In 1929, Knaster et al. [1] proved the well-known theorem for an -simplex. Ky Fan's generalization of the theorem to infinite dimensional topological vector spaces in 1961 [2] proved to be a very versatile tool in modern nonlinear analysis with many far-reaching applications.

Chang and Yen [3] undertook a systematic study of the property, and Chang et al. [4] generalized this property as well as the notion of a family of [4] to the wider concepts of the - property and its related - family.

Among the many contributions in the study of the property and related topics, we mention the work by Amini et al. [5] where the classes of and - mappings have been introduced in the framework of abstract convex spaces. The authors of [5] also define a concept of convexity that contains a number of other concepts of abstract convexities and obtain fixed point theorems for multifunctions verifying the - property on -spaces that extend results of Ben-El-Mechaiekh et al. [6] and Horvath [7], motivated by the works of Ky Fan [2] and Browder [8]. We refer for the study of these notions to Ben-El-Mechaiekh et al. [9], and more recently, to Park [10], and Kim and Park [11].

In this paper, we use a concept of abstract convexity to define the almost - property, the corresponding notion of almost - family as well as the concept of almost -spaces.

Let and be two sets, and let be a set-valued mapping. We will use the following notations in the sequel;

(i)

(ii)

(iii)

(iv) and

(v)if is a nonempty subset of , then denotes the class of all nonempty finite subsets of .

For the case where and are two topological spaces, a set-valued map is said to be closed if its graph is closed. is said to be compact if the image of under is contained in a compact subset of .

Definition 1.1.

An abstract convex space consists of a nonempty topological space , and a family of subsets of such that and belong to and is closed under arbitrary intersection. This kind of abstract convexity was widely studied; see [5, 9, 12, 13].

Suppose that is a nonempty subset of an abstract convex space . Then

(i)a natural definition of -convex hull of is

(ii)we say that is -convex if for each , .

Remark 1.2.

It is clear that if , then is -convex. That is, each member of is -convex.

Definition 1.3.

We list some properties of a uniform space. A uniformity [14] for a set is a nonempty family of subsets of such that

(i)each member of contains the diagonal where the diagonal denotes the set of all pairs for in ;

(ii)if , then ;

(iii)if , then for some ;

(iv)if , then ;

(v)if and , then .

The pair is called a uniform space. Every member in is called an entourage. An entourage is said to be symmetric if whenever .

Definition 1.4.

If is an abstract convex space with a uniformity , then we say that is an abstract convex uniform space.

Definition 1.5.

Let be a nonempty subset of an abstract convex uniform space which has a uniformity , and has a symmetric basis . Then is called almost -convex if, for any and for any , there exists a mapping such that for all and . Moreover, we call the mapping a -convex-inducing mapping.

Remark 1.6.

It is clear that every -convex set must be almost -convex, but the converse is not true. And in general, the -convex-inducing mapping is not unique. If and , then can be regarded as . If , then can be regarded as .

Recently, Amini et al. [5] introduced the class of multifunctions with the property in abstract convex spaces.

Definition 1.7 (see [5]).

Let be a nonempty set, an abstract convex space, and a topological space. If , and are three multifunctions satisfying

then is called a - mapping with respect to . If the multifunction satisfies the requirement that for any - mapping with respect to , the family has the finite intersection property where denotes the closure of , then is said to have the - property with respect to . We define

We extended the property to the almost property, as follows.

Definition 1.8.

Let be a nonempty set, let be an almost -convex subset of an abstract convex uniform space which has a uniformity and has a symmetric basis , and let be a topological space. If , and are three multifunctions satisfying for each , each , and each , there exists a -convex-inducing mapping such that

then is called an almost - mapping with respect to . If the multifunction satisfies the requirement that for any almost - mapping with respect to , the family has the finite intersection property, then is said to have the almost - property with respect to . We define

From the above definitions, we have the following proposition of the family.

Proposition 1.9.

Let be a nonempty set, let be an almost -convex subset of an abstract convex uniform space , let and be two topological spaces, and let be a multifunction. If and if is continuous, then

The -mappings and the -spaces, in an abstract convex space setting, were also introduced by Amini et al. [5].

Definition 1.10 (see [5]).

Let be an abstract convex space, and a topological space. map is called a -mapping if there exists a multifunction such that

(i)for each , implies , and

(ii).

The mapping is called a companion mapping of .

Furthermore, if the abstract convex space which has a uniformity and has a symmetric basis , then is called a -space if for each entourage , there exists a -mapping such that .

Remark 1.11.

(i)If is a -mapping, then for each nonempty subset of , is also a -mapping.

(ii)It is easy to see that if and , then is also a -space.

In order to establish the main result of this paper for the multifunctions with the almost property, we need the following definitions concerning the almost -mappings and the almost -spaces.

Definition 1.12.

Let be an almost -convex subset of an abstract convex uniform space which has a uniformity and has a symmetric base family , and a topological space. A map is called an almost -mapping if there exists a multifunction such that

(i)for each , and , there exists a -convex-inducing such that , and

(ii)

The mapping is called an almost companion mapping of .

Furthermore, is called an almost -space, if, for each entourage , there exists an almost -mapping such that .

Definition 1.13.

Let be an almost -space, and let . We say that has the approximate fixed point property if, for each , there exists such that .