Influence of fear effect on a Holling type III prey-predator system with the prey refuge

1.   Introduction and preliminaries

In the 19th century the study of fixed point theory was initiated by Poincare and in 20th century, it was developed by many mathematicians like Brouwer, Schauder, Banach, Kannan, and others. The theory of fixed point is one of the most powerful subject of functional analysis. Theorems ensuring the existence of fixed points of functions are known as fixed point theorems, see ^{[25,29,30,31,33,36,39]}. Fixed point theory is a beautiful mixture of topology, geometry and analysis which has a large number of applications in many fields such as game theory, mathematics engineering, economics, biology, physics, optimization theory and many others, see ^[8,15,20,26]. In 2000, Hitzler and Seda ^[22] established the notation of dislocated metric space. Dislocated metric space plays very important role in electronics engineering and in logical programming ^[23]. For further results on dislocated metric spaces, see ^[1,6,37].

Arshad et al. ^[6] examined some functions having fixed point but there was no result to guarantee the presence of fixed point of such functions. They defined a restriction and involved a closed ball in his result to guarantee the presence of fixed points of such functions. For further results on closed ball, see ^[3,4,7,38].

Ran and Reurings ^[35] and Nieto et al. ^[32] gave an extension to the results in fixed point theory and obtained results in partially ordered sets, see also ^{[11,12,13,40]}.

Many researchers have used interpolative technique to obtain generalized results by using different form of contractions ^[9,10,18]. Karapınar et al. ^[27] introduced a interpolative Hardy Roger's type contraction mapping and proved a fixed point result. Hardy Roger's theorem has been generalized in different ways by many researchers, see ^{[5,19,21,28,34]}.

In this paper, we obtain common fixed point for a pair of dominated functions satisfying interpolative Hardy Roger's type contraction on a closed ball in ordered dislocated metric spaces. Now, we recall the following definitions and results which will be useful to understand the paper.

Definition 1.1. ^[6] Consider $\Upsilon$ be a nonempty set and $d_{l}:\Upsilon \times \Upsilon \rightarrow \left[ 0, +\infty \right).$ Then $d_{l}$ is known as a $d_{l}-$metric, if the following conditions hold for $m, f, k\in \Upsilon :$

(ⅰ) if $d_{l}\left(m, f\right) = 0,$ then $m = f,$

(ⅱ) $d_{l}\left(m, f\right) = d_{l}\left(f, m\right),$

(ⅲ) $d_{l}\left(m, f\right) \leq d_{l}\left(m, k\right) +d_{l}\left(k, f\right) -d_{l}\left(k, k\right).$

The dislocated metric space is represented by the pair $\left(\Upsilon, d_{l}\right).$ We will use DMS instead of dislocated metric space for now onward. It is evident that if $d_{l}\left(m, f\right) = 0,$ then from (ⅰ) $m = f.$ But the converse is not true in general.

Remark 1.2. ^[6] From (ⅲ) of Definition 1.1, we deduce

for all $m, f, k\in \Upsilon.$

Example 1.3. ^[6] If $\Upsilon = \left[ 0, +\infty \right),$ then $d_{l}\left(m, g\right) = m+g$ define a dislocated metric $d_{l}$ on $\Upsilon.$

Definition 1.4. ^[6] Consider $\left\{ f_{n}\right\}$ be a sequence in a DMS $\left(\Upsilon, d_{l}\right),$ we call $\left\{ f_{n}\right\}$ be a Cauchy sequence if, $\varepsilon > 0,$ there exists $n_{0}\in\mathbb{N}$, so that for all $n, m\geq n_{0}$, we get $d_{l}\left(f_{m}, f_{n}\right) < \varepsilon.$

Definition 1.5. ^[6] Consider $\left\{ f_{n}\right\}$ be a sequence in a DMS $\left(\Upsilon, d_{l}\right)$. We call this sequence to be converges with respect to $d_{l},$ if there exists $f\in \Upsilon$ such that $d_{l}\left(f_{n}, f\right) \rightarrow 0$ as $n\rightarrow +\infty.$ Where, $f$ is known as limit of $\left\{ f_{n}\right\},$ and we write $f_{n}\rightarrow f.$

Definition 1.6. ^[6] A DMS $\left(\Upsilon, d_{l}\right)$ is called complete, if every Cauchy sequence in $\Upsilon$ converges to a point in $\Upsilon$.

Definition 1.7. ^[6] Consider $\Upsilon$ be a nonempty set. The triplet $\left(\Upsilon, \preceq, d_{l}\right)$ is said to be ordered DMS, if:

(ⅰ) if $d_{l}$ to be a dislocated metric of $\Upsilon$,

(ⅱ) if $\preceq$ is a partial order on $\Upsilon$.

Definition 1.8. ^[6] Consider a partial ordered set $\left(\Upsilon, \preceq \right)$. If $m\preceq g$ or $g\preceq m$ holds then $m$ and $g$ are called comparable.

Definition 1.9. ^[2] Consider a partially ordered set $\left(\Upsilon, \preceq \right)$. Let $g$ be self mapping on $\Upsilon.$ Then we call $g$ is dominated mapping, if $gm\preceq m$ for every $m$ in $\Upsilon.$

2.   Main result

Now, we define interpolative dominated contractive condition on a closed ball in ordered dislocated metric space and prove our main result.

Theorem 2.1. Let $\left(\Upsilon, \preceq, d_{l}\right)$ be a complete ordered DMS, $T$ and $S$ are dominated mappings on $\Upsilon$, $f_{0}\in \Upsilon$ and $r > 0$. Assume that $f$ and $y$ are comparable element in $\overline{B\left(f_{0}, r\right) },$ such that

for some $\alpha, \beta, \gamma, \lambda \in \left[ 0, 1\right),$ with $\alpha +2\beta +2\gamma < 1$ and

Then there exists a non increasing sequence $\left\{ f_{n}\right\} \subseteq \overline{B\left(f_{0}, r\right) }$, such that $f_{n}\rightarrow f^{\ast }\in \overline{B\left(f_{0}, r\right) }.$ Also, if $f^{\ast }\preceq f_{n},$ then $f^{\ast } = Tf^{\ast } = Sf^{\ast }$ and $d_{l}\left(f^{\ast }, f^{\ast }\right) = 0.$

Proof. Consider a point $f_{1}$ on $\Upsilon$ such that $f_{1} = Sf_{0}.$ As $Sf_{0}\preceq f_{0}$ so $f_{1}\preceq f_{0}$ and let $f_{2 = }Tf_{1}.$ Now $Tf_{1}\preceq f_{1}$ gives $f_{2}\preceq f_{1},$ continuing this method and choosing $f_{n}$ in $\Upsilon$ such that $f_{2h+1 = }S_{f_{2h}},$ $f_{2h+2} = Tf_{2_{h+1}},$ where $h = 0, 1, 2, \ldots$ clearly, $f_{2h+1 = }S_{f_{2h}}\preceq f_{2h} = Tf_{2h-1}\preceq f_{2h-1},$ and this implies that the sequence $\left\{ f_{n}\right\}$ is non increasing. By using inequality (2.2), we have $d_{l}\left(f_{0}, f_{1}\right) \leq r,$ or $f_{1}\in \overline{B\left(f_{0, }r\right) }.$ Assume that $f_{2}, \ldots, f_{j}\in \overline{B\left(f_{0}, r\right) }$ for some $j\in\mathbb{N}.$ Now, if $2h+1\leq j,$ by using inequality (2.1), we obtain

By Remark 1.2, we have

Suppose that

This implies that

Consequently, the inequality (2.1) yield that

so we conclude that

which is a contradiction, thus we have

This implies that

By simple elimination, the inequality (2.1) becomes

This implies that

Similarly, if $2h\leq j$, we deduce

By the previous inequality, we get

Thus from inequality (2.4), we have

for some $j\in\mathbb{N}.$ Now, using (2.5), and (2.2), we get

Thus $f_{j+1}\in \overline{B\left(f_{0}, r\right) }.$ Therefore $f_{h}\in \overline{B\left(f_{0}, r\right) },$ for all $h\in\mathbb{N}$. Since $f_{h+1}\preceq f_{h}$ for all $h\in\mathbb{N},$ then it follow that

This shows that $\left\{ f_{n}\right\}$ is a Cauchy sequence in $\left(\overline{B\left(f_{0}, r\right) }, d_{l}\right).$ Now, $\left(\overline{ B\left(f_{0}, r\right) }, d_{l}\right)$ is complete because $\overline{ B\left(f_{0}, r\right) }$ is closed. Therefore there exist a point $f^{\ast }\in \overline{B\left(f_{0}, r\right) }$ with

By assumption $f^{\ast }\preceq f_{n}$ as $f_{n}\rightarrow f^{\ast },$ we have

On taking limit $h\rightarrow +\infty$ and by using inequalities (2.4) and (2.6), we obtain $d_{l}\left(f^{\ast }, Sf^{\ast }\right) \leq 0$ which implies,

Similarly from

we can obtain $f^{\ast } = Tf^{\ast }.$ Hence $S$ and $T$ have a common fixed point in $\overline{B\left(f_{0}, r\right) }$. Now,

and this implies that.

In Theorem 2.1, the condition 2.1 is applicable only for all comparable points in a closed ball and the condition 2.2 is used to obtain a sequence in a closed ball and Example 2.10 will show the importance of this restriction. Now, in the next result the condition 2.2 is relaxed and the condition 2.1 is applied for all comparable points in the ground set.

Corollary 2.2. Let $\left(\Upsilon, \preceq, d_{l}\right)$ be a complete ordered DMS, $T$ and $S$ are dominated mappings on $\Upsilon$. Assume that $f$ and $y$ are comparable element in $\Upsilon,$ such that

for some $\alpha, \beta, \gamma, \lambda \in \left[ 0, 1\right),$ with $\alpha +2\beta +2\gamma < 1$. Then there exists a non increasing sequence $\left\{ f_{n}\right\} \subseteq X$ such that $f_{n}\rightarrow f^{\ast }\in X.$ Also, if $f^{\ast }\preceq f_{n},$ then $f^{\ast } = Sf^{\ast } = Tf^{\ast }$ and $d_{l}\left(f^{\ast }, f^{\ast }\right) = 0.$

The metric space version of Corollary 2.2 is given below.

Corollary 2.3. Let $\left(\Upsilon, \preceq, \rho \right)$ be a complete ordered metric space, $T$ and $S$ are dominated mappings on $\Upsilon$. Assume that $f$ and $y$ are comparable elements in $\Upsilon,$ such that

for some $\alpha, \beta, \gamma, \lambda \in \left[ 0, 1\right),$ with $\alpha +2\beta +2\gamma < 1.$ Then there exists a non increasing sequence $\left\{ f_{n}\right\} \subseteq X$ such that $f_{n}\rightarrow f^{\ast }\in X.$ Also, if $f^{\ast }\preceq f_{n},$ then $f^{\ast } = Sf^{\ast } = Tf^{\ast }.$

In Theorem 2.1, if we replace $S$ by $T,$ then the following result is obtained.

Corollary 2.4. Let $\left(\Upsilon, \preceq, d_{l}\right)$ be a complete ordered DMS, $T$ is a dominated mappings on $\Upsilon$, $f_{0}\in \Upsilon$ and $r > 0$. Assume that $f$ and $y$ are comparable element in $\overline{B\left(f_{0}, r\right) },$ such that

for some $\alpha, \beta, \gamma, \lambda \in \left[ 0, 1\right),$ with $\alpha +2\beta +2\gamma < 1$ and

Then there exists a non increasing sequence $\left\{ f_{n}\right\} \subseteq \overline{B\left(f_{0}, r\right) }$, such that $f_{n}\rightarrow f^{\ast }\in \overline{B\left(f_{0}, r\right) }.$ Also, if $f^{\ast }\preceq f_{n},$ then $f^{\ast } = Tf^{\ast }$ and $d_{l}\left(f^{\ast }, f^{\ast }\right) = 0.$

Without closed ball version of Corollary 2.4 is given below.

Corollary 2.5. Let $\left(\Upsilon, \preceq, d_{l}\right)$ be a complete ordered DMS, $T$ are dominated mappings on $\Upsilon$. Assume that $f$ and $y$ are comparable element in $\Upsilon,$ such that

for some $\alpha, \beta, \gamma, \lambda \in \left[ 0, 1\right),$ with $\alpha +2\beta +2\gamma < 1.$ Then there exists a non increasing sequence $\left\{ f_{n}\right\} \subseteq X$, such that $f_{n}\rightarrow f^{\ast }\in X.$ Also, if $f^{\ast }\preceq f_{n},$ then $f^{\ast } = Tf^{\ast }$ and $d_{l}\left(f^{\ast }, f^{\ast }\right) = 0.$

If we put the value of $\alpha$ is equal to zero. Then the following result is obtained.

Corollary 2.6. Let $\left(\Upsilon, \preceq, d_{l}\right)$ be a complete ordered DMS, $T$ and $S$ are dominated mappings on $\Upsilon$, $f_{0}\in \Upsilon$ and $r > 0$. Assume that $f$ and $y$ are comparable element in $\overline{B\left(f_{0}, r\right) },$ such that

for some $\beta, \gamma, \lambda \in \left[ 0, 1\right),$ with $2\beta +2\gamma < 1$. Then there exists a non increasing sequence $\left\{ f_{n}\right\} \subseteq \overline{B\left(f_{0}, r\right) }$, such that $f_{n}\rightarrow f^{\ast }\in \overline{B\left(f_{0}, r\right) }.$ Also, if $f^{\ast }\preceq f_{n},$ then $f^{\ast } = Sf^{\ast } = Tf^{\ast }$ and $d_{l}\left(f^{\ast }, f^{\ast }\right) = 0.$

If we put the value of $\beta$ is equal to zero. Then the following result is obtained.

Corollary 2.7. Let $\left(\Upsilon, \preceq, d_{l}\right)$ be a complete ordered DMS, $T$ and $S$ are dominated mappings on $\Upsilon$, $f_{0}\in \Upsilon$ and $r > 0$. Assume that $f$ and $y$ are comparable element in $\overline{B\left(f_{0}, r\right) },$ such that

for some $\alpha, \gamma, \lambda \in \left[ 0, 1\right),$ with $\alpha +2\gamma < 1$. Then there exists a non increasing sequence $\left\{ f_{n}\right\} \subseteq \overline{B\left(f_{0}, r\right) }$, such that $f_{n}\rightarrow f^{\ast }\in \overline{B\left(f_{0}, r\right) }.$ Also, if $f^{\ast }\preceq f_{n},$ then $f^{\ast } = Sf^{\ast } = Tf^{\ast }$ and $d_{l}\left(f^{\ast }, f^{\ast }\right) = 0.$

If we put the value of $\gamma$ is equal to zero. Then the following result is obtained.

Corollary 2.8. Let $\left(\Upsilon, \preceq, d_{l}\right)$ be a complete ordered DMS, $T$ and $S$ are dominated mappings on $\Upsilon$, $f_{0}\in \Upsilon$ and $r > 0$. Assume that $f$ and $y$ are comparable element in $\overline{B\left(f_{0}, r\right) },$ such that

for some $\alpha, \beta, \lambda \in \left[ 0, 1\right),$ with $\alpha +2\beta < 1$. Then there exists a non increasing sequence $\left\{ f_{n}\right\} \subseteq \overline{B\left(f_{0}, r\right) }$, such that $f_{n}\rightarrow f^{\ast }\in \overline{B\left(f_{0}, r\right) }.$ Also, if $f^{\ast }\preceq f_{n},$ then $f^{\ast } = Sf^{\ast } = Tf^{\ast }$ and $d_{l}\left(f^{\ast }, f^{\ast }\right) = 0.$

If we take complete DMS $\left(\Upsilon, d_{l}\right)$ instead of complete ordered DMS $\left(\Upsilon, \preceq, d_{l}\right)$. Then the following result is obtained.

Corollary 2.9. Let $\left(\Upsilon, d_{l}\right)$ be a complete DMS, $T$ and $S$ are self mappings on $\Upsilon$, $f_{0}\in \Upsilon$ and $r > 0$. Assume that $f$ and $y$ are element in $\overline{B\left(f_{0}, r\right) },$ such that

for some $\alpha, \beta, \gamma, \lambda \in \left[ 0, 1\right),$ with $\alpha +2\beta +2\gamma < 1$ and

Then there exists a sequence $\left\{ f_{n}\right\} \subseteq \overline{ B\left(f_{0}, r\right) }$, such that $f_{n}\rightarrow f^{\ast }\in \overline{B\left(f_{0}, r\right) }$, $f^{\ast } = Tf^{\ast } = Sf^{\ast }$ and $d_{l}\left(f^{\ast }, f^{\ast }\right) = 0.$

Example 2.10. Let $\Upsilon = \left[ 0, +\infty \right) \cap\mathbb{Q}$ be endowed with the order $f\preceq y$ if $d_{l}\left(f, f\right) \leq d_{l}\left(y, y\right),$ and define $d_{l}:\Upsilon \times \Upsilon \rightarrow \Upsilon$ as $d_{l}\left(f, y\right) = f+y.$ Then $\left(\Upsilon, d_{l}\right)$ is an ordered completed dislocated metric space. Let $T, S:\Upsilon \rightarrow \Upsilon$ be defined by,

Clearly $T$ and $S$ are dominated mappings. For $f_{0} = 1,$ $r = 2,$ $\alpha = \frac{1}{7},$ and $\beta = \frac{1}{9},$ $\gamma = \frac{1}{10},$ $\lambda = \frac{3}{7},$ $\overline{B\left(f_{0}, r\right) } = \left[ 0, 1\right] \cap \Upsilon$, and $\left(1-\lambda \right) r = \frac{8}{7} = d_{l}\left(f_{0}, Sf_{0}\right).$ Now if $f = 1,$ $y = 2$ then

and so,

Thus, the contractive condition does not hold on $\Upsilon$. Now if $f, y\in \overline{B\left(f_{0}, r\right) },$ then

Therefore all the condition of theorem are satisfied. Moreover, $0$ is the common fixed point of $T$ and $S.$

3.   Conclusions

Arshad et al. ^[6] analyzed that there are mappings which are contractive only on the subsets of its domain. They deduced the fixed point results satisfying contraction on closed ball to ensure the existence of fixed point of such mappings. On the other hand, Karapınar et al. ^[27] recently gave the concept interpolative contraction and established some result. We extend their findings, and in this paper, fixed point results with interpolative contractive conditions for a pair of generalized locally dominated mappings on closed balls in ordered dislocated metric spaces have been established.

Acknowledgments

This research was funded by National Science, Research and Innovation Fund (NSRF), and King Mongkut's University of Technology North Bangkok with Contract no. KMUTNB-FF-65-24.

Conflict of interest

The authors declare that they do not have any competing interests.