#### Abstract

We introduce the notions of a generalized -contraction, a generalized -weak contraction, a -weak JS-contraction, an integral-type -weak contraction, and an integral-type -weak JS-contraction to establish the fixed point, fixed ellipse, and fixed elliptic disc theorems. Further, we verify these by illustrative examples with geometric interpretations to demonstrate the authenticity of the postulates. The motivation of this work is the fact that the set of nonunique fixed points may include a geometric figure like a circle, an ellipse, a disc, or an elliptic disc. Towards the end, we provide an application of -contraction to chemical sciences.

#### 1. Introduction and Preliminaries

The study of the geometry of the set of nonunique fixed points of a map is a significant area of research. There are numerous examples of a map where the set of nonunique fixed points of the self-map includes some geometric shapes. For example, consider a self-map on the metric space with the usual metric defined on the two-dimensional plane as

Noticeably, the set of nonunique fixed points includes the circle centered at having radius ; that is, is a fixed circle of . It is significant to mention that there exist maps that map the circle to itself but do not fix all the points of the circle . For example, let be a self-map on the two-dimensional plane defined by

Then, , but map fixes only two points and of the circle . Noticeably, does not fix all the points of . For details on this work, one may refer to [1â€“23] and the references therein. A geometric figure (a circle, a disc, an ellipse, and so on) included in the set of nonunique fixed points is called a fixed figure (a fixed circle, a fixed disc, a fixed ellipse, and so on) of the self-map [15].

The aim of the present work is to introduce notions of a generalized -contraction, a generalized -weak contraction, a -weak JS-contraction, a generalized integral-type -weak contraction, and an integral-type -weak JS-contraction to study the geometry of nonunique fixed points. In the sequel, we establish the fixed point, fixed ellipse, and fixed elliptic disc theorems. Further, we verify these by illustrative examples to demonstrate the authenticity of the postulates. Further, we provide an application of Ä†iriÄ‡-type -contraction to obtain an application to chemical sciences. To be specific, we solve a boundary value problem arising when a diffusing material is kept in an absorbing medium between parallel walls of specified concentrations.

Definition 1 [24]. A metric on a nonempty set is a function satisfying(i) iff (ii)(iii)

Definition 2 [5]. An ellipse having foci at and in a metric space is defined as

The midpoint of a line is known as a center of an ellipse. Here, the segment of length on line is the major axis, the line perpendicular to it through the center is the minor axis, and is the length of a semimajor axis of an ellipse. The distance is the linear eccentricity, and the ratio of linear eccentricity and semimajor axis is the eccentricity; that is, . Visibly, the circles are the ellipses of vanishing eccentricity in which both the focal points are the same; that is, . Actually, an ellipse is a compressed circle. Generally, eccentricity is the measure of the deviation of the curve from the circularity of the particular shape.

Example 1. Let and a metric be defined as ; then,That is, an ellipse centered at 7.5 having foci at 5 and 10 is .

Definition 3 [25]. Let symbolize the class of functions such that the subsequent conditions hold:
: is nondecreasing;
: for every sequence , ;
: there exist and such that .

Definition 4 [5]. Let be an ellipse having foci at and in a metric space . Then, is said to be a fixed ellipse of if .

#### 2. Main Results

In this section, we are dealing with maps satisfying some novel contractions which fix one element of the space or more than one element of the space under suitable conditions and a set of nonunique fixed points, including some geometrical shapes, may be either an ellipse or an elliptic disc. First, we define a generalized -contraction to establish a unique fixed point by giving a short and simple proof.

Definition 5. Let , and the map of a metric space is said to be a generalized -contraction with ifwhere .

Remark 6. In the above contraction, if , then is a Ä†iriÄ‡-type -contraction.

Theorem 7. Let be a complete metric space and map be a continuous generalized -contraction. Then, has a unique fixed point. Also, the sequence of iterates converges to a fixed point of in .

Proof. Define a Picard sequence , , , with initial point . If for some , , then is a fixed point of and the proof is complete. So, presume that for each , ; then,where

Case 1. If , thenThat is, , a contradiction.

Case 2. If , thenThat is, .

Following a similar pattern,

Using , .

Using , there exist such that .

If , then for , there exists such that

That is, .

If , then for any , there exists such that

That is, .

Thus, for all and , there exists such that

That is, implies that there exists such that

If ,

Since , series is convergent and exists and is finite; that is, is a Cauchy sequence.

Since is complete, converges to . Since is continuous, . By definition of limit , that is, is a fixed point of .

Let be another fixed point of . So . Now, where

That is, .

Hence, has a unique fixed point in .

Theorem 8. Let be a complete metric space and map be a continuous Ä†iriÄ‡-type -contraction. Then, has a unique fixed point. Also, the sequence of iterates converges to a fixed point of in .

Proof. The proof follows the pattern of Theorem 7 on taking .

The subsequent example appreciates that Theorem 8 gives assurance of the uniqueness of the fixed point.

Example 2. Let and a metric be defined as . Then, is a complete metric space.
Let .
Define a self-map asThen,Now,Clearly, is neither a Ä†iriÄ‡-type contraction [26] nor a Banach contraction [27].
Now, we claim that satisfies Ä†iriÄ‡-type -contraction; that is,

Case 1. When and ,

Case 2. When ,

Thus, is a Ä†iriÄ‡-type -contraction with and has a unique fixed point 1. Further, .

Remark 9. Theorems 7 and 8 are improvements, extensions, and generalizations of Banach [27], Ä†iriÄ‡ [26], Jleli and Samet [25], and references therein. Further, on taking , , in these results, we obtain some novel results which are generalizations of existing results in the literature.

Next, following Joshi et al. [5], we define an elliptic disc and a fixed elliptic disc to study the geometry of nonunique fixed points in a metric space.

Definition 10. An elliptic disc having foci at and in a metric space is defined as .

Remark 11. For defining an ellipse or elliptic disc .

Example 3. Let and a metric be defined as ; then,which is shown by the blue shaded portion in Figure 1.

Definition 12. Let be an elliptic disc having foci at and in a metric space . Then, is said to be a fixed elliptic disc of map if .

We now introduce and exploit a generalized -weak contraction to demonstrate that the set of nonunique fixed points of a map includes an ellipse or an elliptic disc.

Definition 13. Let be an increasing function. A map of a metric space is said to be a generalized -weak contraction with , ifwhere .

Remark 14. In the above contraction, if , then is said to be a Ä†iriÄ‡-type -weak contraction.

Theorem 15. Let be an ellipse in a metric space and . If map is a generalized -weak contraction with and , then is a fixed ellipse of .

Proof. Let be any arbitrary point and . From the definition of , suppose and , so we have , , andSo . Similarly, .
Again, since ,

Case 1. If , then .
By definition of and , , a contradiction.

Case 2. If , then .

Case 3. If , then , a contradiction.

Case 4. If , thenBy definition of and , , a contradiction.

Case 5. If , thenBy definition of and , , a contradiction.

Similarly, we can prove for .

Hence, ; that is, is a fixed ellipse of .

Theorem 16. If in the above theorem , then is a fixed elliptic disc of .

Proof. Now, to show is a fixed elliptic disc of , it is sufficient to demonstrate that fixes an ellipse with . Since is a generalized -weak contraction, then proceeding as in Theorem 15, is a fixed ellipse of as ; that is, . Hence, is a fixed elliptic disc of .

Theorem 17. Theorem 15 remains true if we substitute Ä†iriÄ‡-type -weak contraction in place of generalized -weak contraction.

Proof. The proof follows the pattern of Theorem 15 on taking .

Theorem 18. Theorem 16 remains true if we substitute Ä†iriÄ‡-type -weak contraction in place of generalized -weak contraction.

Proof. The proof follows the pattern of Theorem 16 on taking .

The subsequent examples elucidate Theorems 17 and 18.

Example 4. Let and a metric be defined as .
Let , , , , , and .
The ellipseDefine a self-map as .
Since for , , and for , .

Case 1. For and ,and .

Case 2. For and ,and .
That is, is a Ä†iriÄ‡-type -weak contraction with , , and . Hence, is a fixed ellipse and is a fixed elliptic disc of . One may verify that .

Example 5. Let and a metric be defined as , where and .
Let , , , , , and .
The ellipsewhich is shown by the blue line in Figure 2.
Further, the elliptic disc , which is shown as the blue shaded portion in Figure 2.
Define a self-map as .
Since for , , and for ,