COINCIDENCE AND FIXED POINT THEOREMS IN METRIC AND BANACH SPACES

We prove some coincidence and fixed point theorems for generalized (φ,f )contractive type and f -nonexpansive type multivalued and singlevalued mappings in metric and Banach spaces, respectively. Our results extend, improve, and unify a few known results due to Browder, Browder and Petryshyn, Chang, Huang, and Cho, Husain and Latif, Husain and Tarafdar, and Karlovitz. 2000 Mathematics Subject Classification. 54H25, 47H10.


Introduction. Husain and Latif
introduced the notions of f -contractive type and f -nonexpansive type multivalued mappings and proved fixed point theorems for the mappings in metric and Banach spaces, respectively.Chang, Huang, and Cho [4] introduced the concept of generalized f -contractive type multivalued mappings in metric spaces and established a coincidence point theorem for generalized f -contractive type multivalued mappings.
Motivated and inspired by the above works, we introduce a concept of generalized (φ, f )-contractive type multivalued mappings which contains the concepts of fcontractive type multivalued mappings and generalized f -contractive type multivalued mappings as special cases, and we prove three coincidence and fixed point theorems for such mappings which extend and improve the main results of Chang, Huang, and Cho [4], Husain and Latif [6,7], and others.On the other hand, we study the existence of common fixed points of f -nonexpansive type multivalued mappings which generalizes the main results of Husain and Latif [5,6,7] and contains some results due to Browder [1], Browder and Petryshyn [2], Husain and Tarafdar [8], and Karlovitz [9] as particular cases.

Preliminaries.
Throughout this paper, we denote by R, R + , N the set of real numbers, the set of nonnegative real numbers, and the set of positive integers, respectively.Let Φ be the family of mappings φ : (R + ) 5 → R + such that each φ is nondecreasing for each variable and where Lemma 2.1 (see [3]).Let h : R + → R + be nondecreasing, right continuous, and lim n→∞ h n (t) = 0 for all t ∈ R + .Then  That is, (2.4) holds.This completes the proof.
Let (X, d) be a metric space and M a nonempty subset of X. Define (2.8) Let H denote the Hausdorff metric on CB(X) induced by d.That is, where d(x, A) = inf{d(x, y) : y ∈ A}.
(i) J is said to be a generalized (φ, f )-contractive type multivalued mapping if, for all x ∈ M and for each u x ∈ Jx, there exists ν y ∈ Jy for all y ∈ M satisfying d f y,ν y ,d f x,ν y ,d f y,u x . (2.10) (ii) J is said to be a generalized f -contractive type multivalued mapping if, for all x ∈ M and for each u x ∈ Jx, there exists ν y ∈ Jy for all y ∈ M satisfying (2.11) (iii) J is said to be an f -contractive type multivalued mapping if, for all x ∈ M and for each u x ∈ Jx, there exists ν y ∈ Jy for all y ∈ M satisfying (2.12) (iv) J is said to be an f -nonexpansive type multivalued mapping if, for all x ∈ M and for each u x ∈ Jx, there exists ν y ∈ Jy for all y ∈ M satisfying (2.13) Clearly, each f -contractive type multivalued mapping is both f -nonexpansive type multivalued mapping and generalized f -contractive type multivalued mapping, and each generalized f -contractive type multivalued mapping is a generalized (φ, f )contractive type multivalued mapping.

Coincidence and fixed points of generalized (φ, f )-contractive type multivalued mappings.
In this section, we establish three coincidence and fixed point theorems for generalized (φ, f )-contractive type multivalued mappings in complete metric space.Theorem 3.1.Let M be a nonempty closed subset of a complete metric space (X, d), and let f : M → M be a singlevalued mapping, and J : M → CL(M) a generalized (φ, f )contractive type multivalued mapping.Suppose that f M = M. Then f and J have a coincidence point in M.
Proof.Let x 0 be an arbitrary but fixed element of M and choose and y 2 = f x 2 .Inductively, we can extract a sequence {f is in Jx n and where d n = d(f x n ,f x n+1 ) for all n ≥ 0. We claim that Thus for all n, p ∈ N, we conclude that Observe that f M = M. Hence there exists q ∈ M with f q = z.From f x n+1 ∈ Jx n , f M = M, and (2.10), we can choose that f ν n ∈ Jq satisfying It is easy to see that there exists an increasing sequence {n i } i∈N ⊂ N such that at least one of (3.7), (3.8), and (3.9) holds, We now prove that Suppose that (3.7) is satisfied.If lim i→∞ d(f q, f ν n i ) ≠ 0, then there exist ε > 0 and a subsequence {ν n i j } J∈N of {ν n i } i∈N such that d(f q, f ν n ij ) ≥ ε for all j ∈ N. Note that lim n→∞ f x n = f q and lim n→∞ d(f x n ,f x n+1 ) = 0. Thus there exists k ∈ N satisfying max{d(f x n i j ,f q),(f x n i j ,f x n i j +1 )} < ε for all j ≥ k.Using (3.7), we have for all j ≥ k.Equation (3.11) ensures that which implies that lim j→∞ d(f q, f ν n i j ) = 0.This is a contradiction.Consequently, (3.10) holds.Suppose that (3.8) is fulfilled.In view of (3.8), we obtain that for all i ∈ N, (3.13)By virtue of (3.13) and Lemmas 2.2 and 2.3, we conclude that lim sup (3.14) Using (3.14) and Lemma 2.1, we get that lim sup i→∞ d(f q, f ν n i ) = 0.It is easy to see that (3.10) holds.Suppose that (3.9) is satisfied.Then That is, (3.10) holds.
Since Jq is closed and {f ν n i } i∈N ⊂ Jq, by (3.10) we conclude immediately that f q belongs to Jq.This completes the proof.
As in the proof of Theorem 3.1, we conclude that there exists a subsequence {u n i } i∈N of {u n } n∈N such that lim i→∞ u n i = f z and f z ∈ Jz.This completes the proof.Remark 3.7.In case f = i X (the identity mapping on X), Theorem 3.6 reduces to a result which extends and improves Theorem 2.3 of [6].Theorem 3.8.Let X, M, f , and J be as in Theorem 3.6.Suppose that for all x ∈ M, the following condition is fulfilled: (3.17) Then f and J have a common fixed point in M.
Proof.Theorem 3.6 ensures that there exists z ∈ X such that f z ∈ Jz.Set f n z = a n for all n ∈ N. Equation (3.17) and the closedness of M means that there exists It follows from (2.10) that there exists b n ∈ Jt satisfying Husain and Latif [5] and Kaneko and Sessa [9] proved the following results, respectively.Theorem 3.9 (see [5,Theorem 2.2]).Let M be a nonempty closed subset of a complete metric space (X, d) and f a continuous mapping of M into itself.Suppose J : M → CL(M) is an f -contractive type multivalued mapping such that Jf = f J and JM ⊆ f M.Moreover, assume that one of (3.17) and (3.22) holds for all x ∈ M, where Then f and J have a common fixed point in M.
Theorem 3.10 (see [9,Theorem 3]).Let (X, d) be a complete metric space, and let f : X → X and J : X → CB(X) be compatible continuous mappings such that JX ⊆ f X and (3.23) for any x, y ∈ X, where 0 ≤ r < 1. Assume also that for each x ∈ X either (3.17) or (3.22) holds.Then f and T have a common fixed point in X. Remark 3.11.Note that (3.22) is equivalent to the following It is easy to see that (3.22) implies (3.17).Therefore (3.22) in Theorems 3.9 and 3.10 is superfluous.Thus Theorem 2.2 of [5] is a special case of Theorem 3.8.
Remark 3.12.It follows from [5, Theorem 4.2, Condition (A)] that for any two mappings J i , J j and any x ∈ M, J i x ⊆ J j x, and J j x ⊆ J i x.That is, J i x = J j x for all x ∈ M, and i, j ∈ N. Thus [5, Theorem 4.2] is the same as [5, Theorem 2.2].

Fixed points of f -nonexpansive type multivalued mappings.
In this section, we establish a common fixed point theorem for f -nonexpansive type multivalued mappings in nonempty convex weakly compact subsets of a Banach space under certain conditions.

.3)
Then f and J have a common fixed point in M.
Proof.We may, without loss of generality, assume that 0 ∈ M. For each sequence {h n } n∈N with h n ∈ (0, 1) and lim n→∞ h n = 1, we define J n x = {h n u : u ∈ Jx} for x ∈ M. It follows from (4.3) that As in the proof of Theorem 3.1 in [6], we conclude that J n : M → CL(M) is an fcontractive type multivalued mapping and J n M ⊆ f M for all n ∈ N. Let k ∈ N. We show that if f x is in J k x, then lim n→∞ f n x exists.Assume that f x ∈ J k x = h k Jx.Then there exists u 1 ∈ Jx with f x = h k u 1 .This implies that f u 1 ∈ f Jx ⊆ Jf x = Jh k u 1 = h k Ju 1 and f u 1 = h k u 2 for some u 2 ∈ Ju 1 .Consequently, f 2 x = f h k u 1 = h k f u 1 = h 2 k u 2 .Using (4.3), we can construct a sequence {u n } n∈N ⊆ M with u n+1 ∈ Ju n and f u n = h k u n+1 for all n ≥ 0, where u 0 = x.It is clear that Since M is bounded and {u n } n∈N ⊆ M, it follows that { u n } n∈N is bounded.Hence f n x = h n k u n → 0 as n → ∞.In view of Theorem 3.8, we obtain that f and J n have a common fixed point in M. The remaining part of the proof is as in Theorem 3.1 of [6].This completes the proof.
As a consequence of Theorem 4.1, we have the following corollary.Then f and J have a common fixed point in M.

Corollary 3 . 2 Remark 3 . 3 .Example 3 . 5 .Theorem 3 . 6 .
(see[4, Theorem 2.1]).Let M be a nonempty closed subset of a complete metric space (X, d), f : M → M a singlevalued mapping, and J : M → CL(M) a generalized f -contractive type multivalued mapping.Suppose that f M = M. Then f and J have a coincidence point in M. Theorem 2.2 of[4] and Theorem 2.3 of[6] are special cases of Theorem 3.1.To see that the closedness of M in the hypothesis of Theorem 3.1 is essential, we need the following example.Example 3.4.Let X = R with the usual metric.Take M = (0, 1), Jx ={(1/4)(1 − x 4 )}, and f x = 1−x 2 for all x ∈ M. Obviously, |Jx −Jy| ≤ (1/2)|f x −f y| for all x, y ∈ M. The requirements of Theorem 3.1 are satisfied except that M is closed.But f and J have no coincidence point in M. The following example reveals that the condition of f M = M in Theorem 3.1 is necessary.Let X = R with the usual metric and M = [0, 1].Define Jx = {(1/6)x 2 } and f x = 1−(1/2)x for all x, y ∈ M. Then |Jx−Jy| ≤ (2/3)|f x−f y| for all x, y ∈ M. It is easy to see that the hypotheses of Theorem 3.1 are satisfied except f M = M.However, f and J have no coincidence point in M. Let M be a nonempty closed subset of a complete metric space (X, d), and let f : M → M a singlevalued mapping and J : M → CL(M) a generalized (φ, f )contractive type multivalued mapping.If f is continuous, JM ⊆ f M, and f Jx ⊆ Jf x for all x ∈ M, then f and J have a coincidence point in M. Proof.Let {f x n } n∈N and z be as in the proof of Theorem 3.1.Put u

Corollary 4 . 2
(see[6, Theorem 3.1]).Let M, X, f , J be as in Theorem 4.1.Suppose that f and J satisfy (4.2) and the following relations:f Jx = Jf x, ∀x ∈ M; f x = f 2 x implies tf x ∉ Jx, ∀x ∈ M, t ≥ 1.(4.6) i) h(t) < t for all t > 0; (ii) t = 0 provided that t ≤ h(t) for some t ∈ R + .Let {a n } n∈N and {b n } n∈N be in R + and lim n→∞ a n = 0. Then Let {a n } n∈N be a sequence in R + and h : R + → R + be nondecreasing and right continuous.Then It follows that there exists a subsequence {a n i } i∈N of {a n } n∈N with lim i→∞ h(a n i ) = a.Put lim sup i→∞ a n i = b.Then there exists a subsequence {a n i j } Proof.Set lim supn→∞ h(a n ) = a.j∈N of {a n i } i∈N such that lim j→∞ a n i j = b.Observe that each sequence in R contains a monotone subsequence.Thus {a n i j } j∈N contains a mono- .3) Suppose that d n > d n−1 for some n ∈ N. Thus (3.2) ensures that d n ≤ ψ(d n ).It follows from Lemma 2.1 that d n = 0, which is a contradiction.Therefore (3.3) holds.In view of (3.2) and (3.3), we have = 0. Therefore b n → t as n → ∞.Since Jt is closed and {b n } n∈N ⊂ Jt, we conclude that t belongs to Jt.This completes the proof.
t , 0,d t, b n ,d a n ,t + d t, b n ,d t, a n ≤ ψ d a n ,t + d t, b n , (3.19) which implies that d t, b n ≤ d t, a n + d a n ,b n ≤ d t, a n + ψ d a n ,t + d t, b n .(3.20) Put S = lim sup n→∞ d(b n ,t).Using (3.20) and Lemmas 2.2 and 2.3, we obtain that S ≤ lim sup n→∞ d a n ,t + lim sup n→∞ ψ d a n ,t + d b n ,t ≤ ψ lim sup if for each x ∈ X and each sequence {x n } n∈N weakly converging to x, n→∞ x n − x , ∀y = x.(4.1)Theorem 4.1.Let M be a nonempty weakly compact convex subset of a Banach space X which satisfies Opial's condition.Suppose that f is a continuous affine mapping of M into itself and J