Fixed point theorems for weakly compatible mappings under implicit relations in quaternion valued $G$-metric spaces

1.   Introduction

The fixed point theory plays a major role in mathematics and applied sciences, such as mathematical models, optimization, and economic theories. This is the reason that why the study of metric fixed point theory has been researched extensively in the past decades. Since 1963, many mathematicians tried to generalize the usual notation of metric space and extend some known metric space theorems in more general setting (see ^{[2,3,5,8,9,10,11,12,13,14,15,21,22]}).

In 2005, Mustafa and Sims ^[20] introduced and study a new generalized metric spaces which is called $G$-metric spaces. They found a new fixed point of various mappings in new structure of these spaces. The $G$-metric spaces is defined by the following:

Definition 1 ^[20] Let $X$ be a nonempty set and a function $G$ be defined on the product set $X \times X \times X$ into the interval $[0, +\infty)$ satisfying the following properties:

($G_1$) $\:\: G(a, b, c) = 0 \:\:$ if and only if $\:\:a = b = c;$

($G_2$) $\:\: G(a, a, b) > 0$ for all $a, b \in X \:$ with $a \neq b;$

($G_3$) $\:\: G(a, a, b) \leq G(a, b, c)$ for all $\:\: a, b, c \in X \:$ with $\: c \neq b;$

($G_4$) $\:\: G(a, b, c) = G(a, c, b) = G(b, a, c) = ...\qquad$ (symmetry);

($G_5$) $\:\: G(a, b, c) \leq G(a, d, d) + G(d, b, c)$ for all $\:\: a, b, c, d \in X$ (rectangle inequality).

The function $\: G \:$ is called $G-$metric on $X$ and $\: (X, G) \:$ is called $\: G-$metric space.

In 1843, Irish mathematician, Hamilton ^[18] gave the definition of quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors. The study of quaternion has a lot set of applications such as applying to mechanics in three-dimensional space and practical uses in applied mathematics in particular for calculations involving three-dimensional rotations. There are many features of quaternions are that number system that extends the complex numbers and multiplication of two quaternions is noncommutative.

We denote $\:\textbf{H}\:$ for the skew field of quaternion and $\: q \in \textbf{H}\:$ has the form $\: p = a + bi + cj + dk\:$ where $i^{2} = j^{2} = k^{2} = ijk = -1, \: ij = -ji = k, \: kj = -jk = -i, \: ki = -ik = j$ and the modulus of $p$, $\:\left| p \right| = \sqrt{a^{2} + b^{2} + c^{2} + d^{2}} \:$ where $\: a, b, c \:$ and $\: d \:$ are real numbers, and $\: i, j \:$ and $\: k \:$ are the fundamental quaternion units. Thus a quaternion $p$ may be viewed as a four-dimensional vector $(a, b, c, d)$. By simple treating, quaternion can be written as simply quadruples of real numbers $\: [a, b, c, d], \:$ with addition and multiplication operations that are suitably defined. The components group into the imaginary part $\: (b, c, d) \:$, which we consider this part as a vector and the purely real part $a$ which is called a scalar. Sometimes, we write a quaternion as $\: [V, a] \:$ with $\: V = (b, c, d)$.

$[V, a] = [(b, c, d), a] = [a, b, c, d] = a + bi + cj + dk$ for all $\:\:a, b, c, d \in \mathbb{R}.$

For more properties of quaternion analysis, see ^[7,16,17] and and the references therein.

In order to prove our results, we present some necessary basic notions and concepts in the following.

Let $\textbf{H}$ be the set of quaternion and $p_1, p_2 \in \textbf{H}$. Define a partial order $\precsim$ on $\textbf{H}$ as follows:

$p_{1} \precsim p_{2} \:\:$ iff $\:\: \mathrm{Re}(p_{1}) \leq \mathrm{Re}(p_{2})\:$ and $\: \rm{Im}_{s}(p_{1})\leq \rm{Im}_{s}(p_{2}), \: p_{1}, p_{2} \in \textbf{H}, \: s = i, j, k \:$ where $\rm{Im}_{i} = b, \: \rm{Im}_{j} = c$ and $\rm{Im}_{k} = d.$

(Q$_{1}$) $\:\: \mathrm{Re}(p_{1}) = \mathrm{Re}(p_{2}) \:$ and $\: \rm{Im}_{s_1}(p_{1}) = \rm{Im}_{s_2}pz_{2})\:$ where $s_{1} = j, k, \: Im_{i}(p_{1}) < Im_{i}(p_{2});$

(Q$_{2}$) $\:\: Re(p_{1}) = R(p_{2}), Im_{s_2}(p_{1}) = Im_{s_2}(p_{2})\:$ where $\:s_{2} = i, k, \: Im_{j}(p_{1}) < Im_{j}(p_{2});$

(Q$_{3}$) $\:\: Re(p_{1}) = Re(p_{2}), Im_{s_3}(p_{1}) = Im_{s_3}(p_{2})\:$ where $\:s_{3} = i, j, \: Im_{k}(p_{1}) < Im_{k}(p_{2});$

(Q$_{4}$) $\:\: Re(p_{1}) = R(p_{2}), \: Im_{s_1}(p_{1}) = Im_{s_1}(p_{2}), \: Im_{i}(p_{1}) = Im_{i}(p_{2});$

(Q$_{5}$) $\:\: Re(p_{1}) = R(p_{2}), \: Im_{s_2}(p_{1}) = Im_{s_2}(p_{2}), \: Im_{j}(p_{1}) = Im_{j}(p_{2});$

(Q$_{6}$) $\:\: Re(p_{1}) = R(p_{2}), \: Im_{s_3}(p_{1}) = Im_{s_3}(p_{2}), \: Im_{k}(p_{1}) = Im_{k}(p_{2});$

(Q$_{7}$) $\:\: Re(p_{1}) = R(p_{2}), \: Im_{s}(p_{1}) < Im_{s}(p_{2});$

(Q$_{8}$) $\:\: Re(p_{1}) < R(p_{2}), \: Im_{s}(p_{1}) = Im_{s}(p_{2});$

(Q$_{9}$) $\:\: Re(p_{1}) < R(p_{2}), Im_{s_1}(p_{1}) = Im_{s_1}(p_{2}), \:$ where $\: s_{1} = j, k; \: Im_{i}(p_{1}) < Im_{i}(p_{2});$

(Q$_{10}$) $\:\: Re(p_{1}) < R(p_{2}), \: Im_{s_2}(p_{1}) = Im_{s_2}(p_{2}), \: Im_{j}(p_{1}) < Im_{j}(p_{2});$

(Q$_{11}$) $\:\: Re(p_{1}) < R(p_{2}), \: Im_{s_3}(p_{1}) = Im_{s_3}(p_{2}), \: Im_{k}(p_{1}) < Im_{k}(p_{2});$

(Q$_{12}$) $\:\: Re(p_{1}) < R(p_{2}), \: Im_{s_1}(p_{1}) < Im_{s_1}(p_{2}), \: Im_{i}(p_{1}) = Im_{i}(p_{2});$

(Q$_{13}$) $\:\: Re(p_{1}) < R(p_{2}), \: Im_{s_2}(p_{1}) < Im_{s_2}(p_{2}), \: Im_{j}(p_{1}) = Im_{j}(p_{2});$

(Q$_{14}$) $\:\: Re(p_{1}) < R(p_{2}), \: Im_{s_3}(p_{1}) < Im_{s_3}(p_{2}), \: Im_{k}(p_{1}) = Im_{k}(p_{2});$

(Q$_{15}$) $\:\: Re(p_{1}) < R(p_{2}), \: Im_{s}(p_{1}) < Im_{s}(p_{2});$

(Q$_{16}$) $\:\: Re(p_{1}) = R(p_{2}), \: Im_{s}(p_{1}) = Im_{s}(p_{2}).$

Particularly, we will write $\: p_{1} \precnsim p_{2} \:$ if $\: p_{1} \neq p_{2} \:$ and one from (Q$_{1}$) to (Q$_{16}$) is satisfied and we will write $\: p_{1} \prec p_{2} \:$ if only (Q$_{14}$) is satisfied. It should be noted that

In 2014, Ahmed et al. ^[4] introduced the concept of quaternion metric spaces and established some fixed point theorems in quaternion setting. For presentation the common fixed point of four self-maps which satisfy a general contraction condition was shown in normal cone metric spaces. The quaternion valued metric function is defined as follow.

Definition 2 ^[4] Let $X$ be a nonempty set and $g_{H} : X \times X \longrightarrow \textbf{H}$ be a function satisfying the following properties:

($H_1$) $g_{\textbf{H}}(a, b)\geq 0$ for all $\:\: a, b \in X$;

($H_2$) $g_{\textbf{H}}(a, b) = 0 \quad$ if and only if $\quad a = b$;

($H_3$) $g_{\textbf{H}}(a, b) = g_{\textbf{H}}(a, b) \qquad$ (symmetry);

($H_4$) $g_{\textbf{H}}(a, b)\leq g_{\textbf{H}}(a, c) + g_{\textbf{H}}(c, b)$ for all $\:\: a, b, c \in X$ (triangle inequality).

The function $\:\: g_{\textbf{H}} \:\:$ is called quaternion valued metric on $\: X \:$ and $\: (X, g_{\textbf{H}}) \:$ is called quaternion valued metric space.

Motivated by Ahmed et al.s work in ^[4], Adewale et al. ^[1] introduced the following concept of a quaternion valued $G$-metric spaces and gave some examples of this spaces.

Definition 3 ^[1] Let $\: X \:$ be a nonempty set and let $\: G_{\textbf {H}} : X \times X \times X \longrightarrow \textbf{H} \:$ be a function satisfying the following conditions

($QG_1$) $\:\: G_{\textbf{H}}(a, b, c) = 0_{\textbf{H}} \:$ if $\: a = b = c$;

($QG_2$) $\:\: 0_{\textbf{H}} \prec G_{\textbf{H}}(a, a, b)$ for all $\:\: a, b \in X \:$ with $\: a \neq b$;

($QG_3$) $\:\: G_{\textbf{H}}(a, a, b) \preccurlyeq G_{\textbf{H}}(a, b, c)$ for all $\:\:a, b, c \in X \:$ with $\:b \neq c$;

($QG_4$) $\:\: G_{\textbf{H}}(a, b, c) = G_{\textbf{H}}(a, c, b) = G_{\textbf{H}}(b, c, a) = ... \qquad$ (symmetry);

($QG_5$) $\:\: G_{\textbf{H}}(a, b, c) \preccurlyeq G_{\textbf{H}}(a, d, d) + G_{\textbf{H}}(d, b, c)$ for all $\:\: a, b, c, d \in X \:$ (rectangle inequality).

Therefore, the function $\: G_{\textbf{H}} \:$ is called quaternion valued $\: G_{\textbf{H}}$-metric on $X$ and the pair (X, $G_{\textbf{H}}$) is called quaternion valued $G-$metric space.

We found some gap in one of examples(Example 2, ^[1]), the domain of quaternion valued $\: G_{\textbf{H}}$-metric is not no the product space $X \times X \times X$. So, we now give a new example of $\: G_{\textbf{H}}$-metric as follow:

Example 4 Let $X = \{ \frac{1}{n}:\: n \in \mathbb{N} \}\:$ with

for all $a, b, c \in X.\:\:\: G_{\textbf{H}}:X \times X \times X \longrightarrow \textbf{H} \:$ is defined by

where $\: \triangle \: G_{\textbf{H}} = \sum_{i, j \in \{1, 2, 3\}} \:\:(\left| a_{i}- a_{j}\right|+\left| b_{i}- b_{j}\right|+\left| c_{i}- c_{j}\right|) \:$ and

We see that $\: G_{\textbf{H}} \:$ is quaternion valued $G-$metric on $\: X \:$ but not $G-$metric on $\: X.$

We next recall some definition and basic results that will be used in our subsequent analysis.

Proposition 5 ^[19] Let $\: (X, G_{\textbf{H}}\:)$ be quaternion valued $G-$metric space. Then for all $\: a, b, c, d \in X \:$ the following properties hold:

(1) $\:\: G_{\textbf{H}}(a, b, c) = 0$ implies $\:\:a = b = c$;

(2) $\:\: G_{\textbf{H}}(a, b, c) \preccurlyeq G_{\textbf{H}}(a, a, b) + G_{\textbf{H}}(a, a, c)$;

(3) $\:\: G_{\textbf{H}}(a, b, b) \preccurlyeq 2 \:\: G_{\textbf{H}}(b, a, a)$;

(4) $\:\: G_{\textbf{H}}(a, b, c) \preccurlyeq G_{\textbf{H}}(a, d, c) + G_{\textbf{H}}(d, b, c)$;

(5) $\:\: G_{\textbf{H}}(a, b, c) \preccurlyeq \frac{2}{3}\: (G_{\textbf{H}}(a, b, d) + G_{\textbf{H}}(a, d, c) + G_{\textbf{H}}(d, b, c))$;

(6) $\:\: G_{\textbf{H}}(a, b, c) \preccurlyeq 2 \: (G_{\textbf{H}}(a, a, d) + G_{\textbf{H}}(b, b, d) + G_{\textbf{H}}(c, c, d))$.

Definition 6 ^[6] Let $\: K, L, M \:$ and $\: N \:$ be four self-mappings of a nonempty set $\: X \:$. Then

(i) a point $\: u \in X \:$ is said to be a fixed point of $\: K \:$ if $\: Ku = u$,

(ii) a point $\: u \in X \:$ is said to be a common fixed point of $K$ and $L \:$ if $\: Ku = Lu = u$,

(iii) a point $\: u \in X \:$ is said to be a coincidence point of $\: K \:$ and $\: N$ if $\: Ku = Nu$ and a point $\: t \in X \:$ such that $\: t = Ku = Nu \:$ is called a point of coincidence of $\: K \:$ and $\: N$,

(iv) a point $\: t \in X \:$ is said to be a common point of coincidence of the pairs $\: (K, N) \:$ and $(L, M)$ if there exist $u, v \in X \:$ such that $Ku = Nu = t$ and $Lv = Mv = t$.

In this paper, we proved existence theorems of coincidence of the pairs $(K, N)$ and $(L, M)$ under some suitable conditions. Moreover, the weakly compatible condition of the pairs $(K, N)$ and $(L, M)$ was added for finding a common fixed point of mappings $K, L, M$ and $N$. Finally, we presented examples and conditions which satisfy our main theorems.

2.   Results

We begin with the following definition:

An implicit relation. Let $\digamma$ be the set of all complex valued lower semi-continuous functions $\: F :\textbf{H}^{6}\rightarrow \textbf{H}\:$ satisfying the following conditions: for $u, u^{'}, v \in \textbf{H}$

$(F_{1})$ $F$ is non-increasing in the $5^{th}$ and $6^{th}$ variable,

$(F_{2})$ for $u, v\succsim 0_{\textbf{H}},$ there exists $q\in \lbrack 0, 1)$ such that $\left\vert u\right\vert \leq q\left\vert v\right\vert$ if $F (u, v, u, v, 0_{\textbf{H}}, u+v)\precsim 0_{\textbf{H}}$,

$(F_{3})$ for $u, u^{'} \succ 0_{\textbf{H}},$ there exists $q \in \lbrack 0, 1)$ such that $\left\vert u\right\vert \leq q\left\vert u^{'}\right\vert$ if $F (u, u, 0_{\textbf{H}}, 0_{\textbf{H}}, u^{'}, u)\precsim 0_{\textbf{H}}.$

Now, we present our first main theorems in complete quaternion valued $G-$metric space.

Theorem 7 Let $K, L, M$ and $N$ be four self-mappings on a complete quaternion valued $G-$metric space $(X, {G_{\textbf{H}}})$ such that $K(X)\subseteq M(X)$ and $L(X)\subseteq N(X).$ Assume that there exists $\phi_1, \phi_2\in \digamma$ such that for all $a, b\in X,$ $a\neq b,$

If $M(X)\cup N(X)$ is complete subspace of $X$, then the pairs $(K, N)$ and $(L, M)$ have a unique common point of coincidence. Moreover, if the pairs $(K, N)$ and $(L, M)$ are weakly compatible, then the four mappings have a unique common fixed point.

Proof. Let $x_0$ be arbitrary point in $X$. Since $K(X)\subseteq M(X)$ and $L(X)\subseteq N(X)$, then we can define the sequence $\{{a_n}\}$ in $X$ such that,

Since $\{{b_{n}}\} \subseteq M(X)\cup N(X).$ We then show that $\{{b_{n}}\}$ is a Cauchy sequence. By putting $a = a_{2n+1}\:\: \text{and}\:\: b = a_{2n}$ in ($\phi_1$), we have

This implies that

This means that

From ($F_1$) and ($QG_5$), we get

From $(F_2)$, we obtain

By a similar way, by putting $\: a = a_{2n+2} \:$ and $\: b = a_{2n+1} \:$ in ($\phi_2$), we have successively

that is,

This tends to

From ($F_1$) and ($QG_5$), we obtain

From ($F_2$), we have

Consequently,

Also, for all $n > m$, we get

This means that $\left\lbrace b_n \right\rbrace$ is a Cauchy sequence in $X$. Since $(X, {G_{\textbf{H}}})$ is complete, then there exists $\: u \in X \:$ such that $\: b_{n}\longrightarrow u \:$ as $\: n\longrightarrow +\infty$. Then from Eq (2), we obtain

Since$\:$ $K(X)\subseteq M(X), \:$ if $u\in M(X),$ then there exists $\: \: v \in X \: \:$ such that

We will show that $\:Lv = Mv. \:$ By putting $a = v \:\:\text{and}\:\: b = a_{2n}$ in ($\phi_1$), we have

Putting $n \longrightarrow +\infty$ and using Eqs (3) and (4), we have

This tends to

From ($F_2$), we obtain ${G_{\textbf{H}}}(Lv, Lv, u) = 0_{\textbf{H}} \:$ which tends to $\: Lv = u.$ Hence

Then, $u$ is a point of coincidence of the pair $(L, M).$

Since $L(X) \subseteq N(X),$ there exists $\: w \in X \:$ such that

We will show that $\:Kw = Nw. \:$ By putting $a = w \:\:\text{and}\:\: b = a_{2n+1}$ in ($\phi_2$), we have

Taking $n \longrightarrow +\infty$ and using Eqs (3) and (6), we get

From ($F_2$), we get ${G_{\textbf{H}}}(Kw, Kw, u) = 0_{\textbf{H}} \:$ which implies $\: Kw = u.$ Hence

Then, $u$ is a point of coincidence of the pair $(K, N).$

Hence, $u \in X$ is a common point of coincidence for the four mappings.

To show the uniqueness of a point of coincidence, Let $\: u^* \neq u \:$ be another point of coincidence of the four mappings. Then, there exists $v^{*}, w^{*}$ such that $\: Lv^{*} = Mv^{*} = u^{*} \:$ and $\: Kw^{*} = Nw^{*} = u^{*}.$ Taking $a = v^{*} \:\:\text{and}\:\: b = w$ in $(\phi_1)$, one can write

This tends to

that is,

From ($F_3$), we get

Similarly, Taking $a = v \:\:\text{and}\:\: b = w^{*}$ in $(\phi_2)$, we obtain

From Eqs (2.8) and (2.9), we get

which implies that $\left| {G_{\textbf{H}}}(u^{*}, u^{*}, u) \right| = 0,$ i.e, $u^{*} = u.$ Consequently, the pairs $(K, N)$ and $(L, M)$ have a unique common point of coincidence.

Using Eqs (2.5) and (2.7) and weak compatibility of the pairs $(K, N)$ and $(L, M)$, we obtain that

Then,

This implies that $u$ is a point of coincidence of the pairs $(K, N)$ and $(L, M)$.

Now, we show that $u$ is a common fixed point of $\: K, L, M \:$ and $\:N.\:$ Taking $a = u$ and $b = v$ in ($\phi_1$), we have

This leads us to

that is,

From ($F_3$), we have

Similarly, taking $a = v \:\:\text{and}\:\: b = u$ in $(\phi_2)$, we have

From Eqs (2.12) and (2.13), we obtain

which implies that ${G_{\textbf{H}}}(Lu, Lu, u) = 0_{\textbf{H}},$ i.e., $Tu = u.$ Thus, $Lu = Mu = u.$ Similarly, we can prove $Ku = Nu = u.$ This implies that

i.e., $u$ is a common fixed point of $K, L, M$ and $N$.

The uniqueness of the common fixed point of $K, L, M$ and $N$ is more easy consequence of the uniqueness of the common point of coincidence of the pairs $(K, N)$ and $(L, M)$. Also, the proof is similar in case $u\in N(X)$. This completes the proof.

Remark 8 The conclusion of Theorem 7 remains true if the completeness of $\, M(X) \cup N(X)$ is replaced by the completeness of one of the subspaces $K(X), L(X), M(X)$ or $N(X).$

The following theorem is a new version of Theorem 7 under generalized contractive condition.

Theorem 9 Let $K, L, M$ and $N$ be four self-mappings on a complete quaternion valued $G$-metric space $(X, {G_{\textbf{H}}})$ such that $K(X)\subseteq M(X)$ and $L(X)\subseteq N(X).$ Assume that there exists $\phi_1, \phi_2 \in \digamma$ such that for all $a, b \in X,$ $a\neq b,$

If $M(X)\cup N(X)$ is complete subspace of $X$, then the pairs $(K, N)$ and $(L, M)$ have a unique common point of coincidence. Moreover, if the pairs $(K, N)$ and $(L, M)$ are weakly compatible, then the four mappings have a unique common fixed point.

Proof. Let $x_0$ be arbitrary points in $X$. Since $K(X)\subseteq M(X)$ and $L(X)\subseteq N(X)$, then we can define the sequence $\{{a_n}\}$ in $X$ as (2.2).

Since $\{{b_{n}}\} \subseteq M(X)\cup N(X).$ We then show that $\{{b_{n}}\}$ is a Cauchy sequence. Putting $a = a_{2n+1}\:\: \text{and}\:\: b = a_{2n}$ in ($\phi_1$), we obtain

This leads us to

which implies that

From ($F_1$), ($QG_4$) and ($QG_5$) we have

From $(F_2)$, we get

Similarly, taking $\: a = a_{2n+2} \:$ and $\: b = a_{2n+1}\:$ in ($\phi_2$), we get successively

This implies that

that is,

From ($F_1$), ($QG_4$) and ($QG_5$), one can write

From ($F_2$), we obtain

By a similar way, (step by step) of the proof of Theorem 9, one can complete the proof.

Example 10 Let $X = \{ \frac{1}{n}:\: n \in \mathbb{N} \}\:$ and $\: {G_{\textbf{H}}}:X \times X \times X \longrightarrow \textbf{H} \:$ as defined in Example 4. Let $\: K, L, M, N:X^{3} \longrightarrow X^{3} \:$ be defined by

for all $\: x = (x_{1}, x_{2}, x_{3})\in X^{3}.$

It is easy to see that $K(X)\subseteq M(X)$ and $L(X)\subseteq N(X).$ So there exist $\: \phi_1, \phi_2 :\textbf{H}^{6}\rightarrow \textbf{H}\:$ such that

for all $\: a, b, c, d, e, f \in \textbf{H}. \:$ We see that $\: \phi_1, \phi_2 \in \digamma \:$ and satisfy the following conditions (2.1) and (2.14). Moreover, We see that $\: (0, 0, 0) \:$ is unique common coincidence point of $\: (K, N) \:$ and $\: (L, M), \:$ and also unique common fixed point of mappings $\: K, L, M \:$ and $\: N. \:$ So, Theorem 7 and Theorem 9 are supported by this example.

3.   Conclusions

In this work, we prove existence theorems of unique common points for the pairs (K; N) and (L; M), and unique common fixed points for four mappings K; N; L; M are presented in Theorems 7 and 9. The Example 10 is shown for supporting our main theorems (Theorems 7 and 9).

Acknowledgments

W. Cholamjaik would like to thank University of Phayao, Phayao, Thailand and Thailand Science Research and Innovation under the project IRN62W0007.

Conflict of interest

The authors declare no conflict of interest.