Definition 1) Menger space is defined as triple $\left( S,F,T \right)$ where S is a set , F is a collection of distribution functions and T is a triangular norm function T:$[ 0,1 $]$\times $[ 0,1 $]\to $[ 0,1 ] which has the following property :

$\begin{align} & ( \Delta 1 )T( a,b )=T( b,a )\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,( \text{commutativity} ) \\ & ( \Delta 2 )T( a,b )\le T( c,d )\,\text{whenever}\,a\le c,b\le d\left( \text{monotonicity} \right) \\ & \left( \Delta 3 \right)T\left( a,T\left( b,c \right) \right)=T\left( T\left( a,b \right),c \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( \text{associativity} \right) \\ & \left( \Delta 4 \right)T\left( a,1 \right)=a\left( \text{boundary condition} \right) \\ \end{align}$

A consequence of property the above definition is the following extra property  :

$\begin{align} & \left( \Delta 5 \right)T\left( a,b \right)\le T\left( a,c \right)\,\text{whenever}\,b\le c(\text{obtained from }\Delta \text{2)} \\ & \left( \Delta 6 \right)T\left( a,1 \right)>0\,\text{whenever}\,a>0,T\left( 1,1 \right)=1\left( \text{obtained from }\Delta \text{4} \right) \\ & \left( \Delta 7 \right)T\left( a,b \right)\le a,T\left( a,b \right)\le b\left( \text{obtained from }\Delta \text{4},\Delta 2 \right) \\ \end{align} $

Definition 2) A sequence $\left\{ {{p}_{n}} \right\}\in S$ is called probabilistic fundamental sequence in a probabilistic metric space $\left( S,F,T \right)$ if $\forall \varepsilon ,\lambda >0,\exists M;\forall n,m\ge M;{{F}_{{{p}_{n}},{{p}_{m}}}}$( $\varepsilon $)>1-$\lambda$

Definition 3) We say a Menger space is complete if any probabilistic fundamental sequence in $S$ converges to a member in $S$ .

Definition 4) the probabilistic contraction mapping $G:S\to S$ on complete Menger space is shown to have a fixed point if the following relation holds for all $p,q\in S$ :${{F}_{Gp,Gq}}$( kx $)\ge {{F}_{p,q}}$( x $)\,\,\,\,\text{for all }x>0\,\,$.

Theorem (Sehgal [1]): A contraction on complete Menger space is have a unique fixed point if the triangle norm T has the following property:

${{A}_{1}})$ $T\left( x,x \right)\ge x\forall x\in \left[ 0,1 \right]$

${{A}_{2}})$ $T\text{ is a continous function}$

[1]: Sehgal, V.M. & Bharucha-Reid, A.T. Math. Systems Theory (1972) 6: 97. https://doi.org/10.1007/BF01706080

The question : the only t-norm that satisfies $T\left( x,x \right)\ge x$ is $T\left( x,y \right)=\min \left( x,y \right)$ however other types of t-norm like H-type , archimedean and Lukasiewicz t-norm can not garentee the existence of fixed point regarding this theorem. Is it possible to generalize theorem (Sehgal) by dropping the condition property ${{A}_{1}}$?

Definition 1) A Menger space is defined as a triple $\left( S,F,T \right)$ where $S$ is a set , $F$ is a collection of distribution functions and $T$ is a triangular norm function $T:[ 0,1 ]\times [ 0,1 ]\to [ 0,1 ]$ which has the following properties:

$\begin{align} & ( \Delta 1 ) T( a,b )=T( b,a )\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,( \text{commutativity} ) \\ & ( \Delta 2 )T( a,b )\le T( c,d )\,\text{whenever}\,a\le c,b\le d\left( \text{monotonicity} \right) \\ & \left( \Delta 3 \right)T\left( a,T\left( b,c \right) \right)=T\left( T\left( a,b \right),c \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( \text{associativity} \right) \\ & \left( \Delta 4 \right)T\left( a,1 \right)=a\left( \text{boundary condition} \right) \\ \end{align}$

A consequence of these properties in the above definition is the following extra property:

$\begin{align} & \left( \Delta 5 \right)T\left( a,b \right)\le T\left( a,c \right)\,\text{whenever}\,b\le c(\text{obtained from }\Delta \text{2)} \\ & \left( \Delta 6 \right)T\left( a,1 \right)>0\,\text{whenever}\,a>0,T\left( 1,1 \right)=1\left( \text{obtained from }\Delta \text{4} \right) \\ & \left( \Delta 7 \right)T\left( a,b \right)\le a,T\left( a,b \right)\le b\left( \text{obtained from }\Delta \text{4},\Delta 2 \right) \\ \end{align} $

Definition 2) A sequence $\left\{ {{p}_{n}} \right\}\in S$ is called a probabilistic fundamental sequence in a probabilistic metric space $\left( S,F,T \right)$ if $\forall \varepsilon ,\lambda >0,\exists M;\forall n,m\ge M;{{F}_{{{p}_{n}},{{p}_{m}}}}$( $\varepsilon $)>1-$\lambda$

Definition 3) We say a Menger space is complete if any probabilistic fundamental sequence in $S$ converges to a member of $S$.

Definition 4) The probabilistic contraction mapping $G:S\to S$ on a complete Menger space has a fixed point if the following relation holds for all $p,q\in S$ :${{F}_{Gp,Gq}}$( kx $)\ge {{F}_{p,q}}$( x $)\,\,\,\,\text{for all }x>0\,\,$.

Theorem (Sehgal [1]): A contraction on complete Menger space has a unique fixed point if the triangle norm $T$ has the following property:

${{A}_{1}})$ $T\left( x,x \right)\ge x\forall x\in \left[ 0,1 \right]$

${{A}_{2}})$ $T\text{ is a continuous function}$

[1]: Sehgal, V.M. & Bharucha-Reid, A.T. Math. Systems Theory (1972) 6: 97. https://doi.org/10.1007/BF01706080

The only $t$-norm that satisfies $T\left( x,x \right)\ge x$ is $T\left( x,y \right)=\min \left( x,y \right)$. However other types of $t$-norm like those of $H$-type , archimedean and Lukasiewicz $t$-norms can not guarantee the existence of a fixed point as in this theorem. The question: Is it possible to generalize the theorem of Sehgal by dropping the condition property ${{A}_{1}}$?

Definition 1) Menger space is defined as triple $\left( S,F,T \right)$ where S is a set , F is a collection of distribution functions and T is a triangular norm function T:$[ 0,1 $]$\times $[ 0,1 $]\to $[ 0,1 ] which has the following property :

$\begin{align} & ( \Delta 1 )T( a,b )=T( b,a )\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,( \text{commutativity} ) \\ & ( \Delta 2 )T( a,b )\le T( c,d )\,\text{whenever}\,a\le c,b\le d\left( \text{monotonicity} \right) \\ & \left( \Delta 3 \right)T\left( a,T\left( b,c \right) \right)=T\left( T\left( a,b \right),c \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( \text{associativity} \right) \\ & \left( \Delta 4 \right)T\left( a,1 \right)=a\left( \text{boundary condition} \right) \\ \end{align}$

A consequence of property the above definition is the following extra property :

$\begin{align} & \left( \Delta 5 \right)T\left( a,b \right)\le T\left( a,c \right)\,\text{whenever}\,b\le c(\text{obtained from }\Delta \text{2)} \\ & \left( \Delta 6 \right)T\left( a,1 \right)>0\,\text{whenever}\,a>0,T\left( 1,1 \right)=1\left( \text{obtained from }\Delta \text{4} \right) \\ & \left( \Delta 7 \right)T\left( a,b \right)\le a,T\left( a,b \right)\le b\left( \text{obtained from }\Delta \text{4},\Delta 2 \right) \\ \end{align} $

Definition 2) A sequence $\left\{ {{p}_{n}} \right\}\in S$ is called probabilistic fundamental sequence in a probabilistic metric space $\left( S,F,T \right)$ if $\forall \varepsilon ,\lambda >0,\exists M;\forall n,m\ge M;{{F}_{{{p}_{n}},{{p}_{m}}}}$( $\varepsilon $)>1-$\lambda$

Definition 3) We say a Menger space is complete if any probabilistic fundamental sequence in $S$ converge to a member in $S$ .

Definition 4) the probabilistic contraction mapping $G:S\to S$ on complete Menger space is shown to have a fixed point if the following relation holds for all $p,q\in S$ :${{F}_{Gp,Gq}}$( kx $)\ge {{F}_{p,q}}$( x $)\,\,\,\,\text{for all }x>0\,\,$.

Theorem (Sehgal [1]): a contraction on complete Menger space is have a unique fixed point if the triangle norm T has the following property

${{A}_{1}})$ $T\left( x,x \right)\ge x\forall x\in \left[ 0,1 \right]$

${{A}_{2}})$ $T\text{ is a continous function}$

[1]: Sehgal, V.M. & Bharucha-Reid, A.T. Math. Systems Theory (1972) 6: 97. https://doi.org/10.1007/BF01706080

The question : the only t-norm that satisfy $T\left( x,x \right)\ge x$ is $T\left( x,y \right)=\min \left( x,y \right)$ however other types of t-norm like H-type , archimedean and Lukasiewicz t-norm can not garentee fixed point regarding this theorem. Is it possible to generalize theorem (Sehgal) such by droping the condition property ${{A}_{1}}$?

Definition 1) Menger space is defined as triple $\left( S,F,T \right)$ where S is a set , F is a collection of distribution functions and T is a triangular norm function T:$[ 0,1 $]$\times $[ 0,1 $]\to $[ 0,1 ] which has the following property :

$\begin{align} & ( \Delta 1 )T( a,b )=T( b,a )\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,( \text{commutativity} ) \\ & ( \Delta 2 )T( a,b )\le T( c,d )\,\text{whenever}\,a\le c,b\le d\left( \text{monotonicity} \right) \\ & \left( \Delta 3 \right)T\left( a,T\left( b,c \right) \right)=T\left( T\left( a,b \right),c \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( \text{associativity} \right) \\ & \left( \Delta 4 \right)T\left( a,1 \right)=a\left( \text{boundary condition} \right) \\ \end{align}$

A consequence of property the above definition is the following extra property :

$\begin{align} & \left( \Delta 5 \right)T\left( a,b \right)\le T\left( a,c \right)\,\text{whenever}\,b\le c(\text{obtained from }\Delta \text{2)} \\ & \left( \Delta 6 \right)T\left( a,1 \right)>0\,\text{whenever}\,a>0,T\left( 1,1 \right)=1\left( \text{obtained from }\Delta \text{4} \right) \\ & \left( \Delta 7 \right)T\left( a,b \right)\le a,T\left( a,b \right)\le b\left( \text{obtained from }\Delta \text{4},\Delta 2 \right) \\ \end{align} $

Definition 2) A sequence $\left\{ {{p}_{n}} \right\}\in S$ is called probabilistic fundamental sequence in a probabilistic metric space $\left( S,F,T \right)$ if $\forall \varepsilon ,\lambda >0,\exists M;\forall n,m\ge M;{{F}_{{{p}_{n}},{{p}_{m}}}}$( $\varepsilon $)>1-$\lambda$

Definition 3) We say a Menger space is complete if any probabilistic fundamental sequence in $S$ converges to a member in $S$ .

Definition 4) the probabilistic contraction mapping $G:S\to S$ on complete Menger space is shown to have a fixed point if the following relation holds for all $p,q\in S$ :${{F}_{Gp,Gq}}$( kx $)\ge {{F}_{p,q}}$( x $)\,\,\,\,\text{for all }x>0\,\,$.

Theorem (Sehgal [1]): A contraction on complete Menger space is have a unique fixed point if the triangle norm T has the following property:

${{A}_{1}})$ $T\left( x,x \right)\ge x\forall x\in \left[ 0,1 \right]$

${{A}_{2}})$ $T\text{ is a continous function}$

[1]: Sehgal, V.M. & Bharucha-Reid, A.T. Math. Systems Theory (1972) 6: 97. https://doi.org/10.1007/BF01706080

The question : the only t-norm that satisfies $T\left( x,x \right)\ge x$ is $T\left( x,y \right)=\min \left( x,y \right)$ however other types of t-norm like H-type , archimedean and Lukasiewicz t-norm can not garentee the existence of fixed point regarding this theorem. Is it possible to generalize theorem (Sehgal) by dropping the condition property ${{A}_{1}}$?