Open Access

Generalized uniform spaces, uniformly locally contractive set-valued dynamic systems and fıxed points

Fixed Point Theory and Applications20122012:104

DOI: 10.1186/1687-1812-2012-104

Received: 25 October 2011

Accepted: 21 June 2012

Published: 21 June 2012

Abstract

Motivated by classical Banach contraction principle, Nadler investigated set-valued contractions with respect to Hausdorff distances h in complete metric spaces, Covitz and Nadler (Jr.) investigated set-valued maps which are uniformly locally contractive or contractive with respect to generalized Hausdorff distances H in complete generalized metric spaces and Suzuki investigated set-valued maps which are contractive with respect to distances Q p in complete metric spaces with τ-distances p. Here, we provide more general results which, in particular, include the mentioned ones above. The concepts of generalized uniform spaces, generalized pseudodistances in these spaces and new distances induced by these generalized pseudodistances are introduced and a new type of sequential completeness which extended the usual sequential completeness is defined. Also, the new two kinds of set-valued dynamic systems which are uniformly locally contractive or contractive with respect to these new distances are studied and conditions guaranteeing the convergence of dynamic processes and the existence of fixed points of these uniformly locally contractive or contractive set-valued dynamic systems are established. In addition, the concept of the generalized locally convex space as a special case of the generalized uniform space is introduced. Examples illustrating ideas, methods, definitions, and results are constructed, and fundamental differences between our results and the well-known ones are given. The results are new in generalized uniform spaces, uniform spaces, generalized locally convex and locally convex spaces and they are new even in generalized metric spaces and in metric spaces.

MSC: 54C60; 47H10; 54E15; 46A03.

Keywords

generalized uniform space generalized pseudodistance dynamic system uniformly locally contractivity contractivity dynamic process fixed point generalized locally convex space generalized metric space

Introduction

Let 2 X denotes the family of all nonempty subsets of a space X. Recall that a set-valued dynamic system is defined as a pair (X, T), where X is a certain space and T is a set-valued map T : X → 2 X ; in particular, a set-valued dynamic system includes the usual dynamic system where T is a single-valued map.

Let (X, T) be a set-valued dynamic system. By Fix(T) and End(T) we denote the sets of all fixed points and endpoints (or stationary points) of T, respectively i.e., Fix(T) = {w X : w T(w)} and End(T) = {w X : {w} = T (w)}.

A dynamic process or a trajectory starting at w0 X or a motion of the system (X, T) at w0 is a sequence (w m : m {0} ) defined by w m T(wm-1) for m (see, [1, 2]).

If (X, T) is a dynamic system and w0 X then, by O X , T , w 0 , we denote the set of all dynamic processes of the system (X, T) starting at w0.

A beautiful Banach's contraction principle [3] has inspired a large body of work over the last 50 years and there are several ways in which one might hope to improve this principle.

Theorem 1[3]Let (X, d) be a complete metric space. Let T : XX be a single-valued map satisfying the condition
λ [ 0 , 1 ) x , y X { d ( T ( x ) , T ( y ) ) λ d ( x , y ) } .
(1)

Then: (i) T has a unique fixed point w in X, i.e. Fix(T) = {w}; and (ii) the sequence {T[m](u)} converges to w for each u X.

Let (X, d) be a metric space and let CB(X) denote the class of all nonempty closed and bounded subsets of X. If h : CB(X) × CB(X) → [0, ∞) represents a Hausdorff metric induced by d, it has the form
h ( A , B ) = max { sup a A d ( a , B ) , sup b B d ( b , A ) } , A , B C B ( X ) ,

where d(x, C) = infcCd(x, c), x X, C CB(X).

A natural question to ask is whether the single-valued dynamic system in this principle can be replaced by the set-valued dynamic system. One of the first results in this direction was established in [4].

Theorem 2 [[4], Th. 5] Let (X, d) be a complete metric space. Assume that the set-valued dynamic system (X, T) satisfying T : XCB(X) is (h, λ)-contractive, i.e.,
λ [ 0 , 1 ) x , y X { h ( T ( x ) , T ( y ) ) λ d ( x , y ) } .
(2)

Then T has a fixed point w in X, i.e. w T(w).

There are other important ways of extending the Banach theorem. In particular, many interesting theorems in this setting, proposed by Covitz and Nadler, Jr. [[5], Theorem 1], concern the set-valued dynamic systems in generalized metric spaces.

The concepts of generalized metric spaces and the canonical decompositions of these spaces appeared first in Luxemburg [6] and Jung [7]. Recall that a generalized metric space is a pair (X, d) where X is a nonempty set and d : X2 → [0, ∞] satisfies: (a) x,yX{d(x, y) = 0 iff x = y}; (b) x,yX{d(x, y) = d(y, x); (c) x,y,zX{[d(x, z) < +∞ d(y, z) < +∞] [d(x, y) < + ∞ d(x, y) ≤ d(x, z) + d(z, y)]}. Some characterizations of these spaces were presented by Jung [7] who proved the essential theorems about decomposition of a generalized metric spaces and discovered the way to obtain generalized (complete) metric spaces. Let X β , d β : β B , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq3_HTML.gif -index set, be a family of disjoint metric spaces. If X = β B X β and, for any x, y X,
d ( x , y ) = d β ( x , y ) if x , y X β , β B + if x X β 1 , y X β 2 , β 1 , β 2 B , β 1 β 2
then (X, d) is a generalized metric space. Moreover, if for each β B , (X β , d β ) is complete then (X, d) is a generalized complete metric space. Also, in generalized metric spaces (X, d) he introduced the following equivalence relation on X:
x ~ y i d ( x , y ) < + , x , y X .

Therefore, X is decomposed uniquely into (disjoint) equivalence classes X β : β B , which is called a canonical decomposition. We may read these results as follows.

Theorem 3[7]Let (X, d) be a generalized metric space, let X = β B X β be the canonical decomposition and let β B d β = d | X β × X β . Then: (I) For each β B , (X β , d β ) is a metric space; (II) For any β 1 , β 2 B , with β1β2, d(x, y) = +∞ for any x X β 1 and y X β 2 ; and (III) (X, d) is a generalized complete metric space iff, for each β B , (X β , d β ) is a complete metric space.

Before presenting the results of Covitz and Nadler, Jr. [5] we recall some notations.

Definition 1 Let (X, d) be a generalized metric space.

(a) We say that a nonempty subset Y of X is closed in X if Y = Cl(Y) where Cl(Y), the closure of Y in X, denote the set of all x X for which there exists a sequence (x m : m ) in Y which is d-convergent to x.

(b) The class of all nonempty closed subsets of X is denoted by C(X), i.e. C(X) = {Y : Y 2 X Y = Cl(Y)}.

(c) A generalized Hausdorff distance H : C(X) × C(X) → [0, ∞] induced by d is defined by: for each A, B C(X),
H ( A , B ) = inf { ε > 0 : A N ( ε , B ) B N ( ε , A ) } if is nite + otherwise

where, for each E C(X) and ε > 0, N(ε, E) = {x X : eE{d(x, e) < ε}}.

Theorem 4 [[5], Theorem 1] Let (X, d) be a generalized complete metric space and let w0 X. Assume that a set-valued dynamic system (X, T) satisfying T : XC(X) is (H, ε, λ)-uniformly locally contractive, i.e.
ε ( 0 , ] λ [ 0 , 1 ) x , y X { d ( x , y ) < ε H ( T ( x ) , T ( y ) ) λ d ( x , y ) } .

Then the following alternative holds: either

(A) w m : m { 0 } O ( X , T , w 0 ) m d w m - 1 , w m ε ; or

(B) w m : m { 0 } O ( X , T , w 0 ) w X w F i x ( T ) lim m w m = w .

It is not hard to see that each (H, λ)-contractive set-valued dynamic system defined below is, for each ε (0, + ∞), (H, ε, λ)-uniformly locally contractive.

Theorem 5 [[5], Corollary 1] Let (X, d) be a generalized complete metric space and let w0 X. Assume that the set-valued dynamic system (X, T) satisfying T : XC(X) is (H, λ)-contractive, i.e.,
λ [ 0 , 1 ) x , y X { H ( T ( x ) , T ( y ) ) λ d ( x , y ) } w h e n e v e r d ( x , y ) < .
(3)

Then the following alternative holds: either

(A) w m : m { 0 } O ( X , T , w 0 ) m d w m - 1 , w m = ; or

(B) w m : m { 0 } O ( X , T , w 0 ) w X w F i x ( T ) lim m w m = w .

The following follows from Theorem 5 and generalize Nadler's Theorem 2.

Theorem 6 [[5], Corollary 3] Let (X, d) be a complete metric space and let w0 X. Assume that a set-valued dynamic system (X, T) satisfying T : XC(X) is (h, λ)-contractive, i.e.
λ [ 0 , 1 ) x , y X { h ( T ( x ) , T ( y ) ) λ d ( x , y ) } .
(4)

Then w m : m { 0 } O ( X , T , w 0 ) w X w F i x ( T ) lim m w m = w .

Recall that the investigations of fixed points of maps in complete generalized metric spaces appeared for the first time in Diaz and Margolis [8] and Margolis [9].

Another natural problem is to extend the Nadler's [[4], Th. 5] theorem to set-valued dynamic systems which are contractive with respect to more general distances. In complete metric spaces, this line of research was pioneered by Suzuki [10], who developed many crucial technical tools.

Definition 2[11] Let (X, d) be a metric space. A map p : X × X → [0, ∞) is called a τ-distance on X if there exists a map η : X × [0, ∞) → [0, ∞) and the following conditions hold: (S1) x,y,zX{p(x, z) ≤ p(x, y) + p(y, z)}; (S2) xXt>0{η(x, 0) = 0 η(x, t) ≥ t} and η is concave and continuous in its second variable; (S3) limn→∞x n = x and limn→∞supmnη(z n , p(z n , x m )) = 0 imply that wX{p(w, x) ≤ lim infn→∞p(w, x n )}; (S4) limn→∞supmnp(x n , y m )) = 0 and limn→∞η(x n , t n ) = 0 imply that limn→∞η(y n , t n ) = 0; and (S5) limn→∞η(z n , p(z n , x n )) = 0 and limn→∞η(z n , p(z n , y n )) = 0 imply that limn→∞d(x n , y n ) = 0.

Theorem 7 [[10], Theorem 3.7] Let (X, d) be a complete metric space and let p be a τ-distance on X. Let a set-valued dynamic system (X, T) satisfying T : XC(X) be (Q p , λ)-contractive, i.e.
λ [ 0 , 1 ) x , y X { Q p ( T ( x ) , T ( y ) ) λ p ( x , y ) }
(5)

where Q p (A, B) = supaAinfbBp(a, b). Then there exists w X such that w T(w) and p(w, w) = 0.

Remark 1 Let us observe that this beautiful Suzuki's theorem include Covitz-Nadler's Theorem 6. Indeed, first we see that each metric d is τ-distance (cf. [11]) and next we see that each (h, λ)-contractive set-valued dynamic system (X, T) satisfying T : XC(X) is (Q d , λ)-contractive; in fact, Q d h on C(X) (cf. [12]). Moreover, there exist (Q d , λ)-contractive set valued dynamic systems (X, T) satisfying T : XC(X) which are not (h, λ)-contractive.

It is worth noticing that a number of authors introduce the new various concepts of set-valued contractions of Nadler type in complete metric spaces, study the problem concerning the existence of fixed points for such contractions and obtain the various generalizations of Nadler's result which are different from the mentioned above; see, e.g., Takahashi [13], Jachymski [[14], Theorem 5], Feng and Liu [12], Zhong et al. [15], Mizoguchi and Takahashi [16], Eldred et al. [17], Suzuki [18], Kaneko [19], Reich [20, 21], Quantina and Kamran [22], Suzuki and Takahashi [23], Al-Homidan et al. [24], Latif and Al-Mezel [25], Frigon [26], Klim and Wardowski [27], Ćirić [28] and Pathak and Shahzad [29].

The above are some of the reasons why in nonlinear analysis the study of uniformly locally contractive and contractive set-valued dynamic systems play a particularly important part in the fixed point theory and its applications.

Let us notice that in the proofs of the results of [329], among other things, the following assumptions and observations are essential: (O1) The completeness of metric and generalized metric spaces is necessary; (O2) In Theorems 1, 2 and 4-7, the maps T : (X, d) → (X, d), T : (X, d) → (CB(X), h), T : (X, d) → (C(X), H) and T : (X, p) → (C(X), Q p ) are investigated and the conditions (1)-(5) imply that these maps between spaces (X, d), (X, p), (CB(X), h), (C(X), H) and (C(X), Q p ), respectively, are continuous; (O3) By Theorems 1, 2 and 4-7, for each w Fix(T) the following equalities d(w, w) = 0, h(T(w), T(w)) = 0, H(T(w), T(w)) = 0, Q p (T(w), T(w)) = 0 and p(w, w) = 0 hold, respectively; (O4) The distances h, H, and Q p are defined only on the spaces CB(X) or C(X), respectively.

Also, let us observe that in [3036] we studied some families of generalized pseudodistances in uniform spaces and generalized quasipseudodistances in quasigauge spaces which generalize: metrics, distances of Tataru [37], w-distances of Kada et al. [38], τ- distances of Suzuki [11] and τ-functions of Lin and Du [39] in metric spaces and distances of Vályi [40] in uniform spaces.

Motivated by the comments and observations stated above our main interest of this article is the following:

Question 1 Are there spaces X, new distances on X which are more general than d, h, H, p and Q p , and set-valued dynamic systems (X, T) which are uniformly locally contractive or contractive with respect to new distances, such that the analogous assertions as in Theorems 1, 2 and 4-7 hold but, unfortunately: (M1) Spaces X (metric, generalized metric and more general) are not necessarily complete; (M2) If new distances we replaced by d, h, H, p or Q p then maps T are not necessarily continuous in the sense defined by inequalities (1)-(5), respectively; (M3) For T, w Fix(T) and for new distances the properties in (O3) do not necessarily hold in such generality; (M4) The new distances are defined on 2 X , and thus not only on CB(X) or C(X) as in (O4)?

Our purpose in this article is to answer our question in the affirmative and providing the illustrating examples. More precisely, inspired by ideas of Diaz and Margolis [8], Margolis [9], Luxemburg [6], Jung [7], Nadler [[4], Th. 5], Covitz and Nadler [5] and Suzuki [10] and the above comments and observations, the concepts of the families D = d α : X × X [ 0 , ] , α A ( https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq16_HTML.gif -index set) of generalized pseudometrics on a nonempty set X and the generalized uniform spaces (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) are introduced, the classes L ( X , D ) of https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -families of generalized pseudodistances in (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) are defined and, in (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ), a new type of https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -sequentially completeness with respect to https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -families (which extend the usual sequentially completeness in uniform and locally convex spaces and completeness in metric and generalized metric spaces) are studied (see the following section). Moreover, some partial quasiordered space K A is defined (see Section "Partial quasiordered space K A ") and, using K A , ( i ) L -distances on 2 X (i {1, 2}) with respect to https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -families are introduced (see Section " ( i ) L -distances on 2 X , i {1,2}"). Also, we introduce the definitions of ( i ) L , ϒ , Λ -uniformly locally contractive and ( i ) L , Λ -contractive set-valued dynamic systems (X, T) (i {1, 2}) satisfying T : X → 2 X (see Section " ( i ) L , ϒ , Λ -uniformly locally contractive and ( i ) L , Λ -contractive set-valued dynamic systems (X, T), i {1, 2}") and, for w0 X, we establish the conditions guaranteeing the convergence of dynamic processes O X , T , w 0 and the existence of fixed points for such contractions and, additionally, a special case when T : XC(X) and L = D is studied (see Sections 6-8). Also the concept of the generalized locally convex space as a special case of the generalized uniform space is introduced (see Section "Generalized locally convex spaces X , P "). By generality of spaces and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -families, our results, in particular, include and essentially generalize Theorems 1, 2 and 4-7. The examples illustrating ideas, methods and results are constructed and comparisons of our results with the results of Nadler [[4], Th. 5], Covitz and Nadler [5] and Suzuki [10] are given (see Sections 10-13). Finally, a natural question is formulated (see Section "Concluding remarks"). The results are new in generalized uniform spaces, uniform spaces, generalized locally convex and locally convex spaces and are new even in generalized metric spaces and in metric spaces.

Generalized uniform spaces (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) and the class L ( X , D ) of https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -families of generalized pseudodistances on (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif )

The following terminologies will be much used.

Definition 3 Let X be a nonempty set. (a) The family
D = { d α : X × X [ 0 , ] , α A } , A  - index set ,

is said to be a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -family of generalized pseudometrics on X ( https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -family on X, for short) if the following three conditions hold:

( D 1 ) α A x X d α ( x , x ) = 0 ;

( D 2 ) α A x , y X d α ( x , y ) = d α ( y , x ) ; and

( D 3 ) If α A and x, y, z X and if d α (x, z) and d α (y, z) are finite, then d α (x, y) is finite and d α (x, y) ≤ d α (x, z) + d α (z, y).

(b) If https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -family, then the pair (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) is called a generalized uniform space.

(c) Let (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) be a generalized uniform space. A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -family https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif is said to be separating if

( D 4 ) x , y X x y α A 0 < d α ( x , y ) . .

(d) If a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -family https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif is separating, then the pair (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) is called a Hausdorff generalized uniform space.

(e) Let (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) be a generalized uniform space and let (x m : m ) be a sequence in X. We say that (x m : m ) is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -Cauchy sequence in X if α A lim n sup m > n d α x n , x m = 0 . We say that (x m : m ) is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -convergent in X if there is an x X such that a A lim m d α ( x m , x ) = 0 ( a A lim m x m = x , for short).

(f) If every https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -Cauchy sequence in X is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -convergent sequence in X, then a pair (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) is called a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -sequentially complete generalized uniform space.

Definition 4 Let X be a nonempty set. The family
Q = { q α : X × X [ 0 , ] , α A } , A  - index set ,

is said to be a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq37_HTML.gif -family of generalized quasi pseudometrics on X ( https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq37_HTML.gif -family on X, for short) if the following two conditions hold:

( Q 1 ) α A x X q α ( x , x ) = 0 ;

( Q 2 ) If α A and x, y, z X and if q α (x, z) and q α (z, y) are finite, then q α (x, y) is finite and q α (x, y) ≤ q α (x, z) + q α (z, y).

Definition 5 Let (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) be a generalized uniform space.

(a) The family
L = { L α : X × X [ 0 , ] , α A } , A  - index set ,

is said to be a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -family of generalized pseudodistances on X ( https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -family on X, for short) if the following two conditions hold:

( L 1 ) If α A and x, y, z X and if L α (x, z) and L α (z, y) are finite, then L α (x, y) is finite and L α (x, y) ≤ L α (x, z) + L α (z, y); and

( L 2 ) For any sequences (x m : m ) and (y m : m ) in X such that
α A { lim n sup m > n L α ( x n , x m ) = 0 }
(6)
and
α A { lim m L α ( x m , y m ) = 0 } ,
(7)
the following holds
α A { lim m d α ( x m , y m ) = 0 } .
(8)
(b) Let L ( X , D ) be a class defined as follows
L ( X , D ) = { L : L is L  - family on X } .

Remark 2 Let (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) be a generalized uniform space. (i) L ( X , D ) since D L ( X , D ) . (ii) L ( X , D ) { D } ; see Sections 10-13.

Definition 6 Let (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) be a generalized uniform space, let L L ( X , D ) and let (x m : m ) be a sequence in X.

(a) We say that (x m : m ) is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -Cauchy in X if α A lim n sup m > n L α x n , x m = 0 .

(b) We say that (x m : m ) is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -convergent in X if there exists x X such that a A lim m L α ( x m , x ) = 0 .

(c) We say that (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -sequentially complete if each https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -Cauchy sequence in X is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -convergent in X.

In the following remark, we list some basic properties of https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -families.

Remark 3 Let (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) be a generalized uniform space and let L L ( X , D ) . (i) If α A x X L α ( x , x ) = 0 , then https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif is a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq37_HTML.gif -family on X; examples of L L ( X , D ) which are not https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq37_HTML.gif -families on X are given in Section "Examples of the decompositions of the generalized uniform spaces". (ii) There exist https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -sequentially complete spaces which are not https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -sequentially complete; see Example 15. (iii) If (x m : m ) in X is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -convergent in X, then its limit point is not necessary unique; see Example 1.

Example 1 Let (, |·|) be a metric space. Define the family of L = L : × [ 0 , ] to be
L ( x , y ) = 0 if x y 1 if x > y x , y .

It is obvious that https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -family on and the sequence (1/m : m ) is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -convergent to each point w (0, +∞).

One can prove the following proposition:

Proposition 1 Let (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) be a Hausdorff generalized uniform space and let L L ( X , D ) .

(I) If xy, x, y X, then α A L α ( x , y ) > 0 L α ( y , x ) > 0 .

(II) If (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -sequentially complete and if (x m : m ) is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -Cauchy sequence in X, then (x m : m ) is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -convergent in X.

Proof. (I)) Assume that there are xy, x, y X, such that α A L α ( x , y ) = L α ( y , x ) = 0 . Then, α A L α ( x , x ) = 0 , since, by using ( L 1 ), it follows that α A L α ( x , x ) L α ( x , y ) + L α ( y , x ) = 0 . Defining the sequences (x m : m ) and (y m : m ) in X by x m = x and y m = y for m , and observing that α A L α ( x , y ) = L α ( y , x ) - L α ( x , x ) = 0 , this implies that (6) and (7) for these sequences hold. Then, by ( L 2 ), (8) holds, so it is α A d α ( x , y ) = 0 . On the other hand, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif is separating, so, since xy, it is α A d α ( x , y ) 0 . This leads to a contradiction.

(II) Since α A lim n sup m > n L α ( x n , x m ) = 0 , by Definition 6(c), this proves the existence of x X such that a A lim m L α ( x m , x ) = 0 . We can apply ( L 2 ) to sequences (x m : m ) and (y m = x/ : m ) and then we find that α A lim m d α x m , y m = lim m d α ( x m , x ) = 0 . The uniqueness of the point of x follows from the fact that https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif is separating. □

Partial quasiordered space K A

Proposition 2 Let K A be a set of elements Θ = η α : α A defined by the formula
K A = { Θ = ( η α : α A ) : α A { η α [ - , ] } } , A  -  i n d e x s e t ,
and let Θ = η α : α A K A α A Θ α = η α . The relation K A on K A defined by
Θ = ( η α : α A ) , Ω = ( ω α : α A ) K A { Θ K A Ω α A { η α = [ Θ ] α [ Ω ] α = ω α } }

is a partial quasiordered on K A and the pair K A , K A is a partial quasiordered space.

Proof. For all Θ K A the condition Θ K A Θ holds. For all Θ , Ω , ϒ K A , the conditions Θ K A Ω and Ω K A ϒ imply Θ K A ϒ . For all Θ , Ω K A , the conditions Θ K A Ω and Ω K A Θ imply Θ = Ω. □

Notation. The following notation is fixed throughout the article:

Θ 0 = ( η α = 0 : α A ) ;

Θ + = ( η α = + : α A ) ;

K 0 , + A = { Θ K A : Θ 0 K A Θ Θ K A Θ + } ;

K + A = { Θ = ( η α : α A ) K A : α A { η α ( 0 , + ) } } .

In the sequel, if Θ , Ω K A , then Θ K A Ω will stand for Θ K A Ω and Θ ≠ Ω.

Definition 7 Let S A be a nonempty subset of K A . We say that I S A = I F ( S A ) K A is a infimum of S A if the following two conditions hold:

( I 1 ) Θ S A { I S A K A Θ } ;

( I 2 ) Ω K A { { I S A K A Ω } Θ S A { Θ K A Ω } .

Example 2 Let A = { 1 , 2 , 3 } and let K A = { Θ = ( η 1 , η 2 , η 3 ) : α A { η α [ - , ] } } . If S 1 A = { ( 3 , 5 , 7 ) , ( 4 , 1 , 8 ) } then S 1 A K A and I F ( S 1 A ) does not exist since (3, 5, 7) and (4, 1, 8) are not comparable. If S 2 A = { ( 3 , 5 , 7 ) , ( 4 , 6 , 8 ) } then S 2 A K A and I F ( S 2 A ) = ( 3 , 5 , 7 ) .

( i ) L -distances on 2 X , i {1, 2}

Definition 8 Let (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) be a Hausdorff generalized uniform space and let L L ( X , D ) .

(a) For C 2 X and Θ = η α : α A K + A , let us denote
U L ( Θ , C ) = { u X : c C α A { L α ( u , c ) < η α } } .
(9)
(b) For A, B 2 X let us denote:
H ( 1 ) L ( A , B ) = { Θ K + A : A U L ( Θ , B ) } ,
(10)
H ( 2 ) L ( A , B ) = { Θ K + A : A U L ( Θ , B ) B U L ( Θ , A ) } .
(11)
(c) Let i {1, 2}. The map ( i ) L : 2 X × 2 X K 0 , + A of the form
( i ) L ( A , B ) = I F ( H ( i ) L ( A , B ) ) if I F ( H ( i ) L ( A , B ) ) exists and α A { [ I F ( H ( i ) L ( A , B ) ) ] α < + } Θ + if I F ( H ( i ) L ( A , B ) ) does not exist or if  I F ( H ( i ) L ( A , B ) ) exists and α A { [ I F ( H ( i ) L ( A , B ) ) ] α = + } ,

A, B 2 X , is called a ( i ) L -distance on 2 X generated by https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif ( ( i ) L -distance on 2 X , for short).

Remark 4 For each A, B 2 X , ( 1 ) L ( A , B ) K A ( 2 ) L ( A , B ) .

( i ) L , ϒ , Λ -uniformly locally contractive and ( i ) L , Λ -contractive set-valued dynamic systems (X, T), i {1, 2}

Definition 9 Let (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) be a Hausdorff generalized uniform space, let L L ( X , D ) and let i {1,2}.

(a) Let ( i ) L be a ( i ) L -distance on 2 X and let ϒ = ( ε α : α A ) K A and Λ = ( λ α : α A ) K A be such that α A { ε α ( 0 , ) λ α [ 0 , 1 ) } . We say that a set-valued dynamic system (X, T), T : X → 2 X , is ( i ) L , ϒ , Λ -uniformly locally contractive on X if
α A x , y X { L α ( x , y ) < ε α [ ( i ) L ( T ( x ) , T ( y ) ) ] α λ α L α ( x , y ) } .
(12)
(b) Let ( i ) L be a ( i ) L -distance on 2 X and let Λ = ( λ α : α A ) K A be such that α A { λ α [ 0 , 1 ) } . We say that a set-valued dynamic system (X, T), T : X → 2 X , is ( i ) L , Λ - contractive on X if
α A x , y X { [ ( i ) L ( T ( x ) , T ( y ) ) ] α λ α L α ( x , y ) } .
(13)

Remark 5 Let (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) be a Hausdorff generalized uniform space, let L L ( X , D ) and let ϒ = ( ε α : α A ) K A and Λ = ( λ α : α A ) K A be such that α A { ε α ( 0 , ) λ α [ 0 , 1 ) } .

(i) If (X, T), T : X → 2 X , is ( ( 2 ) L , ϒ , Λ ) -uniformly locally contractive on X then it is ( ( 1 ) L , ϒ , Λ ) -uniformly locally contractive on X.

(ii) If (X, T), T : X → 2 X , is ( ( 2 ) L , Λ ) -contractive on X then it is ( ( 1 ) L , Λ ) -contractive on X.

(iii) Let i {1, 2}. If (X, T), T : X → 2 X , is ( i ) L , Λ -contractive on X then it is ( i ) L , ϒ , Λ -uniformly locally contractive on X.

Statement of results

Definition 10 Let (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) be a Hausdorff generalized uniform space and let x X/We say that a set-valued dynamic system (X, T), T : X → 2 X , is closed at x if whenever (x m : m ) is a sequence https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -converging to x in X and (y m : m ) is a sequence https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -converging to y in X such that y m T(x m ) for all m , then y T(x).

The main existence and convergence result of this article we can now state as follows.

Theorem 8 Assume that (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) is a Hausdorff generalized uniform space, L L ( X , D ) and one of the following properties holds:

(P1) (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -sequentially complete; or

(P2) (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -sequentially complete.

Let i {1, 2}, let ( i ) L : 2 X × 2 X K 0 , + A be a ( i ) L -distance on 2 X and assume that a set-valued dynamic system (X, T), T : X → 2 X , has the property

(C) w 0 X ( w m : m { 0 } ) O ( X , T , w 0 ) w X {limm→∞w m = w T is closed at w}.

(I) If ϒ = ( ε α : α A ) K A and Λ = ( λ α : α A ) K A satisfy α A { ε α ( 0 , ) λ α [ 0 , 1 ) } and (X, T) is ( i ) L , ϒ , Λ -uniformly locally contractive on X then, for each w0 X, the following alternative holds: either

(A1) ( w m : m { 0 } ) O ( X , T , w 0 ) m α 0 A { L α 0 ( w m - 1 , w m ) ε α 0 } ; or

(A2) ( w m : m { 0 } ) O ( X , T , w 0 ) w X {w Fix(T) limm→∞w m = w (w m : m {0} ) is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -Cauchy}.

(II) If Λ = ( λ α : α A ) K A satisfies α A { λ α [ 0 , 1 ) } and (X, T) is ( i ) L , Λ -contractive on X then, for each w0 X, the following alternative holds: either

(B1) ( w m : m { 0 } ) O ( X , T , w 0 ) m α 0 A { L α 0 ( w m - 1 , w m ) = } ; or

(B2) ( w m : m { 0 } ) O ( X , T , w 0 ) w X {w Fix(T) limm→∞w m = w (w m : m {0} ) is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -Cauchy}.

Definition 11 Let (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) be a Hausdorff generalized uniform space.

(a) We say that a nonempty subset Y of X is closed in X if Y = Cl(Y) where Cl(Y), the closure of Y in X, denotes the set of all x X for which there exists a sequence (x m : m ) in Y which is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -convergent to x.

(b) The class of all nonempty closed subsets of X is denoted by C(X), i.e. C(X) = {Y : Y 2 X Y = Cl(Y)}.

Theorem 8 has the following corresponding when L = D and when T : XC(X).

Theorem 9 Let (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) be a Hausdorff https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -sequentially complete generalized uniform space, let i {1, 2} and assume that ( i ) D : C ( X ) × C ( X ) K 0 , + A is a ( i ) D -distance on C(X).

(I) If ϒ = ( ε α : α A ) K A and Λ = ( λ α : α A ) K A satisfy α A { ε α ( 0 , ) λ α [ 0 , 1 ) } and if a set-valued dynamic system (X, T) satisfying T : XC(X) is ( ( i ) D , ϒ , Λ ) -uniformly locally contractive on X then, for each w0 X, the following alternative holds: either

(F1) ( w m : m { 0 } ) O ( X , T , w 0 ) m α 0 A { L α 0 ( w m - 1 , w m ) ε α 0 } ; or

(F2) ( w m : m { 0 } ) O ( X , T , w 0 ) w X { w F i x ( T ) lim m w m = w } .

(II) If Λ = ( λ α : α A ) K A satisfies α A { λ α [ 0 , 1 ) } and a set-valued dynamic system (X, T) satisfying T : XC(X) is ( ( i ) D , Λ ) -contractive on X then, for each w0 X, the following alternative holds: either

(G1) ( w m : m { 0 } ) O ( X , T , w 0 ) m α 0 A { d α 0 ( w m - 1 , w m ) = } ; or

(G2) ( w m : m { 0 } ) O ( X , T , w 0 ) w X { w F i x ( T ) lim m w m = w } .

Proof of Theorem 8

(I) Let i {1, 2}. The proof is divided into three steps.

Step 1. Assume that w0 X and suppose that the assertion (A1) does not hold; that is,
( v m : m { 0 } ) O ( X , T , v 0 = w 0 ) m 0 α A { L α ( v m 0 - 1 , v m 0 ) < ε α } .
(14)
Then there exists ( w m : m { 0 } ) O ( X , T , w 0 ) which is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -Cauchy sequence on X; that is,
α A { lim n sup m > n L α ( w n , w m ) = 0 } .
(15)
Indeed, since (14) holds, thus, by (12), we get
α A { [ ( i ) L ( T ( v m 0 - 1 ) , T ( v m 0 ) ) ] α λ α L α ( v m 0 - 1 , v m 0 ) < λ α ε α } .
(16)
It follows from (16) and Definition 8(c), that there exists I F ( H ( i ) L ( T ( v m 0 - 1 ) , T ( v m 0 ) ) ) and
α A { [ I F ( H L ( i ) ( T ( v m 0 - 1 ) , T ( v m 0 ) ) ) ] α < λ α ε α } .
(17)
From this, denoting Ω = { λ α ε α : α A } K A , we deduce that I F ( H ( i ) L ( T ( v m 0 - 1 ) , T ( v m 0 ) ) ) K A Ω . Consequently, by ( I 2 ), there exists
Θ H ( i ) L ( T ( v m 0 - 1 ) , T ( v m 0 ) )
(18)
such that Θ K A Ω which implies
α A { [ Θ ] α [ Ω ] α = λ α ε α } and Θ Ω .
(19)
If i = 1, then we note that, by (18), (9), and (10), T ( v m 0 - 1 ) U L ( Θ , T ( v m 0 ) ) . Clearly, v m 0 T ( v m 0 - 1 ) . Thus, v m 0 U L ( Θ , T ( v m 0 ) ) and the conclusion
u m 0 + 1 T ( v m 0 ) α A { L α ( v m 0 , u m 0 + 1 ) < [ Θ ] α λ α ε α }

follows directly from (9), (10), (18), and (19).

If i = 2, then we also note that, by (18), (9) and (11), T ( v m 0 - 1 ) U L ( Θ , T ( v m 0 ) ) and T ( v m 0 ) U L ( Θ , T ( v m 0 - 1 ) ) . Clearly, v m 0 T ( v m 0 - 1 ) . Thus, v m 0 U L ( Θ , T ( v m 0 ) ) and the conclusion
u m 0 + 1 T ( v m 0 ) U L ( Θ , T ( v m 0 - 1 ) ) α A { L α ( v m 0 , u m 0 + 1 ) < [ Θ ] α λ α ε α }

follows directly from (9), (11), (18), and (19).

This proves
u m 0 + 1 T ( v m 0 ) α A { L α ( v m 0 , u m 0 + 1 ) < λ α ε α } .
(20)
Since, by (20), α A { L α ( v m 0 , u m 0 + 1 ) < ε α } , it follows, using (12) and (20), that
α A { [ ( i ) L ( T ( v m 0 ) , T ( u m 0 + 1 ) ) ] α λ α L α ( v m 0 , u m 0 + 1 ) < ( λ α ) 2 ε α } .
That is,
α A { [ I F ( H ( i ) L ( T ( v m 0 ) , T ( u m 0 + 1 ) ) ) ] α < ( λ α ) 2 ε α } .
(21)
Denoting Δ = { ( λ α ) 2 ε α : α A } K A , we see that condition (21) implies I F ( H ( i ) L ( T ( v m 0 ) , T ( u m 0 + 1 ) ) ) K A Δ . Hence, by ( I 2 ), there exists
Π H ( i ) L ( T ( v m 0 ) , T ( u m 0 + 1 ) )
(22)
such that Π K A Δ . This means
α A { [ Π ] α [ Δ ] α = ( λ α ) 2 ε α } and Π Δ .
(23)
Let i = 1. Clearly, by (9), (10), and (22), T ( v m 0 ) U L ( Π , T ( u m 0 + 1 ) ) . Moreover, by (20), u m 0 + 1 T ( v m 0 ) . Therefore u m 0 + 1 U L ( Π , T ( u m 0 + 1 ) ) . This, by (9), (10) and (21)-(23), implies
u m 0 + 2 T ( u m 0 + 1 ) α A { L α ( u m 0 + 1 , u m 0 + 2 ) < [ Π ] α < ( λ α ) 2 ε α } .
Let i = 2. Clearly, by (9)-(11) and (22), T ( v m 0 ) U L ( Π , T ( u m 0 + 1 ) ) and T ( u m 0 + 1 ) U L ( Π , T ( v m 0 ) ) . Moreover, u m 0 + 1 T ( v m 0 ) . Therefore u m 0 + 1 U L ( Π , T ( u m 0 + 1 ) ) . This, by (9)-(11) and (21)-(23), implies
u m 0 + 2 T ( u m 0 + 1 ) U L ( Π , T ( v m 0 ) ) α A { L α ( u m 0 + 1 , u m 0 + 2 ) < [ Π ] α < ( λ α ) 2 ε α } .
That is,
u m 0 + 2 T ( u m 0 + 1 ) α A { L α ( u m 0 + 1 , u m 0 + 2 ) < ( λ α ) 2 ε α } .
(24)
By (24), we have α A { L α ( u m 0 + 1 , u m 0 + 2 ) < ε α } and, using (12) and (24), we get
α A { [ ( i ) L ( T ( u m 0 + 1 ) , T ( u m 0 + 2 ) ) ] α λ α L α ( u m 0 + 1 , u m 0 + 2 ) < ( λ α ) 3 ε α } .
This means
α A { [ I F ( H ( i ) L ( T ( u m 0 + 1 ) , T ( u m 0 + 2 ) ) ) ] α < ( λ α ) 3 ε α } .
(25)
By induction, a similar argument as in the proofs of (17)-(25) shows that
( u m 0 + n : n { 0 } ) O ( X , T , u m 0 = v m 0 ) α A n { 0 } { u m 0 + n + 1 T ( u m 0 + n ) L α ( u m 0 + n , u m 0 + n + 1 ) < ( λ α ) n + 1 ε α [ ( i ) ( T ( u m 0 + n ) , T ( v m 0 + n + 1 ) ) ] α λ α L α ( u m 0 + n , u m 0 + n + 1 ) } .
(26)

It is clear that (26) implies that ( w m : m { 0 } ) O ( X , T , w 0 ) where m < m 0 { w m = v m } , w m 0 = u m 0 = v m 0 and m > m 0 { w m = u m } . Additionally, this sequence (w m : m {0} ) is a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -Cauchy sequence on X, i.e., (15) holds.

Step 2. Assume that the condition (C) and the property (P1) hold. If w0 X and the assertion (A1) does not hold, then (A2) holds.

By Step 1, Definition 8(c) and (P1) (note that then (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -sequentially complete), we have that there exists w X satisfying
α A { lim m L α ( w m , w ) = 0 } .
(27)
Applying (15), (27), and ( L 2 ) (where (x m = w m : m ) and (y m = w : m )), we find that
α A { lim m d α ( w m , w ) = 0 } .
(28)

Clearly, since (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) is Hausdorff, condition (28) implies that such a point w is unique.

We observe that w Fix(T). Indeed, we have that a dynamic process (w m : m {0} ) satisfies (28). Hence, by (C), T is closed at w and, since m{w m T(wm-1)}, we get w T(w). This proves that the assertion (A2) holds.

This yields the result when (C) and (P1) hold.

Step 3. Assume that the condition (C) and the property (P2) hold. If w0 X and the assertion (A1) does not hold, then (A2) holds.

If (A1) does not hold, then, by Step 1, there exists a sequence (w m : m {0} ) which satisfies ( w m : m { 0 } ) O ( X , T , w 0 ) and, additionally, this sequence is a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -Cauchy sequence on X, i.e.
α A { lim n sup m > n L α ( w n , w m ) = 0 } .
(29)
We prove that ( w m : m { 0 } ) is a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -Cauchy sequence on X, i.e. that
α A ε > 0 n 0 = n 0 ( α , ε ) s , l , s > l > n 0 { d α ( w s , w l ) < ε } .
(30)
Indeed, by (29), we claim that
α A ε > 0 n 1 = n 1 ( α , ε ) n > n 1 { sup ( L α ( w n , w m ) : m > n } < ε } .
Hence, in particular,
α A ε > 0 n 1 = n 1 ( α , ε ) n > n 1 q { L α ( w n , w q + n ) < ε } .
(31)
Let now r0, j0 , r0 > j0, be arbitrary and fixed. If we define
t m = w r 0 + m and z m = w j 0 + m for m ,
(32)
then (31) implies that
α A { lim m L α ( w m , t m ) = lim m L α ( w m , z m ) = 0 } .
(33)
Therefore, by (29), (33), and ( L 2 ), we get
α A { lim m d α ( w m , t m ) = lim m d α ( w m , z m ) = 0 } .
(34)
From (32)-(34), we then claim that
α A ε > 0 n 2 = n 2 ( α , ε ) m > n 2 { d α ( w m , w r 0 + m ) < ε / 2 }
(35)
and
α A ε > 0 n 3 = n 3 ( α , ε ) m > n 3 { d α ( w m , w j 0 + m ) < ε / 2 } .
(36)
Let now α 0 A and ε0 > 0 be arbitrary and fixed, let n0 = max{n2(α0, ε0), n3(α0, ε0)} + 1 and let s, l be arbitrary and fixed such that s > l > n0. Then s = r0 + n0 and l = j0 + n0 for some r0, j0 such that r0 > j0 and, using (35) and (36), we get
d α 0 ( w s , w l ) = d α 0 ( w r 0 + n 0 , w j 0 + n 0 ) d α 0 ( w n 0 , w r 0 + n 0 ) + d α 0 ( w n 0 , w j 0 + n 0 ) < ε 0 / 2 + ε 0 / 2 = ε 0 .
Hence, we conclude that
α A ε > 0 n 0 = n 0 ( α , ε ) s , l , s > l > n 0 { d α ( w s , w l ) < ε } .

The proof of (30) is complete.

Now we see that there exists a unique w X such that lim m →∞w m = w. Indeed, since (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) is a Hausdorff https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -sequentially complete generalized uniform space and the sequence ( w m : m { 0 } ) is a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -Cauchy sequence on X, thus there exists a unique w X such that limm→∞w m = w.

Moreover, we observe that w Fix(T). Indeed, we have that a dynamic process (w m : m {0} ) satisfies limm→∞w m = w. Hence, by (C), T is closed at w and, since m{w m T(wm-1)}, we get w T(w). We proved that the assertion (A2) holds.

This yields the result when (C) and (P2) hold.

The proof of (I) is complete.

(II) Let i {1, 2}. Let w0 X, let the condition (C) holds and suppose that the assertion (B1) does not hold, i.e. suppose that
( v m : m { 0 } ) O ( X , T , v 0 = w 0 ) m 0 α A { L α ( v m 0 - 1 , v m 0 ) < } .
This implies that there exists the family ϒ = ( ε α : α A ) K A such that α A { ε α ( 0 , ) } and α A { L α ( v m 0 - 1 , v m 0 ) < ε α < } . Consequently,
( v m : m { 0 } ) O ( X , T , v 0 = w 0 ) m 0 α A { L α ( v m 0 - 1 , v m 0 ) < ε α } .

Clearly, (X, T) is ( i ) L , ϒ , Λ -uniformly locally contractive on X since (X, T) is ( i ) L , Λ -contractive on X. From the above and by similar argumentations as in Steps 1-3 of the proof of Theorem 8(I) we conclude that all assumptions of Theorem 8(I) hold and the assertion (A1) of Theorem 8(I) does not hold. Consequently, using Theorem 8(I), we get that the assertion (A2) of Theorem 8(I) holds in the case when the property either (P1) or (P2) holds. Hence, the assertion (B2) of Theorem 8(II) holds.

The proof of Theorem 8 is complete. □

Proof of Theorem 9

(I) Let i {1, 2}. Let w0 X be arbitrary and fixed and suppose that the assertion (F1) does not hold. That is
( v m : m { 0 } ) O ( X , T , v 0 = w 0 ) m 0 α A { d α ( v m 0 - 1 , v m 0 ) < ε α } .
(37)
But then, using analogous considerations as in the Step 1 of the proof of Theorem 8(I), we obtain that
( u m 0 + n : n { 0 } ) O ( X , T , u m 0 = v m 0 ) α A n { 0 } { u m 0 + n + 1 T ( u m 0 + n ) d α ( u m 0 + n , u m 0 + n + 1 ) < ( λ α ) n + 1 ε α α [ ( i ) D ( T ( u m 0 + n ) , T ( u m 0 + n + 1 ) ) ] λ α d α ( u m 0 + n , u m 0 + n + 1 ) } .
(38)
Consequently, the sequence (w m : m {0} ) such that m < m 0 { w m = v m } , w m 0 = u m 0 = v m 0 and m > m 0 { w m = u m } is a dynamic process of T starting at w0 and, additionally, this sequence is a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -Cauchy sequence on X, i.e.
α A { lim n sup m > n d α ( w n , w m ) = 0 } .
(39)
It is clear that (39) implies
α A { lim m d α ( w m , w m + 1 ) = 0 }
(40)
and, since (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) is a Hausdorff https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -sequentially complete generalized uniform space, there exists a unique w X such that
α A { lim m d α ( w m , w ) = 0 } .
(41)
If, for each α A , x X and B Cl(X), we denote
d α ( x , B ) = inf { d α ( x , y ) : y B }
(42)
and
ω α ( x ) = d α ( x , T ( x ) ) ,
(43)
then (42) and (40) implies
α A { lim m ω α ( w m ) = lim m d α ( w m , T ( w m ) ) lim m d α ( w m , w m + 1 ) = 0 } .
(44)
Let m , m > m0, and α A be arbitrary and fixed and let
[ Φ ] α = φ α = [ ( i ) D ( T ( w m ) , T ( w ) ) ] α , Φ K 0 , + A ;
here m0 is defined by (37). Then, by (9)-(11) and definition of ( i ) D ( T ( w m ) , T ( w ) ) , we get that v T ( w m ) c 1 T ( w ) { d α ( v , c 1 ) φ α } and v T ( w ) c 2 T ( w m ) { d α ( v , c 2 ) φ α } . Hence, in particular, if v T(w m ) is arbitrary and fixed, then
d α ( v , T ( w ) ) = inf { d α ( v , z ) : z T ( w ) } d α ( v , c 1 ) φ α .
This implies
sup v T ( w m ) d α ( v , T ( w ) ) φ α = α [ ( i ) D ( T ( w m ) , T ( w ) ) ] .
(45)
Now, by ( D 1 ), (remember that L = D ), for each u T(w) and v T(w m ), we have
d α ( w , u ) d α ( w , w m ) + d α ( w m , v ) + d α ( v , u ) .
Hence, by (42) and ( D 1 ), for each v T(w m ), it follows
d α ( w , T ( w ) ) = ω α ( w ) d α ( w , w m ) + d α ( w m , v ) + d α ( v , T ( w ) ) .
Further, by (38), (43), (44), and (11), we get
d α ( w , T ( w ) ) = ω α ( w ) d α ( w , w m ) + inf v T ( w m ) { d α ( w m , v ) + d α ( v , T ( w ) ) } d α ( w , w m ) + inf v T ( w m ) d α ( w m , v ) + sup v T ( w m ) d α ( v , T ( w ) ) d α ( w , w m ) + ω α ( w m ) + [ ( i ) D ( T ( w m ) , T ( w ) ) ] α d α ( w , w m ) + ω α ( w m ) + λ α d α ( w m , w ) .
Hence, by (41) and (44), α A { ω α ( w ) = d α ( w , T ( w ) ) = 0 } . However, this property of w, i.e.
d α ( w , T ( w ) ) = inf { d α ( w , y ) : y T ( w ) } = 0 ,

and fact that T(w) is closed, gives w T(w). This and (41) yield that (F2) holds.

(II) Let i {1, 2}. Let w0 X and suppose that the assertion (G1) does not hold, i.e. suppose that
( v m : m { 0 } ) O ( X , T , v 0 = w 0 ) m 0 α A { d α ( v m 0 - 1 , v m 0 ) < } .
This implies that there exists the family ϒ = ( ε α : α A ) K A such that α A { ε α ( 0 , ) } and α A { d α ( v m 0 - 1 , v m 0 ) < ε α < } . Consequently,
( v m : m { 0 } ) O ( X , T , v 0 = w 0 ) m 0 α A { d α ( v m 0 - 1 , v m 0 ) < ε α } .

Clearly, (X, T) is ( ( i ) D , ϒ , Λ ) - uniformly locally contractive on X since (X, T) is ( i ) D -contractive on X. Using now similar argumentation as in the proof of Theorem 8(II), we obtain that (G2) holds.

The proof of Theorem 9 is complete. □

Generalized locally convex spaces (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq142_HTML.gif )

We want to show an immediate consequence of the Section "Generalized uniform spaces (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) and the class L ( X , D ) of https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq19_HTML.gif -families of generalized pseu-dodistances on (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif )".

Definition 12 Let X be a vector space over .
  1. (i)
    The family
    P = { p α : X [ 0 , + ] , α A }
     

is said to be a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq142_HTML.gif -family of generalized seminorms on X ( https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq142_HTML.gif -family, for short) if the following three conditions hold:

( P 1 ) a A x X 0 p α ( x ) x = 0 p α ( x ) = 0 ;

( P 2 ) α A λ x X p α ( λ x ) = λ p α ( x ) ; and

( P 3 ) If α A and x, y X and if p α (x) and p α (y) are finite, then p α (x + y) is finite and p α (x + y) ≤ p α (x) + p α (y).
  1. (ii)

    If https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq142_HTML.gif is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq142_HTML.gif -family, then the pair (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq142_HTML.gif ) is called a generalized locally convex space.

     
  2. (iii)

    A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq142_HTML.gif -family https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq142_HTML.gif is said to be separating if

     
( P 4 ) x X x 0 α A 0 < p α ( x ) .
  1. (iv)

    If a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq142_HTML.gif -family https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq142_HTML.gif is separating, then the pair (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq142_HTML.gif ) is called a Hausdorff generalized locally convex space.

     

Remark 6 It is clear that each generalized locally convex space is an generalized uniform space. Indeed, if X is a vector space over and (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq142_HTML.gif ) is a generalized locally convex space, then D = d α : X × X [ 0 , + ] , α A where d α (x,y) = p α (x - y), (x,y) X × X, α A , is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif -family and (X, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-104/MediaObjects/13663_2011_Article_210_IEq17_HTML.gif ) is a generalized uniform space.

Examples of the decompositions of the generalized uniform spaces

Example 3 For each n , let Z n = [2n - 2, 2n - 1] and let q n : Z n × Z n → [0, +∞) where q n (x,y) = |x - y| for x,y Z n . Let Z = n = 1 Z n and define q : Z × Z → [0, +∞] by the formula
q ( x , y ) = q n ( x , y ) if x , y Z n , n + if x Z n , y Z m , n m , n , m .
(46)

Then (Z, q) is a complete generalized metric space.

Example 4 Let Y = = × × be a non-normable real Hausdorff and sequentially complete locally convex space with the family C = { c n , n } of calibrations c n ,n , defined as follows:
c n ( x ) = [ x ] n = x n , x = ( x 1 , x 2 , x 3 , ) Y , n .
For each s , let P s = [2s - 2, 2