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Triple fixed point theorems on FLM algebras
Fixed Point Theory and Applications volume 2013, Article number: 16 (2013)
Abstract
This paper considers tripled fixed point theorems on unital without of order semi-simple fundamental locally multiplicative topological algebras (abbreviated by FLM algebras).
MSC:46H.
1 Introduction
Ansari in [1] introduced the notion of fundamental topological spaces and algebras and proved Cohen’s factorization theorem for these algebras. A topological linear space is said to be fundamental if there exists such that for every sequence of , the convergence of to zero in implies that is Cauchy. A fundamental topological algebra is an algebra whose underlying topological linear space is fundamental.
A fundamental topological algebra is called locally multiplicative if there exists a neighborhood of zero such that for every neighborhood V of zero, the sufficiently large powers of lie in V. The fundamental locally multiplicative topological algebras (FLM) were introduced by Ansari in [2]. Some celebrated theorems in Banach algebras were generalized to FLM algebras in [3], and authors investigated some fixed points theorems for holomorphic functions on these algebras (see Theorems 3.5, 3.6 and 3.7 of [3]).
An algebra is called without of order if for every , , then or .
In [4], Bhaskar and Lakshmikantham introduced the notions of a mixed monotone mapping and a coupled fixed point, proved some coupled fixed point theorems for the mixed monotone mapping and discussed the existence and uniqueness of a solution for a periodic boundary value problem. Also, Samet and Vetro studied a coupled fixed point of N-order in [5]. There are many works on a coupled fixed point of contraction, weak contraction and generalized contraction mappings on various metric spaces such as [6–9].
Let be a metric space and let be a function. An element is said to be a tripled fixed point of the mapping F if , and . Tripled fixed point theorems in partially ordered metric spaces were studied by Berinde and Borcut in [10], and this concept was considered by Aydi et al. for weak compatible mappings in abstract metric spaces [11].
In this paper, at first (Section 2) we obtain some basic results for FLM algebras, and next we consider tripled fixed point theorems on FLM algebras.
2 Some results on FLM algebras
By we mean the set of all elements such that , where is the spectral radius of . We denote the center of topological algebra by such that
Definition 2.1 Let be a metrizable topological algebra. We say is a submultiplicatively metrizable topological algebra if
for each and . For abbreviation, we denote by for any .
Let , ℬ and be metric spaces with meters , and , respectively. Then becomes a metric space with the following meter:
for every , and . When , ℬ and are algebras, then by the usual point-wise definitions for addition, scalar multiplication and product, becomes an algebra.
Proposition 2.2 Let , ℬ and be complete metrizable FLM algebras with submultiplicative meters , and , respectively. Then is a complete metrizable FLM algebra with a submultiplicative meter d.
Proof Let , ℬ and be FLM algebras with meters , and , respectively. By the definition of FLM algebras, obviously, is a complete metrizable FLM algebra with a meter d (the meter defined in (2.1)). For submultiplicativity, we have
for every , and . Also,
Therefore, (2.2) and (2.3) show that d is submultiplicative. □
Similar to Definition 2.1, we write as an abbreviation for . We recall the following theorem from [3].
Theorem 2.3 [[3], Theorem 3.3]
Let be a complete metrizable FLM algebra with a submultiplicative meter . Then .
Lemma 2.4 Let , ℬ and be complete metrizable FLM algebras with submultiplicative meters , and , respectively. Then
for any element .
Proof For given , and , we have , and (Theorem 2.3). From Proposition 2.2, it follows that is a complete metrizable FLM algebra with a submultiplicative meter d. Then again, Theorem 2.3 implies that
for every , and . □
Similar to and , we define these sets for as follows:
and
Clearly, if , then and . Also, if , then , and are in , and by Lemma 2.4 and its proof, we have .
Let be the set of all elements for which can be defined. If is a complete metrizable FLM algebra, then ([[12], Theorem 5.4]). Therefore, in the light of Theorem 5.4 of [12] and Proposition 2.2, we have the following theorem.
Theorem 2.5 Let be a complete metrizable FLM algebra, then .
3 Tripled fixed point theorems
In this section, we consider some results about tripled fixed point theorems on unital complete semi-simple metrizable FLM algebras, and we extend these results to Banach algebras. By , we mean the identity map on .
Theorem 3.1 Let be a unital without of order complete semi-simple metrizable FLM algebra with a submultiplicative meter . If is a holomorphic map that satisfies the conditions , , , , , where , , and , where , , then every is a tripled fixed point for F.
Proof Fix and consider the map with . Clearly, f is a holomorphic function on
Since , , , , , where , , and , where , , then F has a Taylor expansion about :
for every . Therefore,
We claim that
is zero for every . Assume towards a contradiction that there exists such that (3.2) is non-zero. Let be an integer such that
Suppose that q is an element of such that . Now, we consider the following five cases:
-
(1)
, ,
-
(2)
, ,
-
(3)
,
-
(4)
,
-
(5)
.
Case (1). In this case, we have . Let , by (3.1) and (3.3), we have
In (3.4), by , we mean the remaining part of . Since , therefore . Then Lemma 2.4 and Lemma 3.6 of [3] imply
where . Now, we define a holomorphic function H from into as follows:
By (3.4) we conclude that . Vesentini’s theorem ([[13], Theorem 3.4.7]) implies that is a subharmonic function on . Moreover, by the maximum principle, we can write . Then Lemma 3.6 of [3] implies that
The above inequality holds for every . Therefore, if , then
for every with . Hence, Theorem 3.4 of [3] implies that is in radical of . Since is semi-simple, therefore . Since , so , and since is without of order, therefore , a contradiction. Thus, our claim is true, and from (3.1), we conclude that . Similarly, we have , and .
Case (2). In this case, we have . Again, by (3.1) and (3.3), we have
Again, by Lemma 2.4 and Lemma 3.6 of [3], we have
where . Now, we define a holomorphic function H from into as follows:
Then from (3.7) it follows that . Then is a subharmonic function on . Moreover, Lemma 3.6 of [3] implies that
The above inequality holds for every . Therefore, if , then
for every with . Hence, Theorem 3.4 of [3] implies that is in radical of . Since is semi-simple, therefore . Since , so . By using that is without of order, we conclude that , a contradiction. Thus, our claim is true, and from (3.1), we conclude that . Similarly, we have , and .
Case (3). In this case, we suppose that , (without loss of generality, we prove this case for only one i and one j such that ). Again by (3.1) and (3.3), we have
By Lemma 2.4 and Lemma 3.6 of [3], we have
where . Now, we define a holomorphic function H from into as follows:
Then from (3.7) it follows that . Then is a subharmonic function on . Moreover, Lemma 3.6 of [3] implies that
The above inequality holds for every . Therefore, if , then
for every with . Hence, is in radical of , therefore . Since and , so and . Again, by using that is without of order, we conclude that , a contradiction. Thus, our claim is true, and from (3.1), we conclude that . Similarly, we have , and .
Case (4). Let . Then we have . Again, by (3.1) and (3.3), we have
Then
where . Now, we define a holomorphic function H from into as follows:
Then from (3.9) it follows that . Then is a subharmonic function on . Therefore,
Therefore, if , then
for every with . Hence, is in radical of . Since is semi-simple, therefore . Since , and , so , and , we conclude that , a contradiction. Thus, our claim is true, and from (3.1), we conclude that . Similarly, we have , and .
Case (5). Now, let . Then we have . Similar to the previous cases, we have
Then
where . Now, we define a holomorphic function H from into as follows:
Then from (3.11) it follows that . Then is a subharmonic function on , and
Therefore, if , then
for every with . Hence, is in radical of . Therefore, . Since , so , then , a contradiction. Thus, (3.1) implies that our claim is true, and from (3.1), we conclude that . Similarly, we have , and .
By gathering the above five cases, we conclude is a tripled fixed point for F, and since was arbitrary, so every point of is a tripled fixed point for F. □
Corollary 3.2 Let be a unital without of order semi-simple Banach algebra. If is a holomorphic map that satisfies the conditions , , , , , where , , and , where , , then every is a tripled fixed point for F.
In the following theorem, we characterize tripled fixed points of holomorphic functions on FLM algebras.
Theorem 3.3 Let be a unital without of order complete semi-simple metrizable FLM algebra. For given , there is a holomorphic map satisfying the conditions , , , , , where , , and , where , , such that , and .
Proof Let . Then there exist such that
Let , then . Define , then
Now, define as follows:
for every in . Clearly, F is a holomorphic function, , , , , , where , , and , where , , but , and similarly, we can show that there is a holomorphic map with the required conditions such that and . □
Example 3.4 Let be the space of real numbers and let be a function defined by that satisfies the conditions of Theorem 3.1.
Example 3.5 Let X be a unital without of order complete semi-simple Banach algebra and let be a function defined by that satisfies the conditions of Theorem 3.1. For example, let be the measure space on a locally compact Hausdorff space G. Another algebra that we can choose is , where G is a locally compact discrete group.
Corollary 3.6 Let be a unital without of order semi-simple Banach algebra. For given , there is a holomorphic map satisfying the conditions , , , , , where , , and , where , , such that , and .
References
Ansari-Piri E: A class of factorable topological algebras. Proc. Edinb. Math. Soc. 1990, 33: 53–59. 10.1017/S001309150002887X
Ansari-Piri E: Topics on fundamental topological algebras. Honam Math. J. 2001, 23: 59–66.
Zohri A, Jabbari A: Generalization of some properties of Banach algebras to fundamental locally multiplicative topological algebras. Turk. J. Math. 2012, 36: 445–451.
Gnana Bhaskar T, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017
Samet B, Vetro C: Coupled fixed point, f -invariant set and fixed point of N -order. Ann. Funct. Anal. 2010, 1(2):46–56.
Abbas M, Sintunavarat W, Kumam P: Coupled fixed point of generalized contractive mappings on partially ordered G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 31
Sintunavarat W, Cho YJ, Kumam P: Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 81
Sintunavarat W, Cho YJ, Kumam P: Coupled fixed point theorems for weak contraction mapping under F -invariant set. Abstr. Appl. Anal. 2012., 2012: Article ID 324874
Sintunavarat W, Cho YJ, Kumam P: Coupled fixed point theorems for contraction mapping induced by cone ball-metric in partially ordered spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 128
Berinde V, Borcut M: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74(15):4889–4897. 10.1016/j.na.2011.03.032
Aydi H, Abbas M, Sintunavarat W, Kumam P: Tripled fixed point of W -compatible mappings in abstract metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 134
Ansari-Piri E: The linear functionals on fundamental locally multiplicative topological algebras. Turk. J. Math. 2010, 34: 385–391.
Aupetit B: A Primer on Spectral Theory. Springer, New York; 1991.
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Razani, A., Hosseinzadeh, H. Triple fixed point theorems on FLM algebras. Fixed Point Theory Appl 2013, 16 (2013). https://doi.org/10.1186/1687-1812-2013-16
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DOI: https://doi.org/10.1186/1687-1812-2013-16