**ABSTRACT**

In this study, we give some point coincidence theorems for non-linear hybrid contractions, that is, contractive conditions involving single-valued and multi-valued mapping in non-Archimedean Menger probabilistic metric space. By using our results, we can also give common fixed theorem for single-valued and multi-valued mapping in metric space. The results presented in this research generalize and improve many results in metric spaces and probabilistic metric spaces.

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**How to cite this article**

*Journal of Applied Sciences, 8: 854-859.*

**DOI:**10.3923/jas.2008.854.859

**URL:**https://scialert.net/abstract/?doi=jas.2008.854.859

**INTRODUCTION**

Let R denote the set of real numbers and R^{+} the non-negative real. A mapping F: R→R^{+} is called distribution function if its non-decreasing 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 PM-space) 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:

(PM-1) F_{p, q} = 1 for all x>0 iff p = q.

(PM-2) F_{p, q} (0) = 0.

(PM-3) F_{p, q} =F_{q, p}.

(PM-4) If F_{p, q} = 1 and F_{q, r} (y) = 1⇒ F_{p, r} (x+y) = 1.

A probabilistically normed space (briefly a PN-space) 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:

_{x}(t) = 1 for all t>0 if and only if x = 0.

(PN-2) f

_{x}(0) = 0.

If we take F_{x, y} = f_{x-y}, then the PN-space must be a PM-space.

A triangle inequality is said to hold in a PM-space 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:

(PM-5) F_{p, r} (x+y) ≥Δ(F_{p, q} (x) F_{q, r} (y)) for all x, y≥0, where Δ is 2-place function satisfying (Δ-1) to (Δ-5).

A Manger PM-space is a PM-space in which the condition (PM-5) holds universally for some choice of Δ satisfying the conditions:

A triangular norm (briefly, a t-norm) is a 2-place function Δ: [0,1]x [0,1]→[0,1] satisfying the conditions (Δ-2), (Δ-3), (Δ-6) and (Δ-7).

**RESULTS**

We introduce the concept for compatibility for single-valued and multi-valued mapping in non-Archimedean Menger probabilistic metric spaces and give some coincidence point theorems for non-linear hybrid contractions that is, contractive conditions involving single-valued and multi-valued mapping in non-Archimedean Menger probabilistic metric space.

By using our results, we can also give some common fixed-point theorem for single-valued and multi-valued 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 PM-space (X, F, Δ) is said to be of type (C)_{g} if there exists a function g ε G such that:

(1) |

for all x, y, z ε X and t≥0.

**Definition 2:** Nadler (1969); A non-Archimedean Menger PM-space (X, F, Δ) is said to be of type (D)g if there exists a point g ε G such that:

(2) |

for all s, tε[0,1].

**Theorem 1:** Chang *et al*. (1994a); If a non-Archimedean Menger PM-space (X, F, Δ) is of type (D)_{g} and then it is of type (C)_{g}.

(3) |

**Theorem 2:** Chang (1990, 1985, 1984). If (X, F, Δ) is a Menger PM-space with t-norm 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 non-empty closed and bounded subsets of X.

Let δ be the Hausdorff metric on CB(X) induced by the metric d, that is:

(4) |

**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 t-normΔ(a, b) = min{a, b} for all x, y ε [0,1] is a τ -complet Menger PM-space.

**Theorem 5:** Michael (1951); If (X, F, Δ) is τ-complet PM-space with t-normΔ(a, b) = min {a, b} for all x, yε[0,1], then (X, d) is a d-complete metric space, where the metric d is defined as follows:

(5) |

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 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 τ-complete non-Archimedean Menger PM-space of type (D)_{g} with the continuous τ-normΔ(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 non-decreasing for each variable, right-continuous and for any t≥0:

where the function ψ: R^{+} →R^{+} is non-decreasing, right continuous and

**Lemma 1:** Chang (1990). Let ψ: R^{+}→R^{+} be non-decreasing, right continuous and

(6) |

(7) |

**Definition 6:** Chang (1990). Let f be a mapping from X into itself and T be a multi-valued mapping from X into Ω, where Ω is the family of all non-empty τ-closed 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 |

**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 PM-space. Then a mapping from ΩxΩ into D satisfying the following conditions:

(1) |

(2) |

(3) |

(4) |

** Theorem 7:** Let f be τ-continuous mapping from X into itself and be a sequence of τ-continuous multi-valued 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:

(8) |

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),

(9) |

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 τ-complete, 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 τ-complete the sequence {T_{n}x_{n-1}} 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 τ-continuous mapping from X into itself and S, T be τ-continuous multi-valued 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 τ-continuous mapping from X into itself and be a sequence of τ-continuous multi-valued 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 non-Archimedean 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 τ-continuous mapping from X into itself and be a sequence of τ-continuous multi-valued 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 non-Archimedean metric space and C(X) be the family of all non-empty compact subsets of X. Let f be a continuous mapping from X into itself and be a sequence of continuous multi-valued 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 non-Archimedean metric space and C (X) be the family of all non-empty compact subsets of X. Let f be a continuous mapping from X into itself and T be a sequence of continuous multi-valued 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 multi-valued 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 multi-valued 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_{n-1}} 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.