Some new results on expansion mapping and their application

Abstract

In the present article, the concept of expansion is extended in a refined manner by introducing $P$ -expansion defined on a family $F$ of bounded functions. Some fixed function theorems using expansive mappings in complete metric spaces are investigated. These results improve and generalize various results existing in literature. The authenticity of obtained results is verified with the help of some comparative examples. In addition, an application has also been presented which is based on the best approximation of dose distribution for a number of patients (at a same time) getting Tomotherapy.

Keywords

Fixed function complete metric space expansive mappings 47H10 54H25

1 Introduction and preliminaries

The study of fixed points was initiated by H. Poincare [12] in 1886. His result is known as Poincare’s last geometric theorem which ensures the existence of at least two fixed points for an area preserving twist homeomorphism of an annulus. After that, in 1912, L.E.J.Brouwer [3] proved a fixed point theorem for a topological space which states that “A continuous self mapping defined on a closed unit ball in Euclidean space has at least one fixed point”. But a major change in the field of fixed point theory took place in 1922 when the Polish mathematician Stefan Banach [1] proved “Banach Contraction Principle” for a complete metric space.

Theorem 1.1. [1] Let (U, d) be a complete metric space and $D$ be any self mapping defined on U satisfying the inequality $d (D u_{1}, D u_{2}) \leq λ d (u_{1}, u_{2});$ for every pair u₁, u₂ ∈ U where λ is any real number (0 ≤ λ < 1).

Then $D$ has a unique fixed point in U.

After that, this concept attracted many researchers to prove the existence of fixed points by using various metrical contractions. For details, one can refer to ([4, 5, 7]).

On the other hand, in 1984, Wang et al. [15] began a research line in the field of expansive mappings and presented some fixed point results in metric spaces.

Theorem 1.2. [15] Let (U, d) be a complete metric space. Let D : U → U be an onto mapping defined on U. Assume that there exists a constant a > 1 such that $d (D u_{1}, D u_{2}) \geq ad (u_{1}, u_{2});$ for each u₁, u₂ ∈ U. Then $D$ has a unique fixed point in U.

After that, various authors worked in this direction and proved useful results.

In [9], Jungck defined interdependence between commuting mappings and fixed points.

Definition 1.3. [9] Two self mappings D ₁ and D ₂ of a metric space (U, d) are said to be commuting if D ₁ D ₂ u = D ₂ D ₁ u for all u ∈ U.

Theorem 1.4. [9] Let D₁ be a continuous self-mapping defined on a complete metric space (U, d). Then D₁ has a fixed point in U iff there exists a ∈ (0, 1) and a mapping D₂ : U → U which commutes with D₁ such that D₂ (U) ⊂ D₁ (U) and d (D₂ (u) , D₂ (v)) ≤ ad (D₁ (u) , D₂ (v)) for all u, v ∈ U. The mappings D₁ and D₂ have a unique common fixed point if above inequality holds.

Definition 1.5. [10] Let D ₁ and D ₂ be two self mappings defined on a set U. If w = D ₁ u = D ₂ u for some u in U, then u is called a coincidence point of D ₁ and D ₂, and w is called a point of coincidence of D ₁ and D ₂. Self mappings D ₁ and D ₂ are said to be weakly compatible if D ₁ D ₂ u = D ₂ D ₁ u for some u in U such that D ₁ u = D ₂ u.

The concept of weakly compatible mappings is more general as compared to commuting mappings.

Recently, Dhawan et al. [6] presented a new notion of fixed function and obtained some fixed function theorems by using various types of contractions. Following are the definitions and results due to [6].

Definition 1.6. [6] Fixed function: Let $D$ be any self mapping defined on a family of functions $F$ , then $f \in F$ is said to be fixed function of $D$ if $D f = f$ .

Example 1.7. [6] Let $U = ℝ^{+}$ and $D$ be any self mapping on $F$ . Let $f \in F$ be a function defined on U as $f (u) = {\begin{matrix} - 1 & 0 \leq u \leq 1, \\ 0 & otherwise . \end{matrix}$ Then f ³ is a fixed function of $D$ .

Definition 1.8. [6] Let $(U, \hat{d})$ be a complete metric space and let $F$ be the collection of all bounded functions defined on U and d^* be the metric defined on $F$ . Let $D$ be any self mapping on $F$ . Then the given mapping is called $D$ -contraction mapping on $F$ , if for any real number λ ∈ [0, 1), we have $d^{*} (D f, D g) \leq λ d^{*} (f, g) for all f, g \in F;$ where

$\begin{matrix} d^{*} (f, g) & = \sup {\hat{d} (f (u), g (v)) | u, v \in U} \\ = \sup {| f (u) - g (v) | | u, v \in U} . \end{matrix}$ (1.1)

Remark 1.9. [6] Clearly, d ^* is a metric on $F$ as d ^* (f, g) =0 ⇔ f ∼ g for all $f, g \in F$ . Also, for all u, v ∈ U and $f, g, h \in F$ , $\begin{matrix} | f (u) - g (v) | & \leq & | f (u) - h (w) | + | h (w) - g (v) | \\ \leq & sup {| f (u) - h (w) | | u, w \in U} \\ + sup {| h (w) - g (v) | | w, v \in U} \\ \Rightarrow sup {| f (u) - g (v) | | u, v \in U} \\ \leq & sup {| f (u) - h (w) | | u, w \in U} \\ + sup {| h (w) - g (v) | | w, v \in U} \\ \Rightarrow d^{*} (f, g) & \leq & d^{*} (f, h) + d^{*} (h, g) . \end{matrix}$

Theorem 1.10. [6] Let $(U, \hat{d})$ be a complete metric space with metric $\hat{d}$ defined as $\hat{d} (u, v) = | u - v |$ for all u, v ∈ U. Let $F$ be the collection of all bounded functions f defined on U with metric d^*(as defined in (1.1)).

Also, let $D$ be the $D$ -contraction mapping defined on $F$ . Then there exists a unique fixed function $f \in F$ i.e. there exists some $f \in F$ such that $D f = f$ .

Theorem 1.11. [6] Let $(U, \hat{d})$ be a complete metric space (where $\hat{d}$ is the metric as defined earlier) and $F$ be the collection of all bounded functions f defined on U with metric d^*(as defined in (1.1)).

Also, let $D$ be the modified $D$ -contraction mapping on $F$ satisfying $d^{*} (D f, D g) \leq α d^{*} (f, D f) + β d^{*} (g, D g) + γ d^{*} (f, g);$ for all $f, g \in F$ ; α, β, γ non negative with α + β + γ < 1. Then $D$ has a unique fixed function.

The notion of α - ψ contractive mappings and α-admissible mappings was extended for functions in the following manner:

Definition 1.12. [6] The mapping $D : F \to F$ is said to be an α - ψ contractive mapping if there exists two functions α : U × U → [0, + ∞) and ψ ∈ Ψ satisfying $α (f (u), g (v)) d^{*} (D f, D g) \leq ψ (d^{*} (f, g));$ (1.2) for all $f, g \in F$ and u, v ∈ U.

Definition 1.13. [6] Let $D : F \to F$ and α : U × U → [0, + ∞). The mapping $D$ is called an α-admissible mapping if $α (f (u), g (v)) \geq 1 \Rightarrow α (D f (u), D g (v)) \geq 1$ for every $f, g \in F$ and u, v ∈ U.

The result proved in this context is stated as:

Theorem 1.14. [6] Let $(U, \hat{d})$ be a complete metric space and $F$ be the collection of all bounded functions f (defined on U) with metric d^* (as defined in (1.1)). Let $D : F \to F$ be an α - ψ contractive mapping. Also, suppose that

$D$ is α-admissible;

there is some $f_{0} \in F$ for which α (f₀ (u) , $D f_{0} (v)) \geq 1$ for all u, v ∈ U;

$D$ is continuous.

Then $D$ possesses a fixed function in $F$ .

In this article, following the work of Dhawan et al. [6], the concept of fixed function is used to prove some fixed function theorems via expansive mappings in complete metric spaces. Some common fixed function theorems for a pair of weakly compatible mappings are also derived. Moreover, the credibility of obtained results has been verified through examples and a nice application.

2 Main results

Definition 2.1. Let (U, d) be a complete metric space where metric d is defined as d (u, v) = |u - v| ∀ u, v ∈ U and $F$ be the collection of all bounded functions defined on U equipped with metric d ^* (as defined in (1.1)). Let $P : F \to F$ be a self mapping. Then $P$ is called an expansive mapping if for all $f, g \in F$ , there exists a number α > 1 such that $d^{*} (P f, P g) \geq α d^{*} (f, g) .$

Example 2.2. Let U be the set of positive real numbers and the mapping $P$ is defined as $P f = 5 f + 2 \forall f \in F$ . Then $P$ is an expansive mapping as for all u, v ∈ U; $f, g \in F$ , we have $\begin{matrix} d^{*} (P f, P g) & = \sup {| 5 f (u) + 2 - 5 g (v) - 2 | | u, v \in U} \\ = \sup {5 | f (u) - g (v) | | u, v \in U} \\ > \sup {| f (u) - g (v) | | u, v \in U} \\ = α d^{*} (f, g) where α = 2 . \end{matrix}$

The following Lemma will be crucial in proving our main results.

Lemma 2.3. Let (U, d) be a complete metric space with metric d defined as d (u, v) = |u - v| ∀ u, v ∈ U and $F$ be the collection of all bounded functions defined on U equipped with metric d^*(as defined in (1.1)).

Let ${f_{k}}_{(k \in ℕ)}$ be a sequence in $F$ . If there exists a number α ∈ (0, 1) such that $d^{*} (f_{k + 1}, f_{k}) \leq α d^{*} (f_{k}, f_{k - 1}), k = 1, 2, . . . .$ (2.1) Then ${f_{k}}_{(k \in ℕ)}$ is a Cauchy sequence in $F$ .

Proof. By induction, we get from (2.1), $\begin{matrix} d^{*} (f_{k + 1}, f_{k}) & \leq α d^{*} (f_{k}, f_{k - 1}) \leq α^{2} d^{*} (f_{k - 1}, f_{k - 2}) \\ \leq . . . \leq α^{k} d^{*} (f_{1}, f_{0}) . \end{matrix}$ Let $p, q \in ℕ$ be some positive integers with p > q. Let p = q + t where t ≥ 1. Then $\begin{matrix} d^{*} (f_{q}, f_{p}) & = d^{*} (f_{q}, f_{q + t}) \\ \leq d^{*} (f_{q}, f_{q + 1}) + d^{*} (f_{q + 1}, f_{q + 2}) + . . . \\ + d^{*} (f_{q + t - 1}, f_{q + t}) \\ \leq α^{q} d^{*} (f_{1}, f_{0}) + α^{q + 1} d^{*} (f_{1}, f_{0}) + . . . \\ + α^{q + t - 1} d^{*} (f_{1}, f_{0}) \\ \leq \frac{α^{q}}{1 - α} d^{*} (f_{1}, f_{0}) where α < 1 . \end{matrix}$ This shows that ${f_{k}}_{(k \in ℕ)}$ is a Cauchy sequence in $F$ .□

Theorem 2.4. Let (U, d) be a complete metric space with metric d defined as d (u, v) = |u - v| for all u, v ∈ U. Let $F$ be the collection of all bounded functions defined on U with metric d^* (as defined in (1.1)).

Also, let $P$ be the onto expansion self mapping defined on $F$ . Suppose that there exists α, β, γ ≥ 0 with α + β + γ > 1 such that $d^{*} (P f, P g) \geq α d^{*} (f, g) + β d^{*} (f, P f) + γ d^{*} (g, P g);$ (2.2) for all $f, g \in F$ (where f ≠ g). Then $P$ has a fixed function in $F$ .

Proof. Let f ₀ be any arbitrary function in $F$ . Since $P$ is an onto mapping, there exists $f_{1} \in F$ such that $P f_{1} = f_{0}$ . Proceeding in the same way, we can define a sequence ${f_{k}}_{(k \in ℕ)}$ in $F$ such that $f_{k - 1} = P f_{k}, k = 1, 2, . . .$ .

Assume that f _k ≁ f _k-1 ∀ k = 1, 2, . . .(because otherwise f _k is a fixed function of $P$ .)

It follows by (2.2), $\begin{matrix} d^{*} (f_{k - 1}, f_{k}) & = d^{*} (P f_{k}, P f_{k + 1}) \\ \geq α d^{*} (f_{k}, f_{k + 1}) + β d^{*} (f_{k}, P f_{k}) \\ + γ d^{*} (f_{k + 1}, P f_{k + 1}) \\ = α d^{*} (f_{k}, f_{k + 1}) + β d^{*} (f_{k}, f_{k - 1}) \end{matrix}$ $\begin{matrix} + γ d^{*} (f_{k + 1}, f_{k}) \\ \Rightarrow (1 - β) d^{*} (f_{k - 1}, f_{k}) & \geq (α + γ) d^{*} (f_{k}, f_{k + 1}) \\ \Rightarrow d^{*} (f_{k}, f_{k + 1}) & \leq (\frac{1 - β}{α + γ}) d^{*} (f_{k - 1}, f_{k}) \\ = s d^{*} (f_{k - 1}, f_{k}) where s < 1 . \end{matrix}$ By Lemma 2.3, ${f_{k}}_{(k \in ℕ)}$ is a Cauchy sequence in $F$ . Since $(F, d^{*})$ is complete being a collection of functions defined on a complete metric space (U, d), the sequence ${f_{k}}_{(k \in ℕ)}$ is convergent in $F$ .

Let $f^{*} \in F$ be the limit of this sequence. Accordingly, there exists $f \in F$ such that $P f = f^{*}$ .

Now, $\begin{matrix} d^{*} (f_{k}, f^{*}) & = d^{*} (P f_{k + 1}, P f) \\ \geq α d^{*} (f_{k + 1}, f) + β d^{*} (f_{k + 1}, P f_{k + 1}) \\ + γ d^{*} (f, P f) . \end{matrix}$ As k→ ∞, the above inequality becomes $\begin{matrix} 0 = d^{*} (f^{*}, f^{*}) & \geq α d^{*} (f^{*}, f) + β (0) + γ d^{*} (f, f^{*}) \\ \Rightarrow 0 & \geq (α + γ) d^{*} (f, f^{*}); \end{matrix}$ which implies that d ^* (f, f ^*) =0 i.e. f∼f ^*. Thus, f ^* is a fixed function of D.

This completes the proof.□

Remark 2.5. By setting β = γ = 0 in Theorem 2.4, the following result is obtained:

Corollary 2.6. Let (U, d) be a complete metric space with metric d defined as d (u, v) = |u - v| for all u, v ∈ U. Let $F$ be the collection of all bounded functions defined on U with metric d^* (as defined in (1.1)).

Also, let $P$ be the onto expansive self mapping defined on $F$ . Suppose that there exists α > 1 such that $d^{*} (P f, P g) \geq α d^{*} (f, g);$ for all $f, g \in F$ (where f ≠ g). Then $P$ a unique fixed function in $F$ .

Proof. For uniqueness, let g ^* be another fixed function of $P$ . Then $\begin{matrix} d^{*} (f^{*}, g^{*}) & = d^{*} (P f^{*}, P g^{*}) \\ \geq α d^{*} (f^{*}, g^{*}); \end{matrix}$ which is not possible as α > 1. Therefore, the fixed function f ^* is unique.

This completes the proof.□

Corollary 2.7. Let (U, d) be a complete metric space with metric d defined as d (u, v) = |u - v| for all u, v ∈ U and $F$ be the collection of all bounded functions defined on U along with metric d^* (as defined in (1.1)).

Also, let $P$ be an onto expansive self mapping defined on $F$ . Suppose that there exists a +ve integer n and α > 1 such that $d^{*} (P^{n} f, P^{n} g) \geq α d^{*} (f, g);$ for all $f, g \in F$ (where f ≠ g). Then $P$ has a unique fixed function in $F$ .

Proof. By Corollary 2.6, $\begin{matrix} d^{*} (P^{n} f, P^{n} g) & \geq α d^{*} (P^{n - 1} f, P^{n - 1} g) \\ \geq α^{2} d^{*} (P^{n - 2} f, P^{n - 2} g) \\ . . . & \geq α^{n} d^{*} (f, g); \end{matrix}$ and thus $P^{n}$ has a unique fixed function f ^*. But $P^{n} (P f^{*}) = P (P^{n} f^{*}) = P f^{*}$ , therefore, $P f^{*}$ is also a fixed function of $P^{n}$ but the uniqueness implies $P f^{*} = f^{*}$ . Thus, $P$ has a unique fixed function in $F$ .□

Example 2.8. Let $U = ℝ^{+}$ and d be the metric defined on U. Let $F$ be the family of bounded functions defined on U and d ^* be the metric defined on $F$ as $d^{*} (f, g) = \sup {d (f (u), g (v)) | u, v \in U} .$ Note that (U, d) is a complete metric space and thus $(F, d^{*})$ is also complete being collection of bounded functions defined on U.

Let $\begin{matrix} f (u) & = & {\begin{matrix} - 2 & u is rational, \\ 0 & otherwise, \end{matrix} \end{matrix}$ and $\begin{matrix} g (u) & = & {\begin{matrix} 1 & u is rational, \\ 2 & otherwise, \end{matrix} \end{matrix}$ be two functions from the family $F$ . Let the mapping $P$ be defined as $P f = f^{2} + 3 f \forall f \in F$ . It is easy to check that $P$ is an expansive mapping as for $u, v \in U; f, g \in F$ , we have the following cases for $α = \frac{6}{5}$ :

Case-I: If u, v are rationals. Then $\begin{matrix} d^{*} (P f, P g) & = \sup {d (P f (u), P g (v)) | u, v \in U} \\ = \sup {| P f (u) - P g (v) | u, v \in U} \\ = \sup {| f^{2} (u) + 3 f (u) - g^{2} (v) - 3 g (v) | \\ u, v \in U} \\ = 6 \geq \frac{6}{5} \times 3 \\ = α \sup {| f (u) - g (v) | u, v \in U} \\ = α d^{*} (f, g) . \end{matrix}$

Case-II: If u, v are not rationals. Then $\begin{matrix} d^{*} (P f, P g) & = \sup {d (P f (u), P g (v)) | u, v \in U} \\ = \sup {| P f (u) - P g (v) | u, v \in U} \\ = \sup {| f^{2} (u) + 3 f (u) - g^{2} (v) - 3 g (v) | \\ u, v \in U} \\ = 10 \geq \frac{6}{5} \times 2 \\ = α \sup {| f (u) - g (v) | u, v \in U} \\ = α d^{*} (f, g) . \end{matrix}$

Case-III: If u is a rational number and v is not. Then $\begin{matrix} d^{*} (P f, P g) & = \sup {d (P f (u), P g (v)) | u, v \in U} \\ = \sup {| P f (u) - P g (v) | u, v \in U} \\ = \sup {| f^{2} (u) + 3 f (u) - g^{2} (v) - 3 g (v) | \\ u, v \in U} \\ = 12 \geq \frac{6}{5} \times 4 \\ = α \sup {| f (u) - g (v) | u, v \in U} \\ = α d^{*} (f, g) . \end{matrix}$

Case-IV: If v is a rational number and u is not. Then $\begin{matrix} d^{*} (P f, P g) & = \sup {d (P f (u), P g (v)) | u, v \in U} \\ = \sup {| P f (u) - P g (v) | u, v \in U} \\ = \sup {| f^{2} (u) + 3 f (u) - g^{2} (v) - 3 g (v) | \\ u, v \in U} \\ = 4 \geq \frac{6}{5} \times 1 \\ = α \sup {| f (u) - g (v) | u, v \in U} \\ = α d^{*} (f, g) . \end{matrix}$ Thus, all the conditions of Corollary 2.6 are satisfied. So, there exists a unique fixed function. In this example, f is the unique fixed function.

Theorem 2.9. Let (U, d) be a complete metric space with metric space equipped with distance metric d and $(F, d^{*})$ be a metric space where $F$ is the family of all bounded functions defined on U and d^* is the metric as defined in (1.1).

Also, let $P : F \to F$ be a continuous surjective mapping. Assume that there exists a constant α > 1 such that for all $f, g \in F$ (where f ≠ g), $d^{*} (P f, P g) \geq α M (f, g);$ where $M (f, g) = \max {d^{*} (f, g), d^{*} (f, P f), d^{*} (g, P g)} .$ Then $P$ has a fixed function in $F$ .

Proof. Following Theorem 2.4, construct a sequence ${f_{k}}_{k \in ℕ}$ such that $f_{k - 1} = P f_{k} \forall k \geq 1$ where f _k-1 ≠ f _k for all k.

By given condition, $d^{*} (f_{k - 1}, f_{k}) = d^{*} (P f_{k}, P f_{k + 1}) \geq α M (f_{k}, f_{k + 1});$ (2.3) where $\begin{matrix} M (f_{k}, f_{k + 1}) & = \max {d^{*} (f_{k}, f_{k + 1}), d^{*} (f_{k}, P f_{k}), \\ d^{*} (f_{k + 1}, P f_{k + 1})} \\ = \max {d^{*} (f_{k}, f_{k + 1}), d^{*} (f_{k}, f_{k - 1})} . \end{matrix}$ If M (f _k, f _k+1) = d ^* (f _k, f _k-1), then by (2.3) $d^{*} (f_{k - 1}, f_{k}) \geq α d^{*} (f_{k}, f_{k - 1}) where α > 1;$ which implies that d ^* (f _k, f _k-1) =0 i.e. f _k = f _k-1 which is a contradiction.

If M (f _k, f _k+1) = d ^* (f _k, f _k+1), then we have $d^{*} (f_{k - 1}, f_{k}) \geq α d^{*} (f_{k}, f_{k + 1});$ and thus by Lemma 2.3, ${f_{k}}_{k \in ℕ}$ is a Cauchy sequence in $F$ .

Since $(F, d^{*})$ is complete being a collection of bounded functions defined on a complete metric space (U, d), the sequence ${f_{k}}_{k \in ℕ}$ converges in $F$ . Accordingly, there must exist some $f \in F$ such that $\lim_{k \to \infty} d^{*} (f_{k}, f) = 0$ .

As $P$ is continuous, we have $\begin{matrix} \lim_{k \to \infty} d^{*} (P f_{k}, P f) & = 0 \\ \Rightarrow \lim_{k \to \infty} d^{*} (f_{k - 1}, P f) & = 0 \\ \Rightarrow d^{*} (f, P f) & = 0 \end{matrix}$ which shows that f is a fixed function of $P$ .□

Next,we prove a common fixed function theorem for two weakly compatible mappings in complete metric spaces.

Theorem 2.10. Let (U, d) be a complete metric space with metric space equipped with distance metric d and $(F, d^{*})$ be a metric space where $F$ is the family of all bounded functions defined on U and d^* is the metric as defined in (1.1).

Also, let $P_{1} : F \to F$ and $P_{2} : F \to F$ be two weakly compatible self maps such that $P_{2} (F) \subseteq P_{1} (F)$ . Assume that there exists a constant α > 1 such that for all $f, g \in F$ (where f ≠ g), $d^{*} (P_{1} f, P_{1} g) \geq α d^{*} (P_{2} f, P_{2} g)$ (2.4) If one of the subspaces $P_{1} (F)$ or $P_{2} (F)$ is complete, then $P_{1}$ and $P_{2}$ have a unique common fixed function in $F$ .

Proof. Let $f_{0} \in F$ be any arbitrary function. Since $P_{2} (F) \subseteq P_{1} (F)$ , therefore there exists some $f_{1} \in F$ such that $P_{1} f_{1} = P_{2} f_{0}$ . In general, we have some $f_{n + 1} \in F$ such that $g_{n} = P_{1} f_{n + 1} = P_{2} f_{n} .$ By (2.4), $\begin{matrix} d^{*} (P_{2} f_{n}, P_{2} f_{n + 1}) & \leq \frac{1}{α} d^{*} (P_{1} f_{n}, P_{1} f_{n + 1}) \\ = \frac{1}{α} d^{*} (P_{2} f_{n - 1}, P_{2} f_{n}) . \end{matrix}$ By Lemma 2.3, above inequality shows that ${P_{2} f_{n}}$ is a Cauchy sequence in $F$ . Since $(F, d^{*})$ is complete being the collection of bounded functions defined on complete metric space (U, d), therefore, ${P_{2} f_{n}}$ is convergent in $F$ . Let the limit of this sequence be f ^* i.e. $\lim_{n \to \infty} g_{n} = \lim_{n \to \infty} P_{1} f_{n} = \lim_{n \to \infty} P_{2} f_{n} = f^{*} .$ As $P_{1} (F)$ is a complete subspace of $F$ , therefore, there exists a function $\hat{f} \in F$ such that $P_{1} \hat{f} = f^{*}$ .

By (2.4), we obtain $\begin{matrix} d^{*} (P_{2} \hat{f}, P_{2} f_{n}) \leq \frac{1}{α} d^{*} (P_{1} \hat{f}, P_{1} f_{n}) . \end{matrix}$ Taking limit n→ ∞, $\begin{matrix} d^{*} (P_{2} \hat{f}, P_{2} f_{n} f^{*}) & \leq \frac{1}{α} d^{*} (P_{1} \hat{f}, f^{*}) \\ = \frac{1}{α} d^{*} (f^{*}, f^{*}) . \end{matrix}$ This implies that $P_{2} \hat{f} = f^{*}$ . Thus, $P_{1} \hat{f} = P_{2} \hat{f} = f^{*}$ . As $P_{1}$ and $P_{2}$ are weakly compatible mappings, therefore, by definition,

$\begin{matrix} P_{1} P_{2} \hat{f} & = P_{2} P_{1} \hat{f} \\ P_{1} f^{*} & = P_{2} f^{*} . \end{matrix}$ (2.5) Now, it remains to show that f ^* is a fixed point of $P_{1}$ and $P_{2}$ . By (2.4), $\begin{matrix} d^{*} (P_{1} f^{*}, P_{1} f_{n}) & \geq α d^{*} (P_{2} f^{*}, P_{2} f_{n}) . \end{matrix}$ Taking limit n→ ∞, $\begin{matrix} d^{*} (P_{1} f^{*}, f^{*}) & \geq α d^{*} (P_{2} f^{*}, f^{*}) . \end{matrix}$ By (2.5), we have, $P_{1} f^{*} = f^{*}$ which shows that f ^* is a fixed function of $P_{1}$ and $P_{2}$ .

For uniqueness, let g ^* be another fixed function of $P_{1}$ and $P_{2}$ . Then, $\begin{matrix} d^{*} (f^{*}, g^{*}) = d^{*} (P_{1} f^{*}, P_{1} g^{*}) & \geq α d^{*} (P_{2} f^{*}, P_{2} g^{*}) \\ = α d^{*} (f^{*}, g^{*}) . \end{matrix}$ which is a contradiction. This completes the proof.□

Example 2.11. Let U = [0, 2] be any set equipped with usual metric d (u, v) = |u - v| ∀ u, v ∈ U. Let $F$ be the family of bounded functions defined on U and d ^* be the metric defined on $F$ as $d^{*} (f, g) = \sup {d (f (u), g (v)) | u, v \in U} .$ Let the mappings $P_{1}$ and $P_{2}$ be defined as $P_{1} f = f$ and $P_{2} f = \frac{f}{2}$ for all $f \in F$ .

Note that $P_{1}$ and $P_{2}$ are weakly compatible for all $f \in F$ and $P_{2} (F) \subseteq P_{1} (F)$ . It remains to show that the inequality (2.4) in Theorem 2.10 is satisfied. $\begin{matrix} d^{*} (P_{1} f, P_{1} g) & = d^{*} (f, g) \\ = \sup {| f (u) - g (v) | | u, v \in U} \\ \geq α \sup {\frac{| f (u) - g (v) |}{2} | u, v \in U} \\ = α d^{*} (P_{2} f, P_{2} g) where α = 2 > 1 . \end{matrix}$ Thus, all the conditions required for Theorem 2.10 are fulfilled, therefore, there exists a unique fixed function. In this example, Null function is the unique fixed function.

3 Application

The application in this section is based on dose approximation for tumor patients getting Tomotherapy.

Shepard et al. [13], Mohan et al. [11] and T. Bortfeld [2] presented some techniques for the problems encountered in dose calculation during Radiation therapy of tumor patients. In these techniques, a dose deposition coefficient (DDC) matrix is computed to approximate doses that are applied to each voxel in required volume of interest from every beamlet. In 2013, Z. Tian et al. [14] presented a fluence map optimization (FMO) model for dose calculation by splitting the DDC matrix into two components $D_{1}$ and $D_{2}$ on the basis of a threshold intensity value. The matrix $D_{1}$ (major component) consists those values of DDC matrix which are higher than the threshold whereas the minor component $D_{2}$ consists remaining values. In fact, $D_{1}$ represents those doses which correspond to tumor area voxels (specifically) while $D_{2}$ represents scatter doses passing at large distances. But in these calculation, we usually get a large amount of data requiring huge computer memory. To overcome this problem, the values in DDC matrix are truncated and as a result, quality of treatment plan gets affected. Moreover, these methods approximate doses of a patient at a time.

Following this concept, the treatment plan for more than a patient at a time, is presented through our results in a more effective way in this application.

Let us consider the case of two lung cancer patients (with different tumor levels) getting Tomotherapy. Let U denotes the set of all possible intensity values to be given on particular days and in particular sessions. Each patient is getting therapy in two sessions per day. Days and sessions are denoted by D _i and S _j (i, j = 1, 2 .) respectively. $U = {\begin{matrix} (1, D_{1} S_{1}), (\frac{1}{2}, D_{1} S_{2}), (1, D_{2} S_{1}), (\frac{1}{2}, D_{2} S_{2}) & Patient - I, \\ (2, D_{1} S_{1}), (1, D_{1} S_{2}), (2, D_{2} S_{1}), (1, D_{2} S_{2}) & Patient - II . \end{matrix}$ Note that U is complete being a closed and bounded subset of $ℝ^{2}$ . Let $F = {f, g}$ be the family of dose functions and each function represents different doses (to tumor locations) of different tumor patients during Tomotherapy. $\begin{matrix} f (u) = {\begin{matrix} u & Patient - I, \\ \frac{u}{2} & Patient - II . \end{matrix} and g (u) = {\begin{matrix} 2 u & Patient - I, \\ \frac{3}{2} u & Patient - II . \end{matrix} \end{matrix}$ Here, $F$ is the family of bounded functions. Let $P : F \to F$ be the mapping defined as $P f = f^{2} - \frac{f}{2} + \frac{1}{2}$ for all $f \in F$ . Now, it remains to prove that $P$ is an expansion mapping. For u, v ∈ U, we have the following cases:

For Patient-I

Case I- When u = v = 1. Then, $\begin{matrix} | P f - P g | = \frac{5}{2} and | f - g | = 1 . \end{matrix}$

Case II- When $u = 1, v = \frac{1}{2}$ . Then, $\begin{matrix} | P f - P g | = 0 and | f - g | = 0 . \end{matrix}$

Case III- When $u = \frac{1}{2}, v = 1$ . Then, $\begin{matrix} | P f - P g | = 3 and | f - g | = \frac{3}{2} . \end{matrix}$

Case IV- When $u = v = \frac{1}{2}$ . Then, $\begin{matrix} | P f - P g | = \frac{1}{2} and | f - g | = \frac{1}{2} . \end{matrix}$

For Patient-II

Case I- When u = v = 2. Then, $\begin{matrix} | P f - P g | = 7 and | f - g | = 2 . \end{matrix}$

Case II- When u = 2, v = 1. Then, $\begin{matrix} | P f - P g | = 1 and | f - g | = \frac{1}{2} . \end{matrix}$

Case III- When u = 1, v = 2. Then, $\begin{matrix} | P f - P g | = \frac{15}{2} and | f - g | = \frac{5}{2} . \end{matrix}$

Case IV- When u = v = 1. Then, $\begin{matrix} | P f - P g | = \frac{3}{2} and | f - g | = 1 . \end{matrix}$ From all above cases, for Patient-I $\begin{matrix} d^{*} (P f, P g) & = \sup {| P f - P g | u, v \in U} \\ = 3 \geq 2 \times \frac{3}{2} \\ = α d^{*} (f, g); \end{matrix}$ and for Patient-II $\begin{matrix} d^{*} (P f, P g) & = \sup {| P f - P g | u, v \in U} \\ = \frac{15}{2} \geq 2 \times \frac{5}{2} \\ = α d^{*} (f, g); \end{matrix}$ where α = 2 >1.

Thus, all the conditions required for Corollary 2.6 are fulfilled. Therefore, there exists a unique fixed function f of $P$ that yields suitable doses for two patients at the same time.

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