Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 1083-0464 Vol. No. 01, 1996, pp. 30–41. Submitted: October 11, 1995. Published: March 8, 1996.

* SET-VALUED MAPS WITH FIXED AND

COINCIDENCE POINTS

H.K. Pathak*and M.S. Khan

Abstract. In this paper, we prove the existence of solutions of functional equations f_{i}x ∈ Sx ∩ T x and x = f_{i}x ∈ Sx ∩ T x under certain nonlinear hybrid contraction and asymptotic regularity conditions, generalize and improve a recent result due to V. Popa concerning common fixed points of multi-valued mappings weakly commuting with a single-valued mapping and satisfying a generalized contraction type condition. In the sequel, we derive several independent results.

(1991) A.M.S. (MOS) Subject Classification Codes. 54H25, 47H10.

Keywords and Phrases. Coincidence and common fixed points, Hausdorffmetric, orbital complete, proximinal subset & weakly commuting mappings.

I. Introduction

In [16], Nadler proved a fixed point theorem for multi-valued contraction, commonly known as Nadler’s contraction principle. Subsequently, a large number of generalizations of Nadler’s contraction principle appeared in Čirič [1], Khan [10], Kubiak [12], [13], Kaneko [8], [9], Sessa [22]–[24], Meade-Singh [14], Smithon [27], Wegrzyk [28] and in many others.

Recently, non-linear hybrid contractions, that is, contraction types involving single-valued and multivalued mappings have been studied by Khan et al. [11], Mukherjee [15], Naimpally et al. [17], Popa [18], Rhoades et al. [19] and Sessa et al. [24], [25].

In this note, we show the existence of solutions of functional equations f_{i}x ∈ Sx∩T x and x = f_{i}x ∈ Sx∩T x under certain independent non-linear hybrid contraction and asymptotic regularity conditions where {f_{i}} is a family of single-valued mappings on a metric space, S and T are multi-valued mappings on a metric space. Our results are generalizations and improvement of some results of O. Hadžić, H. Kaneko, M.S. Khan et al., T. Kubiak, R.N. Mukherjee, S.A. Naimpally et al., B.E. Rhoades et al. and many others. In the sequel, we also obtain other related results by using the proximinality of sets.

II. Preliminaries

Let (X, d) be a metric space. A subset K of X is said to be proximinal if for each x ∈ X, there exists a point y ∈ K such that d(x, y) = D(x, K), where D(x, A) denotes the ordinary distance between x ∈ X and a non-empty subset A of X.

Typeset by AMS-TEX

*Research partially supported by U.G.C., New Delhi, India.

Department of Mathematics, Kalyan Mahvidyalaya, Bhilai Nagar (M.P.) 490006, India.

Sultan Qaboos University, Department of Mathematics and Computing, College of Science, Al-Khod, Muscat.

Copyright \begin{matrix}c \\ ◯ \end{matrix}1996 by Cameron University

We shall use the following notation and definitions:

CL = {A : A is a non-empty closed subset of X}, CB(X) = {A : A is a non-empty closed and bounded subset of X}, C(X) = {A : A is a non-empty compact subset of X} and P_{x}(X) = {A : A is a non-empty proximinal subset of X}.

For A, B ∈ CL(X) and ε > 0,

N(ε, A) = {x ∈ X : d(x, a) < ε for some a ∈ A}, E_{A,B} = {ε > 0 : A ⊆ N(ε, B) and B ⊆ N(ε, A)}, and

H(A, B) = \{ \begin{matrix}inf E_{A,B}, & if E_{A,B} ≠ ∅ \\ + ∞ & if E_{A,B} = ∅. \end{matrix}

H is called the generalized Hausdorff distance function for CL(X) induced by the metric d, and H defined on CB(X) is said to be the Hausdorff-Pompeiu metric induced by d.

It is well known that P_{x}(X) ⊂ CL(X) and C(X) ⊂ P_{x}(X) [22].

Let {f_{i}} be a family of single-valued mappings from X to itself and S, T be multi-valued mappings from X to the non-empty subsets of X.

Definition 2.1.

If, for x_{∘} ∈ X, there exist sequences {x_{n}} and {y_{n}} in X such that

y_{2n} = f_{2n}x_{2n−1} ∈ Sx_{2n} for every n ∈ N,

and

y_{2n+1} = f_{2n+1}x_{2n} ∈ T x_{2n+1} for every n ∈ N_{∘} = N ∪ {0},

then 0_{f_{i}}(x_{∘}) = {y_{n} : n = 0, 1, 2, … } is said to be the orbit for (S, T : f_{i}) at x_{∘}. Further, 0_{f_{i}}(x_{∘}) is called a regular orbit for (S, T; f) if

d(y_{n}, y_{n+1}) ≤ \{ \begin{matrix}H(Sx_{n−1}, T x_{n}), & if n is odd, \\ H(T x_{n−1}, Sx_{n}), & if n is even. \end{matrix}

Definition 2.2.

If, for x_{∘} ∈ X, there exist sequences {x_{n}} and {y_{n}} in X such that every Cauchy sequence of the form 0_{f_{i}}(x_{∘}) converges in X, then X is called (S, T; f_{i}) -orbitally complete with respect to x_{∘} or simply (S, T; f_{i}, x_{∘}) -orbitally complete. If for each i ∈ N, f_{i} is an identity mapping on X, then 0_{f_{i}}(x_{∘}) is denoted by 0(x_{∘}) and (S, T; f_{i}, x_{∘}) -orbitally completeness by (S, T; x_{∘}) -orbital completeness.

Definition 2.3.

A pair (S, T) is said to be asymptotically regular at x_{∘} ∈ X if for any sequence {x_{n}} in X and each sequence {y_{n}} in X such that

y_{n} ∈ Sx_{n−1} ∪ T x_{n−1}, lim_{n→∞} d(y_{n}, y_{n+1}) = 0.

We remark that Definitions 2.1–2.3 with S = T and f_{i} = f for each i ∈ N reduce to Definitions 4, 6 and 7 of B.E. Rhoades et al. [15], respectively, and orbital completeness need not imply the completeness of the space. Evidently, a complete space is orbitally complete.

Definition 2.4. For each i ∈ N, the mappings f_{i} and S are said to be commuting at a point x ∈ X [3] if f_{i}Sx ⊆ Sf_{i}x. The mappings f_{i} and S are said to be commuting on X if f_{i}Sx ⊆ Sf_{i}x for every point x ∈ X.

In [22], S. Sessa introduced the concept of weak commutativity for single-valued mappings on a metric space, and Sessa et al. [25] extended this concept to the setting of a single-valued mapping and a multi-valued mapping on a metric space.

Definition 2.5.

For each i ∈ N, the mappings f_{i} and S are said to be weakly commuting at x ∈ X if H(f_{i}Sx, Sf_{i}x) ≤ D(f_{i}x, Sx). The mappings f and S are said to be weakly commuting on X if they are weakly commuting at every x ∈ X.

It may be noted that commutativity implies weak commutativity, but the converse need not be true even in the case of single-valued mappings as shown in [20], [22]–[25] by suitable examples.

Let F be the family of all mappings ∅ from the set R^{+} of non-negative real numbers to itself such that ∅ is upper semicontinuous, non-decreasing and ∅(t) < t for any t > 0.

The existence of coincidence points of non-linear hybrid contractions have been shown by Popa [18] by generalizing the result of Hadžić [2]. In doing so, Popa used Hausdorff-Pompeiu metric as used by Hadžić. It may be remarked that the result of Popa [18] is independent of the main result of Kubiak [12].

\underline{Theorem 2.1} [18]. Let (X, d) be a complete metric space, S and T continuous mappings from X into X, A and B closed mappings from X into CB(SX ∩ T X) such that ASx = SAx, BT x = T Bx, for every x ∈ X and

H^{2}(Ax, By) ≤h max{d(Sx, Ax). d(T y, By), d(Sx, By). d(T y, Ax),

\frac{1}{2}[d(Sx, Ax).d(Sx, By) + d(T y, Ax). d(T y, By)], d^{2}(Sx, T y)\} (2.1)

for every x, y ∈ X where h ∈ (0, 1).

Then there exists a sequence {x_{n}}_{n∈N} in X such that:

1. for every n ∈ N, Sx_{2n+1} ∈ Bx_{2n}, T x_{2n} ∈ Ax_{2n−1}.

2. There exists

z = lim_{n→∞} T x_{n} = lim_{n→∞} Sx_{n+1}.

3. Sz ∈ Az, T z ∈ Bz.

On the other hand, Rhoades et al. [19] deal with the same existence problem by using generalized Hausdorff distance function:

\underline{Theorem 2.2} [19].

Let T be a multi-valued mapping from a metric space X into CL(X). If there exists a mapping f from X into itself such that T (X) ⊆ f(X), for each x, y ∈ X and ϕ ∈ F,

(2.2) H(T x, T y) ≤ ϕ(max(D(fx, T x), D(fy, T y), D(fx, T y), D(fy, T x), d(fx, fy)) (2.3) ϕ(t) ≤ qt for each t > 0 and for some 0 < q < 1, (2.4) there exists a point x_{o} ∈ X such that T is asymptotically regular of x_{o} and f(X) is (T; f, x_{o})–orbitally

complete, then f and T have a coincidence point in X.

It is interesting to note that the commuting mappings f (not the identity mapping) and T satisfying the hypothesis of Theorem 2.2 need not have a common fixed point in X.

In [19], Rhoades et al. gave the solutions of the question “What additional conditions will guarantee the existence of a common fixed point of f and T?”

The main objective of this paper is to investigate different sets of conditions under which the fixed point equation

x = f_{i}x ∈ Sx ∩ T x for x ∈ X and each i ∈ N

possesses a solution.

Moreover, we observe that the following conditions are independent:

(a) There exists a constant q, 0 < q < 1 with ϕ(t) < qt for each t > 0 such that

H(Sx, T y) ≤ ϕ(max(D(fx, Sx), D(gy, T y), D(fx, T y), D(gy, Sx), d(fx, gy))).

(b) There exists a constant q, 0 < q < 1 with ϕ(t) < qt for each t > 0 such that

H^{2}(Sx, T y) ≤ϕ(max(D(fx, Sx). D(gy, T y), D(fx, T y). D(gy, Sx),
\frac{1}{2}[D(fx, Sx).D(fx, T y) + D(gy, Sx). D(gy, T y)], d^{2}(fx, gy))).

Example 1.

Let X = {1, 2, 3, 4}. Define a metric d for X and mappings f, g, S and T on X as follows:

d(1, 4) = \frac{5}{2}, d(1, 2) = d(1, 3) = \frac{3}{2}, d(2, 3) = \frac{4}{5},
d(3, 4) = \frac{7}{4}, d(2, 4) = 1,
S(1) = {3}, S(2) = S(3) = S(4) = {4},
T (1) = T (2) = T (4) = {4}, T (3) = {2}, and f = g = id.

The f, g, S and T satisfy condition (a) for each q, \frac{14}{15} ≤ q < 1. For x = 1 and y = 2

H^{2}(S(1), T (2)) = D^{2}(S(1), T (2)) = \frac{49}{16} > ϕ(max(D(1, S(1). D(2, T (2)),

D(1, T (2)). D(2, S(1)), \frac{1}{2}[D(1, S(1)). D(1, T (2)) + D(2, S(1)).

D(2, T (2))], d^{2}(1, 2)))) = \frac{91}{40}.

Thus (b) is not satisfied.

Example 2.

Let X = {1, 2, 3}. Define a metric d for X by

d(1, 2) = 2, d(2, 3) = \frac{3}{2}, d(1, 3) = 2.

Let f, g, S, and T be the mappings on X such that

S(1) = S(2) = S(3) = {1}, T (1) = T (3) = {1},
T (2) = {3}, f = g = id.

Then f, g, S and T satisfy condition (b) for each q, \frac{9}{16} ≤ q < 1. For x = 1 and y = 2 by (a)

H(S(1), T (2)) = D(S(1), T (2)) = 2 ≤ ϕ(max(D(1, S(1)), D(2, T (2)),
D(1, T (2)), D(2, S(1)), d(1, 2))) = ϕ(2) ≤ 2q

which implies q ≤ 1. Thus (a) is not satisfied.

III. Main Results

Now we are ready to give our main theorems:

\underline{Theorem 3.1}.

Let S and T be multi-valued mappings from a metric space X into P_{X}(X) and let {f_{i}} be a family of all continuous mappings from X into itself such that

(3.1) S(X) ∪ T (X) ⊂ f_{i}(X) for each i ∈ N, (3.2)

H^{2}(Sx, T y) ≤ϕ(max(D(f_{i}x, Sx). D(f_{j}y, T y), D(f_{i}x, T y). D(f_{j}y, Sx),
\frac{1}{2}[D(f_{i}x, Sx). D(f_{i}x, T y) + D(f_{j}y, Sx). D(f_{j}y, T y)],
d^{2}(f_{i}x, f_{j}y))),

for each x, y ∈ X, i, j ∈ N, i ≠ j and ϕ ∈ F, (3.3) ϕ(t) ≤ qt for each t > 0 and for some q ∈ (0, 1), (3.4) there exists a point x_{o} ∈ X such that the pair (S, T) is asymptotically regular at x_{o}, and (3.5) for each i ∈ N, f_{i}(x) is (S, T, f_{i}, x_{o})–orbitally complete.

Then (1) f_{i}, S and T have a coincidence point in X. If z is a coincidence point of f_{i}, S and T and f_{i}z is a fixed point of f_{i}, then we have;

(2) f_{i}z is also a fixed point of S provided f_{i} is weakly commuting with S at z for every even i ∈ N for any odd j ∈ N.

(3) f_{i}z is also a fixed point of T provided f_{i} is weakly commuting with T at z.

(4) f_{i}z is a common fixed point of S and T provided f_{i} is weakly commuting with each of S and T at z.

Proof.

(1) Let x_{o} be a point in X satisfying (3.4). Since S(X)∪T (X) ⊂ f_{i}(X), for each i ∈ N, we can find sequences {x_{n}} and {y_{n}} in X such that

y_{2n} = f_{2n}x_{2n−1} ∈ Sx_{2n} for every n ∈ N,
y_{2n+1} = f_{2n+1}x_{2n} ∈ T x_{2n+1} for every n ∈ N_{∘},
d(y_{2n}, y_{2n+1}) ≤ q^{−1/4}H(Sx_{2n}, T x_{2n+1}) and
d(y_{2n+1}, y_{2n+2}) ≤ q^{−1/4} H(T x_{2n+1}, Sx_{2n+2}).

By (3.4), we have

lim_{n→∞} d(y_{n}, y_{n+1}) = 0.

Now we wish to show that {y_{2n}} is a Cauchy sequence in f_{i}(X). Suppose that the sequence {y_{2n}} is not a Cauchy sequence. Then there exists a positive number ∈ such that, for each positive integer 2k, there exist integers 2n(k) and 2m(k) such that

(3.6) 2k ≤ 2n(k) < 2m(k) and (3.7) d[y_{2n(k)}, y_{2m(k)}) ≥ ∈.

Let d_{i,j} = d(y_{i}, y_{j}) and d_{i} = d(y_{i}, y_{i+1}) for each i, j ∈ N. Then, for each integer 2k, we have (3.8)

∈ ≤ d_{2n(k), 2m(k)} ≤ d_{2n(k), 2m(k)−2} + d_{2m(k)−2} + d_{2m(k)−1}.

For each integer 2k, let 2m(k) denote the smallest integer satisfying (2.1) and (2.2). Then we have d_{2m(k), 2m(k)−2} < ∈ and it follows from (2.3) that

(3.9)

lim_{k→∞} d_{2n(k),2m(k)} = ∈ .

Using the triangle inequality, we get

|d_{2n(k), 2m(k)} − d_{2n(k), 2m(k)}| ≤ d_{2m(k)−1} and
|d_{2n(k)+1, 2m(k)−1} − d_{2n(k), 2m(k)}| ≤ d_{2n(k)} + d_{2m(k)−1},

which yield

lim_{k→∞} d_{2n(k), 2m(k)−1} = lim_{k→∞} d_{2n(k)+1, 2m(k)−1} = ∈

in view of (3.4) and (3.9) and so by (3.2), we have

d_{2n(k), 2m(k)} ≤d_{2n(k)} + d_{2n(k)+1,2m(k)}
≤d_{2n(k)} + q^{−1/4} H(Sx_{2m(k)}, T x_{2n(k)+1})
≤d_{2n(k)} + q^{−1/4}{ϕ(max(D(f_{2m(k)+1}x_{2m(k)}, Sx_{2m(k)}).

D(f_{2m(k)+2} x_{2m(k)+1}, T x_{2m(k)+1}), D(f_{2m(k)+1} x_{2m(k)}, T x_{2n(k)+1}). D(f_{2m(k)+2}, x_{2m(k)+1}, Sx_{2m(k)}),

\frac{1}{2}[D(f_{2m(k)+1} x_{2m(k)}, Sx_{2m(k))}. D(f_{2m(k)+1} x_{2m(k)}, T x_{2m(k)+1} + D(f_{2m(k)+2} x_{2m(k)+1}, Sx_{2m(k)}),

D(f_{2m(k)+2} x_{2m(k)+1}, T x_{2m(k)+1})], d^{2}(f_{2m(k)+1} x_{2m(k)},

f_{2m(k)+2} x_{2m(k)+1})))\}^{\frac{1}{2}}

≤d_{2n(k)} + q^{−1/4}{ϕ(max(d_{2m(k)}, d_{2m(k)+1},

d_{2n(k)+1, 2m(k)+1} d_{2m(k)+z, zm(k)},

\frac{1}{2}[d_{2m(k)}. d_{2m(k)+1} + d_{2m(k)+2, 2m(k)}.

d_{2m(k)+1}], d_{2m(k)+1}^{2})))\}^{\frac{1}{2}} .

Using the semicontinuity of ϕ and letting k → ∞, this inequality yields

∈ ≤ q^{−1/4}\{ϕ( ^{2})\}^{\frac{1}{2}} ≤ q^{−1/4} (q^{2})^{\frac{1}{2}} <∈

Since ∈ > 0 and q^{1/4} < 1, which contradicts the choice of ∈ and so the sequence {y_{2n}} is a Cauchy sequence. Similarly, we can prove that {y_{2n+1}} is also a Cauchy sequence in f_{i}(X).

Since f_{i}(X) is (S, T, f_{i}, x_{∘})-orbitally complete, the Cauchy sequence {y_{2n}} has a limit u in f_{1}(X) for each i ∈ N. By condition (3.4), we have

0 = lim_{n→∞} d(y_{2n}, y_{2n+1}) = lim_{n→∞} d(u, y_{2n+1})

and so {y_{2n+1}} also converges to u. Hence there exists a point z ∈ X such that f_{i}z = u. By condition (3.2), for any even i ∈ N,

D(f_{i}z, Sz) ≤d(f_{i}z, y_{2n+1}) = D(y_{2n+1}, Sz)
≤d(f_{i}z, y_{2n+1} + H(Sz, T x_{2n+1})
≤d(f_{i}z, y_{2n+1}) + {ϕ(max(D(f_{i}z, Sz). D(f_{2n+2} x_{2n+1},

T x_{2n+1}), D(f_{i}z, T x_{2n+1}). D(f_{2n+2} x_{2n+1}, Sz),

\frac{1}{2}[D(f_{i}z, Sz). D(f_{i}z, T x_{2n+1}) + D(f_{2n+2} x_{2n+1}, Sz).

D (f_{2n+2} x_{2n+1}, T x_{2n+1})], d^{2}(f_{i}z, f_{2n+2} x_{2n+1})))\}^{\frac{1}{2}}

≤d(f_{i}z, y_{2n+1}) + {ϕ(max(D(f_{i}z, Sz). D(y_{2n+2}, y_{2n+1}),

D(f_{i}z, y_{2n+1}). D(y_{2n+2}, Sz), \frac{1}{2}[D(f_{i}z, Sz). D(f_{i}z, y_{2n+1}) + D(y_{2n+2}, Sz). D(y_{2n+2}, y_{2n+1})],

d^{2}(f_{i}z, y_{2n+2})))}.

Letting n → ∞, this inequality yields

D(f_{i}z, Sz) ≤ \{ϕ(max(0, 0, 0, 0))\}^{\frac{1}{2}} = 0.

If f_{i}z ∈ Sz for any even i ∈ N, then D(f_{i}z, Sz) > 0 and so the above inequality yields a contradiction.

Hence f_{i}z ∈ Sz for every i ∈ N since every proximinal set is closed. Similarly, we have f_{i}z ∈ T z for any odd i ∈ N. Therefore, z is a coincidence point of f_{i}, S and T for each i ∈ N.

(2) If for any even i ∈ N, u = f_{i}z is a fixed point of f_{i}, then u = f_{i}u = f_{i}f_{i}z ∈ f_{i} Sz. If f_{i} is weakly commuting with S at z, then f_{i}Sz = Sf_{i}z since f_{i}z ∈ Sz. Therefore, we have u ∈ f_{i}Sz = Sf_{i}z = Su; that is, u is a fixed point of S.

(3) It follows by the same techniques in (2) and by (2) and (3), we have (4).

Remark.

Theorem 3.1 is true even if we replace P_{X}(X) by CL(X).

By replacing the condition (3.1) of Theorem 3.1 by the orbital regularity, we have the following:

\underline{Theorem 3.2}.

Let S and T be multi-valued mappings from a metric space X into P_{X}(X) and let {f_{i}} be the family of all continuous mappings from X into itself such that the conditions (3.2) holds,

(3.10) ϕ(t) < t for each t > 0 and some ϕ ∈ F. (3.11) for some x_{∘} ∈ X, and each i ∈ N, there exist sequences {x_{n}} and {y_{n}} in X such that the orbit 0_{f_{i}}(x_{∘}) is

regular, the pair (S, T) is asymptotically regular at x_{∘} and f(X) is (S, T, f_{i}, x_{∘})–orbitally complete. Then we have:

(1) for each i ∈ N, f_{i}, S and T have a coincidence point in X,

(2) if the limit of 0_{f_{i}}(x_{∘}) is a fixed point of f_{i}, then all the conclusions of (2) and (3) of Theorem 3.1 are also true. It is well known that if S is a multi-valued mapping from X into C(X), then for every x, y ∈ X and

u ∈ Sx, there exists a point v ∈ Sy such that

d(u, v) ≤ H(Sx, Sy).

It follows that if S and T are multi-valued mappings from X into C(X), then the orbital regularity in Theorem 3.2 can be dropped. Thus we have the following:

\underline{Theorem 3.3}. Let S and T be multi-valued mappings from a metric space X into C(X) and let \{f_{i}\} be a family of continuous mappings from X into itself such that the conditions (3.1), (3.2), (3.4), (3.5) and (3.6) hold. Then all the conclusions of Theorem 3.2 are also true.

In Theorem 3.1, if we replace (3.2) by the following condition

H^{2}(Sx, T y) ≤ϕ(max(D(f_{i}x, Sx). D(f_{i}y, T y), D(f_{i}x, T y). D(f_{j}y, Sx),
\frac{1}{2}[D(f_{i}x, Sx). D(f_{i}x, T y) + D(f_{j}y, Sx). D(f_{i}y, T y)],

d^{2}(f_{i}x, F_{j}y))) (3.8)

for each x, y ∈ X, i, j ∈ N, i ≠ j, and ϕ ∈ F, then the asymptotic regularity of (S, T), (3.4) is not needed. In fact, we have the following:

\underline{Theorem 3.4}. Let S and T be multi-valued mappings from a metric space X into P_{x}(X) and let \{f_{i}\} be a family of continuous mappings from X into itself such that the conditions (3.1), (3.5) and (3.8) hold. Then all the conclusions of Theorem 3.1 are also true.

In Theorem 3.1 and 3.3, taking f = f_{i} on X for each i ∈ N and defining ϕ(t) = qt, o < q < 1, we have the following:

\underline{Corollary 3.5}. Let S and T be multi-valued mappings from a metric space into P_{x}(X) and let f be a continuous mapping from X into itself such that the condition (3.4) holds:

(3.9)  S(X) ∪ T (X) ⊂ f(X)

H^{2}(Sx, T y) ≤q max(D(fx, Sx). D(fy, T y), D(fx, T y). D(fy, Sx),

\frac{1}{2}[D(fx, Sx). D(fx, T y) + D(fy, Sx). D(fy, T y)], d^{2}(fx, fy)) (3.10)

for each x, y ∈ X and for some 0 < q < 1, (3.11) f(X) is (S, T; f, x_{∘})–orbitally complete, and (3.12) f is weakly commuting with each of S and T.

Then f, S, and T have a common fixed point in X.

\underline{Corollary 3.6}. Let S and T be multi-valued mappings from a metric space X into P_{X}(X) and let f be a continuous mapping from X into itself such that the condition (3.9) and (3.12) hold, and

H^{2}(Sx, T y) ≤q max(D(fx, Sx). D(fy, T y), \frac{1}{2}[D^{2}(fx, T y) + D^{2}(fy, Sx)],

\frac{1}{2}[D(fx, Sx). D(fx, T y) + D(fy, Sx). D(fy, T y)], d^{2}(fx, fy))) (3.13)

For each x, y ∈ X and for some 0 < q < 1. Then f, S and T have a common fixed point in X.

By using the technique of Kubiak [13], we also have the following theorem.

\underline{Theorem 3.7}. Let S, T be multi-valued mappings from a metric space X into P_{X}(f(X) ∩ g(X)) and f, g be continuous mappings from X into itself such that the condition (3.12) and (3.13) hold.

Then f, g, S and T have a common fixed point in X.

Remark. Our results differ from the corresponding theorems of Hadžić [3] and Sessa et al. [24] to the version hybrid contraction mappings.

IV. Some More Related Results

Let X be a reflexive Banach space and W C(X) denote the family of non-empty weakly compact subset of X. Recall that every non-empty weakly compact subset of a reflexive Banach space is proximinal [26] and hence closed.

Our main theorem in this section is prefaced with the following lemma:

Lemma 4.1 [14].

Let f ∈ F and t_{∘} > 0. If t_{n+1} ≤ f(t_{n}) for all n ∈ N, then the sequence {t_{n}} converges to 0.

The following theorem is a variant of results due to Kaneko [9] and Khan et al. [11].

\underline{Theorem 4.2}.

Let X be a reflexive Banach space and let {S_{n}}, {T_{n}} be sequences of mappings from X into W C(X) such that

H^{2}(S_{m}x, T_{n}y) ≤ϕ(max(D(x, S_{m}x). D(y, T_{n}y), D(x, T_{n}y). D(y, S_{m}x),

\frac{1}{2}D(x, S_{m}x). D(x, T_{n}y), \frac{1}{2}D(y, S_{m}x). D(y, T_{n}y), d^{2}(x, y)) (4.1)

for each x, y ∈ X and for some ϕ ∈ F. Then there exists a point z ∈ X such that z ∈ S_{m}z ∩ T_{n}z for each m, n ∈ N.

\underline{Proof}.

Firstly, we assume that h = 0. Let x_{∘} ∈ X be an arbitrary point. Then there exists a point x_{1} ∈ S_{1}x_{∘} such that d(x_{∘}, x_{1}) = D(x_{∘}, Sx_{∘}). This is guaranteed due to the fact that S_{1}x_{∘} is a non-empty weakly compact subset of a reflexive Banach space and so it is proximinal. Now, for all n ∈ N, we have

D(x_{1}, T_{n}x_{1}) ≤ H(S_{1}x_{∘}, T_{n}x_{1}) = 0

which implies that x_{1} ∈ T_{n}x_{1}. Similarly, for all n ∈ N,

D(x_{1}, S_{1}x_{1}) ≤ H(T_{n}x_{1}, S_{1}x_{1}) = 0

yields x_{1} ∈ S_{n}x_{1}. Thus x_{1} is a common fixed point of the two sequences {S_{n}} and {T_{n}}.

Next, we assume that h ≠ 0. Let x_{∘} ∈ X be arbitrary but fixed. Then we construct a sequence {x_{n}} in X such that

x_{2n−1} ∈ S_{n}x_{2n−2}, x_{2n} ∈ T_{2n}x_{2n−1}, d(x_{2n−2}, x_{2n−1}) = D(x_{2n−2},
S_{n}x_{2n−2}), and d(x_{2n−1}, x_{2n}) = D(x_{2n−1}, T_{n}x_{2n}).

The existence of such a sequence in X is guaranteed by the proximinality of S_{m}(x) and T_{n}(x) for each m, n ∈ N.

Now, suppose that x_{n} = x_{n+1} for some n ∈ N. If n ∈ N is even, then we have x_{2n} ∈ S_{n+1} x_{2n}. Further, for each m ∈ N, we have

D^{2}(x_{2n}, T_{m}x_{2n}) ≤H^{2}(S_{n+1}x_{2n}, T_{m}x_{2n})
≤ϕ(max(D(x_{2n}, S_{n+1}x_{2n}). D(x_{2n}, T_{m}x_{2n}), D(x_{2n}, T_{m}x_{2n}).
D(x_{2n}, S_{n+1}x_{2n}), \frac{1}{2}D(x_{2n}, S_{2n+1} x_{2n}). D(x_{2n}, T_{m}x_{2n}),
\frac{1}{2}D(x_{2n}, S_{2n+1}x_{2n}). D(x_{2n}, T_{m}x_{2n}) d^{2}(x_{2}n, x_{2n})))
=ϕ(∘),

which yields x_{2n} ∈ T_{m}x_{2n} for each m ∈ N similarly, D^{2}(x_{2n}, S_{m}x_{2n}) ≤ H^{2}(S_{m}x_{2n}, T_{n}x_{2n}) ≤ ϕ(∘) gives x_{2n} ∈ S_{m}x_{2n} for each m ∈ N.

In case, when an odd integer n ∈ N, we have the same results. Therefore, in each case, we have a common fixed point for the sequences {S_{m}} and {T_{n}}. Now suppose that x_{n} ≠ x_{n+1} for all n ∈ N. We shall show that {x_{n}} is a Cauchy sequence in X. We

observe that

d^{2}(x_{2n}, x_{2n+1}) =D^{2}(x_{2n}, S_{n+1}x_{2n})
≤H^{2}(T_{n} x_{2n−1}, S_{n+1}x_{2n})
≤ϕ(max(D(x_{2n}, S_{n+1}x_{2n}). D(x_{2n−1}, T_{n}x_{2n−1}),
D(x_{2n−1}, T_{n}x_{2n−1}). D(x_{2n}, S_{2n+1}x_{2n}),
\frac{1}{2}D(x_{2n−1}, S_{n+1}x_{2n}). D(x_{2n}, S_{n+1}x_{2n}),
\frac{1}{2}D(x_{2n−1}, S_{n+1}x_{2n}). D(x_{2n−1}, T_{n}x_{2n−1}),
d^{2}(x_{2n−1}, x_{2n})))
=ϕ(max(d(x_{2n} x_{2n+1}). d(x_{2n−1}, x_{2n}), d(x_{2n−1}, x_{2n}).
d(x_{2n}, x_{2n+1}), \frac{1}{2}d(x_{2n−1}, x_{2n+1}). d(x_{2n}, x_{2n+1}),
\frac{1}{2}d(x_{2n−1}, x_{2n+1}). d(x_{2n−1}, x_{2n}), d^{2}(x_{2n−1}, x_{2n})))
≤ϕ(max(d(x_{2n−1}, x_{2n}). d(x_{2n}, x_{2n+1}), \frac{1}{2}\{d(x_{2n−1}, x_{2n})
+ d(x_{2n}, x_{2n+1})\} d(x_{2n}, x_{2n+1}), \frac{1}{2}\{d(x_{2n−1}, x_{2n})+
d(x_{2n}, x_{2n+1})} d(x_{2n−1}, x_{m}), d^{2}(x_{2n−1}, x_{2n}))).

If d(x_{2n−1}, x_{2n}) < d(x_{2n}, x_{2n+1}), then the above inequality yields

d^{2}(x_{2n}, x_{2n+1}) ≤ ϕ(d^{2}(x_{2n}, x_{2n+1})) < d^{2}(x_{2n}, x_{2n+1}),

a contradiction. Therefore, we get

d^{2}(x_{2n}, x_{2n+1}) ≤ ϕ(d^{2}(x_{2n−1}, x_{2n})).

.

Similarly, we can show that

d^{2}(x_{2n+1}, x_{2n+2}) ≤ ϕ(d^{2}(x_{2n}, x_{2n+1})).

It follows from the above relation that

d^{2}(x_{n}, x_{n+1}) ≤ ϕ(d^{2}(x_{2n}, x_{2n+1}))

for all n ∈ N. By Lemma 4.1, we have

lim_{n→∞} d^{2}(x_{n}, x_{n+1}) = 0,

which in turn implies that

lim_{n→∞} d(x_{n}, x_{n+1}) = 0.

Thus, {x_{n}} is a Cauchy sequence in X. Since X is a Banach space, there exists z ∈ X such that

lim_{n→∞} x_{n} = z.

Then we have

D^{2}(x_{2n−1}, T_{m}z) ≤H^{2}(S_{n}x_{2n−2}, T_{m}z)
≤ϕ(max(D(x_{2n−2}, S_{n}x_{2n−2}). D(z, T_{m}z), D(x_{2n−2}, T_{m}z).
D(z, S_{n}x_{2n−2}), \frac{1}{2}D(x_{2n−2}, S_{n}x_{2n−2}). D(x_{2n−2}, T_{m}z),
\frac{1}{2}D(z, S_{n}x_{2n−2}). D(z, T_{m}z), d^{2}(x_{2n−2}, z)))
≤ϕ(max(d(x_{2n−2}, x_{2n−1}). D(z, T_{m}z), D(x_{2n−2}, T_{m}z),
d(z, x_{2n−1}), \frac{1}{2}d(x_{2n−2}, x_{2n−1}). D(x_{2n−2}, T_{m}z),
\frac{1}{2}d(z, x_{2n−1}). D(z, T_{m}z), d^{2}(x_{2n−2}z))).

Letting n → ∞, the above inequality yields

D^{2}(z, T_{m}z) = 0

Since T_{m}z is closed, z ∈ T_{m}z for all m ∈ N. Similarly, we have also z ∈ S_{m}z for all m ∈ M. Therefore, z is a common fixed point of the sequences {S_{n}} and {T_{n}}. This completes the proof. In the last two decades, a large number of fixed point theorems appeared for commuting and weakly

commuting mappings (see, for instance, [5]–[7] and [19]–[24]). By using the technique of Khan et al. [11], we have the following results.

Corollary 4.3.

Let f, g, S and T be single-valued mappings from a complete metric space X into itself such that

d^{2}(fx, gy) ≤ϕ(max(d(fx, Sx), d(gy, T y), d(fx, T y). d(gy, Sx),

\frac{1}{2}[d(fx, Sx). d(fx, T y) + d(gy, Sx). d(gy, T y)] d^{2}(Sx, T y))), (4.2)

for each x, y ∈ X and for some ϕ ∈ F, and

(4.3)  {f, S} and {g, T } are weakly commuting pairs.

Then f, g, S and T have a unique common fixed point in X.

Corollary 4.4.

Let S and T be two continuous mappings from a complete metric space X into itself. S and T have a common fixed point in X if and only if there are two self-mappings A and B on X such that

(4.4)  A(X) ∪ B(X) ⊂ S(X) ∩ T (X)

d^{2}(Ax, Bx) ≤ϕ(max(d(Sx, Ax), d(T y, By), d(Sx, By). d(T y, Ax),

\frac{1}{2}[d(Sx, Ax). d(T y, Ax) + d(Sx, By). d(T y, By)], d^{2}(Sx, T y))), (4.5)

for each x, y ∈ X and for some ϕ ∈ F, and

(4.6)  {A, S} and {B, T } are weakly commuting pairs.

Further, z is the unique common fixed point of A, B, S and T.

\underline{Theorem 4.5}.

Let X be a compact metric space and let S, T be multi-valued mappings from X into P_{X}(X) such that either S or T is continuous,

H^{2}(Sx, T y) ≤ϕ(max(d(x, Sx) d(y, T y), d(y, Sx) d(x, T y), \frac{1}{2}d(x, Sx)

d(x, T y), \frac{1}{2}d(y, Sx) d(y, T y), d^{2}(x, y))) (4.7)

for each x, y ∈ X, x ≠ y, and for some ϕ ∈ F. Then either S or T has a fixed point in X.

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