Coincidence Point Theorem for Non-Linear Hybrid Contractions in Non-Archimedean Menger PM-Space

Rateb Al-Btoush

Rateb Al-Btoush, 2008. Coincidence Point Theorem for Non-Linear Hybrid Contractions in Non-Archimedean Menger PM-Space. Journal of Applied Sciences, 8: 854-859.

DOI: 10.3923/jas.2008.854.859

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

RESULTS

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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 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 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 {xn} 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 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 Tn x, n = 1,2, ... there exists a point b ε Tn+1 a Such that for all t>0. Then there exists a point z ε X such that fzεTn for n = 1,2,3, ..., that is, z is a coincidence point of f and Tn.

Proof: Since T_n(X)⊂f(X) for n = 1,2, ... by condition (4) and g ε G for an arbitrary x₀ ε X, we can choose x₁ ε X such that fx₁ ε T₁x₀ ε Ω.

For this point x₁ there exists a point x₂ ε X such that fx₂εT₂x₁εΩ and

for all t≥0. Similarly, there exists a point x3εX such that

for all t≥0. Inductively, we can obtain a sequence {xn} 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 t₀>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_nx_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_nzεΩ, we have fzεT_nz 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 aεTnx, n = 1,2, ........ there exists a point bεTn+1a 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_nz 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 Tn are compatible for n = 1,2,3, ... .
•	For any xεX and aεTnx, n = 1,2, ... , there exists a point bεTn+1a 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_nz 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:

•	Tn(X)⊂f(X) for n = 1,2, ......
•	f and Tn 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_nZ 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_nx for n = 1,2, ... , there exists a point bεT_n+1a 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 ≠ f² 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 x₀εX, we can choose a point x₁εX such that fx₁εTx₀. For this point x₁, there exists a point x₂εX such that fx₂εTx₁ and

For all t≥0 Inductively, we can obtain a sequence in X such that fx_nεTx_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.