Some Fixed Point Results in Menger Space Using the Notion of CLR and JCLR Property

INTRODUCTION

As of now, an intriguing territory of exploration is demonstrating brings about Menger space. They presented the idea of probabilistic Menger space. They demonstrated fascinating outcomes with regards to summed up the consequences of Kumar and Pant. They presented the idea of CLR-property and this is additionally summed up as JCLR-property. Utilizing these ideas, we summed up the previously mentioned results. We saw that the states of closedness of the subspaces and coherence of the mappings are not required in building up our outcomes. As normal R represents the arrangement of every single genuine number, R+ represents the arrangement of all non-negative genuine numbers and N represents the arrangement of every regular number.

We hereunder give the accompanying definitions and the outcome needed in ensuing area. Definition 2.1. ([2]) A mapping F : R → R+ is called a distribution if and only if it is nondecreasing, left continuous with inf{F (t) : t ∈ R} = 0 and sup{F (t) : t ∈ R} = 1. The set of all distribution functions are denoted by L. For example, Heaviside function H : R → R+, defined by is a distribution function. Definition 2.2. ([2]) Probabilistic metric space (PM-space) is an ordered pair (X, F ), where X is a non empty set and F : X × X → L is defined by (p, q) → Fp,q where {Fp,q : p, q ∈ X} ⊆ L, and the functions Fp,q satisfy the following: (a) Fp,q(t) = 1 for all t > 0 if and only if p = q; (b) Fp,q(0) = 0; (c) Fp,q(t) = Fq,p(t); (d) Fp,q(t) = 1 and Fq,r(s) = 1, then Fp,r(t + s) = 1. Definition 2.3. ([2]) A mapping T: [0, 1] X [0, 1] → [0, 1] is called a triangular norm (or t-norm) if (a) T (0, 0) = 0 and T (a, 1) = a for all a ∈ [0, 1]; (b) T (a, b) = T (b, a), for all a, b ∈ [0, 1]; (c) T (a, b) ™ T (c, d) for all a, b, c, d ∈ [0, 1] with a ™ c and b ™ d;

DEFINITION 2.4. ([2]) A Menger space is a triplet (X, F, T ), where (X, F ) is a Probabilistic metric space and T is a t-norm such that for all p, q, r∈ X and all t, s > 0,

DEFINITION 2.5. ([9]) Self mappings f and g of a Menger space (X, F, T ) are said to be weakly compatible if and only if for any t > 0, Ffx,gx(t) = 1 for some x ∈ X implies Ffgx,gfx(t) = 1; i.e, fx = gx for some x ∈ X implies fgx = gfx. DEFINITION 2.6. ([1]) A function is said to be an implicit relation if

for x, y >0, ϕ(x, y, x, y) > 0 or ϕ(x, y, y, x) > 0 implies x >y,

EXAMPLE 2.1. Define ϕ : (R+)4 R by ϕ(x1, x2, x3, x4) = ax1+bx2+cx3+dx4 with a + b + c + d = 0,a + b > 0, a + c > 0 and a + d > 0. Clearly, ϕ is an implicit relation. In particular,

Notation: Let Φ be the class of all implicit relations. DEFINITION 2.7. ([8]) Let (X, F, T ) be a Menger space, where T is continuous t-norm. 1. A sequence {pn} in X is said to converge to a point p in X (written as pn → p) if for every ∈ > 0 and λ > 0, there exists a positive integer M (∈, λ) such that Fpn,p(∈) > 1 − λ for all n >M (∈, λ). 2. A sequence {pn} in X is said to be Cauchy if for each every ∈> 0 and λ > 0, there is a positive integer M (∈, λ) such that Fpn,pm (∈) > 1 − λ for all n, m ∈ N with n, m >M (∈, λ). 3. A Menger space (X, F, T ) is said to be complete if every Cauchy sequence in X converges to a point of it. Fx,y(kt) > Fx,y(t)

DEFINITION 2.8. Let (X, F, T ) be a Menger space, where T denotes a continuous t-norm and f , g, h, k be self-mappings on X. (a) The ordered pairs (f, g) and (h, k) are said to satisfy the ―common limit in the range of g‖ (CLRg−) property if and only if there exist sequences {xn} and {yn} in X such that for some x ∈ X and for all t > 0. (b) The ordered pairs (f, g) and (h, k) are said to satisfy the ‖ joint common limit in the ranges of g and k‖ (JCLRgh−) property if and only if there exist sequences {xn} and {yn} in X such that ku = gu and for some u ∈ X and for all t > 0.

such that Since p(X) fg(X), there is a v X such that pu = fgv. By taking x = xn and y = v in (iv), we get that

As n → ∞, the above becomes

So, by the property of ϕ, Fpu,qv(αt) ― Fpu,qv(t). By Lemma(2.1), pu = qv. Since q(X) hk(X), there is a w X such that qv = hkw. By taking x = w and y = v in (iv), we get that ϕ(Fpw,qv(αt), Fhkw,fgv(t), Fpw,hkw(t), Fqv,fgv(αt)) >0 i.e, ϕ(Fpw,qv=hkw(αt), 1, Fpw,hkw(t), 1) >0 o, by the property of ϕ, we have

Thus pw = hkw = pu = fhv = qu = z(say).

and q(fg)v = fg(q)v. i.e, pz = hkz and qz = fgz. By putting x = z and y = v in (iv), we get that ϕ(Fpz,qv=z (αt), Fhkz=pz,fgv=z (t), Fpz,hkz=pz (t), Fqv=z,fgv=z (αt)) > 0 i.e, ϕ(Fpz,z(αt), Fpz,z(t), 1, 1) > 0 ⇒ ϕ(Fpz,z(t), Fpz,z(t), 1, 1) > 0 ⇒ Fpz,z(t) ― 1 ⇒ pz = z. Similarly, by taking x = w and y = z in (iv), we get that qz = z. Thus pz = hkz = z = qz = fgz. Since h commutes with k, we have hk(hz) = h(hkz) = hz. Suppose p commutes with h, so p(hz) = h(pz) = hz; by taking x = hz and y = z in (iv), we get that ϕ(Fphz=hz,qz=z (αt), Fhkhz=hz,fgz=z (t), Fphz=hz,hkhz=hz (t), Fqz=z,fgz=z (αt)) > 0

Since hkz = z, follows that kz = z. Thus hz = kz = pz = z. Suppose p commutes with k, so p(kz) = k(pz) = kz. Since h commutes with k, we have hk(kz) = k(hkz) = kz. By taking x = kz and y = z in (iv), we get that kz = z. Since hkz = z, follows that hz = z. Thus hz = kz = pz = z. Now, (vi) is similar to (v) when p, h, k are replaced by q, f, g respectively. Hence as above, we get z = fz = gz = qz. Thus fz = gz = hz = kz = pz = qz = z. Hence z is a common fixed point of f , g, h, k, p and q. Case II: Suppose (iii)(b) holds. By definition, there exist sequences {xn} and {yn} in X such that for some v ∈ X. Since q(X) ⊆ hk(X), there is a u ∈ X such that qv = hku. By taking x = u and y = yn in (iv), we get that qv = pu. Since p(X) fg(X), there is a w X such that pu = fgw. By taking x = u and y = w in (iv), we get that qw = fgw. Thus pu = hku = qv = fgw = qw = z(say). Since (p, hk) and (q, fg) are weakly compatible, p(hk)u = hk(p)u andq(fg)w = fg(q)w. i.e, pz = hkz and qz = fgz. From this stage, the proof is the same given in the previous case. Thus, z is a common fixed point of f , g, h, k, p and q. Uniqueness: If w is also a common fixed point of f , g, h, k, p and q. By taking x = z and y = w in (iv), we get that ϕ(Fpz,qw(αt), Fhkz,fgw(t), Fpz,hkz(t), Fqw,fgw(αt)) > 0 i.e, ϕ(Fz,w(αt), Fz,w(t), Fz,z(t), Fw,w(αt)) > 0 So, by property of ϕ, Fz,w(αt) ― Fz,w(t). By lemma(2.1), we get that w = z. Hence z is the unique common fixed point of f , g, h, k, p and q. This completes the proof of the theorem. NOTE 3.1. Theorem (3.1) is also valid if

Now we give the following example in support of our Theorem (3.1). EXAMPLE 3.1. Let X = [0, ∞), a ∈ b = min{a, b} for all a, b ∈ [0, 1] and for all x, y ∈ X and for all t > 0. Then (X, F, ∈ ) is a Menger space. Define self mappings f , g, h, k, p and q on X by fx = x , gx = x1/2, hx = x3, kx = x,

qx = 0, for all x ∈ X. Define ϕ : (R+)4 → R by

Then ϕ is an implicit relation. For x ™ 1 and y ∈ X, we have ϕ(F0,0(αt), Fx3 ,y (t), F0,x3 (t), F0,y(αt))

Different states of the Theorem are inconsequentially fulfilled. Unmistakably ′0′ is the remarkable regular fixed purpose of f , g, h, k, p and q in X. Presently, taking g = k = I(the character planning on X) in Theorem (3.1), we have the accompanying:

Now, we prove the following:

(iii) ϕ(Fpx,qy(αt), Fhkx,fgy(t), Fpx,hkx(t), Fqy,fgy(αt)) > 0,for all x, y ∈ X & t > 0 and for some ϕ ∈ Φ & α ∈ (0, 1);

Proof. Suppose (p, hk) and (q, fg) share JCLR(hk)(fg)-property, by definition, there exist sequences {xn} and {yn} in X such that for some u ∈ X. By taking x = u and y = yn in (iii), we get that ϕ(Fpu,qyn (αt), Fhku,fgyn (t), Fpu,hkx(t), Fqyn,fgyn (αt)) >0 As n → ∞, we get that ϕ(Fpu,hku(αt), Fhku,hku(t), Fpu,hku(t), Fhku,hku(αt)) >0 i.e, ϕ(Fpu,hku(αt), 1, Fpu,hku(t), 1)> 0 ⇒ ϕ(Fpu,hku(t), 1, Fpu,hku(t), 1) >0 ⇒ Fpu,hku(t) ― Fpu,hku(t) ⇒ pu = hku (by lemma(2.1)). By taking x = xn and y = u in (iii), we get that ϕ(Fpxn,qu(αt), Fhkxn,fgu(t), Fpxn,hkxn (t), Fqu,fgu(αt)) > 0 As n → ∞, we get that ϕ(Ffgu,qu(αt), Ffgu,fgu(t), Ffgu,fgu(t), Fqu,fgu(αt)) > 0 ⇒ Ffgu,qu(αt) ― 1 ⇒ fgu = qu. stage, the verification is a similar given in the Theorem (3.1). Subsequently, we get that z is a typical fixed purpose of f , g, h, k, p and q. Uniqueness follows inconsequentially. Note 3.2. Theorem (3.2) is also valid if (iii) is replaced by ϕ(Fpx,qy(αt), Fhkx,fgy(t), Fpx,hkx(αt), Fqy,fgy(t)) >0. We now give the following example in support of Theorem (3.2). EXAMPLE 3.2. Let X = [0, ∞), a ∗ b = min{a, b} for all a, b ∈ [0, 1] and Fx,y(t) for all x, y ∈ X and for all t > 0. Then (X, F, ∗) is a Menger space. Define self mappings f , g, h, k, p and q on X by fx = x , gx = x 2 , hx = x5, kx = x,

qx = 0, for all x ∈ X. Define ϕ : (R+)4 → R by ϕ(x1, x2, x3, x4) = 5x1 − 3x2 − 2x4. Then ϕ is an implicit relation. For x ™ 3 and y ∈ X, we have ϕ(F0,0(αt), Fx5 ,y2 (t), F0,x5 (t), F0,y2 (αt))

For x > 3 and y ∈ X, we have

Different states of the Theorem are inconsequentially fulfilled. Unmistakably ′0′ is the one of a kind normal fixed purpose of f , g, h, k, p and q in X. By taking g = k = I (the personality planning on X) in Theorem (3.2), we have the accompanying: Corollary 3.2. Let (X, F, T ) be a Menger space, where T means a continuous t-standard and f, h, p and q act naturally mappings of X, fulfilling:

Presently we demonstrate the accompanying: THEOREM 3.3. Let (X, F, T ) be a Menger space, where T means a ceaseless t-standard and f, g, h, k, p and q act naturally mappings of X, fulfilling: (i) the sets {p, hk} and {q, fg} are feebly viable; (ii) the requested sets (p, hk) and (q, fg) share JCLRpq-property;

Proof. Since (p, hk) and (q, fg) share JCLRpq-property, by definition, there exist arrangements {xn} and {yn} in X such that

Case I: Assume (iii)(a) holds. By taking x = xn and y = u in (iii)(a), we

As n → ∞, we get that ϕ(Fpu=qu,fgu(αt), Fpu=qu,fgu(t), Fpu=qu,qu(t), Fpu=qu,qu(αt)) > 0

⇒ ϕ(Fqu,fgu(t), Fqu,fgu(t), 1, 1) >0

By taking x = u and y = yn in (iii)(a), we get that

As n → ∞, we get that ϕ(Fhku,pu=qu(αt), Fpu,pu=qu(t), Fhku,pu=qu(t), Fpu,pu=qu(αt)) >0 i.e, ϕ(Fhku,pu(αt), Fpu,pu(t), Fhku,pu(t), Fpu,pu(αt))> 0 ⇒ ϕ(Fhku,pu(αt), 1, Fhku,pu(t), 1)> 0 ⇒ ϕ(Fhku,pu(t), 1, Fhku,pu(t), 1) >0 (by the property of ϕ) ⇒ Fhku,pu(t) > 1 ⇒ hku = pu. Thus fgu = qu = hku = pu = z(say) Since {p, hk} and {q, fg} are weakly compatible,

i.e, pz = hkz and qz = fgz. From this stage, the confirmation is a similar given in the Theorem(3.1). Consequently, we get that z is a typical fixed purpose of f , g, h, k, p and q.

As n → ∞, we get that ϕ(Fpu=qu,fgu(αt), Fpu=qu,fgu(t), Fpu=qu,qu(αt), Fpu=qu,qu(t)) > 0 i.e, ϕ(Fqu,fgu(αt), Fqu,fgu(t), 1, 1)> 0 ⇒ ϕ(Fqu,fgu(t), Fqu,fgu(t), 1, 1) >0

By taking x = u and y = yn in (iii)(b), we get that

Fpu,pu=qu(t)) >0 i.e, ϕ(Fhku,pu(αt), Fpu,pu(t), Fhku,pu(αt), Fpu,pu(t)) > 0 ⇒ ϕ(Fhku,pu(αt), 1, Fhku,pu(αt), 1) > 0 ⇒ Fhku,pu(αt) ― 1 (by the property of ϕ) ⇒ hku = pu. Thus fgu = qu = hku = pu = z(say). Since {p, hk} and {q, fg} are weakly compatible,

i.e, pz = hkz and qz = fgz. From this stage, the evidence is a similar given in the Theorem(3.1). Subsequently, we get that z is a typical fixed purpose of f , g, h, k, p and q. Uniqueness follows inconsequentially.

1. V. M. Sehgal and A. T. Bharucha-Reid (1972). ―Fixed points of contraction mappings on probabilistic metric spaces,‖ Mathematical Systems Theory, vol. 6, pp. 97–102. View at: Google Scholar | Zentralblatt MATH | MathSciNet 2. O. Hadžić (1982). ―On common fixed point theorems in 2-metric spaces,‖ Univerzitet u Novom Sadu, Zbornik Radova Prirodno-Matematičkog Fakulteta, vol. 12, pp. 7–18. 3. O. Hadžić (1982). ―Some theorems on the fixed points in probabilistic metric and random normed spaces,‖ Bollettino dell'Unione Matematica Italiana B, vol. 1, no. 6, pp. 381–391. 4. O. Hadžić (1989). ―Fixed point theorems for multivalued mappings in some classes of fuzzy metric spaces,‖ Fuzzy Sets and Systems, vol. 29, no. 1, pp. 115–125. 5. O. Hadzic (1994). ―A fixed point theorem for multivalued mappings in 2-Menger spaces,‖ Univerzitet u Novom Sadu, Zbornik Radova Prirodno-Matematičkog Fakulteta, vol. 24, pp. 1–7. 7. O. Hadžić and E. Pap (2001). Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic Publishers, Dordrecht, The Netherlands. 8. T. L. Hicks (1983). ―Fixed point theory in probabilistic metric spaces,‖ Univerzitet u Novom Sadu, Zbornik Radova Prirodno-Matematičkog Fakulteta, vol. 13, pp. 63–72. 9. S. Gähler (1963). ―2-metrische Räume und ihre topologische Struktur,‖ Mathematische Nachrichten, vol. 26, pp. 115–148. 10. K. Iseki (1975). ―Fixed point theorems in 2-metric spaces,‖ Mathematics Seminar Notes, vol. 3, pp. 133–136.

Department of Applied Mathematics, JSS Academy of Technical Education, C-20/1, Sector 62, Noida 201301, Uttar Pradesh, India