Grabiec [1] demonstrated the Banach contraction theorem in the fuzzy metric spaces introduced by Kramosil and Michalek [2]. Park [3-5], Park and Kim [6] also proved a fixed-point theorem in a fuzzy metric space.
Recently, Park et al. [7] defined an intuitionistic fuzzy metric space while Park et al. [8] proved a fixed-point Banach theorem for the contractive mapping of a complete intuitionistic fuzzy metric space. Park et al. [9] defined a type(
This paper proves some common fixed points for four self-mappings that satisfy type(
First, some definitions and properties of the intuitionistic fuzzy metric space
Let us recall ([11]) that a continuous
Similarly, a continuous
[12] The 5？tuple (
(a) M(x, y, t) > 0,
(b) M(x, y, t) = 1 ⇔ x = y,
(c) M(x, y, t) = M(y, x, t),
(d) M(x, y, t) * M(y, z, s) ≤ M(x, z, t + s),
(e) M(x, y, ？) : (0,∞) → (0, 1] is continuous,
(f) N(x, y, t) > 0,
(g) N(x, y, t) = 0 ⇔ x = y,
(h) N(x, y, t) = N(y, x, t),
(i) N(x, y, t) ？ N(y, z, s) ≥ N(x, z, t + s),
(j) N(x, y, ？) : (0,∞) → (0, 1] is continuous.
Note that (
[13] Let (
for
[13] Let
(a) {
for all
(b) {
for all
(c)
In this paper,
for all
[6] Let {
for all
[14] Let
M(x, y, kt) ≥ M(x, y, t),
N(x, y, kt) ≤ N(x, y, t),
then
This section introduces type(
[14] Let
for all
([10]) Let
for all
[10] Let
[10] Let
[10] Let
(a)limn→∞ BBxn = Ax if A is continuous at x ∈ X,
(b)limn→∞ AAxn = Bx if B is continuous at x ∈ X,
(c)ABx = BAx and Ax = Bx if A and B are continuous at x ∈ X.
Let
for all
Thus,
Next, we have lim
Furthermore,
and
Therefore,
This section proves the main theorem and presents an example using the given conditions in an intuitionistic fuzzy metric space.
Let
(a) AT(X) ∪ BS(X) ⊂ ST(X);
(b) there exists k ∈ (0, 1) so for all x, y ∈ X and t > 0,
M2(Ax,By,kt) * [M(Sx, Ax, kt)M(Ty,By, kt)]
*M2(Ty,By, kt) + aM(Ty,By, kt)M(Sx,By, 2kt)
≥ [pM(Sx, Ax, t) + qM(Sx, Ty, t)]M(Sx,By, 2kt),
N2(Ax,By,kt) ？ [N(Sx, Ax, kt)N(Ty,By, kt)] ？N2(Ty,By, kt) + aM(Ty,By, kt)N(Sx,By, 2kt) ≤ [pN(Sx, Ax, t) + qN(Sx, Ty, t)]N(Sx,By, 2kt),
where 0 <
(c) S and T are continuous and ST = TS;
(d) the pairs (A, S) and (B, T) are type(β) compatible.
Thus,
M2(ATx2n,BSx2n+1, kt) * [M(STx2n, ATx2n, kt) ×M(TSx2n+1,BSx2n+1, kt)] *M2(TSx2n+1, BSx2n+1, kt) + aM(TSx2n+1,BSx2n+1, kt) ×M(STx2n,BSx2n+1, 2kt) ≥ [pM(STx2n+1, ATx2n, t) +qM(STx2n, TSx2n+1, t)] ×M(STx2n,BSx2n+1, 2kt),
N2(ATx2n,BSx2n+1, kt) ？ [N(STx2n, ATx2n, kt) ×N(TSx2n+1,BSx2n+1, kt)] ？ N2(TSx2n+1, BSx2n+1, kt) + aN(TSx2n+1,BSx2n+1, kt) ×N(STx2n,BSx2n+1, 2kt) ≤ [pN(STx2n+1, ATx2n, t) +qN(STx2n, TSx2n+1, t)] ×N(STx2n,BSx2n+1, 2kt)
and
M2(STx2n+1, STx2n+2, kt) * [M(z2n, STx2n+1, kt) ×M(z2n+1, STx2n+2, kt)] *M2(z2n+1, STx2n+2, kt) +aM(z2n+1, STx2n+2, kt)M(z2n, STx2n+2, 2kt) ≥ [pM(z2n, STx2n+1, t) + qM(z2n, z2n+1, t)] ×M(z2n, STx2n+2, 2kt),
N2(STx2n+1, STx2n+2, kt) ？ [N(z2n, STx2n+1, kt) ×N(z2n+1, STx2n+2, kt)] ？ N2(z2n+1, STx2n+2, kt) +aN(z2n+1, STx2n+2, kt)N(z2n, STx2n+2, 2kt) ≤ [pN(z2n, STx2n+1, t) + qN(z2n, z2n+1, t)] ×N(z2n, STx2n+2, 2kt).
Then,
M2(z2n+1, z2n+2, kt) *[M(z2n, z2n+1, kt)M(z2n+1, z2n+2, kt)] +aM(z2n+1, z2n+2, kt)M(z2n, z2n+2, 2kt) ≥ [p + q]M(z2n, z2n+1, t)M(z2n, z2n+2, 2kt),
N2(z2n+1, z2n+2, kt) ？[N(z2n, z2n+1, kt)N(z2n+1, z2n+2, kt)] +aN(z2n+1, z2n+2, kt)N(z2n, z2n+2, 2kt) ≤ [p + q]N(z2n, z2n+1, t)N(z2n, z2n+2, 2kt),
and
M2(z2n+1, z2n+2, kt)M(z2n+1, z2n+2, kt)] +aM(z2n+1, z2n+2, kt)M(z2n, z2n+2, 2kt) ≥ [p + q]M(z2n, z2n+1, t)M(z2n, z2n+2, 2kt),
N2(z2n+1, z2n+2, kt)N(z2n+1, z2n+2, kt)] +aN(z2n+1, z2n+2, kt)N(z2n, z2n+2, 2kt) ≤ [p + q]N(z2n, z2n+1, t)N(z2n, z2n+2, 2kt).
Therefore, it follows that
z2n+1M(z2n+1, z2n+2, kt) ≥ M(z2n, z2n+1, t),
N(z2n+1, z2n+2, kt) ≤ N(z2n, z2n+1, t)
for all
M(zm+1, zm+2, kt) ≥ M(zm, zm+1, t),
N(zm+1, zm+2, kt) ≤ N(zm, zm+1, t)
Thus, {
Let
M(AAy2n, SSy2n, t) → 1,
M(BBu2n+1, TT2n+1, t) → 1,
N(AAy2n, SSy2n, t) → 0,
N(BBu2n+1, TT2n+1, t) → 0
as
Next, by taking
M2(z, Tz, kt) * [M(z, z, kt)M(Tz, Tz, kt)] *M2(Tz, Tz, kt) + aM(Tz, Tz, kt)M(z, Tz, 2kt) ≥ [pM(z, z, t) + qM(z, Tz, t)]M(z, Tz, 2kt),
N2(z, Tz, kt) ？ [N(z, z, kt)N(Tz, Tz, kt)] ？N2(Tz, Tz, kt) + aN(Tz, Tz, kt)N(z, Tz, 2kt) ≤ [pN(z, z, t) + qN(z, Tz, t)]N(z, Tz, 2kt),
then
M2(z, Tz, kt) + aM(z, Tz, 2kt) ≥ [p + qM(z, Tz, t)]M(z, Tz, 2kt), N2(z, Tz, kt) ≤ qN(z, Tz, t)N(z, Tz, 2kt).
Since
M(z, Tz, kt) + a ≥ p + qM(z, Tz, t),
N(z, Tz, kt) ≤ qN(z, Tz, t)
and
Thus,
Next, by taking
M(z,Bz, kt) *M(z,Bz, kt) +aM(z,Bz, kt)M(z,Bz, 2kt) ≥ (p + q)M(z,Bz, 2kt),
N(z,Bz, kt) ？ N(z,Bz, kt) +aN(z,Bz, kt)N(z,Bz, 2kt) ≤ 0.
Thus,
M(z,Bz, kt) + aM(z,Bz, kt) ≥ p + q,
N(z,Bz, kt) + aN(z,Bz, kt) ≤ 0.
Therefore,
M(z,Bz, kt) ≥ 1,
N(z,Bz, kt) ≤ 0
for all
Let
Using condition (b), we have
M2(z,w, kt) * [M(z, z, kt)M(w,w, kt)] *M2(w,w, kt) + aM(w,w, kt)M(z,w, 2kt) ≥ [pM(z, z, t) + qM(z,w, t)]M(z,w, 2kt),
N2(z,w, kt) ？ [N(z, z, kt)N(w,w, kt)] ？N2(w,w, kt) + aN(w,w, kt)N(z,w, 2kt) ≤ [pN(z, z, t) + qN(z,w, t)]M(z,w, 2kt).
Thus,
M2(z,w, kt) +M(z,w, 2kt) ≥ (p + qM(z,w, t))M(z,w, 2kt), N2(z,w, kt) ≤ qM(z,w, t)M(z,w, 2kt),
Therefore,
M(z,w, kt) ≤ M(z,w, 2kt),
N(z,w, kt) ≥ N(z,w, 2kt),
so
Thus,
Let
(e) A(X) ⊂ S(X),
(f) there exists k ∈ (0,1) so for all x, y ∈ X and t > 0,
M2(Ax, Ay, kt) * [M(Sx, Ax, kt)M(Sy, Ay, kt)] M2(Sy, Ay, kt) + aM(Sy, Ay, kt)M(Sx, Ay, 2kt) ≥ [pM(Sx, Ax, t) + qM(Sx, Sy, t)]M(Sx, Ay, 2kt),
N2(Ax, Ay, kt) ？ [N(Sx, Ax, kt)N(Sy, Ay, kt)kt)] ？N2(Sy, Ay, kt) + aM(Sy, Ay, kt)N(Sx, Ay, 2kt) ≤ [pN(Sx, Ax, t) + qN(Sx, Sy, t)]N(Sx, Ay, 2kt),
where 0 <
(g) S is continuous,
(h) A and S are type(β) compatible.
Thus,
Let
with the metric
for all
for all
Furthermore,
and
Similarly,
M(0,B0, kt) + aM(0,B0, kt) ≥ p + q,
N(0,B0, kt) + aN(0,B0, kt) ≤ 0.
Therefore,
Let w be another common fixed point of
M2(0,w,kt) +M(0,w, 2kt) ≥ (p + qM(0,w,t))M(0,w,2kt),
M2(0,w, kt) ≤ qM(0,w, t)M(0,w, 2kt).
Therefore, because
M(0,w, kt) ≤ M(0,w, 2kt),
N(0,w, kt) ≥ N(0,w, 2kt),
Thus,
Therefore, 0 =
Park et al. [7] defined an intuitionistic fuzzy metric space and Park et al. [8] proved a fixed-point Banach theorem for the contractive mapping of a complete intuitionistic fuzzy metric space. Park et al. [9] defined a type(
This paper attempted to develop a method to provide a proof based on the fundamental concepts and properties defined in the new space. I think that the results of this paper will be extended to the intuitionistic M-fuzzy metric space and other spaces. Further research should be conducted to determine how to combine the collaborative learning algorithm with our proof method in the future.