INTRODUCTION
Let R denote the set of real numbers and R^{+} the nonnegative
real. A mapping F: R→R^{+} is called distribution function
if its nondecreasing and left continuous with inf (F) = 0 and sup(F)
= 1. We will denote D by the set of all distribution functions.
Let H denote the specific distribution function defined by:
We shall also, for convenience, adhere to the convention that for any
distribution function F and for any
A probabilistic metric space (briefly, a PMspace) is an ordered pair
(X, F), where, X is a set and F is a mapping of FxF into D. i.e., F associate
a distribution function F(p, q) by F_{p, q}, where the symbol
F_{p, q}, (x) will denote the value of F_{p, q}, for the
argument x. the functions F_{p, q}, are assumed to satisfy the
following conditions:
(PM1) F_{p, q} = 1 for all x>0 iff p = q.
(PM2) F_{p, q} (0) = 0.
(PM3) F_{p, q} =F_{q, p}.
(PM4) If F_{p, q} = 1 and F_{q, r} (y) = 1⇒ F_{p,
r} (x+y) = 1.
A probabilistically normed space (briefly a PNspace) is an ordered pair
(X, F) where X is a real linear space, F is a mapping of X into D. (We
shall denote the distribution functions by F (x) by f_{x}) satisfying
the following conditions:
(PN1) f
_{x} (t) = 1 for all t>0 if and only if x = 0.
(PN2) f
_{x} (0) = 0.
If we take F_{x, y} = f_{xy}, then the PNspace must
be a PMspace.
A triangle inequality is said to hold in a PMspace if and only if it
holds for all triple of points, distinct not, in the space.
Let Δ: [0, 1] be a 2 place function satisfying the following conditions:
Menger (1942) introduced, as the generalized triangle inequality, the
following condition:
(PM5) F_{p, r} (x+y) ≥Δ(F_{p, q} (x) F_{q,
r} (y)) for all x, y≥0, where Δ is 2place function satisfying
(Δ1) to (Δ5).
A Manger PMspace is a PMspace in which the condition (PM5) holds universally
for some choice of Δ satisfying the conditions:
A triangular norm (briefly, a tnorm) is a 2place function Δ: [0,1]x
[0,1]→[0,1] satisfying the conditions (Δ2), (Δ3), (Δ6)
and (Δ7).
RESULTS
We introduce the concept for compatibility for singlevalued and multivalued
mapping in nonArchimedean Menger probabilistic metric spaces and give
some coincidence point theorems for nonlinear hybrid contractions that
is, contractive conditions involving singlevalued and multivalued mapping
in nonArchimedean Menger probabilistic metric space.
By using our results, we can also give some common fixedpoint theorem
for singlevalued and multivalued mapping in metric space.
The results presented in this study generalize and improve many results
of (Kaneko and Sessa, 1989), Nadler and many others in metric spaces and
probabilistic metric spaces.
Let G be the family of functions g: [0,1]→[0,∞] such that
g is continuous, strictly decreasing, g (1) = 0 and g (0)<∞
Definition 1: Menger (1942); A Menger PMspace (X, F, Δ)
is said to be of type (C)_{g} if there exists a function g ε
G such that:
for all x, y, z ε X and t≥0.
Definition 2: Nadler (1969); A nonArchimedean Menger PMspace
(X, F, Δ) is said to be of type (D)g if there exists a point g ε
G such that:
for all s, tε[0,1].
Theorem 1: Chang et al. (1994a); If a nonArchimedean Menger
PMspace (X, F, Δ) is of type (D)_{g} and then it is of type
(C)_{g}.
Theorem 2: Chang (1990, 1985, 1984). If (X, F, Δ) is a Menger
PMspace with tnorm Δ(a, b) ≥Δ_{m} (a, b) for all
a, b ε[0,1] and then it`s of type (D)_{g} for gε G type
(D)g.
Definition 3: Menger (1942). Let (X, d) be a metric space and
CB (X) be the family of all nonempty closed and bounded subsets of X.
Let δ be the Hausdorff metric on CB(X) induced by the metric d,
that is:
Theorem 3: Menger (1942); (a) (CB(X), d) is a metric space.
(b) If (X, d) is complete then (CB (X), d) is complete.
Theorem 4: Chang (1985); Let (X, d) be a complete metric space.
If we define F: XxX→D as follows:
for all x, y ε X and t ε R then the space (X, F, Δ) with
the tnormΔ(a, b) = min{a, b} for all x, y ε [0,1] is a τ
complet Menger PMspace.
Theorem 5: Michael (1951); If (X, F, Δ) is τcomplet
PMspace with tnormΔ(a, b) = min {a, b} for all x, yε[0,1],
then (X, d) is a dcomplete metric space, where the metric d is defined
as follows:
for all x, y ε X
Definition 4: Chang (1985). Let AεCB(X) and xεA. We
define the probabilistic distance F_{x,A} between the point x
and the set A as follows:
For all tεR.
For all A, B ε CB (X) and t ε R., then
is the Menger Hausdorf metric induced by F.
Definition 5: Michael (1951); Let (X, F, Δ) be a τcomplete
nonArchimedean Menger PMspace of type (D)_{g} with the continuous
τnormΔ(a, b) = min {a, b} for all a, b ε [0,1]. Let Φ
be the family of mappings φ: (R^{+})^{5}→R^{+}
such that each φ is nondecreasing for each variable, rightcontinuous
and for any t≥0:
where the function ψ: R^{+} →R^{+} is nondecreasing,
right continuous and
Lemma 1: Chang (1990). Let ψ: R^{+}→R^{+}
be nondecreasing, right continuous and
Definition 6: Chang (1990). Let f be a mapping from X into itself
and T be a multivalued mapping from X into Ω, where Ω is the
family of all nonempty τclosed and probabilistically bounded subsets
of X. Then:
• 
The mappings f and T are said to be commuting if fTxεΩ
and fTx = Tfx for all xεX. 
• 
The mappings f and T are said to be compatible if fTxεΩ
and .

For all t>0, whenever {x_{n}} is a sequence
in X such that

Theorem 6: Chang (1990). Any commuting mappings are compatible,
but the converse is not true.
Lemma 2: Chang et al. (1994a). Let (Ω,,
Δ) be a Menger PMspace. Then a mapping
from ΩxΩ into D satisfying the following conditions:
(1) 

(2) 

(3) 

(4) 

Theorem 7: Let f be τcontinuous mapping from X into itself
and be
a sequence of τcontinuous multivalued mappings from X into Ω
satisfying the following conditions:
For all x, yεX, t≥0 and i ≠ j, i, j = 12
where, g ε G and ψ ε Ψ.
• 
Suppose further that for any x ε X and a ε T_{n}
x, n = 1,2, ... there exists a point b ε T_{n+1} a Such
that
for all t>0. Then there exists a point z ε X such that
fzεT_{n} for n = 1,2,3, ..., that is, z is a coincidence
point of f and T_{n}. 
Proof: Since T_{n}(X)⊂f(X) for n = 1,2, ... by condition
(4) and g ε G for an arbitrary x_{0} ε X, we can choose
x_{1} ε X such that fx_{1} ε T_{1}x_{0}
ε Ω.
For this point x_{1} there exists a point x_{2} ε
X such that fx_{2}εT_{2}x_{1}εΩ
and

for all t≥0. Similarly, there exists a point x_{3}εX
such that 
for all t≥0. Inductively, we can obtain a sequence {x_{n}}
in X such thatand 
for all t≥0.
Now, we show that the sequence {fx_{n}} is a cauchy sequence
in X. In fact by lemma 2 and conditions (3), (4), since gεG we have:
If
for some t_{0}>0, from Eq. 8 and lemma (1),
it follows that:
Which is a contradiction. Thus, for any t>0, we have
For n = 1,2, ... and so, by (8),
For all t>0. Hence, for any positive integers m,n with m>n and
t>0,
as n→∞, which implies that
for any positive integer m, that is, {fx_{n}} is a cauchy sequence
in X.
Since (X,F,Ω) is τcomplete, the sequence {fx_{n}}
converges to a point z in X. On the other hand, by condition (7) and (9),
since we have:
Letting
(t)→1 as n→∞ for all t>0, that is,
a cauchy sequence in
is τcomplete the sequence {T_{n}x_{n1}} converge
to a set AεΩ.
Next, we shall show that zεA. Indeed, we
have:
as n→∞ which implies F_{z,A}(t)→1, as n→∞
for all t>0. Thus, since AεΩ, zεA. Therefore, since
f and T_{n} for n = 1,2, ... , we have
as n→∞ that is,
as n→∞. Since T_{n}zεΩ, we have fzεT_{n}z
for n = 1,2, ... . This completes the proof.
Corollary 1: Let f be τcontinuous mapping from X into itself
and S, T be τcontinuous multivalued mappings from X into Ω
satisfying the following conditions:
• 
S(X)∪T(X)⊂f(X), 
• 
The pair f, S and f, T are compatible, 
• 
for all x,y ε X and t≥0 where gεG and φεΦ.

• 
Suppose further that for any xεX and zεX there exists
a point bεTa such that:
for all, then there exists a point zεX such that fzεSx∩Tz,
that is, z is a coincidence point of the pairs f, S and f,T. t>0.

Proof: Taking T_{2n+1} = S and T_{2n+2} = T, n
= 0,1, ..., in theorem 7, the result follows immediately.
Corollary 2: Let f be τcontinuous mapping from X into itself
and
be a sequence of τcontinuous multivalued mappings from X into Ω
satisfying the following conditions:
• 
For any xεX and aεT_{n}x, n = 1,2, ........ there
exists a point bεT_{n+1}a such that
for all t>0. 
• 
for all x,yεX and t≥0 where gεG and φεΦ. 
Then there exists a point z ε X such that zεT
_{n}z for n = 1,2,
... , that is; the point z is a common fixed point of T
_{n}.
Proof: Taking f = I_{X} (the identity mapping on X) in Theorem
7, the proof follows immediately.
Definition 7: Menger (1942). A metric space (X, d) is said to
be nonArchimedean if the following condition holds:
d(x, y)≤max {d(x,z), d(z,y)}, for all x,y,zεX 
Theorem 8: Chang et al. (1994b). Let f be a τcontinuous
mapping from X into itself and
be a sequence of τcontinuous multivalued mappings from X into Ω
satisfying the conditions:
• 

• 
f and T_{n} are compatible for n = 1,2,3, ... . 
• 
For any xεX and aεT_{n}x, n = 1,2, ... , there
exists a point bεT_{n+1}a such that for
all t>0. 
• 
There exists a constant k> such that :
for all x,yεX and t≥0.

Then there exists a point zεX such that fzεT_{n}z for
n = 1,2,3 ... , That is, z is a coincidence point of f and T_{n}.
Theorem 9: Let (X,d) be a complete nonArchimedean metric space
and C(X) be the family of all nonempty compact subsets of X. Let f be
a continuous mapping from X into itself and
be a sequence of continuous multivalued mappings T_{n} from X
into C(x) satisfying the following conditions:
• 
T_{n}(X)⊂f(X) for n = 1,2, ...... 
• 
f and T_{n} are commuting for n = 1,2, ... . 
• 
There exists a point αε(0,1) such that
For all x,yεX and I≠j,i,j = 1,2,, ... . 
Then there exists a point zεX such that fzεT
_{n}Z for
n = 1,2, ... , that is, z is a coincidence point of f and T
_{n}.
Proof: By definition (4) and condition (3), we have:
For all x, yεX and i ≠ j,i,j = 1,2, ... .
Moreover, for any xεX and aεT_{n}x for n = 1,2, ...
, there exists a point bεT_{n+1}a such that:
and so we have
for all t≥0.
Therefore all conditions of Theorem 8 are satisfied and hence this theorem
follows immediately. This completes the proof.
Corollary 3: Let (X,d) be a complete nonArchimedean metric space
and C (X) be the family of all nonempty compact subsets of X. Let f be
a continuous mapping from X into itself and T be a sequence of continuous
multivalued mappings T from X into C (x) satisfying the following conditions:
• 
T(X)⊂f(X). 
• 
f, T are commuting. 
• 
There exists an hε(0,1) such that: 
δ(Tx,Ty)≤hd (fx, fy), for all x,yεX. Then there exists
a point zεX such that fzεTz.
Proof: By Definition 4 and condition 3, we have:
for all x,y ε X.
Moreover, for any xεX and aεTx, there exists a point bεTa
such that:
And so we have
for all t≥0. Therefore, by theorem 8 there exists a point zεX
such that fzεTz. This complete the proof.
Lemma 3: Kaneko and Sessa (1989). Let (X,d) be metric space. Let
f be a mapping from X into itself and T be a multivalued mapping from
X into C(X) such that the mappings f and T are compatible. If fzεTz
for some zεTz for some zεX, then fTz = Tfz.
Theorem 10: Let (X, d) and C(X) be as in Theorem 9. Let f be a
continuous mapping from X into itself and T be a continuous multivalued
mapping from X into C(X) satisfying the following conditions:
• 
T(X)⊂f(X), 
• 
f and T are compatible, 
• 
There exists an αε(0,1) such that

for all x,yεX. Suppose also that, for each xεX either (a) fx
≠ f^{2} x implies fx
Tx or (b) fxεTx implies
for some zεX. Then f and T have a common fixed point in X.
Proof: Taking T_{n} = T in Theorem 9 then there exists
a point zεX such that fzεTz, i.e., z is a coincidence point
of f and T. Then by Lemma 3, we have fTz = Tfz.
Now, by definition 4 and condition (3), we have:
For all x,yεX.
Moreover, for any xεX and αεX for n = 1,2, ... , there
exists a point bεTa such that:
and so we have
Since T(X)⊂f(X), for arbitrary x_{0}εX, we can choose
a point x_{1}εX such that fx_{1}εTx_{0}.
For this point x_{1}, there exists a point x_{2}εX
such that fx_{2}εTx_{1} and
For all t≥0 Inductively, we can obtain a sequence in X such that fx_{n}εTx_{n+1}
and
for all t≥0.
Then {x_{n}} is a cauchy sequence in X. But (X,d) is complete,
then x_{n}→x.
Also the subsequences {fx_{n}}, {Tx_{n1}} converges
to x.
Now, since f, T are continuous then fx_{n}→fx, Tx fx_{n}→Tx.
This shows that x is a common fixed point of f, T.