Common fixed point theorems for multimaps in locally convex spaces and some applications
© Goudarzi et al.; licensee Springer. 2013
Received: 2 September 2012
Accepted: 9 January 2013
Published: 22 February 2013
In this paper some fixed point theorems for multimaps on locally convex spaces have been introduced. As their applications, some well-known fixed point theorems will be deduced. Also, we see a notable result in differential equations.
AMS Subject Classification:46A55, 46B99.
Fixed point theorems have been studied by many authors. Probably the oldest fixed point theorem is the contraction mapping principle. From this theorem, the study of fixed point theorems arose in different settings. More than a thousand of fixed point theorems have been studied and proved by many authors. Among them, there are some notable cases such as Brower’s, Schauder-Tychonov’s, Markov-Kakutani’s and many others. Although many fixed point theorems have been flourishing for both single maps and groups of maps in metric and Banach spaces, only a few have been reported in general topological vector spaces (for example, see [1, 2] and ). Recently, set-valued maps (or multimaps) have been very important tools in non-linear analysis; e.g., see  and . Some existence results have been generalized and extended to multimaps in general spaces (as an example, see ).
In this paper, some new generalized fixed point theorems for multimaps in locally convex spaces have been proved, and as their results, we imply some notable theorems such as Markov-Kakutani and Schauder fixed point theorems. Finally, we see an application of the proved results in differential equations.
Let be a Hausdorff locally convex space. A family of seminorms on X is called an associated family for τ if the family forms a base of neighborhoods of zero for τ, where and . A family of seminorms on X is called an augmented associated family for τ if is an associated family such that for any . We will denote by and the associated and augmented associated seminorms , respectively. As a well-known result, there always exists a family of seminorms on X such that (see [, p.203]). A subset M of X is τ-bounded in X if and only if each is bounded on M.
A subset is said to be star-shaped if there exists an element such that for all and for all . Such an element is called a star-point of A. The set of all star-points of A is called the star-core of A.
for all and . There is an example of an affine mapping with respect to a point which is not affine (see ).
T is said to be upper semicontinuous if for each and each open set V in Y with , there exists an open neighborhood U of x such that for some . Let be a family of multimaps on X into . is a commutating family of multimaps if for each and every pair S, T in , we have . Also, we denote the fixed point set as .
Theorem 2.1 ()
Let K be a compact and star-shaped subset of a topological linear space X. Then every decreasing chain of nonempty, compact and star-shaped subsets of K has a nonempty intersection that is compact and star-shaped.
Lemma 2.2 ()
Suppose that K is a compact star-shaped subset of a topological linear space X and A is the corresponding star-core of K. Then A is a compact convex subset of A.
Lemma 2.3 ()
Let X be a nonempty, compact and convex subset of a Hausdorff locally convex space and let be an upper semicontinuous multimap such that for each , is nonempty, closed and convex. Then there exists a point such that .
Theorem 2.4 (Schauder )
Let X be a nonempty, compact and convex subset of a Hausdorff locally convex space V. If is a continuous map, then T has a fixed point.
3 Main results
3.1 Fixed point results
for all and .
To establish the main results, we need the following lemma.
Lemma 3.2 Let C be a star-shaped subset of a topological vector space X and be onto and q-convex with each q in the star-core of C. If S is the star-core of C, then S is invariant under T.
Hence S is invariant under T. □
Now the main result of this section will be proved.
Then there is a point such that for each , we have .
which is a contradiction. So, the multimap is upper semicontinuous, and for each x, is nonempty, closed and convex. Therefore, there exists such that . By induction, we obtain a family in X which has the finite intersection property. We know that each ℱ is nonempty and compact. So, there exists such that . □
In the sequel, we see some important corollaries that can be implied as the results of the above theorem.
In Theorem 3.3, when is a commutating family of single-valued maps, then the convexity assumption on is actually the affine condition. Also, we can easily see that equation (1) automatically holds. So, we have the following results.
Corollary 3.4 (Markov-Kakutani)
Let X be a nonempty, compact and convex subset of a Hausdorff locally convex space and let be a commutating family of affine mappings. Then there exists an element such that for each .
Corollary 3.5 ()
Let K be a compact star-shaped subset of a locally convex space V. Suppose F is a commutative family of continuous affine mappings of K into itself. Then F has a common fixed point in K.
If we apply the technique introduced in the first part of proof in Theorem 3.3, we have the following extension of Lemma 2.3.
Corollary 3.6 Let X be a nonempty, compact and star-shaped subset of a Hausdorff locally convex space V and let be an upper semicontinuous multimap such that for each , is nonempty, closed and convex. Then there exists a point such that .
Also, if T is a single-valued map, we have the following extension of the Schauder fixed point theorem.
Corollary 3.7 Let X be a nonempty, compact and star-shaped subset of a Hausdorff locally convex space V. If is a continuous map, then T has a fixed point.
3.2 Applications for the Cauchy problem
for each and for all , in the domain of T.
be a continuous operator such that the family is equicontractive. Then operator equation (2) admits a solution in A.
In the sequel, an application of the above theorem will be presented, but first we remind some preliminaries.
in the real scalar case, where . We will show the existence of a solution for equation (5). Suppose that and let stand for the usual -(semi)-norm. Also, assume that . For any fixed , set and . By these assumptions, we have the following theorem.
Theorem 3.10 Let the above conditions hold. If and , then equation (5) admits a (global) solution .
where . This proves the equicontractivity condition and hence the proof is complete. □
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