Common Fixed-Point Theorems for Families of Compatible Mappings in Neutrosophic Metric Spaces
Abstract
:1. Introduction
2. Preliminaries
- (T1)
- is associative and commutative;
- (T2)
- is continuous;
- (T3)
- (T4)
- whenever and and
- (T5)
- (f1)
- ;
- (f2)
- for all , iff ;
- (f3)
- (f4)
- for all
- (f5)
- is left continuous and
- (a)
- (b)
- (c)
- (d)
- (e)
- (f)
- is left continuous;
- (g)
- (h)
- (i)
- (j)
- (k)
- (l)
- is right continuous;
- (m)
- 1.
- 2.
- 3.
- Either is continuous.
- 4.
- is compatible of type and is semi-compatible.
- 5.
- There exists such that for every
- (NMS1)
- (NMS2)
- (NMS3)
- (NMS4)
- (NMS5)
- (NMS6)
- is left continuous;
- (NMS7)
- (NMS8)
- (NMS9)
- (NMS10)
- (NMS11)
- (NMS12)
- is right continuous;
- (NMS13)
- (NMS14)
- (NMS15)
- (NMS16)
- (NMS17)
- (NMS18)
- is right continuous;
- (NMS19)
- (a)
- A sequence in is said to a Cauchy sequence if for each and ,
- (b)
- A NMS is only called complete if every Cauchy sequence is convergent.
3. Main Results
- (1)
- (2)
- (3)
- either or is continuous;
- (4)
- is compatible, and is weakly compatible;
- (5)
- such that
- (a)
- Putting and with in condition (5), we have
- (b)
- If and . With in condition (5), we have
- (c)
- If , and with in condition (5). Using the conditions in condition (2), we have
- (d)
- As there is such that If and with the in condition (5), we have
- (e)
- If and are with in condition (5), we have
- (f)
- If and are With in condition (5), we have
- (g)
- If and are with in condition (5), we have
- (h)
- As there is such that . If and are with in condition (5), we have