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

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.

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} ∪ ℕ) defined by w^m ∈ T(w^m-1) for m ∈ ℕ (see, [1,2]).

If (X, T) is a dynamic system and w⁰ ∈ X then, by , 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 : X → X be a single-valued map satisfying the condition

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

where d(x, C) = inf_c∈C d(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 : X → CB(X) is (h, λ)-contractive, i.e.,

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,y∈X {d(x, y) = 0 iff x = y}; (b) ∀_x,y∈X {d(x, y) = d(y, x); (c) ∀_x,y,z∈X {[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 , -index set, be a family of disjoint metric spaces. If and, for any x, y ∈ X,

then (X, d) is a generalized metric space. Moreover, if for each , (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:

Therefore, X is decomposed uniquely into (disjoint) equivalence classes , 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 be the canonical decomposition and let . Then: (I) For each , (X_β, d_β) is a metric space; (II) For any , with β₁ ≠ β₂, d(x, y) = +∞ for any and ; and (III) (X, d) is a generalized complete metric space iff, for each , (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 ∈ ℕ) 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),

where, for each E ∈ C(X) and ε > 0, N(ε, E) = {x ∈ X : ∃_e∈E {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 : X → C(X) is (H, ε, λ)-uniformly locally contractive, i.e.

Then the following alternative holds: either

(A) ; or

(B) .

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 : X → C(X) is (H, λ)-contractive, i.e.,

Then the following alternative holds: either

(A) ; or

(B) .

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 : X → C(X) is (h, λ)-contractive, i.e.

Then .

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,z∈X {p(x, z) ≤ p(x, y) + p(y, z)}; (S2) ∀_x∈X ∀_t>0{η(x, 0) = 0 ⋀ η(x, t) ≥ t} and η is concave and continuous in its second variable; (S3) lim_n→∞ x_n = x and lim_n→∞ sup_m≥n η(z_n, p(z_n, x_m)) = 0 imply that ∀_w∈X {p(w, x) ≤ lim inf_n→∞ p(w, x_n)}; (S4) lim_n→∞ sup_m≥n p(x_n, y_m)) = 0 and lim_n→∞ η(x_n, t_n) = 0 imply that lim_n→∞ η(y_n, t_n) = 0; and (S5) lim_n→∞ η(z_n, p(z_n, x_n)) = 0 and lim_n→∞ η(z_n, p(z_n, y_n)) = 0 imply that lim_n→∞ 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 : X → C(X) be (Q_p, λ)-contractive, i.e.

where Q_p(A, B) = sup_a∈A inf_b∈B p(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 : X → C(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 : X → C(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 [3-29], 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 [30-36] 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 (-index set) of generalized pseudometrics on a nonempty set X and the generalized uniform spaces (X, ) are introduced, the classes of -families of generalized pseudodistances in (X, ) are defined and, in (X, ), a new type of -sequentially completeness with respect to -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 is defined (see Section "Partial quasiordered space ") and, using , -distances on 2^X (i ∈ {1, 2}) with respect to -families are introduced (see Section "-distances on 2^X, i ∈ {1,2}"). Also, we introduce the definitions of -uniformly locally contractive and -contractive set-valued dynamic systems (X, T) (i ∈ {1, 2}) satisfying T : X → 2^X (see Section "-uniformly locally contractive and -contractive set-valued dynamic systems (X, T), i ∈ {1, 2}") and, for w⁰ ∈ X, we establish the conditions guaranteeing the convergence of dynamic processes and the existence of fixed points for such contractions and, additionally, a special case when T : X → C(X) and 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 "). By generality of spaces and -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.

The following terminologies will be much used.

Definition 3 Let X be a nonempty set. (a) The family

is said to be a -family of generalized pseudometrics on X (-family on X, for short) if the following three conditions hold:

() ;

() ; and

If 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 is -family, then the pair (X, ) is called a generalized uniform space.

(c) Let (X, ) be a generalized uniform space. A -family is said to be separating if

() .

(d) If a -family is separating, then the pair (X, ) is called a Hausdorff generalized uniform space.

(e) Let (X, ) be a generalized uniform space and let (x_m : m ∈ ℕ) be a sequence in X. We say that (x_m : m ∈ ℕ) is -Cauchy sequence in X if . We say that (x_m : m ∈ ℕ) is -convergent in X if there is an x ∈ X such that (, for short).

(f) If every -Cauchy sequence in X is -convergent sequence in X, then a pair (X, ) is called a -sequentially complete generalized uniform space.

Definition 4 Let X be a nonempty set. The family

is said to be a -family of generalized quasi pseudometrics on X (-family on X, for short) if the following two conditions hold:

() ;

() If 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, ) be a generalized uniform space.

(a) The family

is said to be a -family of generalized pseudodistances on X (-family on X, for short) if the following two conditions hold:

() If 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

() For any sequences (x_m : m ∈ ℕ) and (y_m : m ∈ ℕ) in X such that

and

the following holds

(b) Let be a class defined as follows

Remark 2 Let (X, ) be a generalized uniform space. (i) since . (ii) ; see Sections 10-13.

Definition 6 Let (X, ) be a generalized uniform space, let and let (x_m : m ∈ ℕ) be a sequence in X.

(a) We say that (x_m : m ∈ ℕ) is -Cauchy in X if .

(b) We say that (x_m : m ∈ ℕ) is -convergent in X if there exists x ∈ X such that .

(c) We say that (X, ) is -sequentially complete if each -Cauchy sequence in X is -convergent in X.

In the following remark, we list some basic properties of -families.

Remark 3 Let (X, ) be a generalized uniform space and let . (i) If , then is a -family on X; examples of which are not -families on X are given in Section "Examples of the decompositions of the generalized uniform spaces". (ii) There exist -sequentially complete spaces which are not -sequentially complete; see Example 15. (iii) If (x_m : m ∈ ℕ) in X is -convergent in X, then its limit point is not necessary unique; see Example 1.

Example 1 Let (ℝ, |·|) be a metric space. Define the family of to be

It is obvious that is -family on ℝ and the sequence (1/m : m ∈ ℕ) is -convergent to each point w ∈ (0, +∞).

One can prove the following proposition:

Proposition 1 Let (X, ) be a Hausdorff generalized uniform space and let .

(I) If x ≠ y, x, y ∈ X, then .

(II) If (X, ) is -sequentially complete and if (x_m : m ∈ ℕ) is -Cauchy sequence in X, then (x_m : m ∈ ℕ) is -convergent in X.

Proof. (I)) Assume that there are x ≠ y, x, y ∈ X, such that . Then, , since, by using (), it follows that . Defining the sequences (x_m : m ∈ ℕ) and (y_m : m ∈ ℕ) in X by x_m = x and y_m = y for m ∈ ℕ, and observing that , this implies that (6) and (7) for these sequences hold. Then, by (), (8) holds, so it is . On the other hand, is separating, so, since x ≠ y, it is . This leads to a contradiction.

(II) Since , by Definition 6(c), this proves the existence of x ∈ X such that . We can apply () to sequences (x_m : m ∈ ℕ) and (y_m = x/ : m ∈ ℕ) and then we find that . The uniqueness of the point of x follows from the fact that is separating. □

Proposition 2 Let be a set of elements defined by the formula

and let . The relation on defined by

is a partial quasiordered on and the pair is a partial quasiordered space.

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

Notation. The following notation is fixed throughout the article:

;

;

;

.

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

Definition 7 Let be a nonempty subset of . We say that is a infimum of if the following two conditions hold:

() ;

() .

Example 2 Let and let . If then and does not exist since (3, 5, 7) and (4, 1, 8) are not comparable. If then and .

Definition 8 Let (X, ) be a Hausdorff generalized uniform space and let .

(a) For C ∈ 2^X and , let us denote

(b) For A, B ∈ 2^X let us denote:

(c) Let i ∈ {1, 2}. The map of the form

A, B ∈ 2^X, is called a -distance on 2^X generated by (-distance on 2^X, for short).

Remark 4 For each A, B ∈ 2^X, .

Definition 9 Let (X, ) be a Hausdorff generalized uniform space, let and let i ∈ {1,2}.

(a) Let be a -distance on 2^X and let and be such that . We say that a set-valued dynamic system (X, T), T : X → 2^X, is -uniformly locally contractive on X if

(b) Let be a -distance on 2^X and let be such that . We say that a set-valued dynamic system (X, T), T : X → 2^X, is - contractive on X if

Remark 5 Let (X, ) be a Hausdorff generalized uniform space, let and let and be such that .

(i) If (X, T), T : X → 2^X, is -uniformly locally contractive on X then it is -uniformly locally contractive on X.

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

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

Definition 10 Let (X, ) 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 ∈ ℕ) is a sequence -converging to x in X and (y_m : m ∈ ℕ) is a sequence -converging to y in X such that y_m ∈ T(x_m) for all m ∈ ℕ, then y ∈ T(x).

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

Theorem 8 Assume that (X, ) is a Hausdorff generalized uniform space, and one of the following properties holds:

(P1) (X, ) is -sequentially complete; or

(P2) (X, ) is -sequentially complete.

Let i ∈ {1, 2}, let be a -distance on 2^X and assume that a set-valued dynamic system (X, T), T : X → 2^X, has the property

(C) {lim_m→∞ w^m = w ⇒ T is closed at w}.

(I) If and satisfy and (X, T) is -uniformly locally contractive on X then, for each w⁰ ∈ X, the following alternative holds: either

(A1) ; or

(A2) {w ∈ Fix(T) ⋀ lim_m→∞ w^m = w ⋀ (w^m : m ∈ {0} ∪ ℕ) is -Cauchy}.

(II) If satisfies and (X, T) is -contractive on X then, for each w⁰ ∈ X, the following alternative holds: either

(B1) ; or

(B2) {w ∈ Fix(T) ⋀ lim_m→∞ w^m = w ⋀ (w^m : m ∈ {0} ∪ ℕ) is -Cauchy}.

Definition 11 Let (X, ) 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 ∈ ℕ) in Y which is -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 and when T : X → C(X).

Theorem 9 Let (X, ) be a Hausdorff -sequentially complete generalized uniform space, let i ∈ {1, 2} and assume that is a -distance on C(X).

(I) If and satisfy and if a set-valued dynamic system (X, T) satisfying T : X → C(X) is -uniformly locally contractive on X then, for each w⁰ ∈ X, the following alternative holds: either

(F1) ; or

(F2) .

(II) If satisfies and a set-valued dynamic system (X, T) satisfying T : X → C(X) is -contractive on X then, for each w⁰ ∈ X, the following alternative holds: either

(G1) ; or

(G2) .

(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,

Then there exists which is -Cauchy sequence on X; that is,

Indeed, since (14) holds, thus, by (12), we get

It follows from (16) and Definition 8(c), that there exists and

From this, denoting , we deduce that . Consequently, by (), there exists

such that which implies

If i = 1, then we note that, by (18), (9), and (10), . Clearly, . Thus, and the conclusion

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

If i = 2, then we also note that, by (18), (9) and (11), and . Clearly, . Thus, and the conclusion

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

This proves

Since, by (20), , it follows, using (12) and (20), that

That is,

Denoting , we see that condition (21) implies . Hence, by (), there exists

such that . This means

Let i = 1. Clearly, by (9), (10), and (22), . Moreover, by (20), . Therefore . This, by (9), (10) and (21)-(23), implies

Let i = 2. Clearly, by (9)-(11) and (22), and . Moreover, . Therefore . This, by (9)-(11) and (21)-(23), implies

That is,

By (24), we have and, using (12) and (24), we get

This means

By induction, a similar argument as in the proofs of (17)-(25) shows that

It is clear that (26) implies that where and . Additionally, this sequence (w^m : m ∈ {0} ⋂ ℕ) is a -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, ) is -sequentially complete), we have that there exists w ∈ X satisfying

Applying (15), (27), and () (where (x_m = w^m : m ∈ ℕ) and (y_m = w : m ∈ ℕ)), we find that

Clearly, since (X, ) 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} ∪ ℕ) satisfies (28). Hence, by (C), T is closed at w and, since ∀_m∈ℕ{w^m ∈ T(w^m-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} ⋂ ℕ) which satisfies and, additionally, this sequence is a -Cauchy sequence on X, i.e.

We prove that is a -Cauchy sequence on X, i.e. that

Indeed, by (29), we claim that

Hence, in particular,

Let now r₀, j₀ ∈ ℕ, r₀ > j₀, be arbitrary and fixed. If we define

then (31) implies that

Therefore, by (29), (33), and (), we get

From (32)-(34), we then claim that

and

Let now and ε₀ > 0 be arbitrary and fixed, let n₀ = max{n₂(α₀, ε₀), n₃(α₀, ε₀)} + 1 and let s, l ∈ ℕ be arbitrary and fixed such that s > l > n₀. Then s = r₀ + n₀ and l = j₀ + n₀ for some r₀, j₀ ∈ ℕ such that r₀ > j₀ and, using (35) and (36), we get

Hence, we conclude that

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, ) is a Hausdorff -sequentially complete generalized uniform space and the sequence is a -Cauchy sequence on X, thus there exists a unique w ∈ X such that lim_m→∞ w^m = w.

Moreover, we observe that w ∈ Fix(T). Indeed, we have that a dynamic process (w^m : m ∈ {0} ∪ ℕ) satisfies lim_m→∞ w^m = w. Hence, by (C), T is closed at w and, since ∀_m∈ℕ{w^m ∈ T(w^m-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

This implies that there exists the family such that and . Consequently,

Clearly, (X, T) is -uniformly locally contractive on X since (X, T) is -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. □

(I) Let i ∈ {1, 2}. Let w⁰ ∈ X be arbitrary and fixed and suppose that the assertion (F1) does not hold. That is

But then, using analogous considerations as in the Step 1 of the proof of Theorem 8(I), we obtain that

Consequently, the sequence (w^m : m ∈ {0} ∪ ℕ) such that and is a dynamic process of T starting at w⁰ and, additionally, this sequence is a -Cauchy sequence on X, i.e.

It is clear that (39) implies

and, since (X, ) is a Hausdorff -sequentially complete generalized uniform space, there exists a unique w ∈ X such that

If, for each , x ∈ X and B ⊂ Cl(X), we denote

and

then (42) and (40) implies

Let m ∈ ℕ, m > m₀, and be arbitrary and fixed and let

here m₀ is defined by (37). Then, by (9)-(11) and definition of , we get that and . Hence, in particular, if v ∈ T(w^m) is arbitrary and fixed, then

This implies

Now, by (), (remember that ), for each u ∈ T(w) and v ∈ T(w^m), we have

Hence, by (42) and (), for each v ∈ T(w^m), it follows

Further, by (38), (43), (44), and (11), we get

Hence, by (41) and (44), . However, this property of w, i.e.

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

This implies that there exists the family such that and . Consequently,

Clearly, (X, T) is - uniformly locally contractive on X since (X, T) is -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. □

We want to show an immediate consequence of the Section "Generalized uniform spaces (X, ) and the class of -families of generalized pseu-dodistances on (X, )".

Definition 12 Let X be a vector space over ℝ.

(i) The family

is said to be a -family of generalized seminorms on X (-family, for short) if the following three conditions hold:

() ;

() ; and

() If 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).

(ii) If is -family, then the pair (X, ) is called a generalized locally convex space.

(iii) A -family is said to be separating if

() .

(iv) If a -family is separating, then the pair (X, ) 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 ℝ and (X, ) is a generalized locally convex space, then where d_α(x,y) = p_α(x - y), (x,y) ∈ X × X, , is -family and (X, ) is a generalized uniform space.

Example 3 For each n ∈ 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 and define q : Z × Z → [0, +∞] by the formula

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

Example 4 Let Y = ℝ^ℕ = ℝ × ℝ × ⋯ be a non-normable real Hausdorff and sequentially complete locally convex space with the family of calibrations c_n,n ∈ ℕ, defined as follows:

For each s ∈ ℕ, let P_s = [2s - 2, 2s - 1]^ℕ be a Hausdorff sequentially complete uniform space with uniformity defined by the saturated family {p_s,n : n ∈ ℕ} of pseudometrics p_s,n : P_s × P_s → [0, +∞), n ∈ ℕ, defined as follows:

Let and define p_n:P × P → [0, +∞], n ∈ ℕ, as follows

Then (P, {p_n:P × P → [0, +∞], n ∈ ℕ}) is a Hausdorff {p_n:P × P→ [0, +∞], n ∈ ℕ}-sequentially complete generalized uniform space.

In this section we describe some elements of the class .

Example 5 Let (X, ) be a Hausdorff generalized uniform space where , -index set, is a -family. Let the set E ⊂ X, containing at least two different points, be arbitrary and fixed and, for each , let L_a : X × X → [0, +∞] be defined by the formula:

We show that the family is -family on (X, ).

First, we observe that the condition () holds. Indeed, let and x, y, z ∈ X be arbitrary and fixed and such that L_α(x, z) < +∞ and L_α(z, y) < + ∞. By (48), this implies that: x, y, z ∈ E; d_α(x, z) = L_α(x, z) < +∞; and d_α(z, y) = L_α(z,y) < +∞. Then, by , we get that d_α(x,y) < +∞ and d_α(x, y) ≤ d_α(x,z) + d_a(z,y). Consequently, since x,y,z ∈ E, this mean that L_α(x,y) = d_a(x,y) < +∞ and L_a(x,y) ≤ L_a(x,z) + L_a(z,y). Therefore, the condition () holds.

To prove that () holds, we assume that the sequences (x_m : m ∈ ℕ) and (y_m : m ∈ ℕ) in X satisfy (6) and (7). Then, in particular, (7) is of the form

By definition of , this implies that

Therefore, we obtain that

This means that the sequences (x_m : m ∈ ℕ) and (y_m : m ∈ ℕ) satisfy (8). Hence we conclude that the condition is satisfied.

Example 6 Let (X, ) be a generalized metric space where is a -family. Let the set E ⊂ X, containing at least two different points, be arbitrary and fixed and let L : X × X → [0, +∞] be defined by the formula (see (48)):

By Example 5, the family is -family on X.

Example 7 Let (X, ) be a Hausdorff generalized uniform space where , -index set, is a -family. Let the sets E and F satisfying E ⊂ F ⊂ X be arbitrary and fixed and such that E contains at least two different points and F contains at least three different points. Let 0 < a_α < b_α < c_α < +∞, , and let, for each , L_α : X × X → [0, +∞] be defined by the formula:

We show that the family is -family on X.

First, we observe that the condition ( holds. Indeed, let and x, y, z ∈ X satisfying L_α(x, z) < + ∞ and L_α(z, y) < + ∞ be arbitrary and fixed. Clearly, by definition of L_α, this implies that x, y, z ∈ F. We consider the following cases:

Case 1. If L_α(x, y) = d_α(x, y) + b_α, then by (50) we conclude that, {x, y} ⋂ F\E = {x, y}. Now, if z ∈ E, then L_α(x, z) = d_α(x, z) + c_a; L_α(z, y) = d_α(z, y); and consequently, since b_α < c_α, by , we get

If z ∈ F \ E, then L_α(x, z) = d_α(x, z) + b_α; L_α(z, y) = d_α(z, y) + b_α; and consequently, by , we get

Case 2. If L_α(x, y) = d_α(x, y) + c_α, then by (50) we conclude that, x ∈ F \ E∧y ∈ E. Now, if z ∈ E then L_α(x, z) = d_α(x, z) + c_α; L_α(z, y) = d_α(z, y) + α_α; and consequently, by , we get

If z ∈ F \ E, then L_α(x, z) = d_α(x, z) + b_α; L_α(z, y) = d_α(z, y) + c_α; and consequently, by , we get

Case 3. If L_α(x, y) = d_α(x, y), then by (50) we conclude that, x ∈ E∧y ∈ F\E. Now, if z ∈ E then L_α(x, z) = d_α(x, z) + a_α; L_α(z, y) = d_α(z, y); and consequently, by , we get

If z ∈ F\E, then L_α(x, z) = d_α(x, z); L_α(z, y) = d_α(z, y) + b_α; and consequently, by , we get

Case 4. If L_α(x, y) = d_α(x, y) + a_α, then by (50) we conclude that, x ∈ E∧y ∈ E. Now, if z ∈ E then L_α(x, z) = d_α(x, z) + a_α; L_α(z, y) = d_α(z, y) + a_α; and consequently, by , we get

If z ∈ F\E, then L_α(x, z) = d_α(x, z); L_α(z, y) = d_α(z, y) + c_α; and consequently, since a_α < c_α, by , we get

Consequently, the condition holds.

To prove that holds, we assume that the sequences (x_m : m ∈ ℕ) and (y_m : m ∈ ℕ) in X satisfy (6) and (7). Then, in particular, (7) is of the form

By definition of , this implies that

As a consequence of this, we get

This means that the sequences (x_m : m ∈ ℕ) and (y_m : m ∈ ℕ) satisfy (8). Therefore, the property ( holds.

It is worth noticing that, there exists x, y ∈ X such that, for each , L_α(x, y) = L_α(y, x) does not hold. Indeed, if x ∈ E and y ∈ F \ E, then

Example 8 Let X, be a generalized metric space where is a -family. Let the sets E and F satifying E ⊂ F ⊂ X be arbitrary and fixed and such that E contains at least two different points and F contains at least three different points. Let L : X × X → [0, +∞] be defined by the formula:

By Example 7, the family is -family on X.

Example 9 Let (X, ) be a Hausdorff generalized uniform space where , -index set, is a -family. Let the sets E and F satisfying E ⊂ F ⊂ X be arbitrary and fixed and such that E contains at least two different points and F contains at least three different points. Let 0 < b_α < c_α < +∞, , and let, for each , L_α : X × X → [0, +∞] be defined by the formula:

We show that the family is -family on X.

First, we observe that the condition ( holds. Indeed, let and x, y, z ∈ X satisfying L_α(x, z) < +∞ and L_α(z, y) < +∞ be arbitrary and fixed. Clearly, by definition of L_α, this implies that x, y, z ∈ F. We consider the following cases:

Case 1. If L_α(x, y) = d_α(x, y) + b_α, then by (52) we conclude that, {x, y} ⋂ F\E = {x, y}. Now, if z ∈ E, then

and consequently, since b_α < c_α, by , we get

If z ∈ F \ E, then

and consequently, by , we get

Case 2. If L_α(x, y) = d_α(x, y) + c_α, then by (52) we conclude that, x ∈ F\E∧y ∈ E. Now, if z ∈ E then

and consequently, by , we get

If z ∈ F \ E, then

and consequently, by , we get

Case 3. If L_α(x, y) = d_α(x, y), then by (52) we conclude that, x ∈ E ∧ y ∈ E or x ∈ E ∧ y ∈ F\E. First, assume that x ∈ E∧y ∈ E. Now, if z ∈ E then

and consequently, by , we get

If z ∈ F \ E, then

and consequently, by , we get

Next, we assume that x ∈ E ∧ y ∈ F\E. Now, if z ∈ E then L_α(x, z) = d_α(x, z); L_α(z, y) = d_α(z, y); and consequently, by , we get

If z ∈ F \ E, then

and consequently, by , we get

Consequently, the condition ( holds.

To prove that ( holds, we assume that the sequences (x_m : m ∈ ℕ) and (y_m : m ∈ ℕ) in X satisfy (6) and (7). Then, in particular, (7) is of the form

By definition of , this implies that

As a consequence of this, we get

This means that the sequences (x_m : m ∈ ℕ) and (y_m : m ∈ ℕ) satisfy (8). Therefore, the property ( holds.

It is worth noticing that, there exists x, y ∈ X such that, for each , L_α(x, y) = L_α(y, x) does not hold. Indeed, if x ∈ E and y ∈ F \ E, then

Example 10 Let (X, ) be a generalized metric space where is a -family. Let the sets E and F satisfying E ⊂F ⊂ X be arbitrary and fixed, and such that E contains at least two different points and F contains at least three different points. Let L : X × X → [0, +∞] be defined by the formula:

By Example 9, the family is -family on X.

The following example illustrates the Theorem 8(I) in the case when (X, ) is -sequentially complete and (X, T) is -uniformly locally contractive on X where and .

Example 11 Let P and {p_n : P × P → [0,+∞], n ∈ ℕ} be as in Example 4. Let X = P ⋂ [0,9]^ℕ and let , d_n : X × X → [0, +∞], n ∈ ℕ, where, for each n ∈ ℕ, we define . Then (X, ) is a Hausdorff -sequentially complete generalized uniform space. This gives that the property (P2) of Theorem 8 holds.

The elements of ℝ^ℕ we denote by x = (x₁,x₂,...). In particular, the element (x,x,...) ∈ ℝ^ℕ we denote by .

Let and let a set-valued dynamic system (X, T) be given by the formula

Let and let be a family of the maps given by the formula:

By Example 4, the family is -family on X.

Now, we show that, for and , (X, T) is -uniformly locally contractive on X, i.e. that

where

Indeed, let x, y ∈ X be arbitrary and fixed. Since, by (55), this family is symmetric on X, we may consider only the following four cases:

Case 1. Let x ∈ F and let y ∈ X\F.

If , then, since , by (55), for each n ∈ ℕ, we have

By (47), from this, for each n ∈ ℕ, we get

If , then, since , by (55), we obtain that for each y ∈ X \ F. Consequently, for each n ∈ ℕ, x ∈ F and y ∈ X\F, inequality L_n(x, y) < 1/2 in (56) does not hold and this case we do not have to consider this case.

Case 2. Let x, y ∈ F be such that x ≠ y or . Then, by definition of F, or . But, , therefore, by (55), we get ∀_n∈ℕ{L_n(x,y) = +∞}. Therefore, by (56), this case we can also be omitted.

Case 3. Let x, y ∈ F be such that . Then, since , by (55) and (47), we get

and, consequently, for each n ∈ ℕ, the inequality L_n(x, y) < 1/2 holds. In virtue ofthis, we show that the inequalities in (56) hold. With this aim, we see that:

(3_i) By (54), we have ;

(3_ii) Next, if , then, by (3_i),

(3_iii) Now, by (3i), (3ii), (58), and (59), we get

(3_iv) Therefore, by (3_iii), we have ;

(3_v) The consequence of (57) and (3_iv) is

Hence, by (60), we conclude that

holds.

Case 4. Let x, y ∉ F. Then we see that:

(4_i) By (54), we have ;

(4_ii) Next, if , then

(4_iii) Now, by (4_i) and (4_ii), we get

(4_iv) Therefore, by (4_iii), ;

(4_v) According to (57) and (4_iv), we have

Consequently, by (60),

We proved that (X, T) is -uniformly locally contractive on X. We see also that (C) holds.

Finally, we see that . Hence, for each w⁰ ∈ X, there exists a dynamic process (w^m : m ∈ {0} ∪ ℕ) such that: (i) ; (ii) ; and (iii) .

The following example illustrates the Theorem 8(I) in the case when (X, ) is -sequentially complete for some , , but not -sequentially complete and (X, T) is -uniformly locally contractive on X.

Example 12 Let X and {p_n : P × P → [0, +∞], n ∈ ℕ} be as in Example 4. Let and let , d_k : X × X → [0,∞], k ∈ ℕ, where, for each k ∈ ℕ, we define . Then (X, ) is a Hausdorff generalized uniform space.

We observe that (X, ) is not a -sequentially complete space. Indeed, we consider the sequence (x_m : m ∈ ℕ) defined as follows: , m ∈ ℕ. Of course, the sequence (x_m : m ∈ ℕ) is -Cauchy sequence on X. Indeed, we have which implies that

Consequently,

However, there does not exist x ∈ X such that lim_m→∞ x_m = x. Therefore, X is not -sequentially complete.

Let and let be a family of the maps given by the formula:

By (47), this gives

where N = {0,1, 2, 3, 4, 5}, x, y ∈ X and k ∈ ℕ.

By Example 4, the familly is -family on X.

We show that X is -sequentially complete space. Indeed, let (x_m : m ∈ ℕ) be arbitrary and fixed -Cauchy sequence in X, i.e.

This implies that

Hence, in particular, we conclude that

Now, (63) and (61) gives that

Of course, since (x_m : m ∈ ℕ) is arbitrary nad fixed, then there exists a unique s₀ ∈ N for all k ∈ ℕ. Now, putting l₀ = min_k∈ℕ{n₀(k)} we obtain that

The property (61) and (64) gives that

Using (65), (64) and definition of E, we may consider only the following two cases:

Case 1. If or or or , then in each of these situations the sequence, as a constant sequence, is, by (61), -convergent to , respectively.

Case 2. If , then

so by (65) and (62), we obtain

This gives that (x_m : m ∈ ℕ) is a -Cauchy sequence in X, so also the sequence is a -Cauchy sequence in [4,5]^ℕ. Since [4,5]^ℕ is a -complete uniform space, so there exists x ∈ X such that

i.e (x_m : m ∈ ℕ) is -convergent. In consequence, X is -sequentially complete generalized uniform space.

Now, let and let (X, T) be given by the formula

By the same reasoning as in Example 11, we obtain that, for and is -uniformly locally contractive on X, for each w⁰ ∈ X there exists a dynamic process (w^m : m ∈ {0} ∪ ℕ) such that and .

Now, in Example 13, for given (X, ) and (X, T), we study the assertions of Theorem 8(I) with respect to changing of the family of and of the point w⁰ ∈ X.

Example 13 Let (X, ) be a complete metric space where X = [0,1] and let , d:X × X→ [0,∞), d(x, y) = |x-y|, x, y ∈ X. Let a dynamic system (X, T) be given by the formula:

Question 2 For these (X, D) and (X, T) and for ε = 1/2 and λ = 1/2, what are the assertions of our theorems with respect to changing of the family L and of the point w⁰ ∈ X?

Answer 1 We show that there exists -family on X such that: (a) (X,T) is not -uniformly locally contractive on X; and (b) (X, T) is -uniformly locally contractive on X and for each w⁰ ∈ X the assertion (A1) holds.

(a) Let E = (1/2,1) and F = (1/2,1] ⊂ X (we see that E ⊂ F ⊂ X)and let L : X × X → [0,+∞] be defined by (51). It follows from Example 8 that the family is -family on X.

We see that (X, T) is not -uniformly locally contractive on X. Otherwise, , where

We note, by (51), (66) and definitions of E and F, that the condition

implies, in particular,

and, for η > 0, then the following hold

Indeed, if x, y ∈ X satisfying (67) are arbitrary and fixed, then from (51) we conclude that (67) holds only if x ∈ E and y ∈ F \ E. Hence, we get that x ∈ (1/2,1), y = 1 and d(x,y) < 1/2, which, by (66), gives (68). Of course, by (51), the equality (69) holds. Now, if η > 0, then, by (68),

and

Thus, (70) holds.

Now, by (67)-(70), we see that

that is, for x ∈ (1/2,1) and y = 1, we have .

Therefore,

Consequently, we proved that (X, T) is not -uniformly locally contractive on X.

This gives that the assumptions of Theorem 8(I) for i = 2 and for defined by (51) where X = [0,1], E = (1/2,1) and F = (1/2,1] does not hold.

(b) However, by (67)-(70), we get

that is, for x ∈ (1/2,1) and y = 1, we have . Therefore,

Consequently, we proved that (X, T) is -uniformly locally contractive on X.

This gives that the assumptions of Theorem 8(I) for i = 1 and for defined by (51) where X = [0,1], E = (1/2, 1) and F = (1/2, 1] hold.

Now, we see that, for each w⁰ ∈ X, the assertion (A1) holds. Indeed, we have:

Case 1. Let w⁰ ∈ [0,1/4). Then, for each dynamic process (w^m : m ∈ {0}∪ ℕ) of (X, T) starting at w⁰, by (66), we have: (i) if w¹ ≠ 1, then ∀_m∈ℕ{w^m ∈ E} and, by (51), L(w⁰,w¹) = +∞ > 1/2 and

or (ii) if w¹ = 1, then ∀_m∈ℕ{w^m = 1 ∈ F\E} and, by (51), L(w⁰, w¹) = +∞ > 1/2 and

Consequently, for each w⁰ ∈ [0,1/4), each a dynamic process (w^m : m ∈ {0} ∪ ℕ) of (X, T) starting at w⁰ satisfies ∀_m∈ℕ{L(w^m-1, w^m) > 1/2}, i.e. for each w⁰ ∈ [0,1/4), the assertion (A1) holds.

Case 2. Let w⁰ ∈ [1/4,1). Then, for each dynamic process (w^m : m ∈ {0} ∪ ℕ) of (X,T) starting at w⁰, by (66), we have that ∀_m∈ℕ{w^m ∈ E} and, by (51),

and

Consequently, for each w⁰ ∈ [1/4,1), each a dynamic process (w^m : m ∈ {0} ∪ ℕ) of (X,T) starting at w⁰ satisfies ∀_m∈ℕ{L(w^m-1, w^m) > 1/2}, i.e. for each w⁰ ∈ [1/4,1), the assertion (A1) holds.

Case 3. Let w⁰ = 1. Then, for a dynamic process (w^m : m ∈ {0} ∪ ℕ) of (X, T) starting at w⁰, by (66), we have that ∀_m∈ℕ{w^m = 1 ∈ F\E} and, by (51),

Consequently, if w⁰ = 1, a dynamic process (w^m : m ∈ {0} ∪ ℕ) of (X, T) starting at w⁰ satisfies ∀_m∈ℕ{L(w^m-1, w^m) > 1/2}, i.e. for w⁰ = 1, the assertion (A1) holds.

Remark 7 Let us observe that, for each w⁰ ∈ X, there exists a dynamic process (w^m : m ∈ {0} ∪ ℕ) starting at w⁰ such that lim_m→∞ w^m = 1, lim_m→∞ L(w^m, 1) = lim_m→∞ d(w^m, 1) = 0 and 1 ∈ Fix(T). However, assertion (A2) does not hold since from Cases 1-3 it follows that, for each w⁰ ∈ X, each dynamic process (w^m : m ∈ {0} ∪ ℕ) starting at w⁰ is not -Cauchy.

Answer 2 We show that there exists -family on X such that (X, T) is -uniformly locally contractive on X and, for each w⁰ ∈ X, the assertion (A2) holds.

Let E = [1/2,1] ⊂ X and let L : X × X → [0, +∞] be defined by the formula:

It follows, from Example 6, that the family is -family on X.

We see that (X, T) is -uniformly locally contractive on X, i.e.

where

Indeed, first, we see that, by (66) and (71),

implies

and, for η > 0,

Indeed, if x, y ∈ X satisfying (72) are arbitrary and fixed, then from (71) we conclude that (72) holds only if x, y ∈ E. Hence, we get that x, y ∈ [1/2,1] and d(x, y) < 1/2, which, by (66), gives (73). Of course, by (49), (74) holds. Now, if η > 0, then, by (73),

and

Thus, (75) holds.

Now, by (72)-(75), we see that

that is, for x, y ∈ [1/2,1], we have . Therefore,

Consequently, we proved that (X, T) is -uniformly locally contractive on X.

This gives that the assumptions of Theorem 8(I) for defined by (71) and for i = 2 hold.

We see that, for each w⁰ ∈ X, the assertion (A2) holds. Indeed, we have that: 1 ∈ Fix(T); for each w⁰ ∈ X and for each dynamic processes (w^m : m ∈ {0} ∪ ℕ) of (X,T) starting at w⁰, by (66), we have that , so lim_m→∞ L(w^m, 1) = lim_m→∞ d(w^m, 1) = 0 and lim_n→∞ sup_m>n L(wⁿ,w^m) = lim_n→∞ sup_m>n d(wⁿ, w^m) = 0. Therefore, the sequence (w^m : m ∈ {0} ∪ ℕ) is -Cauchy.

Remark 8 We see that . Indeed, by (71), L(1,1) = d(1,1) = 0 and

Answer 3 We show that there exists -family on X such that: (i) (X, T) is -uniformly locally contractive on X; (ii) There exists w ∈ X such that End(T) = {w}; (iii) For each w⁰ ∈ X\End(T) the assertion (A2) holds; and (iv) For w⁰ = w the assertion (A1) holds (since L(w, w) = 3 where ).

Define E = (1/2,1) and F = (1/2,1] ⊂ X (we see that E ⊂ F ⊂ X) and let L:X × X→[0, +∞] be defined by (53). It follows from Example 10 that the family is -family on X.

First, we show that (X,T) is not -uniformly locally contractive on X. Otherwise,

where

Let us notice that, by (53) and (66),

implies

and, for η > 0,

Indeed, if x, y ∈ X satisfying (76) are arbitrary and fixed, then from (53) we conclude that (76) holds only in two following cases: (i) (x, y) ∈ E × (F\E) or (ii) (x,y) ∈ E × E.

Now we see that, in particular, if x ∈ E and y ∈ F \ E, then we get that x ∈ (1/2,1), y = 1 and d(x, y) < 1/2, which, by (66), gives (77). Of course, by (53), (78) holds. Now, if η > 0, then, by (77),

and

Thus, (79) holds. Further, by (76)-(79), we see that

that is, for x ∈ (1/2,1) and y = 1, we have .

Therefore,

Consequently, we proved that (X,T) is not -uniformly locally contractive on X. This gives that the assumptions of Theorem 8(I) for such and for i = 2 do not hold.

Next, to prove that (X, T) is -uniformly locally contractive on X, we assume that x, y ∈ X satisfying (76) are arbitrary and fixed. Then, by (53), we conclude that (76) holds only in the following two cases:

Case 1. Let x ∈ E and let y ∈ F\E. By (76)-(79), we get

that is, for x ∈ (1/2,1) and y = 1, we have .

Therefore,

Case 2. Let x, y ∈ E. By (66), T(x) = {x/2 + 1/2}, T(y) = {y/2 + 1/2}, and, consequently, we get

that is, for x, y ∈ (1/2,1), we have . Therefore,

From Cases 1 and 2 it follows that (X, T) is -uniformly locally contractive on X.

It is clear that the assumptions of Theorem 8(I) for such and for i = 1 hold.

Now we prove that if w⁰ ∈ [0,1), then the assertion (A2) holds and if w⁰ = 1 then the assertion (A1) holds. Indeed, we have the following three cases:

Case 1. Let w⁰ ∈ [0,1/4). Then, by (66), there exists a dynamic process (w^m : m ∈ {0} ∪ ℕ) of (X, T) starting at w⁰ of the form: w¹ ≠ 1 and . Then, by (53), L(w⁰, w¹) = +∞ and . Consequently, a dynamic process (w^m : m ∈ {0} ∪ ℕ) is -Cauchy on X, lim_m→∞ w^m = 1 and 1 ∈ Fix(T), i.e. for each w⁰ ∈ [0,1/4), the assertion (A2) holds.

Case 2. Let w⁰ ∈ [1/4,1). Then, for each a dynamic process (w^m : m ∈ {0} ∪ ℕ) of (X,T) starting at w⁰, by (66), we have that ∀_m∈ℕ{w^m ∈ E} and, by (53),

and . Consequently, for each w⁰ ∈ [1/4,1), each a dynamic process (w^m : m ∈ {0}∪ ℕ) of (X, T) starting at w⁰ is -Cauchy on X, lim_{m →∞} w^m = 1 and 1 ∈ Fix(T), i.e., for each w⁰ ∈ [1/4,1), the assertion (A2) holds.

Case 3. Let w⁰ = 1. Then, for a dynamic process (w^m : m ∈ {0} ∪ ℕ) of (X, T) starting at w⁰, by (66), we have that ∀_m∈ℕ{w^m = 1 ∈ F\E} and, by (53), ∀_m∈ℕ{L(w^m-1, w^m) = d(w^m-1, w^m) + 3 > 1/2}. Consequently, if w⁰ = 1, a dynamic process (w^m : m ∈ {0} ∪ ℕ) of (X, T) starting at w⁰ satisfies ∀_m∈ℕ{L(w^m-1, w^m) > 1/2}, i.e. for w⁰ = 1, the assertion (A1) holds.

Finally, we see that, for each w⁰ ∈ X, there exists a dynamic process (w^m : m ∈ {0} ∪ ℕ) such that lim_m→∞ w^m = 1, lim_m→∞ L(w^m, 1) = lim_m→∞ d(w^m, 1) = 0 and 1 ∈ Fix(T).

Remark 9 Let us point out that . Indeed, by (53), L(1, 1) = d(1, 1) + 3 = 3 and

It is worth noticing that our results in metric spaces and in generalized metric spaces include Banach's [3], Nadler's [[4], Th. 5], Covitz-Nadler's [[5], Theorem 1] and Suzuki's [[10], Theorem 3.7] results.

Clearly, it is not otherwise. More precisely: (a) In Example 14 we construct -complete generalized metric space (X, ), a -family on X satisfying and a set-valued dynamic system (X, T) which is -uniformly locally contractive on X and next we show that the assertion (A2) holds; (b) In Example 15 we show that, for each ε ∈ (0, ∞), λ ∈ [0, 1) and i ∈ {1, 2}, the set-valued dynamic system (X, T) defined in Example 14 is not -uniformly locally contractive on X and thus we cannot use Theorems 1, 2 and 4-7; (c) In Example 16 we construct a complete metric space which is -family on X and -uniformly locally contractive set-valued dynamic system (X, T) such that, for each w⁰ ∈ X, the assertion (A2) holds and, additionally, L(w, w) > 0 for w ∈ Fix(T) which gives that our theorems are different from Theorem 7.

Example 14 Let Z and q be as in Example 3. Let X = Z ⋂ [0,9] and let where d = q|_[0,9]. Then (X, ) is a -complete generalized metric space. Let F = {1, 7} and let (X, T) be given by the formula

we see that T : X → C(X). Let E = {0, 1, 2} ∪ [4,5] ∪ {6, 8} and let L be of the form

By Example 6, the family is -family on X. By the similar reasoning as in Example 11, we show that (X, T) is -uniformly locally contractive on X. We see that for each w⁰ ∈ X there exists a dynamic process (w^m : m ∈ {0} ∪ ℕ) such that lim_m→∞ w^m = 2 and 2 ∈ Fix(T).

Remark 10 We notice that .

Example 15 Let X, and T be such as in Example 14. We show that, for any ε ∈ (0, ∞), λ ∈ [0, 1) and i ∈ {1, 2}, T is not -uniformly locally contractive on X.

Otherwise, there exist ε₀ ∈ (0, ∞), λ₀ ∈ [0, 1) and i ∈ {1, 2} such that

We consider the following three cases:

Case 1. If ε₀ = 1, then, in particular, for x₀ = 1 and y₀ = 1/2, since x₀, y₀ ∈ [0,1], by formula (46), we get

However, T(x₀) = {4, 5}, T(y₀) = {1, 2}, and, by (46),

Hence

Consequently,

and (80) gives

This leads to a contradiction.

Case 2. If ε₀ ∈ (1, ∞), then by a similar reasoning as in Case 1 we prove that (80) does not hold.

Case 3. If ε₀ ∈ (0, 1), then, in particular, for x₀ = 1 and y₀ = ((1 - ε₀)/2), we obtain that x₀, y₀ ∈ [0,1] and by a similar reasoning as in Case 1 we prove that (80) does not hold.

Example 16 Let X = [0,1] and where d : X × X → [0, ∞) is defined by the formula d(x, y) = |x - y|, x, y ∈ X. Then (X, ) is a complete metric space. Let E = [1/2, 1) and F = [1/2, 1] ⊂ X (we see that E ⊂ F ⊂ X) and let L : X × X → [0, +∞] be defined by (53). It follows from Example 10 that the family is -family on X. Let (X, T) be given by the formula:

First, we show that (X, T) is -uniformly locally contractive on X. Assume that x, y ∈ X satisfying L(x, y) < 1/2 are arbitrary and fixed. Then from (53) we conclude that L(x, y) < 1/2 implies (x, y) ∈ E × (F \ E) or (x, y) ∈ E × E. Consequently, the following two cases hold:

Case 1. Let x ∈ E and y ∈ F\E. Then, by (81) we get: T(x) = {x/2+1/2};

that is, for x ∈ [1/2, 1) and y = 1, we have 1/2 - x/2 ≤ 1/4 ≤ x/2 and . Therefore,

Case 2. Let x, y ∈ E. Then, by (81), T(x) = {x/2 + 1/2}, T(y) = {y/2 + 1/2} and, consequently, we get

that is, for x, y ∈ (1/2, 1), we have . Therefore,

Consequently, we proved that (X, T) is -uniformly locally contractive on X. We also see that all assumptions of Theorem 8(I) for this and for i = 1 hold.

Now, we show that, for each w⁰ ∈ X, the assertion (A2) holds. Indeed, we have the following three cases:

Case 1. Let w⁰ ∈ [0, 1/4). Then, there exists a dynamic process (w^m : m ∈ {0} ∪ ℕ) of (X, T) starting at w⁰ of the form: w¹ ≠ 1, and ∀_m∈ℕ{w^m ∈ [1/2, 1) = E}. Then, by (53), L(w⁰, w¹) = +∞ and ∀_m≥2{L(w^m-1, w^m) = d(w^m-1, w^m)}. Consequently, a dynamic process (w^m : m ∈ {0} ∪ ℕ) is -Cauchy on X, lim_m→∞ w^m = 1 and 1 ∈ Fix(T), i.e. for each w⁰ ∈ [0, 1/4), the assertion (A2) holds.

Case 2. Let w⁰ ∈ [1/4, 1). Then, for each a dynamic process (w^m : m ∈ {0} ∪ ℕ) of (X, T) starting at w⁰, by (81), we have that ∀_m∈ℕ{w^m ∈ E} and, by (53),

and ∀_m≥2{L(w^m-1, w^m) = d(w^m-1, w^m)}. Consequently, for each w⁰ ∈ [1/4, 1), each a dynamic process (w^m : m ∈ {0} ∪ ℕ) of (X, T) starting at w⁰ is -Cauchy on X, lim_m→∞ w^m = 1 and 1 ∈ Fix(T), i.e. for each w⁰ ∈ [1/4, 1), the assertion (A2) holds.

Case 3. Let w⁰ = 1. Then, there exists a dynamic process (w^m : m ∈ {0} ∪ ℕ) of (X, T) starting at w⁰, of the form: w⁰ = 1, w¹ = 1/2, ∀_m≥2{w^m ∈ E}, and, by (53), L(w⁰, w¹) = d(w⁰, w¹)+4 and ∀_m≥2{L(w^m-1, w^m) = d(w^m-1, w^m)}. Consequently, this dynamic process (w^m : m ∈ {0} ∪ ℕ) is -Cauchy on X, lim_m→∞ w^m = 1 and 1 ∈ Fix(T), i.e. for w⁰ = 1, the assertion (A2) holds.

Remark 11 One can also notice that L(1, 1) = 3 > 0 and . Indeed, we have L(1, 1) = d(1, 1) + 3 = 3 and

The Caristi [41] and Ekeland [42] results can be read, respectively, as follows.

Theorem 10 [41]Let (X, d) be a complete metric space. Let T : X → X be a single-valued map. Let φ : X → (-∞, +∞] be a map which is proper lower semicontinuous and bounded from below; we say that a map φ : X → (-∞, +∞] is proper if its effective domain, dom(φ) = {x : φ(x) < +∞}, is nonempty. Assume ∀_x∈X{d(x, T(x)) ≤ φ(x) - φ(T(x))}. Then T has a fixed point w in X, i.e. w = T(w).

Theorem 11 [42]Let (X, d) be a complete metric space. Let φ : X → (-∞, +∞] be a proper lower semicontinuous and bounded from below. Then, for every ε > 0 and for every x₀ ∈ dom(φ), there exists w ∈ X such that: (i) φ(w)+εd(x₀, w) ≤ φ(x₀); and (ii) ∀_x∈X\{w}{φ(w) < φ(x) +εd(x, w)}.

The Banach [3], Nadler [[4], Th. 5], Caristi [41], and Ekeland [42] results have extensive applications in many fields of mathematics and applied mathematics, they have been extended in many different directions and a number of authors have found their simpler proofs. Caristi's and Nadler's results yield Banach's result and Caristi's and Ekeland's results are equivalent. Jachymski [[14], Theorem 5], using a similar idea as in Takahashi [13], proved that Caristi's result yields Nadler's result.

Regarding this, we raise a question:

Question 3 Is it possible to find some analogons of Caristi's and Ekeland's theorems in generalized uniform spaces (or in generalized locally convex spaces or in generalized metric spaces) with generalized pseudodistances, and without lower semicontinuity assumptions as in [30]?

It is also natural to ask the following question:

Question 4 What additional assumptions in Theorems 8 and 9 (and thus also in Theorems 2 and 4-7) guarantee the uniqueness of fixed points?

The authors declare that they have no competing interests.

The authors have equitably contributed in obtaining the new results presented in this article. All authors read and approved the final manuscript.

1. Aubin, JP, Siegel, J: Fixed points and stationary points of dissipative multivalued maps. Proc Am Math Soc. 78, 391–398 (1980). Publisher Full Text

2. Yuan, GX-Z: KKM Theory and Applications in Nonlinear Analysis. Marcel Dekker, New York (1999)

3. Banach, S: Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales. Fund Math. 3, 133–181 (1922)

4. Nadler, SB: Multi-valued contraction mappings. Pacific J Math. 30, 475–488 (1969)

5. Covitz, H, Nadler, SB Jr.: Multi-valued contraction mappings in generalized metric spaces. Israel J Math. 8, 5–11 (1970). Publisher Full Text

6. Luxemburg, WAJ: On the convergence of successive approximations in the theory of ordinary differential equations. II. Nederl Akad Wetensch Proc Ser A Indag Math. 20, 540–546 (1958)

7. Jung, CFK: On a generalized complete metric spaces. Bull Am Math Soc. 75, 113–116 (1969). Publisher Full Text

8. Diaz, JB, Margolis, B: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull Am Math Soc. 74, 305–309 (1968). Publisher Full Text

9. Margolis, B: On some fixed points theorems in generalized complete metric spaces. Bull Am Math Soc. 74, 275–282 (1968). Publisher Full Text

10. Suzuki, T: Several fixed point theorems concerning τ-distance. Fixed Point Theory Appl. 2004(3), 195–209 (2004)

11. Suzuki, T: Generalized distance and existence theorems in complete metric spaces. J Math Anal Appl. 253, 440–458 (2001). Publisher Full Text

12. Feng, Y, Liu, S: Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. J Math Anal Appl. 317, 103–112 (2006). Publisher Full Text

13. Takahashi, W: Existence theorems generalizing fixed point theorems for multivalued mappings. In: Baillon, JB, Théra, M (eds.) Fixed Point Theory and Applications (Marseille, 1989), Pitman Res Notes Math Ser, vol. 252, pp. 397–406. Longman Sci. Tech., Harlow (1991)

14. Jachymski, J: Caristi's fixed point theorem and selections of set-valued contractions. J Math Anal Appl. 227, 55–67 (1998). Publisher Full Text

15. Zhong, CH, Zhu, J, Zhao, PH: An extension of multi-valued contraction mappings and fixed points. Proc Am Math Soc. 128, 2439–2444 (1999)

16. Mizoguchi, N, Takahashi, W: Fixed point theorems for multivalued mappings on complete metric spaces. J Math Anal Appl. 141, 177–188 (1989). Publisher Full Text

17. Eldred, A, Anuradha, J, Veeramani, P: On the equivalence of the Mizoguchi-Takahashi fixed point theorem to Nadler's theorem. Appl Math Lett. 22, 1539–1542 (2009). Publisher Full Text

18. Suzuki, T: Mizoguchi-Takahashi's fixed point theorem is a real generalization of Nadler's. J Math Anal Appl. 340, 752–755 (2008). Publisher Full Text

19. Kaneko, H: Generalized contractive multi-valued mappings and their fixed points. Math Japonica. 33, 57–64 (1988)

20. Reich, S: Fixed points of contractive functions. Boll Unione Mat Ital. 4, 26–42 (1972)

21. Reich, S: Some problems and results in fixed point theory. Contemp Math. 21, 179–187 (1983)

22. Quantina, K, Kamran, T: Nadler's type principle with hight order of convergence. Nonlinear Anal. 69, 4106–4120 (2008). Publisher Full Text

23. Suzuki, T, Takahashi, W: Fixed point theorems and characterizations of metric completeness. Topol Meth Nonlinear Anal. 8, 371–382 (1997)

24. Al-Homidan, S, Ansari, QH, Yao, JC: Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. Nonlinear Anal. 69, 126–139 (2008). Publisher Full Text

25. Latif, A, Al-Mezel, SA: Fixed point results in quasimetric spaces. Fixed Point Theory Appl. 2011, 8 Article ID 178306 (2011)

    Article ID 178306

    BioMed Central Full Text

26. Frigon, M: Fixed point results for multivalued maps in metric spaces with generalized inwardness conditions. Fixed Point Theory Appl. 2010, 19 Article ID 183217 (2010)

27. Klim, D, Wardowski, D: Fixed point theorems for set-valued contractions in complete metric spaces. J Math Anal Appl. 334, 132–139 (2007). Publisher Full Text

28. Ćirić, L: Multi-valued nonlinear contraction mappings. Nonlinear Anal. 71, 2716–2723 (2009). Publisher Full Text

29. Pathak, HK, Shahzad, N: Fixed point results for set-valued contractions by altering distances in complete metric spaces. Nonlinear Anal. 70, 2634–2641 (2009). Publisher Full Text

30. Włodarczyk, K, Plebaniak, R: Maximality principle and general results of Ekeland and Caristi types without lower semicontinuity assumptions in cone uniform spaces with generalized pseudodistances. Fixed Point Theory Appl. 2010, 35 Article ID 175453 (2010)

31. Włodarczyk, K, Plebaniak, R: Periodic point, endpoint, and convergence theorems for dissipative set-valued dynamic systems with generalized pseudodistances in cone uniform and uniform spaces. Fixed Point Theory Appl. 2010, 32 Article ID 864536 (2010)

32. Włodarczyk, K, Plebaniak, R, Doliński, M: Cone uniform, cone locally convex and cone metric spaces, endpoints, set-valued dynamic systems and quasi-asymptotic contractions. Nonlinear Anal. 71, 5022–5031 (2009). Publisher Full Text

33. Włodarczyk, K, Plebaniak, R: A fixed point theorem of Subrahmanyam type in uniform spaces with generalized pseudodistances. Appl Math Lett. 24, 325–328 (2011). Publisher Full Text

34. Włodarczyk, K, Plebaniak, R: Quasigauge spaces with generalized quasipseudodistances and periodic points of dissipative set-valued dynamic systems. Fixed Point Theory Appl. 2011, 23 Article ID 712706 (2011)

    Article ID 712706

    BioMed Central Full Text

35. Włodarczyk, K, Plebaniak, R: Contractivity of Leader type and fixed points in uniform spaces with generalized pseudodistances. J Math Anal Appl. 387, 533–541 (2012). Publisher Full Text

36. Włodarczyk, K, Plebaniak, R: Kannan-type contractions and fixed points in uniform spaces. Fixed Point Theory Appl. 2011, 90 (2011). BioMed Central Full Text

37. Tataru, D: Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms. J Math Anal Appl. 163, 345–392 (1992). Publisher Full Text

38. Kada, O, Suzuki, T, Takahashi, W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math Japonica. 44, 381–391 (1996)

39. Lin, LJ, Du, WS: Ekeland's variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces. J Math Anal Appl. 323, 360–370 (2006). Publisher Full Text

40. Vályi, I: A general maximality principle and a fixed point theorem in uniform spaces. Period Math Hungar. 16, 127–134 (1985). Publisher Full Text

41. Caristi, J: Fixed point theorems for mappings satisfying inwardness conditions. Trans Am Math Soc. 215, 241–151 (1976)

42. Ekeland, I: On the variational principle. J Math Anal Appl. 47, 324–353 (1974). Publisher Full Text