# Using Dual Double Fuzzy Semi-Metric to Study the Convergence
## 2 Fuzzy Semi-Metric Space
Then:_ \[M\left(x_{1},x_{2},t_{1}\right)*M\left(x_{2},x_{3},t_{2}\right)*\cdots*M\left( x_{p},x_{p+1},t_{p}\right)\leq M\left(x_{1},x_{p+1},t_{1}+t_{2}+\cdots+t_{p} \right).\]0 \[M\left(x_{1},x_{2},t_{1}\right)*M\left(x_{2},x_{3},t_{2}\right)*\cdots*M\left( x_{p},x_{p+1},t_{p}\right)\leq M\left(x_{1},x_{p+1},t_{1}+t_{2}+\cdots+t_{p} \right).\]1 (4) _and:_ \[M\left(x_{1},x_{2},t_{1}\right)*M\left(x_{2},x_{3},t_{2}\right)*\cdots*M\left( x_{p},x_{p+1},t_{p}\right)\leq M\left(x_{1},x_{p+1},t_{1}+t_{2}+\cdots+t_{p} \right).\]2 \[M\left(x_{1},x_{2},t_{1}\right)*M\left(x_{2},x_{3},t_{2}\right)*\cdots*M\left( x_{p},x_{p+1},t_{p}\right)\leq M\left(x_{1},x_{p+1},t_{1}+t_{2}+\cdots+t_{p} \right).\]3 (5) _From Equation (_4_), we also have:_ \[M\left(x_{1},x_{2},t_{1}\right)*M\left(x_{2},x_{3},t_{2}\right)*\cdots*M\left( x_{p},x_{p+1},t_{p}\right)\leq M\left(x_{1},x_{p+1},t_{1}+t_{2}+\cdots+t_{p} \right).\]4 \[M\left(x_{1},x_{2},t_{1}\right)*M\left(x_{2},x_{3},t_{2}\right)*\cdots*M\left( x_{p},x_{p+1},t_{p}\right)\leq M\left(x_{1},x_{p+1},t_{1}+t_{2}+\cdots+t_{p} \right).\]5 (6) _which implies:_ \[M\left(x_{1},x_{2},t_{1}\right)*M\left(x_{2},x_{3},t_{2}\right)*\cdots*M\left( x_{p},x_{p+1},t_{p}\right)\leq M\left(x_{1},x_{p+1},t_{1}+t_{2}+\cdots+t_{p} \right).\]6 (7) _by referring to Equation (_5_). In general, we have the following cases:_ 1. _If_ \(p\) _is even, then:_ \[M\left(x_{1},x_{2},t_{1}\right)*M\left(x_{2},x_{3},t_{2}\right)*\cdots*M\left( x_{p},x_{p+1},t_{p}\right)\leq M\left(x_{1},x_{p+1},t_{1}+t_{2}+\cdots+t_{p} \right).\]7 \[M\left(x_{1},x_{2},t_{1}\right)*M\left(x_{2},x_{3},t_{2}\right)*\cdots*M\left( x_{p},x_{p+1},t_{p}\right)\leq M\left(x_{1},x_{p+1},t_{1}+t_{2}+\cdots+t_{p} \right).\]8 _and:_ \[M\left(x_{1},x_{2},t_{1}\right)*M\left(x_{2},x_{3},t_{2}\right)*\cdots*M\left( x_{p},x_{p+1},t_{p}\right)\leq M\left(x_{1},x_{p+1},t_{1}+t_{2}+\cdots+t_{p} \right).\]9 \[M(a,b,t_{1})*M(c,b,t_{2})\leq\min\left\{M(a,c,t_{1}+t_{2}),M(c,a,t_{1}+t_{2}) \right\},\]0 2. _If_ \(p\) _is odd, then:_ \[M(a,b,t_{1})*M(c,b,t_{2})\leq\min\left\{M(a,c,t_{1}+t_{2}),M(c,a,t_{1}+t_{2}) \right\},\]1 \[M(a,b,t_{1})*M(c,b,t_{2})\leq\min\left\{M(a,c,t_{1}+t_{2}),M(c,a,t_{1}+t_{2}) \right\},\]2 _and:_ \[M(a,b,t_{1})*M(c,b,t_{2})\leq\min\left\{M(a,c,t_{1}+t_{2}),M(c,a,t_{1}+t_{2}) \right\},\]3 \[M(a,b,t_{1})*M(c,b,t_{2})\leq\min\left\{M(a,c,t_{1}+t_{2}),M(c,a,t_{1}+t_{2}) \right\},\]4 Let \((X,M)\) be a fuzzy semi-metric space. * We say that \(M\) is nondecreasing if and only if, given any fixed \(x,y\in X\), \(M(x,y,t_{1})\geq M(x,y,t_{2})\) for \(t_{1}>t_{2}>0\). * We say that \(M\) is increasing if and only if, given any fixed \(x,y\in X\), \(M(x,y,t_{1})>M(x,y,t_{2})\) for \(t_{1}>t_{2}>0\). * We say that \(M\) is symmetrically nondecreasing if and only if, given any fixed \(x,y\in X\), \(M(x,y,t_{1})\geq M(y,x,t_{2})\) for \(t_{1}>t_{2}>0\). * We say that \(M\) is symmetrically increasing if and only if, given any fixed \(x,y\in X\), \(M(x,y,t_{1})>M(y,x,t_{2})\) for \(t_{1}  
The following interesting results were modified from Wu [15] using the similar argument, which will be used in further discussion. **Proposition 1**. : _(Wu [15]) Let \((X,M)\) be a fuzzy semi-metric space. Then we have the following properties:_  
1. _If_ \(M\) _satisfies the_ \(\bowtie\)_-triangle inequality, then_ \(M\) _is nondecreasing. If_ \(M\) _satisfies the strict_ \(\bowtie\)_-triangle inequality, then_ \(M\) _is increasing._
2. _If_ \(M\) _satisfies the_ \(\triangleright\)_-triangle inequality or the_ \(\triangleleft\)_-triangle inequality, then_ \(M\) _is both nondecreasing and symmetrically nondecreasing. If_ \(M\) _satisfies the strict_ \(\triangleright\)_-triangle inequality or the strict_ \(\triangleleft\)_-triangle inequality, then_ \(M\) _is both increasing and symmetrically increasing._
3. _If_ \(M\) _satisfies the_ \(\diamond\)_-triangle inequality, then_ \(M\) _is symmetrically nondecreasing. If_ \(M\) _satisfies the strict_ \(\diamond\)_-triangle inequality, then_ \(M\) _is symmetrically increasing._