Abstract
In this paper, we present common fixed point theorems for commuting operators which generalize Darbo’s and Sadovski’s fixed point theorems. As example and application, we study the existence of common solutions of equations in Banach spaces using the measure of noncompactness.
MSC:47H10, 47H09.
Similar content being viewed by others
1 Introduction
In 1955, Darbo [1] proved the fixed point property for α-set contraction (i.e., with ) on a closed, bounded and convex subset of Banach spaces. Since then many interesting works have appeared. For example, in 1967, Sadovski [2] proved the fixed point property for condensing functions (i.e., with ) on closed, bounded and convex subset of Banach spaces. It should be noted that any α-set contraction is a condensing function, but the converse is not true (see [[3], p.160]). In 2007, we have proved in [4] the existence of a common fixed point for commuting mappings satisfying
where α is the measure of noncompactness on a closed, bounded and convex subset Ω of a locally convex space X, and and S are continuous functions from Ω to Ω with and, in addition, are affine or linear. Furthermore, if for every , the in (1) and (2) are equal to the identity function, then we obtain in particular Darbo’s (see [1]) as well as Sadovski’s fixed point theorems (see [2]), which are used to study the existence of solutions of one equation.
The aim of this paper is to prove the existence of a common fixed point of the operators T and S satisfying
where T is affine, S and T are continuous functions, and T commutes with S. As application, we study the existence of common solutions of the following equations:
under appropriate assumptions on functions f, , and .
2 Preliminaries
We begin by recalling some needed definitions and results. Let be a Banach space and denote by ℬ the family of all bounded subsets of X.
Definition 2.1 The function defined, for every , by
is called the measure of noncompactness in X.
The measure of noncompactness α satisfies the following properties.
Let . Then
-
(1)
For , we have .
-
(2)
For any , we have .
-
(3)
.
-
(4)
.
-
(5)
if and only if is compact.
-
(6)
.
-
(7)
, where is the convex hull of A in X.
Definition 2.3 A mapping T on a convex set ℳ is affine if it satisfies the identity
whenever , .
Theorem 2.4 (see [4])
Let X be a Hausdorff complete and locally convex space, whose topology is defined by a family of semi-norms . Let Ω be a convex closed bounded subset of X, I be a set of index, and and S be two continuous functions from Ω into Ω such that:
-
(a)
For any , commutes with S.
-
(b)
For any and , we have , where is the convex hull of A in Ω.
-
(c)
There exists such that for any
Then we have:
-
(1)
The set is nonempty and compact.
-
(2)
For any , has a fixed point and is a closed set and invariant by S.
-
(3)
If is affine and is a commuting family, then and S have a common fixed point and the set is compact.
-
(4)
If is a commuting family and S is affine, then there exists a common fixed point for the mappings .
Theorem 2.5 (see [4])
Let X be a Hausdorff complete and locally convex space, whose topology is defined by a family of semi-norms . Let Ω be a convex, closed and bounded subset of X, I be a given set of index, and , S be continuous functions from Ω into Ω such that:
-
(a)
For every , commutes with S.
-
(b)
For every , is linear.
-
(c)
There exists such that for every and , with , we have
Then
-
(1)
and S have a fixed point, and is a nonempty and compact set.
-
(2)
If is a commuting family and S is affine, then there exists a common fixed point for the mappings in .
3 Fixed point theorems
It is well known that if ST has a fixed point, for given operators T and S, then S and T not necessarily have a fixed point or a common fixed point. Thus, it will be of interest to establish some results showing the existence of a common fixed point for T and S when the operator ST has a fixed point. The last fact can be used to study the existence of common solutions of equations.
Theorem 3.1 Let X be a Banach space and Ω be a nonempty convex, closed and bounded subset of X. Let T and S be two continuous functions from Ω into Ω such that:
-
(a)
.
-
(b)
T is affine.
-
(c)
There exists such that for any we have
Then the set is nonempty and compact.
Proof Consider the operator . It is clear that H maps Ω into Ω, commutes with T and is continuous. Moreover, by (2) and (3) of Proposition 2.2, we have
for any . Furthermore, since we have and therefore . Hence by making appeal to Theorem 2.4, we conclude that is nonempty and compact. Furthermore, for any , we have
Whence, S and T have a common fixed point. Let , then
which implies . Now, since S and T are continuous, then ℱ is compact. □
Observe that Theorem 3.1 above shows the following:
(∗) If the operator ST has a fixed point, then T and S have a common fixed point.
(∗∗) If the operator T equals the identity function, then we obtain Darbo’s fixed point theorem.
Theorem 3.2 Let X be a Banach space and Ω be a nonempty convex, closed and bounded subset of X. Let , , and S be continuous functions from Ω into Ω such that:
-
(a)
and for any .
-
(b)
, are affine.
-
(c)
There exists such that for any we have
Then the set is nonempty and compact.
Proof Consider the following operator . It is clear that H maps Ω into Ω, commutes with , and is continuous. Moreover, we have for any
Hence, by Theorem 2.4, H and have a common fixed point which is a fixed point with S. Thus, the nonempty set is convex, closed and bounded subset of Ω, for being continuous and affine. Moreover, by (a) we have and . Therefore, for any , we have
By the same argument as before, we consider for . It follows that the set is nonempty and compact. □
Theorem 3.2 can be used to show that any affine operator T that commutes with S and satisfies Darbo’s condition [1] has a common fixed point with S.
Theorem 3.3 Let X be a Banach space, Ω be a nonempty convex, closed and bounded subset of X, T and S be continuous functions from Ω into Ω such that:
-
(a)
T commutes with S.
-
(b)
T is linear.
-
(c)
For any with , we have
Then the set is nonempty.
Proof Let and consider the operator
for any . It is clear that H maps Ω into Ω, commutes with T, and is continuous. Moreover, for any such that , we have
Hence, by Theorem 2.5, H and T have a common fixed point which is also a fixed point of S. □
As a consequence of Theorem 3.3 above, one can recover Sadovski’s fixed point theorem when T is equal to the identity function.
4 Example and application
4.1 Example
Let be the space of Lebesgue integrable functions on the measurable subset of ℝ with the standard norm
Let defined by
Therefore, we can see that for any and for all , the following condition is satisfied:
Then using Krzyz’s theorem (see [6]), the linear operator
transforms the set of non-increasing functions from into itself. Furthermore, we can see that the norm of the convolution operator K satisfies
The Hausdorff measure of noncompactness on the Banach space E, noted by (see [5, 7]), is defined as
where is a closed ball in E centered at zero and of radius r.
Note that there is another measure γ on the space [8]. Indeed, for any , let
where denotes the Lebesgue measure of the subset D (see [7, 8]), and
We have
Then we have the following theorem.
Theorem 4.1 (see [8])
Let X be a nonempty, bounded and compact in measure subset of . Then
Now, let us consider the following operator defined by
for any , where and the function is increasing and absolutely continuous such that for some constant and for almost all .
Using the same argument as in [7], we can show that for any , we have
Hence, for , we have
and if we take
then
Moreover,
This implies that S and K map the ball into , where with .
Further, let be the subset of consisting of all functions that are a.e. positive and non-increasing on , which is a compact in measure (see [9]), bounded, closed and convex subset of . Therefore, K and S map into (see [[7], p.454]). Whence, for any , we have
Next, by making use of Theorem 4.1, we get
This implies that S is a χ-contraction with the constant or . If we take , then the constant . Hence, neither Darbo’s fixed point nor Sadovski’s fixed point theorem are applied to the operator S. On the other hand, we have , indeed the function for any is a fixed point of K (see [4]), SK maps into and
Thus,
which implies that SK is a χ-contraction with the constant . Now, by Theorem 3.1, S and K have a common fixed point satisfying
This gives rise to a solution of the following equation:
4.2 Common solutions of equations in Banach spaces
Let be a Banach space and B be a convex, closed and bounded subset of X. Denote by the space of all continuous functions from ; , into B endowed with the norm
Assume that
-
(a)
for given fixed , there exists such that
for all , ;
-
(b)
are linear continuous, satisfying for any and .
Theorem 4.2 Under hypotheses (a) and (b), equations (5), (6), (7), and (8) have at least one common solution in .
Proof First, it is clear that is a closed, bounded and convex subset of . On the other hand, by considering , for , we have
This implies that
for any . Furthermore, since any contraction with the constant k is an α-contraction with the same constant k (α is the measure of noncompactness in ), then
Finally, since S and commute, we conclude from Theorem 3.2 that , , and S have a common fixed point. Therefore, equations (5), (6), (7), and (8) have at least one common solution in . □
References
Darbo G: Punti unitti in transformazioni a condominio non compatto. Rend. Semin. Mat. Univ. Padova 1955, 24: 84–92.
Sadovski BN: On a fixed point principle. Funct. Anal. Appl. 1967, 1: 74–76.
Istratescu VI: Fixed Point Theory. Reidel, Boston; 1981.
Hajji A, Hanebaly E: Commuting mappings and α -compact type fixed point theorems in locally convex spaces. Int. J. Math. Anal. 2007, 1(14):661–680.
Banas J, Goebel K Lect. Notes in Math. 60. In Measure of Noncompactness in Banach Spaces. Dekker, Now York; 1980.
Krzyz J: On monotonicity-preserving transformations. Ann. Univ. Mariae Curie-Skl̄odowska, Sect. A 1952, 6: 91–111.
El-Sayed WG: Nonlinear functional integral equations of convolution type. Port. Math. 1997, 54(4):449–456.
Banas J, El-Sayed WG: Measure of noncompactness and solvability of an integral equation in class of functions of locally bounded variation. J. Math. Anal. Appl. 1992, 167: 133–151. 10.1016/0022-247X(92)90241-5
Dunford N, Schwartz JT: Linear Operators I. Interscience, Leyden; 1963.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that he has no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Hajji, A. A generalization of Darbo’s fixed point and common solutions of equations in Banach spaces. Fixed Point Theory Appl 2013, 62 (2013). https://doi.org/10.1186/1687-1812-2013-62
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-62