Study of multivalued fixed point problems for generalized contractions in double controlled dislocated quasi metric type spaces

1.   Introduction and preliminaries

Fixed point theory is an important branch of functional analysis and its applications are used in various fields of pure and applied mathematics. Nadler ^[18] was the first author who showed the contraction principle for multivalued mappings in a complete metric space. This proof basically uses the concept of Hausdorff's distance between sets. Nadler's result proved as a source of inspiration and a large number of researchers have started their research works in this field. Multivalued mappings are generalizitions of single valued mappings. Fixed point results for multivalued mappings have a lot of applications in engineering, economics, Nash equilibria and game theory ^[7,10,11,16]. Due to its important applications in various subjects, many authors have showed interesting results for multivalued mappings, which can be seen in ^{[8,26,27,31,33]}.

By removing one and a half restriction from out of three restriction of a metric space, we get dislocated quasi metric space ^[33]. Complete dislocated quasi metric space is a generalization of 0-complete and complete quasi-partial metric space ^[3,12,15]. Dislocated quasi metric also generalizes dislocated metric and partial metric. Fixed point results established by various researchers in dislocated quasi metric space can be seen in ^[4,6,30,37].

For the solution of matrix equations, Ran and Reurings ^[21] showed a significant result with an order. Lateral, Nieto et al. ^[20] extended the result in ^[21] governed a solution for ODE with periodic boundary conditions for ordered mappings. In ^[2] Abdeljawad et al., proved a significant result for best proximity points for cyclical contraction mappings. Altun et al. ^[5] renovated the approach to common fixed point of mappings, satisfying a generalized contraction with new order condition in a complete ordered metric space. For more results with order see ^[9,13,19].

On the other hand Kamran et al. ^[14] introduced a new concept of generalized b-metric spaces, named as extended b-metric spaces see also ^[29]. They replaced the parameter $b\geq 1$ in the triangle inequality by the control function $\theta :X\times X\rightarrow \lbrack 1, \infty).$ Recently, Mlaiki et al. ^[17] conceptualized the triangle inequality in b-metric spaces using a different styled controlled function and introduced controlled metric type spaces. After this, Abdeljawad et al. ^[1] auditioned the concept of controlled metric type spaces by introducing two control functions $\alpha (w, g)$ and $\mu (w, g)$ and set up double controlled metric type spaces. Recently Shoaib et al. ^[36] introduced the notion of double controlled metric type spaces which is a generalization of ^[1] and proved fixed point results for multivalued mappings. Furthermore, some recent useful results on this setting can be seen in ^[34,35]. The study recollects rudimentary concepts worth important for it to employ the definition in order to prove upcoming some new generalized results.

Definition 1.1. ^[33] Let $\vartheta$ is a nonempty set. Then, $\xi _{q}:\vartheta \times \vartheta \rightarrow \lbrack 0, \infty)$ is called a dislocated quasi metric (or simply $\xi _{q}$-metric) if the following conditions hold for any $l, g, z\in \vartheta :$

(i) If $\xi _{q}(l, g) = \xi _{q}(g, l) = 0,$ then $l = g;$

(ii) $\xi _{q}(l, g)\leq \xi _{q}(l, z)+\xi _{q}(z, g).$

The pair $(\vartheta, \xi _{q})$ is called a dislocated quasi metric space.

It is clear that if $\xi _{q}(l, g) = \xi _{q}(g, l) = 0$, then from (i), $l = g$. But if $l = g$, then $\xi _{q}(l, g)$ may not be $0.$ It is observed that if $\xi _{q}(l, g) = \xi _{q}(g, l)$ for all $l, g\in \vartheta,$ then $(\vartheta, \xi _{q})$ becomes a dislocated metric space (metric-like space).

Example 1.2. ^[33] Let $\vartheta =$ $R^{+}\cup \{0\}$ and $\xi _{q}(l, g) = l+\max \{l, g\}$ for any $l, g\in \vartheta.$ Then, $(\vartheta, \xi _{q})$ is a dislocated quasi metric space.

Definition 1.3. ^[1] Given non-comparable functions $\alpha, \mu :\vartheta \times \vartheta \rightarrow \lbrack 1, \infty)$. If $q:\vartheta \times \vartheta \rightarrow \lbrack 0, \infty)$ satisfies:

(q1) $q(l, g) = 0$ if and only if $l = g$,

(q2) $q(l, g) = q(g, l),$

(q3) $q(l, g)\leq \alpha (l, z)q(l, z)+\mu (z, g)q(z, g),$ for all $l, z, g\in \vartheta$. Then, $q$ is called double controlled metric type with the functions $\alpha$, $\mu$ and the pair $(\vartheta, q)$ is called double controlled metric type space with the functions $\alpha, \mu$.

Theorem 1.4. ^[1] Let $(\vartheta, q)$ be a complete double controlled metric type space with the functions $\alpha, \mu$ $:\vartheta \times \vartheta$ $\rightarrow \lbrack 1, \infty)$ and let $T:\vartheta \rightarrow \vartheta$ be a given mapping. Suppose that the following conditions are satisfied:

There exists $t\in (0, 1)$ such that

For $\upsilon _{0}\in \vartheta,$ choose $\upsilon _{\acute{n}} = T^{\acute{n }}\upsilon _{0}.$ Assume that

In addition, for every $\upsilon \in \vartheta$, we have

Then $T$ has a unique fixed point $\upsilon ^{\ast }\in \vartheta$.

Definition 1.5. ^[36] Given non-comparable functions $\alpha, \mu :$$\vartheta$$\times$$\vartheta$$\rightarrow \lbrack 1, \infty)$. If $\xi _{q}:$$\vartheta$$\times$$\vartheta$$\rightarrow \lbrack 0, \infty)$ satisfies:

($\xi _{q}$1) $\xi _{q}(l, g) = \xi _{q}(g, l) = 0,$ then $l = g$,

($\xi _{q}$2) $\xi _{q}(l, g)\leq \alpha (l, z)q(l, z)+\mu (z, g)q(z, g),$

for all $l, z, g\in$$\vartheta$. Then, $\xi _{q}$ is called a double controlled dislocated quasi metric type space with the functions $\alpha$ and $\mu$ and $($$\vartheta$$, \xi _{q})$ is called a double controlled dislocated quasi metric type space. If $\mu (z, g) = \alpha (z, g)$ then $($$\vartheta$$, \xi _{q})$ is called a controlled quasi metric type space.

Remark 1.6. Any dislocated quasi metric space or double controlled metric type space is double controlled dislocated quasi metric type space but, the converse is not true in general. Also, a controlled dislocated quasi metric type space is double controlled quasi metric type space. The converse is not true in general (see Examples (1.7 and 2.4)).

Example 1.7. Let $\vartheta = \left\{ 0, 1, 2, 3\right\}.$ Define $\xi _{q}:\vartheta \times \vartheta \rightarrow \lbrack 0$, $\infty)$ by $\xi _{q}(0, 1) = 0,$ $\xi _{q}(0, 2) = 1,$ $\xi _{q}(0, 3) = \frac{1}{4}, \ \xi _{q}(1, 0) = \frac{1}{2},$ $\xi _{q}\left(1, 2\right) = 2,$ $\xi _{q}(1, 3) = \frac{1}{3}, \ \xi _{q}(2, 0) = \frac{1}{2},$ $\xi _{q}(2, 1) = 1,$ $\xi _{q}(2, 3) = \frac{1}{3}, \ \xi _{q}(3, 0) = \frac{3}{2}, \ \xi _{q}(3, 1) = 2, \ \xi _{q}(3, 2) = \frac{1}{4}, \ \xi _{q}(0, 0) = \frac{1}{2},$ $\xi _{q}(1, 1) = 0,$ $\xi _{q}(2, 2) = 2,$ $\xi _{q}(3, 3) = 0.$ Define $\alpha, \mu :\vartheta \times \vartheta \rightarrow \lbrack 1$, $\infty)$ as $\alpha (0, 1) = \alpha \left(1, 2\right) = \alpha (2, 1) = \alpha (0, 2) = 1,$ $\alpha (2, 0) = \alpha (3, 2) = 2,$ $\alpha (3, 1) = \alpha (1, 0) = \alpha (3, 0) = \alpha (0, 3) = \frac{4}{3},$ $\alpha (1, 3) = \alpha (2, 3) = 3, \ \alpha (0, 0) = \alpha (1, 1) = \alpha (2, 2) = \alpha (3, 3) = 1,$

$\mu (1, 2) = \mu \left(2, 1\right) = \frac{3}{2},$ $\mu (2, 0) = 2,$ $\mu (3, 0) = \mu (0, 3) = \mu (1, 0) = \mu (0, 1) = \mu (1, 3) = \mu (3, 1) = \mu (0, 0) = \mu (1, 1) = \mu (2, 2) = \mu (3, 3) = 1,$ $\ \mu (3, 2) = 4, \ \mu (2, 3) = 1, \ \mu (0, 2) = 2.$ It is obvious that $\xi _{q}$ is double controlled dislocated quasi metric type for all $l, g, z\in X.$ It is clear that $\xi _{q}$ is not double controlled metric type space. Also, it is not controlled dislocated quasi metric type. Indeed,

Definition 1.8. ^[36] Let $(\vartheta, \xi _{q})$ be a double controlled dislocated quasi metric type space.

(i) A sequence $\{l_{n}\}$ in $(\vartheta, \xi _{q})$ is called left $K$ -Cauchy if for all $\varepsilon > 0$, there exists $n_{0}$ $\in \mathbb{N}$ such that $\xi _{q}(l_{m}, l_{n}) < \varepsilon$, $\forall$ $n > m\geq n_{0}.$

(ii) A sequence $\{l_{n}\}$ is double controlled dislocated quasi-converges to $l$ if $\lim\limits_{n\rightarrow \infty }\xi _{q}(l_{n}, l) = \lim\limits_{n\rightarrow \infty }\xi _{q}(l, l_{n}) = 0$ or for any $\varepsilon > 0$, there exists $n_{0}$ $\in \mathbb{N},$ such that for all $n > n_{0},$ $\xi _{q}(l, l_{n}) < \varepsilon$ and $\xi _{q}(l_{n}, l) < \varepsilon$. In this case $l$ is called a $\xi _{q}$-limit of $\{l_{n}\}.$

(iii) $(\vartheta, \xi _{q})$ is called left $K$-sequentially complete if every left $K$-Cauchy sequence in $\left(\vartheta, \xi _{q}\right)$ convergent to a point $l\in \vartheta$ such that $\xi _{q}(l, l) = 0$.

Definition 1.9. ^[34] Let $($$\vartheta$$, \xi _{q})$ be a double controlled dislocated quasi metric type space. Let $K$ be a nonempty subset of $\vartheta$ and let $l\in \vartheta.$ An element $g_{0}\in K$ is called a best approximation in $K$ if

If each $l\in \vartheta$ has at least one best approximation in $K,$ then $K$ is called a proximinal set. We denote the set of all proximinal subsets of $\vartheta$ by $P(\vartheta).$

Definition 1.10. ^[34] The function $H_{\xi _{q}}:P(\vartheta)\times P(\vartheta)\rightarrow \lbrack 0, \infty),$ defined by

is called double controlled dislocated quasi Hausdorff metric type on $P(\vartheta).$ Also $(P(\vartheta), H_{\xi _{q}})$ is known as double controlled dislocated quasi Hausdorff metric type space.

Lemma 1.11. ^[35] Let $(\vartheta, \xi _{q})$ be a double controlled dislocated quasi metric type space. Let $(P(\vartheta), H_{\xi _{q}})$ be a double controlled dislocated quasi Hausdorff metric type space on $P(\vartheta).$ Then, for all $A, B\in P(\vartheta)$ and for each $a\in A,$ there exists $b_{a}\in B,$ such that $H_{\xi _{q}}(A, B)\geq \xi _{q}(a, b_{a})$ and $H_{\xi _{q}}(B, A)\geq \xi _{q}(b_{a}, a)$.

Lemma 1.12. ^[35] Let $(\vartheta, \xi _{q})$ be a double controlled dislocated quasi metric type space. For $A, B\in P(\vartheta)$ and $a, b, z\in \vartheta,$ then

where

2.   Results and discussion

Let $($$\vartheta$$, \xi _{q})$ be a double controlled dislocated quasi metric type space$,$ $g_{0}\in$$\vartheta$ and $T:\vartheta \rightarrow P(\vartheta)$ be a multifunction on $\vartheta$. Let $g_{1}\in Tg_{0}$ be an element such that $\xi _{q}(g_{0}, {T}g_{0}) = \xi _{q}(g_{0}, g_{1}),$ $\xi _{q}({T}g_{0}, g_{0}) = \xi _{q}(g_{1}, g_{0})$. Let $g_{2}\in Tg_{1}$ be such that $\xi _{q}(g_{1}, Tg_{1}) = \xi _{q}(g_{1}, g_{2}),$ $\xi _{q}(Tg_{1}, g_{1}) = \xi _{q}(g_{2}, g_{1}).$ Let $g_{3}\in {T}g_{2}$ be such that $\xi _{q}(g_{2}, {T}g_{2}) = \xi _{q}(g_{2}, g_{3})$ and so on. Thus, we construct a sequence $g_{n}\ $of points in $\vartheta$ such that $g_{2n+1}\in {T}g_{2n}$ and $g_{2n+2}\in Tg_{2n+1},$ with $\xi _{q}(g_{2n}, {T} g_{2n}) = \xi _{q}(g_{2n}, g_{2n+1}),$ $\xi _{q}({T}g_{2n}, g_{2n}) = \xi _{q}(g_{2n+1}, g_{2n}),$ and $\xi _{q}(g_{2n+1}, Tg_{2n+1}) = \xi _{q}(g_{2n+1}, g_{2n+2}),$ $\xi _{q}(Tg_{2n+1}, g_{2n+1}) = \xi _{q}(g_{2n+2}, g_{2n+1}),$ where $n = 0, 1, 2, \cdots.$ We denote this iterative sequence by $\{\vartheta T(g_{n})\}.$ We say that $\{\vartheta T(g_{n})\}$ is a sequence in $\vartheta$ generated by $g_{0}$ under double controlled dislocated quasi metric $\xi _{q}.$ If $\xi _{q}$ is dislocated quasi b-metric, then we say that $\{\vartheta T(g_{n})\}$ is a sequence in $\vartheta$ generated by $g_{0}$ under dislocated quasi b-metric $\xi _{q}$. We can define $\{\vartheta T(g_{n})\}$ in other metrics in a similar way. Let $M\subseteq$ $\vartheta$, define $\xi ^{\ast }\left(w, M\right) = \inf \left\{ \xi \left(w, a\right), \text{ }a\in M\right\}$ and $\xi ^{\ast }\left(M, g\right)$ $= \inf \left\{ \xi \left(b, g\right), \text{ }b\in M\right\}.$ Let us introduce the following definition:

Definition 2.1. Let $\vartheta$ be a nonempty set and $\xi :$$\vartheta$$\times$$\vartheta$$\rightarrow \lbrack 0, +\infty)$ be a mapping such that $\xi \left(w, g\right) \geq 1$ and $\xi \left(g, w\right) \geq 1$ imply $w = g$. Let $\Omega, T:\vartheta \rightarrow P\left(\vartheta \right)$ be a multivalued mappings and $\{\vartheta T(g_{n})\}$ is a sequence in $\vartheta$ generated by $g_{0}$ under double controlled dislocated quasi metric $\xi _{q}$, then $\Omega, T$ are said to be $\xi ^{\ast }$-$\xi _{q}$ multivalued mappings, if for each $w\in \{\vartheta T(g_{n})\},$ then we have

where $\xi _{q}\left(w, Tw\right) = \xi _{q}\left(w, g\right)$ and $\xi _{q}\left(Tw, w\right) = \xi _{q}\left(g, w\right).$

Definition 2.2. Let $(\vartheta, \xi _{q})$ be a complete double controlled dislocated quasi metric type space and $\Omega, T$ be $\xi ^{\ast }$-$\xi _{q}$ multivalued mappings. Then the pair $\left(\Omega, T\right)$ is called $\xi ^{\ast }$ Kannan type double controlled contraction, if for every two consecutive points $w, g$ belonging to the range of an iterative sequence $\{\vartheta T(g_{n})\}$ with $\xi ^{\ast }\left(\Omega g, g\right) \geq 1,$ $\xi ^{\ast }\left(w, \Omega w\right) \geq 1$ or $\xi ^{\ast }\left(\Omega w, w\right) \geq 1,$ $\xi ^{\ast }\left(g, \Omega g\right) \geq 1$ and $\xi _{q}(w,$$g$$) > 0,$ we have

whenever, $t\in \lbrack 0, \frac{1}{2})$ and for $g_{0}\in \{\vartheta T(g_{n})\}$,

Theorem 2.3. Let $(\vartheta, \xi _{q})$ be a left $K$ -sequentially complete double controlled dislocated quasi metric type space. Let a pair $\left(\Omega, T\right)$ be a $\xi ^{\ast }$ Kannan type double controlled contraction$.$ Assume that:

(i) The set $G(\Omega) = \left\{ w:\xi ^{\ast }\left(w, \Omega w\right) \geq 1\right\}$ is closed and contained $g_{0}.$

(ii) For every $g\in \{\vartheta T(g_{n})\}$, we have

Then $\{\vartheta T(g_{n})\}\rightarrow u\in \vartheta$. Also, if (2.1) holds for each $w, g$ $\in \{u\},$ then $\Omega$ and $T$ have a common fixed point $u$ in $\vartheta$ and $\xi _{q}(u, u) = 0$.

Proof. Since $g_{0}$ is an arbitrary element of $G\left(\Omega \right)$, from condition (i) $\xi ^{\ast }\left(g_{0}, \Omega g_{0}\right) \geq 1.$ Let $\{\vartheta T(g_{n})\}$ be the iterative sequence in $\vartheta$ generated by a point $g_{0}\in \vartheta$.

Since $\xi ^{\ast }\left(g_{0}, \Omega g_{0}\right) \geq 1$, $\xi _{q}\left(g_{0}, {T}g_{0}\right) = \xi _{q}\left(g_{0}, g_{1}\right)$ and $\xi _{q}\left({T}g_{0}, g_{0}\right) = \xi _{q}\left(g_{1}, g_{0}\right)$. As $(\Omega, T)$ is $\xi ^{\ast }$ multivalued mapping, so $\xi ^{\ast }\left({ \Omega }g_{1}, g_{1}\right) \geq 1.$ Now, $\xi ^{\ast }\left(\Omega g_{1}, g_{1}\right) \geq 1,$ $\xi _{q}\left(g_{1}, Tg_{1}\right) = \xi _{q}\left(g_{1}, g_{2}\right)$ and $\xi _{q}\left(Tg_{1}, g_{1}\right) = \xi _{q}\left(g_{2}, g_{1}\right)$ imply that $\xi ^{\ast }\left(g_{2}, \Omega g_{2}\right) \geq 1.$ By induction we deduce that $\xi ^{\ast }\left(g_{2p}, \Omega g_{2p}\right) \geq 1$ and $\xi ^{\ast }\left(\Omega g_{2p+1}, g_{2p+1}\right) \geq 1$, for all $p = 0, 1, 2, \cdots$. Now, by Lemma 1.11, we have

Since $g_{2p}, g_{2p-1}\in \{\vartheta T(g_{n})\},$ $\xi ^{\ast }\left(g_{2p}, \Omega g_{2p}\right) \geq 1$ and $\xi ^{\ast }\left(\Omega g_{2p-1}, g_{2p-1}\right) \geq 1,$ by the condition (2.1), we get

Now, by Lemma 1.11, we have

Since $g_{2p-2}, g_{2p-1}\in \{\vartheta T(g_{n})\},$ $\xi ^{\ast }\left(g_{2p-2}, \Omega g_{2p-2}\right) \geq 1$ and $\xi ^{\ast }\left(\Omega g_{2p-1}, g_{2p-1}\right) \geq 1,$ by the condition (2.1), we have

Using (2.6) in (2.5), we have

Now, by (2.4) we have

Since $g_{2p-2}, g_{2p-1}\in \{\vartheta T(g_{n})\},$ $\xi ^{\ast }\left(\Omega g_{2p-1}, g_{2p-1}\right) \geq 1\ $and$\ \xi ^{\ast }\left(g_{2p-2}, \Omega g_{2p-2}\right) \geq 1,$ by the condition (2.1), we get

From (2.7) and (2.8), we have

Using (2.9) in (2.5), we have

Similarly, we can obtain

Continuing in this way, we get

Now$,$ we can write (2.10) as

Now, to prove that $\left\{ g_{n}\right\}$ is a Cauchy sequence, for all natural numbers $n < m,$ we have

We used $\alpha (w, g)\geq 1$. Let

Hence, we have

The ratio test together with (2.2) implies that the limit of the real number sequence $\left\{ \Omega _{n}\right\}$ exists, and so $\left\{ \Omega _{n}\right\}$ is left Cauchy. Indeed, the ratio test is applied to the term $a_{i} = \left(\prod\limits_{j = 0}^{i}\mu \left(g_{j}, g_{m}\right) \right) \alpha \left(g_{i}, g_{i+1}\right).$ Letting $n, m\rightarrow \infty$ in (2.12), we get

So the sequence $\left\{ \vartheta T(g_{n})\right\}$ is a left Cauchy. Since $(\vartheta, \xi _{q})$ is left $K$-sequentially double controlled complete dislocated quasi metric type space, $\left\{ \vartheta T(g_{n})\right\} \rightarrow u,$ that is,

Since $G(\Omega)$ is a closed set, $G(\Omega)$ is left $K$-sequentially complete. Since $\{g_{2p}\}$ is a subsequence of $\{\vartheta T(g_{n})\}$ contained in $G(\Omega),$ $\{g_{2p}\}\rightarrow u.$ Completeness of $G(\Omega)$ implies $u\in G(\Omega),$ that is,

Now,

This implies $\xi _{q}(u, u) = 0$ as $n\rightarrow \infty.$ Now, we show that $u$ is a common fixed point. We claim that $q_{b}\left(u, Tu\right) = 0$. On contrary suppose $\xi _{q}(u, {T}u) > 0.$ Now by Lemma 1.11, we have

Since, $\xi ^{\ast }\left(\Omega g_{2n}, g_{2n}\right) \geq 1$ and $\xi ^{\ast }\left(u, \Omega u\right) \geq 1,$ by (2.1), we get

Taking the lim as $n\rightarrow \infty$ on both sides of (2.16), we get

Now by Lemma 1.12, we have

Taking the lim as $n\rightarrow \infty$ and using (2.3) and (2.14), we get

By (2.3) and (2.17), we get

It is a contradiction. Therefore

Thus $u\in Tu.$ Now, suppose $\xi _{q}(Tu, u) > 0.$ By Lemma 1.11, we have

Since, $\xi ^{\ast }\left(\Omega g_{2n}, g_{2n}\right) \geq 1$ and $\xi ^{\ast }\left(u, \Omega u\right) \geq 1,$ by (2.1), we get

Taking the lim as $n\rightarrow \infty$ on both sides of the above inequality, we get

Now by Lemma 1.12, we have

Taking the lim as $n\rightarrow \infty$ and inequality (2.3) and (2.14), we get

It is a contradiction. Hence $u\in Tu.$ As $\xi ^{\ast }\left(u, \Omega u\right) \geq 1$ and $\xi _{q}\left(u, Tu\right) = \xi _{q}\left(Tu, u\right) = 0,$ Definition 2.1 implies

From (2.15) and (2.19) $\xi ^{\ast }\left(u, \Omega u\right) \geq 1,$ $\xi ^{\ast }\left(\Omega u, u\right) \geq 1.$ This implies $\xi \left(u, g\right) \geq 1,$ $\xi \left(g, u\right) \geq 1, \ $for all $g\in \Omega u.$ Thus $u = g.$ Hence, $u$ is a common fixed point for $\Omega$ and $T.$

Example 2.4. Let $\vartheta = \left[ 0, 4\right] \cap \mathbb{Q} ^{+}.$ Define the function$\ \xi _{q}:\vartheta \times \vartheta \rightarrow \lbrack 0, +\infty)$ by $\xi _{q}(w, g) = \left(w+2g\right) ^{2}$ if $w\neq g$ and $\xi _{q}(w, g) = 0,$ if $w = g.$ Then $(\vartheta, \xi _{q})$ is a complete double controlled dislocated quasi metric type space with

Define the mappings $\Omega, T:\vartheta \rightarrow P(\vartheta)$ as follows:

Let

Then $\xi _{q}$ is not a controlled dislocated quasi metric type space for the function $\alpha$. Indeed,

Now, $\xi _{q}\left(w_{0}, \Omega w_{0}\right) \ = \xi _{q}\left(1, \Omega 1\right) = \xi _{q}\left(1, \frac{1}{8}\right) = \left(1+\frac{2}{8}\right) ^{2} = \left(\frac{5}{4}\right) ^{2}.$ We define the sequence $\{\vartheta T(g_{n})\} = \{1, \frac{1}{8}, \frac{1}{64}, \frac{1}{512}, \frac{1}{4096}, \cdots \}$ in $\vartheta$ generated by $g_{0} = 1.$ Let $w_{0} = 1.$ Then we have

So (i) is satisfied.

Take $\frac{1}{64}\in \left\{ \vartheta T\left(g_{n}\right) \right\}.$ Then we have

Also,

Note that $\xi ^{\ast }\left(w, \Omega w\right) \geq 1,$ for all $w\in A$ implies $\xi ^{\ast }\left(\Omega g, g\right) \geq 1$, for all $g\in B.$ Also, $\xi ^{\ast }\left(\Omega w, w\right) \geq 1$, for all $w\in B$ implies $\xi ^{\ast }\left(g, \Omega g\right) \geq 1$, for all $g\in A.$ So the pair $(\Omega, T)$ is $\xi ^{\ast }$-$\xi _{q}$ multivalued mapping on $\left\{ \vartheta T\left(g_{n}\right) \right\}.$

Now, for all $w, g\in \vartheta \cap \left\{ \vartheta Tw_{n}\right\}$ with $\xi ^{\ast }\left(\Omega g, g\right) \geq 1,$ $\xi ^{\ast }\left(w, \Omega w\right) \geq 1$ and $t = \frac{2}{5},$ we have the following cases:

In the case: $w < g.$ Let $w = \frac{1}{64},$ $g = \frac{1}{8}.$ Then we have

Now, we have

In the case $w > g.$ Take $w = 1,$ $g = \frac{1}{8}.$ Then we have

Now,

In the case: $w = 0,$ $g > 0.$ Let $w = 0,$ $g = \frac{1}{8}.$ Then we have

Now,

In the case vi: Take $w = \frac{1}{8},$ and $g = 0.$ Then we have

Now,

And the case $w = 0$ and $g = 0$ is trivially true. So all cases are satisfied. Let $g_{0} = 1.$ Then we have $g_{1} = Tg_{0} = \frac{1}{8},$ $g_{2} = Tg_{1} = \frac{1 }{64}$, $g_{3} = Tg_{2} = \frac{1}{512}, ..._{\cdot }$

That is, the pair $\left(\Omega, T\right)$ is $\xi ^{\ast }$ Kannan type double controlled contraction. Finally, for every $g\in \{\vartheta T(g_{n})\}$, we have

Hence all the hypothesis of Theorem 2.3 are satisfied and $0$ is a common fixed point of $\Omega$ and $T$.

The next Theorem 2.6 is a special case of our main Theorem 2.3 if we use in Theorem 2.6 $\alpha (w, g) = \mu ((w, g) = b$ for all $w, g\in \vartheta$ then we get the result on dislocated quasi-b metric type space instead double controlled dislocated quasi type metric spaces.

Definition 2.5. Let $(\vartheta, \xi _{q})$ be a left $K$ -sequentially complete dislocated quasi b-metric type space and $\left(\Omega, T\right)$ be a $\xi ^{\ast }$-$\xi _{q}$ multivalued mapping under dislocated quasi $b$-metric $\xi _{q}$, with $b > 1$ then the pair $\left(\Omega, T\right)$ is called $\xi ^{\ast }$ Kannan type $b$-contraction, if for every two consecutive points $w, g$ belonging to the range of an iterative sequence $\{\vartheta T(g_{n})\}$ with $\xi ^{\ast }\left(\Omega g, g\right) \geq 1,$ $\xi ^{\ast }\left(w, \Omega w\right) \geq 1$ or $\xi ^{\ast }\left(\Omega w, w\right) \geq 1,$ $\xi ^{\ast }\left(g, \Omega g\right) \geq 1$ and $\xi _{q}(w,$$g$$) > 0,$ we have

whenever, $t\in \lbrack 0, \frac{1}{2})$ and $b < \frac{1-t}{t}.$

Theorem 2.6. Let $(\vartheta, \xi _{q})$ be a left $K$ -sequentially complete dislocated quasi b-metric space and a pair $\left(\Omega, T\right)$ be a $\xi ^{\ast }$ Kannan type b-contraction$.$ Assume that:

The set $G(\Omega) = \left\{ w:\xi ^{\ast }\left(w, \Omega w\right) \geq 1\right\}$ is closed and contained in $g_{0}.$ Then $\{\vartheta T(g_{n})\}\rightarrow u\in \vartheta$. Also, if (2.20) holds for each $w, g$ $\in \{u\},$ then $\Omega$ and $T$ have a common fixed point $u$ in $\vartheta$ and $\xi _{q}(u, u) = 0$.

Remark 2.7. In Example 2.4, $\xi _{q}(w, g) = \left(w+2g\right) ^{2}$ is dislocated quasi b-metric with $b\geq 2,$ but we can not apply Theorem 2.6 because the pair $\left(\Omega, T\right)$ is not $\xi ^{\ast }$ Kannan type b-contraction, indeed $b\nleq \frac{3}{2} = \frac{1-t}{t}.$

Definition 2.8. Let $(\vartheta, d)$ be a complete metric space and $\left(\Omega, T\right)$ be a $\xi ^{\ast }$- $d$ multivalued mapping under metric $d$, then the pair $\left(\Omega, T\right)$ is called $\xi ^{\ast }$ Kannan type contraction, if for every two consecutive points $w, g$ belonging to the range of an iterative sequence $\{\vartheta T(g_{n})\}$ with $\xi ^{\ast }\left(\Omega g, g\right) \geq 1,$ $\xi ^{\ast }\left(w, \Omega w\right) \geq 1$ and $d(w,$$g$$) > 0,$ we have

whenever, $t\in \lbrack 0, \frac{1}{2}).$

Theorem 2.9. Let $(\vartheta, d)$ be a complete metric space. Let a pair $\left(\Omega, T\right)$ be a $\xi ^{\ast }$ Kannan type contraction $.$ Assume that the set $G(\Omega) = \left\{ w:\xi ^{\ast }\left(w, \Omega w\right) \geq 1\right\}$ is closed and contained $g_{0}.$ Then $\{\vartheta T(g_{n})\}\rightarrow u\in \vartheta$. Also, if (2.21) holds for each $w, g$ $\in \{u\},$ then $\Omega$ and $T$ have a common fixed point $u$ in $\vartheta.$

Motivated by the result Samet et al. ^[28] we have obtained the upcoming results for order. Further results which generalized partial order can be seen in (^{[22,23,24,25]}). Recall that if ${X}$ is a nonempty set, $\preceq$ is a partial order on $X,$ then $\left(X, \preceq \right)$ is called non empty partially ordered set. Let $a\in X$ and $B\subseteq X$. We say that $a\preceq B$ whenever for all $b\in B,$ we have $a\preceq b.$

Definition 2.10. Let $\left(\vartheta, \preceq \right)$ is a non empty partially ordered set. Let $\Omega, T:\vartheta \rightarrow P\left(\vartheta \right)$ be the multivalued mappings and $\{\vartheta T(g_{n})\}$ is a sequence in $\vartheta$ generated by $g_{0}$ under metric $d$, then $\left(\Omega, T\right)$ is said to be $\preceq$-$d$ multivalued mapping, if $w\in \{\vartheta T(g_{n})\},$ we have

where $d\left(w, Tw\right) = d\left(w, g\right).$

Definition 2.11. Let $(\vartheta, \preceq, d)$ is an ordered complete metric space and $\left(\Omega, T\right)$ be a $\preceq$-$d$ multivalued mapping. Let $\alpha :\vartheta \times \vartheta \rightarrow \lbrack 0, \infty)$ be a function, then the pair $\left(\Omega, T\right)$ is called $\preceq$ Kannan type contraction, if for every two consecutive points $w, g$ belonging to the range of an iterative sequence $\{\vartheta T(g_{n})\}$ with $\Omega g\succeq g,$ $w\preceq \Omega w$ or $\Omega w\succeq w,$ $g\preceq \Omega g$ and $d(w,$$g$$) > 0,$ we have

whenever, $t\in \lbrack 0, \frac{1}{2}).$

Theorem 2.12. Let $(\vartheta, d)$ is a complete metric space. Let a pair $\left(\Omega, T\right)$ be a $\preceq$ Kannan type contraction $.$ Let $\alpha :\vartheta \times \vartheta \rightarrow \lbrack 0, \infty)$ be a function, and assume that the set $G(\Omega) = \left\{ w:w\preceq \Omega w\right\}$ is closed and contained $g_{0}.$ Then $\{\vartheta T(g_{n})\}\rightarrow u\in \vartheta$. Also, if (2.22) holds for each $w, g$ $\in \{u\},$ then $\Omega$ and $T$ have a common fixed point $u$ in $\vartheta$.

Proof. Let $\alpha :\vartheta \times \vartheta \rightarrow \lbrack 0, +\infty)$ be a mapping such that

As $(\Omega$ $T)$ is $\preceq$-$d$ multivalued mapping so $d(w, Tw) = d(w, g)$ then $w\preceq \Omega w$ implies $\Omega g\succeq g$ or $w\preceq b$ for all $b\in \Omega w$ implies $e\succeq g$ for all $e\in \Omega g$ or $\alpha (w, b) = 1$ $\forall$ $b\in \Omega w$ implies $\alpha (e, g) = 1$ $\forall$ $e\in \Omega g$ or inf$\left\{ \alpha (w, b):b\in \Omega w\right\} = 1$ implies inf$\left\{ \alpha (e, g):e\in \Omega g\right\} = 1$ or

Similarly case (b) $\Omega w\succeq w$ implies $g\preceq \Omega g$ gives $\alpha ^{\ast }(\Omega w, w)\geq 1$ implies $\alpha ^{\ast }(g, \Omega g)\geq 1.$ So $(\Omega$ $T)$ is a $\alpha ^{\ast }$-$d$ multivalued mapping. As $(\Omega$ $T)$ is $\preceq$ Kannan type contraction. By using definition of $\alpha$ we can easily prove that $(\Omega$ $T)$ is $\alpha ^{\ast }$ Kannan type contraction. The $G(\Omega) = \{w:w\preceq \Omega w\}$ is closed and contained $g_{0}$ implies $G(\Omega) = \{w:\alpha ^{\ast }(w, \Omega w)\geq 1\}$ is closed and contained $g_{0}.$ Then, by Theorem 2.9, we have $\{\vartheta T(g_{n})\}$ is a sequence in $\vartheta$ and $\{\vartheta T(g_{n})\}\rightarrow u\in \vartheta.$ Also, (2.22) holds for $w, g\in \{u\}$ implies (2.21) holds for $w, g\in \{u\}$. Hence, by Theorem 2.9 $\Omega$ and $T$ have a common fixed point $u$ in $\vartheta.$

3.   Conclusions

In this research, we have achieved sufficient conditions to prove the existence of common fixed point for a pair of multivalued mappings satisfying a new generalized Kannan type double controlled contraction with a sequence in left $K$-sequentially complete double controlled dislocated quasi type metric spaces. An example is given to show the variety of our results. Moreover, we investigated our results in a better framework of double controlled dislocated quasi-metric spaces. New results in dislocated quasi b-metric spaces, controlled quasi metric spaces, dislocated quasi metric spaces and metric spaces can be obtained as corollaries of our results.

Conflict of interest

The authors declare that they have no competing interests.