Vol 4, No 11 (2013)

Available online through

ISSN 2229 – 5046

RESULTS ON FIXED POINT THEOREM AND MENGER SPACES

Rajesh Shrivastava

_{, Rajendra Kumar Dubey}

_{and Pankaj Tiwari}

1

_{Govt. Science & Comm. College Benjeer, Bhopal - (M.P.), India.}

2

_{Govt. Science P. G. College, Rewa - (M.P.), India.}

(Received on: 07-10-13; Revised & Accepted on: 31-10-13)

ABSTRACT

I

Keywords: Menger space, t-norm, Common fixed point, Compatible maps, Weak - compatible maps.

AMS subject classification (2000): 45H10, 54H25.

I.INTRODUCTIONANDPRELIMINARIES

There have been a number of generalizations of metric space. One such generalization is Menger space in which, used distribution functions instead of nonnegative real numbers as value of metric. A Menger space is a space in which the concept of distance is considered to be a probabilistic, rather than deterministic. For detail discussion of Menger spaces and their applications we refer to Schweizer and Sklar [16]. The theory of Menger space is fundamental importance in probabilistic functional analysis.

The important development of fixed point theory in Menger spaces were due to Sehgal and Bharucha-Reid [13]. A probabilistic metric space shortly 𝑃𝑃𝑃𝑃-Space, is an ordered pair (𝑋𝑋,𝐹𝐹) consisting of a non empty set 𝑋𝑋 and a mapping 𝐹𝐹 from 𝑋𝑋 × 𝑋𝑋 to 𝐿𝐿, where 𝐿𝐿 is the collection of all distribution functions (a distribution function 𝐹𝐹 is non decreasing and left continuous mapping of reals in to [0,1] with properties, 𝑖𝑖𝑖𝑖𝑖𝑖 𝐹𝐹(𝑥𝑥) = 0 and 𝑠𝑠𝑠𝑠𝑠𝑠 𝐹𝐹(𝑥𝑥) = 1).

The value of 𝐹𝐹 at (𝑥𝑥,𝑦𝑦)∈ 𝑋𝑋 × 𝑋𝑋 is represented by 𝐹𝐹_{𝑥𝑥,𝑦𝑦}. The function 𝐹𝐹_{𝑥𝑥,𝑦𝑦} are assumed satisfy the following conditions:

(FM-0) 𝐹𝐹_{𝑥𝑥,𝑦𝑦}(𝑡𝑡) = 1, for all 𝑡𝑡> 0, iff 𝑥𝑥=𝑦𝑦; (FM-1) 𝐹𝐹_{𝑥𝑥,𝑦𝑦}(0) = 0, if 𝑡𝑡= 0;

(FM-2) 𝐹𝐹_{𝑥𝑥,𝑦𝑦}(𝑡𝑡) =𝐹𝐹_{𝑦𝑦,𝑥𝑥}(𝑡𝑡);

(FM-3) 𝐹𝐹_{𝑥𝑥,𝑦𝑦} (𝑡𝑡) = 1 and 𝐹𝐹_{𝑦𝑦,𝑧𝑧} (𝑠𝑠) = 1 then𝐹𝐹_{𝑥𝑥,𝑧𝑧}(𝑡𝑡 + 𝑠𝑠) = 1.

A mapping 𝑇𝑇: [0,1] × [0,1]→[0,1] is a 𝑡𝑡-norm, if it satisfies the following conditions: (FM-4) 𝑇𝑇(𝑎𝑎, 1) =𝑎𝑎 for every 𝑎𝑎 ∈[0,1];

(FM-5) 𝑇𝑇(0, 0) = 0,

(FM-6) 𝑇𝑇(𝑎𝑎,𝑏𝑏) =𝑇𝑇(𝑏𝑏,𝑎𝑎) for every 𝑎𝑎,𝑏𝑏 ∈[0,1]; (FM-7) 𝑇𝑇(𝑐𝑐,𝑑𝑑) ≥ 𝑇𝑇(𝑎𝑎,𝑏𝑏)for 𝑐𝑐 ≥ 𝑎𝑎 and 𝑑𝑑 ≥ 𝑏𝑏

(FM-8) 𝑇𝑇(𝑇𝑇(𝑎𝑎,𝑏𝑏),𝑐𝑐) =𝑇𝑇(𝑎𝑎,𝑇𝑇(𝑏𝑏,𝑐𝑐)) where 𝑎𝑎,𝑏𝑏,𝑐𝑐,𝑑𝑑 ∈[0,1].

A Menger space is a triplet (𝑋𝑋,𝐹𝐹,𝑇𝑇), where (𝑋𝑋,𝐹𝐹) is a 𝑃𝑃𝑃𝑃-Space, 𝑋𝑋 is a non-empty set and a 𝑡𝑡 − norm satisfying instead of (FM-8) a stronger requirement.

(FM-9) 𝐹𝐹_{𝑥𝑥,𝑧𝑧}(𝑡𝑡 + 𝑠𝑠)≥ 𝑇𝑇 �𝐹𝐹_{𝑥𝑥,𝑦𝑦}(𝑡𝑡),𝐹𝐹_{𝑦𝑦,𝑧𝑧}(𝑠𝑠)� for all 𝑥𝑥 ≥0,𝑦𝑦 ≥0.

For a given metric space (𝑋𝑋,𝑑𝑑) with usual metric 𝑑𝑑, one can put 𝐹𝐹_{𝑥𝑥,𝑦𝑦} (𝑡𝑡) = 𝐻𝐻 (𝑡𝑡 − 𝑑𝑑(𝑥𝑥,𝑦𝑦)) for all 𝑥𝑥,𝑦𝑦 ∈ 𝑋𝑋 and t > 0. where 𝐻𝐻 is defined as:

𝐻𝐻(𝑡𝑡) = � _{0 𝑖𝑖𝑖𝑖 𝑠𝑠 ≤}1 𝑖𝑖𝑖𝑖 𝑠𝑠> 0,_0.

and 𝑡𝑡-norm 𝑇𝑇 is defined as 𝑇𝑇(𝑎𝑎,𝑏𝑏) = 𝑚𝑚𝑖𝑖𝑖𝑖 {𝑎𝑎,𝑏𝑏}.

IJMA- 4(11), Nov.-2013.

For the proof of our result we required the following definitions.

Definition: 1.1 [11] Let (𝑋𝑋,𝐹𝐹,∗) be a Menger space and ∗ be a continuous 𝑡𝑡-norm.

a) A sequence {𝑥𝑥_𝑖𝑖} in 𝑋𝑋 is said to be converge to a point 𝑥𝑥 in 𝑋𝑋 (written 𝑥𝑥_𝑖𝑖→𝑥𝑥) iff for every 𝜀𝜀> 0 and λ ∈ (0, 1), there exists an integer 𝑖𝑖₀= 𝑖𝑖₀(𝜀𝜀,𝜆𝜆) such that 𝐹𝐹_{𝑥𝑥𝑖𝑖,𝑥𝑥}(𝜀𝜀) > 1− λ for all 𝑖𝑖 ≥ 𝑖𝑖₀.

b) (b) A sequence {𝑥𝑥_𝑖𝑖} in 𝑋𝑋 is said to be Cauchy if for every ε > 0 and 𝜆𝜆 ∈(0, 1), there exists an integer 𝑖𝑖₀= 𝑖𝑖0(ε,λ) such that 𝐹𝐹𝑥𝑥𝑖𝑖,𝑥𝑥𝑖𝑖+𝑠𝑠(ε) > 1−λ for all 𝑖𝑖 ≥ 𝑖𝑖0 and 𝑠𝑠 > 0.

c) A Menger space in which every Cauchy sequence is convergent is said to be complete.

Remark: 1.2 If ∗ is a continuous t-norm, it follows from (𝐹𝐹𝑃𝑃 −4) that the limit of sequence in Menger space is uniquely determined.

Definition: 1.3[15] Self maps 𝐴𝐴 and 𝐵𝐵 of a Menger space (𝑋𝑋,𝐹𝐹,∗) are said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence points, i.e. if 𝐴𝐴𝑥𝑥=𝐵𝐵𝑥𝑥 for some 𝑥𝑥 ∈ 𝑋𝑋 then 𝐴𝐴𝐵𝐵𝑥𝑥=𝐵𝐵𝐴𝐴𝑥𝑥.

Definition: 1.4[11] Self maps 𝐴𝐴 and 𝐵𝐵 of a Menger space (𝑋𝑋,𝐹𝐹,∗) are said to be compatible if F_{𝐴𝐴𝐵𝐵𝑥𝑥𝑚𝑚,𝐵𝐵𝐴𝐴𝑥𝑥𝑖𝑖,}(𝑡𝑡) → 1 for all 𝑡𝑡 > 0, whenever {𝑥𝑥_𝑖𝑖} is a sequence in 𝑋𝑋 such that 𝐴𝐴𝑥𝑥_𝑖𝑖→ 𝑥𝑥, 𝐵𝐵𝑥𝑥_𝑖𝑖 → 𝑥𝑥 for some 𝑥𝑥 in 𝑋𝑋 as 𝑖𝑖 → ∞.

Remark: 1.5 If self maps 𝐴𝐴 and 𝐵𝐵 of a Menger space (𝑋𝑋,𝐹𝐹,∗) are compatible then they are weakly compatible.

The following is an example of pair of self maps in a Menger space which are weakly compatible but not compatible.

Example: 1.6 Let (𝑋𝑋,𝑑𝑑) be a metric space where 𝑋𝑋= [0, 2] and (𝑋𝑋,𝐹𝐹,∗) be the induced Menger space with 𝐹𝐹𝑥𝑥,𝑦𝑦(𝑡𝑡) = 𝐻𝐻�𝑡𝑡 − 𝑑𝑑(𝑥𝑥,𝑦𝑦)�,∀ 𝑥𝑥,𝑦𝑦 ∈ 𝑋𝑋 and ∀ 𝑡𝑡 > 0.

Define self maps A and B as follows:

𝐴𝐴𝑥𝑥 = �2 _{2 𝑖𝑖𝑖𝑖 1}− 𝑥𝑥, 𝑖𝑖𝑖𝑖 0_{≤ 𝑥𝑥 ≤}≤ 𝑥𝑥< 1,_2, and 𝐵𝐵𝑥𝑥 = �𝑥𝑥, 𝑖𝑖𝑖𝑖 0≤ 𝑥𝑥< 1, 2 𝑖𝑖𝑖𝑖 1≤ 𝑥𝑥 ≤2,

Take 𝑥𝑥_𝑖𝑖= 1−1/𝑖𝑖. Then F_{𝐴𝐴𝑥𝑥𝑖𝑖+1,}(𝑡𝑡)= 𝐻𝐻 (𝑡𝑡 −(1/𝑖𝑖)) and lim⁡𝑖𝑖 →∞F𝐴𝐴𝑥𝑥 𝑖𝑖+1,(𝑡𝑡)= 𝐻𝐻 (𝑡𝑡) = 1.

Hence 𝐴𝐴𝑥𝑥_𝑖𝑖 → ∞ as 𝑖𝑖 → ∞. Similarly, 𝐵𝐵𝑥𝑥_𝑖𝑖 → ∞ as 𝑖𝑖 → ∞. Also 𝐹𝐹_{𝐴𝐴𝐵𝐵𝑥𝑥𝑖𝑖,𝐵𝐵𝐴𝐴𝑥𝑥𝑖𝑖} (𝑡𝑡) = 𝐻𝐻 (𝑡𝑡 −(1 −1/𝑖𝑖)) and lim_{𝑖𝑖→∞}F_{𝐴𝐴𝐵𝐵𝑥𝑥𝑖𝑖,𝐵𝐵𝐴𝐴𝑥𝑥𝑖𝑖,}(𝑡𝑡)→ 1 =𝐻𝐻(𝑡𝑡 − 1) ≠ 1,∀ 𝑡𝑡 > 0.

Hence the pair (𝐴𝐴,𝐵𝐵) is not compatible. Set of coincidence points of 𝐴𝐴 and 𝐵𝐵 is [1, 2]. Now for any 𝑥𝑥 ∈ [1, 2], 𝐴𝐴𝑥𝑥 =𝐵𝐵𝑥𝑥= 2, and 𝐴𝐴𝐵𝐵(𝑥𝑥) =𝐴𝐴 (2) = 2 =𝑆𝑆(2) =𝑆𝑆𝐴𝐴(𝑥𝑥). Thus 𝐴𝐴 and 𝐵𝐵 are weakly compatible but not compatible.

Lemma: 1.7 Let {𝑥𝑥_𝑖𝑖} be a sequence in a Menger space(𝑋𝑋,𝐹𝐹,∗) with continuous 𝑡𝑡-norm ∗ and 𝑡𝑡 ∗ 𝑡𝑡 𝑡𝑡. If there exists a constant 𝑘𝑘 ∈(0, 1) such that

𝐹𝐹𝑥𝑥𝑖𝑖,𝑥𝑥𝑖𝑖+1(𝑘𝑘𝑡𝑡)≥ 𝐹𝐹𝑥𝑥𝑖𝑖−1,𝑥𝑥𝑖𝑖(𝑡𝑡)

for all 𝑡𝑡> 0 and 𝑖𝑖= 1, 2. . ., then {𝑥𝑥_𝑖𝑖} is a Cauchy sequence in 𝑋𝑋.

Lemma: 1.8[15] Let (𝑋𝑋,𝐹𝐹,∗) be a Menger space. If there exists 𝑘𝑘 ∈(0, 1) such that

𝐹𝐹𝑥𝑥,𝑦𝑦(𝑘𝑘𝑡𝑡) ≥ 𝐹𝐹𝑥𝑥,𝑦𝑦 (𝑡𝑡)

for all 𝑥𝑥,𝑦𝑦 ∈ 𝑋𝑋 and 𝑡𝑡 > 0, then 𝑥𝑥=𝑦𝑦.

2. MAIN RESULTS

Theorem: 2.1 Let 𝐴𝐴,𝐵𝐵,𝑆𝑆,𝑇𝑇 and 𝑃𝑃 be self maps on a complete Menger space (𝑋𝑋,𝐹𝐹,∗) with 𝑡𝑡 ∗ 𝑡𝑡 ≥ 𝑡𝑡 for all 𝑡𝑡 ∈ [0, 1], satisfying:

(a) 𝑃𝑃(𝑋𝑋) ⊆ 𝐴𝐴𝐵𝐵(𝑋𝑋),𝑃𝑃(𝑋𝑋) ⊆ 𝑆𝑆𝑇𝑇(𝑋𝑋);

𝑃𝑃𝑃𝑃𝑥𝑥,𝑃𝑃𝑦𝑦, (𝑘𝑘𝑡𝑡 )≥ 𝑃𝑃𝐴𝐴𝐵𝐵𝑥𝑥,𝑃𝑃𝑥𝑥,(𝑡𝑡)∗ 𝑃𝑃𝑃𝑃𝑥𝑥,𝑆𝑆𝑇𝑇𝑦𝑦,(𝑡𝑡)∗ 𝑃𝑃𝐴𝐴𝐵𝐵𝑥𝑥,𝑆𝑆𝑇𝑇𝑦𝑦,(𝑡𝑡)∗𝑃𝑃𝑃𝑃𝑥𝑥,𝐴𝐴𝐵𝐵𝑥𝑥,_𝑃𝑃(𝑡𝑡)∗ 𝑃𝑃𝑃𝑃𝑥𝑥,𝑆𝑆𝑇𝑇𝑦𝑦,(𝑡𝑡)

𝑆𝑆𝑇𝑇𝑦𝑦,𝐴𝐴𝐵𝐵𝑥𝑥,(𝑡𝑡) ∗ 𝑃𝑃𝐴𝐴𝐵𝐵𝑥𝑥,𝑃𝑃𝑦𝑦,(3− 𝛼𝛼)𝑡𝑡 for all 𝑥𝑥,𝑦𝑦 ∈ 𝑋𝑋,𝛼𝛼 ∈(0,3) and 𝑡𝑡> 0,

(c) 𝑃𝑃𝐵𝐵=𝐵𝐵𝑃𝑃,𝑃𝑃𝑇𝑇=𝑇𝑇𝑃𝑃,𝐴𝐴𝐵𝐵=𝐵𝐵𝐴𝐴 and 𝑆𝑆𝑇𝑇=𝑇𝑇𝑆𝑆,

(d) 𝐴𝐴 and 𝐵𝐵 are continuous,

(e) the pair (𝑃𝑃,𝐴𝐴𝐵𝐵) is compatible (if compatible then it is weak compatible)

Then 𝐴𝐴,𝐵𝐵,𝑆𝑆,𝑇𝑇 and 𝑃𝑃 have a common fixed point in 𝑋𝑋.

Proof: Since 𝑃𝑃(𝑋𝑋)⊂ 𝐴𝐴𝐵𝐵(𝑋𝑋), for 𝑥𝑥₀𝜖𝜖 𝑋𝑋, we can choose a point 𝑥𝑥₀𝜖𝜖 𝑋𝑋 such that 𝑃𝑃𝑥𝑥₀=𝐴𝐴𝐵𝐵𝑥𝑥₁. Since 𝑃𝑃(𝑋𝑋)⊂ST(X), for this point 𝑥𝑥₁, we can choose a point 𝑥𝑥₂𝜖𝜖 𝑋𝑋 such that 𝑃𝑃𝑥𝑥₁=𝑆𝑆𝑇𝑇𝑥𝑥₂. Thus by induction, we can define a sequence 𝑦𝑦_𝑖𝑖𝜖𝜖𝑋𝑋 as follows:

𝑦𝑦2𝑖𝑖 =𝑃𝑃𝑥𝑥2𝑖𝑖 =𝐴𝐴𝐵𝐵𝑥𝑥2𝑖𝑖+1and 𝑦𝑦2𝑖𝑖+1=𝑃𝑃𝑥𝑥2𝑖𝑖+1=𝑆𝑆𝑇𝑇𝑥𝑥2𝑖𝑖+1

for 𝑖𝑖= 1,2, … . from (b),

For all 𝑡𝑡> 0 and 𝛼𝛼= 2− 𝑞𝑞 with 𝑞𝑞 ∈(0, 2), we have

𝑃𝑃𝑦𝑦2𝑖𝑖+1,𝑦𝑦2𝑖𝑖+2,(𝑘𝑘𝑡𝑡) =𝑃𝑃𝑃𝑃𝑥𝑥2𝑖𝑖+1,𝑃𝑃𝑥𝑥2𝑖𝑖+2,(𝑘𝑘𝑡𝑡)≥ 𝑃𝑃𝑦𝑦2𝑖𝑖+1,𝑦𝑦2𝑖𝑖+1,(𝑡𝑡)∗ 𝑃𝑃𝑦𝑦2𝑖𝑖,𝑦𝑦2𝑖𝑖+1,(𝑡𝑡) ∗ 𝑃𝑃_{𝑦𝑦2𝑖𝑖,𝑦𝑦2𝑖𝑖+1,}(𝑡𝑡)∗𝑃𝑃𝑦𝑦2𝑖𝑖+1,𝑦𝑦2𝑖𝑖,(𝑡𝑡)∗𝑃𝑃𝑦𝑦2𝑖𝑖+1,𝑦𝑦2𝑖𝑖+1,(𝑡𝑡)

𝑃𝑃𝑦𝑦2𝑖𝑖+1,𝑦𝑦2𝑖𝑖,(𝑡𝑡) ∗ 𝑃𝑃_{𝑦𝑦2𝑖𝑖,𝑦𝑦2𝑖𝑖+2,}(1 +𝑞𝑞)𝑡𝑡,

𝑃𝑃𝑦𝑦2𝑖𝑖+1,𝑦𝑦2𝑖𝑖+2,(𝑘𝑘𝑡𝑡)≥ 𝑃𝑃𝑦𝑦2𝑖𝑖,𝑦𝑦2𝑖𝑖+1,(𝑡𝑡)∗ 𝑃𝑃𝑦𝑦2𝑖𝑖,𝑦𝑦2𝑖𝑖+2,(1 +𝑞𝑞)𝑡𝑡

≥ 𝑃𝑃_{𝑦𝑦2𝑖𝑖,𝑦𝑦2𝑖𝑖+1,}(𝑡𝑡)∗ 𝑃𝑃_{𝑦𝑦2𝑖𝑖,𝑦𝑦2𝑖𝑖+1,}(𝑡𝑡)∗ 𝑃𝑃_{𝑦𝑦2𝑖𝑖+1,𝑦𝑦2𝑖𝑖+2,}(𝑞𝑞𝑡𝑡)

≥ 𝑃𝑃_{𝑦𝑦2𝑖𝑖,𝑦𝑦2𝑖𝑖+1,}(𝑡𝑡)* 𝑃𝑃_{𝑦𝑦2𝑖𝑖+1,𝑦𝑦2𝑖𝑖+2,}(t)

as 𝑞𝑞 →1.Since∗is continuous and 𝑃𝑃_{𝑥𝑥,𝑦𝑦}(∗) is continuous, letting 𝑞𝑞 →1 in above eq., we get

𝑃𝑃𝑦𝑦2𝑖𝑖+1,𝑦𝑦2𝑖𝑖+2,(𝑘𝑘𝑡𝑡)≥ 𝑃𝑃𝑦𝑦2𝑖𝑖,𝑦𝑦2𝑖𝑖+1,(𝑡𝑡)∗ 𝑃𝑃𝑦𝑦2𝑖𝑖+1,𝑦𝑦2𝑖𝑖+2,(𝑡𝑡) … … (1)

Similarly, we have

𝑃𝑃𝑦𝑦2𝑖𝑖+2,𝑦𝑦2𝑖𝑖+3,(𝑘𝑘𝑡𝑡)≥ 𝑃𝑃𝑦𝑦2𝑖𝑖+1,𝑦𝑦2𝑖𝑖+2,(𝑡𝑡)∗ 𝑃𝑃𝑦𝑦2𝑖𝑖+2,𝑦𝑦2𝑖𝑖+2,(𝑡𝑡) … … (2)

Thus from (1) and (2), it follows that

𝑃𝑃𝑦𝑦𝑖𝑖+1,𝑦𝑦𝑖𝑖+2,(𝑘𝑘𝑡𝑡)≥ 𝑃𝑃𝑦𝑦𝑖𝑖,𝑦𝑦𝑖𝑖+1,(𝑡𝑡)∗ 𝑃𝑃𝑦𝑦𝑖𝑖+1,𝑦𝑦𝑖𝑖+2,(𝑡𝑡)

for 𝑖𝑖= 1,2, … and then for positive integers 𝑖𝑖 and 𝑠𝑠,

𝑃𝑃𝑦𝑦𝑖𝑖+1,𝑦𝑦𝑖𝑖+2,(𝑘𝑘𝑡𝑡)≥ 𝑃𝑃𝑦𝑦𝑖𝑖,𝑦𝑦𝑖𝑖+1,(𝑡𝑡)∗ 𝑃𝑃𝑦𝑦𝑖𝑖+1,𝑦𝑦𝑖𝑖+2,� 𝑡𝑡 𝑘𝑘𝑠𝑠�.

Thus, since 𝑃𝑃_{𝑦𝑦𝑖𝑖+1,𝑦𝑦𝑖𝑖+1,}�𝑡𝑡

𝑘𝑘𝑠𝑠� →1 as 𝑠𝑠 → ∞ we have

𝑃𝑃𝑦𝑦𝑖𝑖+1,𝑦𝑦𝑖𝑖+2,(𝑘𝑘𝑡𝑡)≥ 𝑃𝑃𝑦𝑦𝑖𝑖,𝑦𝑦𝑖𝑖+1,(𝑡𝑡).

𝑦𝑦𝑖𝑖is Cauchy sequence in 𝑋𝑋 and since 𝑥𝑥 is complete,𝑦𝑦𝑖𝑖converges to a point 𝑧𝑧 𝜖𝜖 𝑋𝑋.

Since 𝑃𝑃𝑥𝑥_𝑖𝑖,𝐴𝐴𝐵𝐵𝑥𝑥_2𝑖𝑖+1and 𝑆𝑆𝑇𝑇𝑥𝑥_2𝑖𝑖+2are subsequences of 𝑦𝑦_𝑖𝑖, they also converge to the point 𝑧𝑧, since 𝐴𝐴,𝐵𝐵 are continuous and pair {𝑃𝑃,𝐴𝐴𝐵𝐵} is compatible and also weak compatible, we have

IJMA- 4(11), Nov.-2013.

From (𝑏𝑏) with 𝛼𝛼= 2, we get

𝑃𝑃𝑃𝑃𝐴𝐴𝐵𝐵𝑥𝑥2𝑖𝑖+1, 𝑃𝑃𝑥𝑥2𝑖𝑖+2,,(𝑘𝑘𝑡𝑡)≥ 𝑃𝑃(𝐴𝐴𝐵𝐵)2𝑥𝑥2𝑖𝑖+1,(𝑡𝑡)∗ 𝑃𝑃𝑃𝑃𝐴𝐴𝐵𝐵𝑥𝑥2𝑖𝑖+1, 𝑆𝑆𝑇𝑇𝑥𝑥2𝑖𝑖+2,,(𝑡𝑡)

∗ 𝑃𝑃_{(𝐴𝐴𝐵𝐵)}2_{𝑥𝑥2𝑖𝑖+1, 𝑆𝑆𝑇𝑇𝑥𝑥}

2𝑖𝑖+2,,(𝑡𝑡)∗

𝑃𝑃_{𝑃𝑃𝐴𝐴 𝐵𝐵𝑥𝑥2𝑖𝑖+1,(𝐴𝐴𝐵𝐵)2 𝑥𝑥2𝑖𝑖+1,}(𝑡𝑡)∗𝑃𝑃_{𝑃𝑃𝐴𝐴𝐵𝐵𝑥𝑥2𝑖𝑖+1, 𝑆𝑆𝑇𝑇𝑥𝑥} 2𝑖𝑖+2,,(𝑡𝑡) 𝑃𝑃_{𝑆𝑆𝑇𝑇𝑥𝑥2𝑖𝑖+2,(𝐴𝐴𝐵𝐵)2 𝑥𝑥2𝑖𝑖+1,} (𝑡𝑡)

∗ 𝑃𝑃_{(𝐴𝐴𝐵𝐵)}2_{𝑥𝑥2𝑖𝑖+1, 𝑃𝑃𝑥𝑥} 2𝑖𝑖+2,,(𝑡𝑡) which implies that

𝑃𝑃𝐴𝐴𝐵𝐵𝑧𝑧,𝑧𝑧(𝑘𝑘𝑡𝑡) =𝑙𝑙𝑖𝑖𝑚𝑚𝑖𝑖→∞ 𝑃𝑃𝑃𝑃𝐴𝐴𝐵𝐵𝑥𝑥2𝑖𝑖+2, (𝑘𝑘𝑡𝑡)

≥1∗ 𝑃𝑃_{𝐴𝐴𝐵𝐵𝑧𝑧,𝑧𝑧,} (𝑡𝑡)∗ 𝑃𝑃_{𝐴𝐴𝐵𝐵𝑧𝑧,𝑧𝑧,} (𝑡𝑡)∗1∗𝑃𝑃𝐴𝐴𝐵𝐵𝑧𝑧,𝑧𝑧 (𝑡𝑡)

𝑃𝑃𝑧𝑧,𝐴𝐴𝐵𝐵𝑧𝑧 (𝑡𝑡) ∗ 𝑃𝑃𝐴𝐴𝐵𝐵𝑧𝑧,𝑧𝑧,𝑧𝑧 (𝑡𝑡)

We have

𝐴𝐴𝐵𝐵𝑧𝑧=𝑧𝑧, since 𝑃𝑃_{𝑧𝑧,,𝑆𝑆𝑇𝑇𝑧𝑧} (𝑡𝑡)≥ 𝑃𝑃_{𝑧𝑧,𝐴𝐴𝐵𝐵𝑧𝑧}(𝑡𝑡) = 1 for all 𝑡𝑡> 0,

We get 𝑆𝑆𝑇𝑇𝑧𝑧=𝑧𝑧. again by (b) with 𝛼𝛼= 2,

We have

𝑃𝑃𝑃𝑃𝐴𝐴𝐵𝐵𝑥𝑥2𝑖𝑖+1, 𝑃𝑃𝑧𝑧(𝑘𝑘𝑡𝑡)≥ 𝑃𝑃(𝐴𝐴𝐵𝐵)2𝑥𝑥2𝑖𝑖+1, 𝑃𝑃𝐴𝐴𝐵𝐵𝑥𝑥2𝑖𝑖+1,,(𝑡𝑡)∗ 𝑃𝑃𝑃𝑃𝐴𝐴𝐵𝐵𝑥𝑥2𝑖𝑖+1, 𝑆𝑆𝑇𝑇𝑧𝑧, ,,(𝑡𝑡)

∗ 𝑃𝑃_{(𝐴𝐴𝐵𝐵)}2_{𝑥𝑥2𝑖𝑖+1, 𝑆𝑆𝑇𝑇𝑧𝑧} , ,,(𝑡𝑡)∗

𝑃𝑃_{𝑃𝑃𝐴𝐴𝐵𝐵𝑥𝑥2𝑖𝑖+1,(𝐴𝐴𝐵𝐵)2 𝑥𝑥2𝑖𝑖+1,}(𝑡𝑡)∗𝑃𝑃𝑃𝑃𝐴𝐴𝐵𝐵𝑥𝑥_{2𝑖𝑖+1, 𝑆𝑆𝑇𝑇𝑧𝑧, ,,}(𝑡𝑡)

𝑃𝑃_{𝑆𝑆𝑇𝑇𝑧𝑧,(𝐴𝐴𝐵𝐵)2 𝑥𝑥2𝑖𝑖+1,} (𝑡𝑡) ∗ 𝑃𝑃_{(𝐴𝐴𝐵𝐵)}2_{𝑥𝑥2𝑖𝑖+1, 𝑃𝑃𝑧𝑧}

,,(𝑡𝑡)

which implies that

𝑃𝑃𝐴𝐴𝐵𝐵𝑧𝑧,𝑃𝑃𝑧𝑧,𝑃𝑃𝑧𝑧(𝑘𝑘𝑡𝑡) =𝑙𝑙𝑖𝑖𝑚𝑚𝑖𝑖→∞ 𝑃𝑃𝑃𝑃𝐴𝐴𝐵𝐵𝑥𝑥2𝑖𝑖+1, 𝑃𝑃𝑧𝑧, (𝑘𝑘𝑡𝑡)

≥1∗1∗1∗1∗ 𝑃𝑃_{𝐴𝐴𝐵𝐵𝑧𝑧,𝑃𝑃𝑧𝑧,}(𝑡𝑡)

≥ 𝑃𝑃_{𝐴𝐴𝐵𝐵𝑧𝑧,𝑃𝑃𝑧𝑧,}(𝑡𝑡).

We have 𝐴𝐴𝐵𝐵𝑧𝑧=𝑃𝑃𝑧𝑧.Now, we show that 𝐵𝐵𝑧𝑧=𝑧𝑧.Infact, from (b) with 𝛼𝛼= 2, and (𝑐𝑐) we get,

𝑃𝑃𝐵𝐵𝑧𝑧,𝑧𝑧,(𝑘𝑘𝑡𝑡) =𝑃𝑃𝐵𝐵𝑃𝑃𝑧𝑧,𝑃𝑃𝑧𝑧,(𝑘𝑘𝑡𝑡)

=𝑃𝑃_{𝑃𝑃𝐵𝐵𝑧𝑧,𝑃𝑃𝑧𝑧,}(𝑘𝑘𝑡𝑡)

𝑃𝑃𝑃𝑃𝐵𝐵𝑧𝑧,𝑃𝑃𝑧𝑧,(𝑘𝑘𝑡𝑡)≥ 𝑃𝑃𝑃𝑃𝐵𝐵𝑧𝑧,𝑆𝑆𝑇𝑇𝑧𝑧,(𝑡𝑡)∗ 𝑃𝑃𝐴𝐴𝐵𝐵𝐵𝐵𝑧𝑧,𝑆𝑆𝑇𝑇𝑧𝑧,(𝑡𝑡)∗𝑃𝑃𝑃𝑃𝐵𝐵𝑧𝑧,𝐴𝐴𝐵𝐵𝐵𝐵𝑧𝑧_𝑃𝑃,(𝑡𝑡)∗ 𝑃𝑃𝑃𝑃𝐵𝐵𝑧𝑧,𝑧𝑧,𝑧𝑧 (𝑡𝑡)

𝑧𝑧,𝑃𝑃𝐵𝐵𝑧𝑧, (𝑡𝑡) ∗ 𝑃𝑃𝑃𝑃𝐵𝐵𝑧𝑧,𝑧𝑧(𝑡𝑡)

= 1∗ 𝑃𝑃_{𝐵𝐵𝑧𝑧,𝑧𝑧,} (𝑡𝑡)∗ 𝑃𝑃_{𝐵𝐵𝑧𝑧,𝑧𝑧,} (𝑡𝑡)∗1∗ 𝑃𝑃_{𝐵𝐵𝑧𝑧,𝑧𝑧,} (𝑡𝑡)

=𝑃𝑃_{𝐵𝐵𝑧𝑧,𝑧𝑧,} (𝑡𝑡).

which implies that 𝐵𝐵𝑧𝑧=𝑧𝑧.Since 𝐴𝐴𝐵𝐵𝑧𝑧=𝑧𝑧,

we have 𝐴𝐴𝑧𝑧=𝑧𝑧.Next, we show that 𝑇𝑇𝑧𝑧=𝑧𝑧. Indeed from (𝑏𝑏) with 𝛼𝛼= 2, and (𝑐𝑐) we get

𝑃𝑃𝑇𝑇𝑧𝑧,𝑧𝑧, (𝑘𝑘𝑡𝑡) =𝑃𝑃𝑇𝑇𝑃𝑃𝑧𝑧,𝑃𝑃𝑧𝑧,(𝑘𝑘𝑡𝑡)=𝑃𝑃𝑃𝑃𝑧𝑧,𝑃𝑃𝑧𝑧, (𝑘𝑘𝑡𝑡)

≥1∗ 𝑃𝑃_{𝑧𝑧,𝑇𝑇𝑧𝑧,} (𝑡𝑡)∗ 𝑃𝑃_{𝑧𝑧,𝑇𝑇𝑧𝑧,} (𝑡𝑡)∗1∗ 𝑃𝑃_{𝑧𝑧,𝑇𝑇𝑧𝑧,} (𝑡𝑡)

≥ 𝑃𝑃_{𝑇𝑇𝑧𝑧,𝑧𝑧,} (𝑡𝑡),

which implies that 𝑇𝑇𝑧𝑧=𝑧𝑧.Since 𝑆𝑆𝑇𝑇𝑧𝑧=𝑧𝑧, we have 𝑆𝑆𝑧𝑧=𝑆𝑆𝑇𝑇𝑧𝑧=𝑧𝑧. Therefore, by combining the above results we obtain,

𝐴𝐴𝑧𝑧=𝐵𝐵𝑧𝑧=𝑆𝑆𝑧𝑧=𝑇𝑇𝑧𝑧=𝑃𝑃𝑧𝑧.

Finally, the uniqueness of the fixed point of 𝐴𝐴,𝐵𝐵, S, T and P.

Corollary: 2.2 Let 𝐴𝐴,𝐵𝐵,𝑆𝑆,𝑇𝑇 and 𝑃𝑃 be self maps on a complete Menger space (𝑋𝑋,𝐹𝐹,∗) with 𝑡𝑡 ∗ 𝑡𝑡 ≥ 𝑡𝑡 for all 𝑡𝑡 ∈[0, 1], satisfying;

(a) 𝑃𝑃(𝑋𝑋)⊆ 𝐴𝐴(𝑋𝑋) and 𝑃𝑃(𝑋𝑋)⊆ 𝑆𝑆(𝑋𝑋)

(b) there exists a constant 𝑘𝑘 ∈(0, 1) such that

𝑃𝑃_{𝑃𝑃𝑥𝑥,𝑃𝑃𝑦𝑦,} (𝑘𝑘𝑡𝑡 )≥ 𝑃𝑃_{𝐴𝐴𝑥𝑥,𝑃𝑃𝑥𝑥,}(𝑡𝑡)∗ 𝑃𝑃_{𝑃𝑃𝑥𝑥,𝑆𝑆𝑦𝑦,}(𝑡𝑡)∗ 𝑃𝑃_{𝐴𝐴𝑥𝑥,𝑆𝑆𝑦𝑦,}(𝑡𝑡)∗𝑃𝑃𝑃𝑃𝑥𝑥,𝐴𝐴𝑥𝑥,(𝑡𝑡)∗𝑃𝑃𝑃𝑃𝑥𝑥,𝑆𝑆𝑦𝑦,(𝑡𝑡)

𝑃𝑃𝑆𝑆𝑦𝑦,𝐴𝐴𝑥𝑥,(𝑡𝑡) ∗ 𝑃𝑃𝐴𝐴𝑥𝑥,𝑃𝑃𝑦𝑦,(3− 𝛼𝛼)𝑡𝑡 for all 𝑥𝑥,𝑦𝑦 ∈ 𝑋𝑋,𝛼𝛼 ∈(0,3) and 𝑡𝑡> 0,

(c) 𝐴𝐴 or 𝑃𝑃 are continuous,

(d) the pair {𝑃𝑃,𝐴𝐴} is compatible,

Then 𝐴𝐴,𝑆𝑆 and 𝑃𝑃 have a common fixed point in 𝑋𝑋.

CONCLUSION

In this present article we prove a common fixed point theorem in Menger spaces by using five compatible mappings. In fact our main result is more general then other previous known results.

ACKNOWLEDGEMENT

The authors thank the referees for their careful reading of the manuscript and for their valuable suggestions.

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