Open Access

Fixed point theorems of convex-power 1-set-contraction operators in Banach spaces

Fixed Point Theory and Applications20122012:56

https://doi.org/10.1186/1687-1812-2012-56

Received: 7 July 2011

Accepted: 5 April 2012

Published: 5 April 2012

Abstract

In this article, we give the definition of a class of new operators, namely, convex-power 1-set-contraction operators in Banach spaces, and study the existence of fixed points of this class of operators. By using methods of approximation by operators, we obtain fixed point theorems of convex-power 1-set-contraction operators, which generalize fixed point theorems of 1-set-contraction operators in Banach spaces. By using the fixed point theorem, the existence of solutions of nonlinear Sturm-Liouville problems in Banach spaces is investigated under more general conditions than those used in former literatures.

Mathematics Subject Classification 2010: 47H10.

Keywords

convex-power 1-set-contractionfixed point theoremBanach spacesSturm-Liouville problems

0 Introduction

For the need of studying differential equations and integral equations, Sun and Zhang [1] gave the definition of convex-power condensing operators and obtained the fixed point theorem of this class of operators. Li [2] gave the fixed point theorem of semi-closed 1-set-contraction operators.

In this article, by combinating the definitions of convex-power condensing operators and 1-set-contraction operators, we give the definition of convex-power 1-set-contraction operators in Banach spaces and study the existence of fixed points of this class of new operators. The results in this article generalize the ones in [13]. By using the fixed point theorem, the existence of solutions of nonlinear Sturm-Liouville problems in Banach spaces is investigated under more general conditions than those used in former literatures.

1 Preliminaries

Before providing the main results, we introduce some basic definitions and results (see [16]).

In this article, we always assume that E is a Banach space, D E, and α(S) denotes the Kuratowski measure of noncompactness of a bounded set S E.

Let A: DE be continuous. If there exists a constant k ≥ 0, such that for any bounded subset S D,
α A S k α S .

Then A is said to be k-set-contraction in D.

Let A: DE be continuous and bounded. If there exist x0 D and a positive integer n0, such that for any bounded and nonprecompact subset S D,
α A n 0 , x 0 S < α S ,

where A ( 1 , x 0 ) ( S ) = A ( S ) , A n , x 0 S = A c o - A n - 1 , x 0 S , x 0 , n = 2, 3,....

Then A is said to be convex-power condensing.

Lemma 1.1[1]. Let D E be bounded, convex, and closed. Suppose that A: DD is convex-power condensing, then A has at least one fixed point in D.

Definition 1.1[2]. A: DE is said to be semi-closed if for any closed set F D, (I - A)F is closed.

Definition 1.2[3]. Let A: DE, {x n } D bounded, {x n -Ax n } strongly convergent. A is said to be semi-compact if {x n } has a strongly convergent subsequence.

2 Main results

Next, we will give the definition of convex-power 1-set-contraction operators in Banach spaces.

Definition 2.1. Let A : DE be continuous and bounded. If there exist x0 D and a positive integer n0, such that for any bounded subset S D,
α A n 0 , x 0 S α S ,

where A ( 1 , x 0 ) ( S ) = A ( S ) , A n , x 0 S = A c o - A n - 1 , x 0 S , x 0 , n = 2, 3,....

Then A is said to be convex-power 1-set-contraction in Banach spaces.

Remark 2.1. Obviously, 1-set-contraction operators are convex-power 1-set-contraction operators. Convex-power 1-set-contraction operators are more general.

Now, we establish the main theorem as follows:

Theorem 2.1. Let E be a Banach space, D E bounded, convex, and closed. Suppose that A: DD is semi-closed and convex-power 1-set-contraction, then A has at least one fixed point in D.

Proof. Since A is convex-power 1-set-contraction, there exist x0 D and a positive integer n0, such that for any bounded subset S D,
α A n 0 , x 0 S α S .

x D , let A n x = 1 - 1 n A x + 1 n x 0 n = 2 , 3 , , then A n : DD. y D - x0 = {x - x0|x D}, let By = A(y + x0)-x0.

For any bounded subset S D,
B 1 , 0 S - x 0 = B S - x 0 = A S - x 0 = A 1 , x 0 S - x 0 , A n 1 , x 0 S = A n S = 1 - 1 n A S + 1 n x 0 = 1 - 1 n A 1 , x 0 S + 1 n x 0 ; B 2 , 0 S - x 0 = B c o - B 1 , 0 S - x 0 , 0 = B c o - A 1 . x 0 S - x 0 , 0 = A c o - A 1 , x 0 S - x 0 , 0 + x 0 - x 0 = A c o - A 1 , x 0 S , x 0 - x 0 = A 2 , x 0 S - x 0 ,
A n 2 , x 0 S = A n c o - A n 1 , x 0 S , x 0 = A n c o - 1 - 1 n B 1 , 0 S - x 0 + x 0 , x 0 = A n c o - 1 - 1 n B 1 , 0 S - x 0 , 0 + x 0 A n c o - B 1 , 0 S - x 0 , 0 + x 0 = A n c o - B 1 , 0 S - x 0 + x 0 , x 0 = A n c o - A 1 , x 0 S , x 0 = 1 - 1 n A 2 , x 0 S + 1 n x 0 ;
and generally,
B n 0 , 0 S - x 0 = B c o - B n 0 - 1 , 0 S - x 0 , 0 = B c o - A n 0 - 1 , x 0 S - x 0 , 0 = A c o - A n 0 - 1 , x 0 S - x 0 , 0 + x 0 - x 0 = A n 0 , x 0 S - x 0 ,
A n n 0 , x 0 S = A n c o - A n n 0 - 1 , x 0 S , x 0 A n c o - 1 - 1 n A n 0 - 1 , x 0 S + 1 n x 0 , x 0 = A n c o - 1 - 1 n B n 0 - 1 , 0 S - x 0 + x 0 , x 0 A n c o - B n 0 - 1 , 0 S - x 0 , 0 + x 0 = A n c o - A n 0 - 1 , x 0 S , x 0 = 1 - 1 n A n 0 , x 0 S + 1 n x 0 .
By the definition of the convex-power 1-set-contraction operator and the properties of the measure of noncompactness, we have
α A n n 0 , x 0 S 1 - 1 n α A n 0 , x 0 S 1 - 1 n α S < α S , n = 2 , 3 , .
Therefore, A n : DD is convex-power condensing. By Lemma 1.1, A n has a fixed point x n in D, i.e., A n x n = x n (n = 2, 3,...). Since | | A x - A n x | | = 1 n | | A x - x 0 | | , x D, and A is bounded in D, then for any x D, ||Ax - A n x|| → 0 (n → +∞). Obviously,
| | A x n - x n | | = | | A x n - A n x n | | 0 n + .

i.e., Ax n - x n → 0 (n → +∞). Since A is semi-closed and D is closed, 0 (I - A)D. Therefore, there exists x0 D, such that x0 = Ax0. The proof is completed.

Remark 2.2. In Theorem 2.1, let n0 = 1, the fixed point theorem of semiclosed 1-set-contraction operators in[2]is obtained. Therefore, Theorem2.1. generalizes the fixed point theorem of semi-closed 1-set-contraction operators.

Theorem 2.2. Let E be a Banach space, D E bounded, convex and closed. Suppose that A: DD is semi-compact and convex-power 1-set-contraction, then A has at least one fixed point in D.

Proof. x D , let A n x = 1 - 1 n A x + 1 n x 0 n = 2 , 3 , . By the proof of Theorem 2.1, A n : DD has a fixed point x n , and {A n } is uniformly convergent to A in D. By x n D, then ||A n x n - Ax n || → 0. i.e., ||x n - Ax n || → 0. Therefore, (I - A)(x n ) → 0 (n → +∞).

Since A is semi-compact and {x n } D is bounded, {x n } has a convergent subsequence x n i . Let x n i x 0 n i + . Since D is closed, x0 D. Since A is continuous in D, x n i - A x n i x 0 - A x 0 n i + . By x n i - A x n i 0 n i + , we have x0 - Ax0 = 0. The proof is completed.

3 Application

Let E be a Banach space. Consider the existence of solutions of nonlinear Sturm-Liouville problems in E as follows:
- L x t = f t , x , t 0 , 1 ; a x 0 - b x 0 = 0 , c x 1 + d x 1 = 0
(3.1)

where (Lx)(t) = (p(t)x')'+q(t)x, f C[I × E, E](I = [0, 1]).

Assume that
H 1 p t C 1 I , R , p t > 0 , q t C I , R , q t 0 , a 0 , b 0 , c 0 , d 0 , a 2 + b 2 0 , c 2 + d 2 0 ,
and the homogeneous equations of (3.1)
- L x t = 0 , t 0 , 1 ; a x 0 - b x 0 = 0 , c x 1 + d x 1 = 0
(3.2)

has only zero solution in C2 [I, R].

Let G(t, s) be Green function of (3.2), i.e.,
G t , s = 1 ρ u t v s , 0 t s 1 ; 1 ρ u s v t , 0 s t 1 .
(3.3)

Lemma 3.1[6]. Assume that (H1) holds, then Green function G(t, s) of (3.3) has the following properties:

(i) G(t, s) is continuous and symmetric in [0, 1] × [0, 1];

(ii) u(t) C2[0, 1] is monotonically increasing, and u(t) > 0, t (0, 1];

(iii) v(t) C2[0, 1] is monotonically decreasing, and v(t) > 0, t [0, 1);

(iv) (Lu)(t) ≡ 0, u(0) = b, u'(0) = a;

(v) (Lv)(t) ≡ 0, v(0) = d, v'(0) = -c;

(vi) ρ is a positive constant.

Let
T x t = 0 1 G t , s f s , x s d s , t 0 , 1 , x C I , E , K φ t = 0 1 G t , s φ s d s , t 0 , 1 , φ C I , R .

We can prove that the solution in C2[I, E] of (3.1) is equivalent to the fixed point of T (see [7]).

Since G(t, s) is continuous, it can be easily proved that K: C [I, R] → C [I, R] is linear and completely continuous. By Lemma 3.1, t, s [0, 1],
u s v s u 1 v 0 G t , t G t , s G t , t .

Therefore, by Krein-Rutman Theorem [6], the first characteristic value of K is λ1 > 0, and λ1 = (r(K))-1.

Now we give some conditions:

(H 2 ) f C[I × E, E], for any bounded subset B in E, f is uniformly continuous in I × B, and there exists k [0, λ1), such that
α f t , B t k α B t , t 0 , 1 ,

where λ1 is the first characteristic value of K.

(H 3 ) there exist M (0, λ1) and h(t) C[I , R+], such that for any (t, x) I × E,
| | f t , x | | M | | x | | + h t .

Theorem 3.1. Suppose that (H1), (H2), (H3) hold, then Sturm-Liouville problems (3.1) has at least one solution in C2[I, E].

To prove Theorem 3.1, here we introduce some lemmas.

Lemma 3.2[7]. For M < λ1as above, let K1 = M K, then there exists a norm | | | | C I , R * which is equivalent to || · ||C[I, R]and satisfies:

(1) | | K 1 φ | | C I , R * σ | | φ | | C I , R * , where σ = M + λ 1 2 λ 1 ,

(2) if 0 ≤ φ(t) ≤ ψ(t), t I, then | | φ | | C I , R * | | ψ | | C I , R * , where ||φ||C[I, R]= maxtI|φ(t)|.

Lemma 3.3[7]. If B C [I, E] is equicontinuous, u0 C[I, E], then c o - B , u 0 is also equicontinuous in C[I, E].

Lemma 3.4[7]. If B C[I, E] is equicontinuous and bounded, then α(B) = maxtIα(B(t)).

Lemma 3.5[7]. If B C[I, E] is equicontinuous and bounded, then α(B(t)) C[I, R+], and
α t 0 t B s d s t 0 t α B s d s , t I .

Proof of Theorem 3.1. Set R 1 > 2 λ 1 λ 1 - M | | K h | | C I , R * , where K h t = 0 1 G t , s h s d s .

Let D = x C I , E | φ t = | | x t | | and | | φ | | C I , R * R 1 . Since ||·||C[I, R]is equivalent to | | | | C I , R * , D is bounded, convex, and closed in C[I, E].

First x D, ||x||C[I, E]= maxtI||x(t)|| = maxtIφ(t) = maxtI|φ(t)| = ||φ||C[I, R], then D is bounded.

Second, x n D, x n x0, n → +∞. Therefore, φ n t = | | x n t | | , | | φ n | | C I , R * R 1 , | | x n - x 0 | | C I , E 0 , i.e., maxtI||x n (t) - x0(t)|| → 0.

Let φ ̃ n t = | | x n t - x 0 t | | , φ0(t) = ||x0(t)||, then φ 0 t φ ̃ n t + φ n t , | | φ ̃ n | | C I , R 0 . By Lemma 3.2,
| | φ 0 | | C I , R * | | φ ̃ n + φ n | | C I , R * | | φ ̃ n | | C I , R * + | | φ n | | C I , R * | | φ ̃ n | | C I , R * + R 1

Let n → +∞, then | | φ 0 | | C I , R * R 1 , i.e., x0 D, D is closed.

Finally, x1, x2 D, 0 ≤ α ≤ 1. Let φ i (t) = ||x i (t)||, i = 1,2; φ3(t) = ||αx1(t)+(1-α)x2(t)||. Obviously, φ3αφ1(t)+(1-α)φ2(2). By Lemma 3.2,
| | φ 3 | | C I , R * | | α | | φ 1 | | C I , R * | | + 1 - α | | φ 2 | | C I , R * R 1 .

Then D is convex. Therefore, D is bounded, convex, and closed.

By (H2), f is uniformly continuous in I × D, then T: DC[I, E] is continuous.

First, we prove that T: DD. For any given x in D, let φ(t) = ||Tx(t)||ψ(t) = ||x(t)||. By (H3),
φ t = | | T x t | | = | | 0 1 G t , s f s , x s d s | | 0 1 G t , s | | f s , x s | | d s 0 1 G t , s M ψ s d s + 0 1 G t , s h s d s = K 1 ψ t + K h t .
By Lemma 3.2,
| | φ | | C I , R * | | K 1 ψ + K h | | C I , R * | | K 1 ψ | | C I , R * + | | K h | | C I , R * σ | | ψ | | C I , R * + | | K h | | C I , R * σ R 1 + λ 1 - M 2 λ 1 R 1 = R 1 .

Therefore, T : DD is continuous

Next, we prove that T(D) is equicontinuous in C[I, E]. By (H2), M1 > 0, ||f(t, x)|| ≤ M1, (t, x) I × D. Then,
| | T x t 1 - T x t 2 | | M 1 0 1 | G t 1 , s - G t 2 , s | d s , t 1 , t 2 I , x D .

Therefore, T(D) is equicontinuous.

Let F = c o - T D D . Obviously, F is bounded, convex, and closed, and T c o - T D T D c o - T D , i.e., T: FF. By Lemma 3.3, F is equicontinuous in C[I, E].

Next, we prove that T: FF is convex-power 1-set-contraction. Obviously, T is bounded and continuous. Set x0 F, we'll prove that there exists n0, such that for any bounded B F,
α T n 0 , x 0 B α B .
By B F D, T (B) is equicontinuous. Then T 2 , x 0 B is equicontinuous from T 2 , x 0 B = T c o - T B , x 0 T D Generally, n N, T n , x 0 B is equicontinuous. Since T n , x 0 B is bounded, By Lemma 3.4,
α T n , x 0 B = max t I α T n , x 0 B t n = 2 , 3 , . . . .
(3.4)
Since G(t, s) is continuous in I ×I, f is uniformly continuous in I ×D, then
| | G t , s 1 f s 1 , x s 1 - G t , s 2 f s 2 , x s 2 | | | | G t , s 1 - G t , s 2 | | | | f s 1 , x s 1 | | + | | G t , s 2 | | | | f s 1 , x s 1 - f s 2 , x s 2 | | s 1 , s 2 I , x B
Therefore G(t, s)f(s, B(s))( s, t I) is equicontinuous in C[I, E]. By (H2), Lemmas 3.4 and 3.5,
α T 1 , x 0 B t = α T B t = α 0 1 G t , s f s , B s d s 0 1 G t , s α f s , B s d s k 0 1 G t , s α B s d s k α B 0 1 G t , s d s = k α B K φ 0 t

where φ0(t) ≡ 1, t I.

By the equicontinuity of T 1 , x 0 B = T B and the uniform continuity of f, G t , s f s , c o - T 1 , x 0 B s , x 0 s , t I is equicontinuous. Therefore,
α T 2 , x 0 B t = α T c o - T 1 , x 0 B t , x 0 = α 0 1 G t , s f s , c o - T 1 , x 0 B s , x 0 d s 0 1 G t , s α f s , c o - T 1 , x 0 B s , x 0 d s k 0 1 G t , s α c o - T 1 , x 0 B s , x 0 d s = k 0 1 G t , s α T 1 , x 0 B s d s k 2 α B 0 1 G t , s K φ 0 s d s = k 2 α B K 2 φ 0 t .
Generally,
α T n , x 0 B t k n α B K n φ 0 t
We have r k K = k r K = k λ 1 - 1 < λ 1 λ 1 - 1 = 1 . By the definition of spectral radius, let ε = 1 - r k K 2 , then m0 > 0, when n > m0,
max t I | k n K n φ 0 t | = | | k n K n φ 0 | | | | k n K n | | | | φ 0 | | = | | k n K n | | r k K + ε n = 1 + r k K 2 n < 1 .
Set n0 > m0, then t I,
α T n 0 , x 0 B t k n 0 α B K n 0 φ 0 t | | k n 0 K n 0 φ 0 | | α B 1 + r k K 2 n 0 α B α B .

By (3.4), α T n 0 , x 0 B α B . Therefore, T: FF is convex-power 1-set-contraction. Since f is uniformly continuous, T is semi-closed. By Theorem 2.1, T has one fixed point in C[I, E], i.e., Sturm-Liouville problems (3.1) has at least one solution in C2[I, E].

Declarations

Acknowledgements

This study was supported by NNSF-CHINA (10971179) (China).

Authors’ Affiliations

(1)
Department of Mathematics, Xuzhou Normal University, Xuzhou, China

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© Lvhuizi and Jingxian; licensee Springer. 2012

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