ON COINCIDENCE AND COMMON FIXED POINTS OF NEARLY DENSIFYING MAPPINGS

Coincidence and commonfixed point theorems for certain new classes of nearly densifying mappings are established. Our results extend, improve, and unify a lot of previously known theorems.

Throughout this paper, (X, d) denotes a metric space, N, R + , and R denote the sets of positive integers, nonnegative real numbers, and real numbers, respectively, and ω = N {0}.Define = F | F : X × X → R + and F(x,y) = 0 if and only if x = y , 1 = {F | F ∈ and F is upper semicontinuous in X × X}, 2 = {F | F ∈ and F is lower semicontinuous in X × X}. (2.1) Let G be a family of self-mappings in and G * be the semigroup generated by G under composition.Clearly, CIS G ⊇ G * ⊇ {g n : n ∈ ω} for any g ∈ G.For A, B ⊆ X, x, y ∈ X, f ∈ G, and F ∈ , define Ā denotes the closure of A. f is said to have diminishing orbital diameter if lim n→∞ δ d (O f (f n x)) < δ d (O f (x)) for all x ∈ X with δ d (O f (x)) > 0. f is called contractive with respect to d if d(f x, f y) < d(x, y) for all distinct x, y ∈ X.
Definition 2.1.Let G be a semigroup of self-mappings on a metric space (X, d) and F ∈ .G is said to have F -diminishing orbital diameter, if for any x ∈ X with δ F (Gx) > 0 there is s ∈ G such that δ F (Gsx) < δ F (Gx).Definition 2.2 (see [15]).Let A be a bounded subset of a metric space (X, d).Then α(A), the measure of noncompactness of A, is the infimum of all ε > 0 such that A admits a finite covering consisting of subsets with diameters less than ε.
The following properties of α are well known.Lemma 2.3.Let (X, d) be a metric space and A, B be bounded subsets of X.Then α A B = max α(A), α(B) ; (2.4) α(A) = 0 ⇐⇒ A is pre-compact, i.e., A is totally bounded; (2.5) Definition 2.4 (see [4]).A continuous self-mapping f in a metric space (X, d) is said to be densifying if α(f (A)) < α(A) for every bounded subset A of X with α(A) > 0. Definition 2.5 (see [27]).A self-mapping f in a metric space (X, d) is said to be nearly densifying if α(f (A)) < α(A) for every bounded and f -invariant subset A of X with α(A) > 0.
Obviously, each densifying mapping is nearly densifying, but the converse is false.Definition 2.6 (see [23]).Let X be a topological space, f a self-mapping in X, and M a nonempty subset of X. M is an attractor for compact sets under f if (i) M is compact and f M ⊆ M, (ii) given any compact set C ⊆ X and any open neighborhood U of M, there exists Let G be a left reversible semigroup.We define a relation ≥ on G by a ≥ b if and only if a ∈ bG {b}.It is easy to check that (G, ≥) is a directed set.Lemma 2.7.Let G be a left reversible semigroup of continuous self-mappings in a compact metric space (X, d), A = f ∈G f x, and F ∈ 1 .Then ( We now prove that f X is a compact subset of X for each f ∈ G. Let x be in X and x n n∈N ⊆ X with lim n→∞ f x n = x.The compactness of X ensures that there exists a subsequence {x n k } k∈N of {x n } n∈N such that it converges to some point t ∈ X.Since From the compactness of X we can choose two subnets {x f k } and {y f k } of {x f } and {y f }, respectively, such that x f k → x and y f k → y for some x, y ∈ X.For every g ∈ G and f k ≥ g, we get that By virtue of closedness of gX, we infer that x, y ∈ gX.This means that x, y ∈ A. Consequently, We finally prove that f A = A for all f ∈ G. Let f ∈ G and x ∈ A. For any g ∈ G there exist a, b ∈ G with f a = gb.Note that x ∈ A ⊆ aX.Thus there is y ∈ X with x = ay.It follows that f x = f ay = gby ∈ gX.This implies that f A ⊆ g∈G gX = A for f ∈ G.For the reverse inclusion, let f ,g ∈ G and y ∈ A. It follows from y ∈ f gX that there exists x g ∈ gX with f x g = y.The compactness X ensures that there exists a convergent subnet x g k of x g such that x g k → x for some x ∈ X. Therefore y = f x.For any h, g ∈ G with g ≥ h, we obtain that hX is closed and x g belongs to hX.Thus the limit point x of x g lies in hX.That is, x ∈ A. Note that y = f x ∈ f A. Therefore, A ⊆ f A for f ∈ G.This completes the proof.

Common fixed point theorems for nearly densifying mappings
Theorem 3.1.Let G and H be finite families of continuous and nearly densifying self-mappings in a complete metric space (X, d).
Therefore, there exist x ∈ A and y ∈ B such that a = gx and b = hy.Using (3.1), we have which is a contradiction.Consequently, A = B = a singleton, say, {w} for some w ∈ X.Thus w = f w for all f ∈ G H. That is, G and H have a common fixed point w ∈ X.If G has another fixed point v ∈ X and v ≠ w, by (3.1) we infer that which is absurd.Hence G has a unique fixed point w.Similarly, we conclude that H has also a unique fixed point w.
It follows from Lemma 2.7 that (Y , w).In view of (3.1) and Theorem 3.1(i).We obtain that Let G and H be finite families of continuous and nearly densifying self-mappings in a complete bounded metric space (X, d) satisfying (3.1).Assume that G * , H * are near commutative.Then Theorem 3.1(i), (iii), and the following statements hold: (i) Suppose that δ F G * sx > 0. In view of (3.6) and (3.8) there exists s ∈ G * such that δ F G * sx < δ F G * x .That is, G * has F -diminishing orbital diameter.Analogously, H * has F -diminishing orbital diameter also.This completes the proof.
We now state without proof analogues of Theorems 3.1 and 3.2.

Theorem 3.3. Let G be a finite family of continuous and nearly densifying selfmappings in a complete metric space (X, d). If there exist
G * x 0 is bounded and G * is left reversible.Then the following statements hold: (i) G has a unique common fixed point w ∈ X, and Theorem 3.4.Let f and g be continuous self-mappings in a complete metric space (X, d).Assume that there exist i, j, p, q ∈ N, F ∈ 1 , x 0 ,y 0 ∈ X such that (i) F(f p x, g q y)<δ F s∈CIS f sO f (x), t∈CIS g tO g (y) , ∀x, y ∈ X with f p x ≠ g q y; (ii) f i and g j are nearly densifying; (iii) O f (x 0 ) and O g (y 0 ) are bounded.Then the following statements hold: (1) f and g have a unique common fixed point w ∈ X, and w is also the only fixed point of f and g, respectively; (3.11) ).In view of Theorem 3.4(ii), (iii) and we conclude easily that A ∈ NCI f and f A = A. Similarly, B ∈ NCI g and gB = B.The rest of the proof is the same as that of Theorem 3.1.This completes the proof.
Remark 3.5.Theorem 3.4 extends Theorems 3 and 4 of Liu [19], the theorem of Sharma and Srivastava [29].Akin to Theorem 3.4, we have the following.Theorem 3.6.Let f be continuous self-mapping in a complete metric space (X, d).
Assume that there exist i, p, q ∈ N, Then the following statements hold: (1) f has a unique fixed point w∈X, and Remark 3.7.Theorem 4 of Khan [10] and Theorem 4 of Rao [22] are special cases of Theorem 3.6.Theorem 3.8.Let f and g be continuous self-mappings in a complete bounded metric space (X, d).Assume that there exist i, j, p, q ∈ N satisfying Theorem 3.4(ii) and (3.13) Then Theorem 3.4( 1) and (3.11) and the following statements hold: Proof.It follows from Theorem 3.4 that Theorem 3.4(1), (3.11), and Theorem 3.8(i) hold.By the definitions of CIS f and CIS g , we conclude easily that Theorem 3.8(iii) holds.Since f n O f (x) = O f (f n x) and g n O g (y) = O g (g n y), so Theorem 3.6(iv) is satisfied.Now we prove that Theorem 3.8(ii) holds.Assume that B be any nonempty compact subset of X.Using Lemma 2.3, we have This shows that {w} is an attractor for compact sets under f .Thus Theorem 3.8(ii) follows from theorem of [9] and Remark 1 of [9].This completes the proof.
Similarly, we have the following theorem.Theorem 3.9.Let f be a continuous self-mapping in a complete bounded metric space (X, d).Assume that there exist i, p, q ∈ N satisfying Theorem 3.6(ii) and (3.15) Then Theorem 3.6(2) and the following statements hold: (i) f has a unique fixed point w ∈ X, and has diminishing orbital diameter and (ii) there exists a bounded complete metric d 1 on X which is equivalent to d such that f is contractive with respect to d 1 ; (iii) CIS f has a unique common fixed point w ∈ X. Remark 3.10.Theorem 3.8 generalizes Theorem 4 of [2] and Theorem 4 of [22].Theorem 3.9 extends and improves Theorem 3 of [1], Corollary 2 of [9], Theorem 3.1 of [17], and Theorems 1 and 2 of [18]

Coincidence point theorems for two pairs of nearly densifying mappings
Theorem 4.1.Let f , g, s, and t be a continuous and nearly densifying mappings from a complete metric space (X, d) into itself satisfying Let G = {f ,g,s,t}.Assume that there exist F 1 ,F 2 ∈ and x 0 ∈ X such that  It is evident to see that α(A) = 0. Thus Ā is compact by completeness of X. Set B h∈G * h Ā. Lemma 2.7 ensures that f B = gB = sB = tB = B ≠ ∅ and B is compact.Let F 1 be in 2 .Define r : B → R + by putting r (x) = F 1 (tx, gx).Since r is a lower semi-continuous function on the compact set B, so there exists b ∈ B with Suppose that neither f and s nor g and t have a coincidence point.Then where b = stc ∈ B. In view of (4.1), (4.3), (4.4), (4.7) and (4.8), we have  which is a contradiction.Hence f and s or g and t must have a coincidence point.The argument is similar for F 2 ∈ 2 .This completes the proof.
Theorem 4.2.Let f ,g,s, and t be continuous and nearly densifying mappings from a complete metric space (X, d) into itself satisfying f ,g ∈ C s C t .Let G = {f ,g,s,t} and H = {s, t}.Assume that there exist F 1 ,F 2 ∈ and x 0 ∈ X such that (4.2), (4.3), (4.4), and the following statement hold: for all x, y ∈ X with sx ≠ ty, f x ≠ gy, and for all x, y ∈ X with tx ≠ sy, gx ≠ f y.Thus the conditions of Theorems 4.1 and 4.2 are satisfied.However, f and s have two coincidence points 1 and 3, while f ,g,s, and t have none.
Theorem 4.6.Let f ,g,s, and t be continuous and nearly densifying mappings from a complete metric space (X, d) into itself satisfying f ,g,s ∈ C t and g ∈ C s .Assume that there exist F 1 ,F 2 ∈ and x 0 ∈ X satisfying (4.2), (4.3), and (4.4).If b is a common coincidence point of f ,g,s, and t, then tb is a unique common fixed point of f ,g,s, and t.
Proof.Since f ,g,s ∈ C t , g ∈ C s , and f b = gb = sb = tb, we have t 2 b = tf b = f tb = tgb = gtb = tsb = stb.Suppose that t 2 b ≠ tb.From (4.3) and (4.4) we conclude that which is a contradiction.Therefore tb = t 2 b = f tb = gtb = stb.That is, tb is a common fixed point of f ,g,s, and t.The uniqueness of a common fixed point follows from (4.3) and (4.4).This completes the proof.
Theorem 4.8.Let f ,g,s, and t be continuous and nearly densifying mappings from a complete metric space (X, d) into itself and G = {f ,g,s,t}.Suppose that there exist F ∈ 2 and x 0 ∈ X such that (4.5) and the following hold: which is a contradiction.Hence f a = sa.This completes the proof.
Theorem 4.9.Let f and g be continuous and nearly densifying mappings from a complete metric space (X, d) into itself and G = {f ,g}.Suppose that there exist F ∈ 2 and x 0 ∈ X satisfying (4.Then f or g has a fixed point in X.
Proof.It may be completed following the proof of Theorem 4.8.

Remark 4 . 3 . 2 . 4 . 4 .Example 4 . 5 .
and H * is left reversible.(4.11)Then f and s or g and t have a coincidence point in X.Proof.Put A = G * x 0 and B = h∈H * h Ā.As in the proof of Theorem 4.1, we infer that B is nonempty compact subset of Ā and sB = tB = B ⊇ f B gB.The remaining part of the proof is as in Theorem 4.1.This completes the proof.Theorem 3.1 of[12] and Theorem 3.1 of[13] are special cases of Theorem 4.RemarkThe following example reveals that f ,g,s, and t in Theorems 4.1 and 4.2 do not necessarily have a coincidence point and that if either f and s or g and t have a coincidence point, then the coincidence point may not be unique.Let X = {1, 3, 6} with the usual metric d and F 1 = F 2 = d.Define f ,g,s,t : X → X by f 1 = g3 = g6 = 1, f3 = f 6 = g1 = 3 and s = t = i X -the identity mapping on X.Take G = {f ,g,s,t} and H = {s, t}.Clearly,g 2 = f = f 2 , g = f g = gf = g 3 , G = G * , H = H * ,and G * and H * are left reversible.It is easy to verify that d(f x, gy) = 2 < 3 = d(sx, ty) (4.12)
* x 0 ,H * y 0 are bounded and G * ,H * are left reversible.* y ∈ X; (ii) there exist bounded complete metrics d 1 ,d 2 on X which are equivalent to d such that f ,g are contractive with respect to d 1 and d 2 , respectively; (iii) CIS f and CIS g have a unique common fixed point w ∈ X, and w is also the only fixed point of CIS f and CIS g , respectively; (iv) f and g have diminishing orbital diameter.