
Moreover, by Parseval equality, we have that
u 22 =
∞ k=1
|hk |2 |k 4 π 4 − α − k 2 π 2 β|2
(19)
and
u 22 =
∞ k=1
|k 4 π 4
k 4 π 4 |hk |2 . − α − k 2 π 2 β|2
(20)
Let a, b > 0. We choose an equivalent norm in Sobolev space H 2 (I ) by
u a,b = a 2 u 22 + b2 u 22 and denote the Banach space of H 2 (I ) reendowed norm · a,b by Ea,b . Lemma 1. Let (α, β) ∈ / L. Then the solution operator of LBVP (6), T : L2 (I ) → Ea,b is a compact linear operator and its norm satisfies
T B(L2 (I ),Ea,b ) max k∈N
a + bk 2 π 2 . |k 4 π 4 − α − k 2 π 2 β|
(21)
Proof. We need to prove that (21) holds only. For every h ∈ L2 (I ), let u = T h. From (19), (20) and (16), we have that
T h 2a,b = u 2a,b = a 2 u 22 + b2 u 22 =
∞ (a 2 + b2 k 4 π 4 )|hk |2 |k 4 π 4 − α − k 2 π 2 β|2 k=1 ∞ k=1
(a + bk 2 π 2 )2 |hk |2 |k 4 π 4 − α − k 2 π 2 β|2
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Y. Li / J. Math. Anal. Appl. 308 (2005) 121–128
2 ∞ a + bk 2 π 2 max 4 4 |hk |2 k∈N |k π − α − k 2 π 2 β| k=1 2 2 2 a + bk π = max 4 4
h 22 . k∈N |k π − α − k 2 π 2 β|
This implies that (21) holds. Proof of Theorem 1. We define mapping F : Ea,b → L2 (I ) by F (u)(t) = f t, u(t), u (t) − αu(t) + βu (t), ∀u ∈ Ea,b .
(22)
From (8) it follows that F : Ea,b → L2 (I ) is continuous and it satisfies F (u) u a,b + c, ∀u ∈ Ea,b . 2
(23)
Therefore the mapping defined by Q = T ◦ F : Ea,b → Ea,b
(24)
is a completely continuous mapping. By the definition of T , the solution of BVP (1) is equivalent to fixed point of Q. From (7) and (21) it follows that T B(L2 (I ),Ea,b ) < 1. We choose R
c · T B(L2 (I ),Ea,b ) 1 − T B(L2 (I ),Ea,b )
(25)
¯ ¯ and let B(θ, R) = {u ∈ Ea,b | u a,b R}. Then for ∀u ∈ B(θ, R), from (23) and (25) we have Q(u) = T F (u) T 2 B(L (I ),Ea,b ) · F (u) 2 a,b a,b T B(L2 (I ),Ea,b ) u a,b + c T B(L2 (I ),Ea,b ) (R + c) R, ¯ ¯ ¯ namely, Q(u) ∈ B(θ, R). Therefore, Q(B(θ, R)) ⊂ B(θ, R). By Schauder fixed theorem, ¯ Q has a fixed point in B(θ, R), which is a solution of BVP (1). The proof of Theorem 1 is completed. 2 Let r > 0. Now we choose another equivalent norm in Sobolev space H 2 (I ) by
u r = r u 22 + u 22 and denote the Banach space of H 2 (I ) reendowed norm · r by Er . Similarly to Lemma 1, we have Lemma 2. Let (α, β) ∈ / L. Then T : L2 (I ) → Er is a compact linear operator and its norm satisfies r . (26)
T B(L2 (I ),Er ) dist((α, β), L)
Y. Li / J. Math. Anal. Appl. 308 (2005) 121–128
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Proof. For all h ∈ L2 (I ), letting u = T h, from (19), (20) and (16) we have that
T h 2r = u 2r = r 2 u 22 + u 22 ∞
(1 + k 4 π 4 )|hk |2 |k 4 π 4 − α − k 2 π 2 β|2 k=1 √ 2 ∞ 1 + k4π 4 2 r max 4 4 |hk |2 k∈N |k π − α − k 2 π 2 β| k=1 √ 2 4 4 1+k π = r 2 max 4 4
h 22 . k∈N |k π − α − k 2 π 2 β| = r2
From this it follows that
√
1 + k4π 4 k∈N |k 4 π 4 − α − k 2 π 2 β| 1 = r · max k∈N dist((α, β), Lk ) r = . dist((α, β), L)
T B(L2 (I ),Er ) r · max
The proof of Lemma 2 is completed.
2
¯ Proof of Theorem 2. From (9) we see that B(α, β; r) ∩ L = ∅. This implies that dist((α, β), L) > r. Thus, from (26) we obtain that T B(L2 (I ),Er ) < 1. Let F be the mapping defined by (22). Then from (10) it follows that F : Er → L2 (I ) is continuous and it satisfies F (u) u r + c, ∀u ∈ Er . (27) 2
Thus Q = T ◦ F : Er → Er is a completely continuous mapping. Using (27) and the same argument in the proof of Theorem 1, we easily see that Q has a fixed point, which is a solution of BVP (1). The proof of Theorem 2 is completed. 2
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