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 [1–23] 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 Ćirić-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 Ćirić-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, .
That is, , a contradiction.
Hence, has a unique fixed point in .
Theorem 8. Let be a complete metric space and map be a continuous Ćirić-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 Ćirić-type contraction [26] nor a Banach contraction [27].
Now, we claim that satisfies Ćirić-type -contraction; that is,
Case 1. When and ,
Case 2. When ,
Thus, is a Ćirić-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], Ćirić [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 Ćirić-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 .
If , , a contradiction.
If , , a contradiction.
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 Ćirić-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 Ćirić-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 Ćirić-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 , .
Case 1. For and ,and .
Case 2. For and ,That is, is a Ćirić-type -weak contraction with , , and . Hence, is a fixed ellipse and is a fixed elliptic disc of . One may verify that .
Now, we introduce -weak JS-contraction to study the geometry of nonunique fixed points.
Definition 19. Let be an increasing function with ; then, a map of a metric space is said to be a -weak JS-contraction with , ifwhere , , , , and are nonnegative and .
Theorem 20. Let be an ellipse in a metric space and . If map is a -weak JS-contraction with and , then is a fixed ellipse of .
Proof. Let be any arbitrary point and . From the definition of , since is -weak JS-contraction for , suppose and , so we have , , andSo . Similarly, .
Again, since , soSince and is increasing, , a contradiction.
Similarly, we can prove for .
Hence, ; that is, is a fixed ellipse of .
Theorem 21. If in Theorem 20, , then is a fixed elliptic disc of .
Proof. The proof follows the pattern of Theorem 16.
The subsequent example elucidates Theorems 20 and 21.
Example 6. Let and a metric be defined as . Let , , , , , and .
The ellipseDefine a self-map asThen, .
Then, .
Case 1. For and ,for , , , and , satisfying ; that is,
Case 2. For and ,for , , , and , satisfying ; that is,That is, is a -weak JS-contraction with , , and , , , and . Hence, is a fixed ellipse and is a fixed elliptic disc of . One may verify that . Noticeably, a fixed ellipse or a fixed elliptic disc is not essentially unique, as is also a fixed ellipse and is also a fixed elliptic disc of .
Now, we prove the fixed ellipse and fixed elliptic disc conclusions of the integra-type. Let be a locally integrable function so that, for each , .
Definition 22. Let be an increasing function. A map of a metric space is said to be a generalized integral-type -weak contraction with , ifwhere .
Theorem 23. Let be an ellipse in a metric space and . If map is a generalized integral-type -weak contraction with and , then is a fixed ellipse of .
Proof. The proof follows the pattern of Theorem 15.
Theorem 24. If in Theorem 23, , then is a fixed elliptic disc of .
Proof. The proof follows the pattern of Theorem 16.
Next, we introduce an integral-type -weak JS-contraction.
Definition 25. Let be an increasing function with ; then, map of a metric space is said to be an integral-type -weak JS-contraction with , if implies thatwhere , , , , and are nonnegative and .
Theorem 26. Let be an ellipse in a metric space and . If map is an integral-type -weak JS-contraction with and , then is a fixed ellipse of .
Proof. The proof follows the pattern of Theorem 15.
Theorem 27. If in Theorem 26, , then is a fixed elliptic disc of .
Proof. The proof follows the pattern of Theorem 16.
Remark 28. (1)It is interesting to mention here that if a self-map fixes an elliptic disc, then it also fixes an ellipse (see Examples 4, 5, and 6). However, the reverse may not be true(2)The ellipses (elliptic discs) in metric spaces change their shapes on changing the center, lengths of a semiminor or semimajor axis, or metric under consideration. Also, it is not necessary that an ellipse or an elliptic disc defined in a metric space be the same as an ellipse or an elliptic disc in a Euclidean space. Noticeably, ellipses discussed in Examples 1, 4, and 6 and elliptic discs in Examples 4 and 6 are different from the ellipse and elliptic disc in a Euclidean space(3)If there exists a self-map that maps the ellipse (elliptic disc) to itself, then that ellipse (elliptic disc) may not be a fixed ellipse (elliptic disc); that is, the self-map may not fix all the points of the ellipse (elliptic disc)(4)All the elliptic discs inside a fixed elliptic disc are also fixed elliptic discs of a self-map. Further, a fixed ellipse is not essentially unique (see Example 6)(5)If both the focuses coincide, then fixed ellipse results, as well as fixed elliptic disc results, reduce to analogous fixed circle and fixed disc results, respectively. Noticeably, if (say), then and , with center and radius (6)For work on the set of nonunique fixed points forming a fixed figure, one may refer to [1–23] and references therein
3. Applications
Inspired by the fact that the fixed point theory is applicable in many real-world problems, we solve a boundary value problem arising when a diffusing matter is kept in an absorbing medium between parallel walls of fixed concentrations and . The concentration of the substance at time is given bywhere is the absorption coefficient and is the source density.
The Green function associated with an initial value problem (48) is
Problem (48) is equivalent to
Let be a set of Riemann integrable functions on ; that is, . Define bywhere . Clearly, is a complete metric space.
Theorem 29. Consider the boundary value problem (48). Let be a self-map in a complete metric space , satisfyingThen, the boundary value problem (48) has a solution.
Proof. Define a map byClearly, a fixed point of map is a solution to the problem (48).
Since ,If , thenwhere and . Therefore, all the conditions of Theorem 8 are verified. Hence, has a unique fixed point, which is indeed a unique solution to a boundary value problem (48).
4. Conclusion
We have explored new directions as a fixed ellipse and a fixed elliptic disc to the geometry of the set of nonunique fixed points of a map on a metric space via novel contractions. Further, we have utilized a generalized -contraction and Ćirić-type -contraction to establish a unique fixed point. Also, we have verified established results by illustrative examples to demonstrate the authenticity of the postulates and substantiated utility of our results by solving a boundary value problem of chemical sciences. The study of a set of unique and nonunique fixed points in the current context would be an interesting area for future study.
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
All authors contributed equally to this research.