On Unique and Nonunique Fixed Points and Fixed Circles in <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-5.229601pt" id="M1" height="21.4104pt" version="1.1" viewBox="-0.0657574 -16.1808 25.0168 21.4104" width="25.0168pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g198-14" d="M1048 672C1036 675 1028 674 1020 674C822 646 666 318 554 134C521 81 495 39 468 39C457 39 455 46 455 57C455 79 469 109 472 118C520 227 653 461 794 660L779 672C768 675 766 674 762 674C654 674 544 586 344 250C241 78 187 15 119 15C104 15 91 21 88 36H91C96 33 101 32 106 32C133 32 149 53 149 76C149 103 128 128 97 128C65 128 38 102 38 63C38 20 71 -15 118 -15C235 -15 307 132 473 397C542 508 638 615 712 644L713 643C646 565 574 455 521 373C460 281 378 144 378 66C378 23 399 -4 434 -4C518 -4 559 76 647 231C792 488 891 603 977 635L978 633C779 366 670 179 670 70C670 18 708 -12 756 -12C819 -12 883 42 959 134L940 150C868 66 810 18 770 18C751 18 737 28 737 53C737 111 865 359 1056 659L1048 672Z"/></g><g transform="matrix(.012,0,0,-0.012,18.396,-7.578)"><path id="g50-99" d="M460 334C460 396 434 451 378 451C330 451 241 408 148 300H146L237 679C241 697 241 710 232 710C213 710 153 684 67 675L66 646H95C141 646 145 642 134 595L39 170C23 97 31 54 46 33C64 8 100 -12 137 -12C178 -12 234 3 298 43C391 101 460 222 460 334ZM371 320C371 204 316 89 248 51C230 41 208 37 192 37C143 37 102 72 119 166C124 194 129 215 135 235C202 323 298 392 335 392C353 392 371 372 371 320Z"/></g><g transform="matrix(.012,0,0,-0.012,18.396,4.995)"><path id="g185-40" d="M219 86C216 168 211 250 206 337C201 410 189 448 163 448C131 448 79 396 43 344L60 322C91 359 110 375 118 375S132 358 136 298C141 238 152 81 155 -12H182C242 62 331 177 390 258C435 321 451 360 451 391C450 424 432 448 408 448C390 448 372 435 366 419C362 410 362 401 366 394C373 383 376 367 376 350C376 283 262 138 221 86H219Z"/></g></svg>-</span>Metric Space and Application to Cantilever Beam Problem

Joshi, Meena; Tomar, Anita; Nabwey, Hossam A.; George, Reny

doi:https://doi.org/10.1155/2021/6681044

Journal of Function Spaces

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Research Article | Open Access

Volume 2021 | Article ID 6681044 | https://doi.org/10.1155/2021/6681044

On Unique and Nonunique Fixed Points and Fixed Circles in -Metric Space and Application to Cantilever Beam Problem

Meena Joshi,¹Anita Tomar,²Hossam A. Nabwey,^3,4and Reny George^3,5

Academic Editor: Huseyin Isik

Received03 Jan 2021

Revised28 Jan 2021

Accepted05 Feb 2021

Published08 Mar 2021

Abstract

We introduce -metric to generalize and improve -metric and unify numerous existing distance notions. Further, we define topological notions like open ball, closed ball, convergence of a sequence, Cauchy sequence, and completeness of the space to discuss topology on -metric space and to create an environment for the survival of a unique fixed point. Also, we introduce a notion of a fixed circle and a fixed disc to study the geometry of the set of nonunique fixed points of a discontinuous self-map and establish fixed circle and fixed disc theorems. Further, we verify all these results by illustrative examples to demonstrate the authenticity of the postulates. Towards the end, we solve a fourth order differential equation arising in the bending of an elastic beam.

1. Introduction

A Greek mathematician Euclid of Alexandria (323-283 BC) was the first to communicate the notion of distance. Indeed, Euclidean distance is a noteworthy measure of closeness among two quantities, which is one of the initial conceptions appreciated by mankind. Fréchet [1] deliberated the general and more axiomatic form of distance as “-space.” Hausdorff [2] reexamined it as a metric space in the setting of points which has been refined, discussed, and generalized in numerous ways. Bakhtin [3], Branciari [4], Asadi et al. [5], George et al. [6], Mitrović and Radenović [7], Özgür et al. [8], Karahan and Isik [9], and Asim et al. [10] introduced the notions of a -metric, a rectangular metric and a -generalized metric, an -metric, a rectangular -metric, a -metric, a rectangular -metric, a generalized -partial metric, and an -metric, respectively. Further, one may also refer to Fernandez et al. [11] and Kanwal et al. [12, 13] for work in -cone metric spaces over Banach algebra, orthogonal -metric spaces, and weak partial -metric spaces, respectively. One may allude to Kirk and Shahzed [14], to study in detail about the generalizations of the metric notion.

The aim of this work is to introduce a novel distance structure called an -metric space which is an improvement and generalization of an -metric space. Also, we introduce topological notions like open ball, closed ball, convergence of a sequence, Cauchy sequence, and completeness of the space to discuss topology and to create an environment for the survival of a unique fixed point in an -metric space. Further, we demonstrate that the collection of open balls, which forms a basis on -metric space, generates a topology on it. In the sequel, with the help of examples and remarks, we demonstrate that an -metric space unifies and combines numerous distance conceptions and marks supremacy over all those spaces wherein the continuity of a map is required for the survival of a fixed point. We also introduce a notion of a fixed circle and a fixed disc to study the geometry of the set of nonunique fixed points of a discontinuous self-map in an -metric space and establish fixed circle and greatest fixed disc theorems. We verify these results by illustrative examples with geometric interpretations to establish the authenticity of the postulates. Towards the end, we solve fourth order differential equation arising in the Cantilever beam problem.

2. Preliminaries

In this section, we use notations and .

In 2017, Mitrović and Radenović [7] introduced -metric space.

Definition 1. A -metric on a nonempty set with is a map satisfying and are distinct. A pair is a -metric space.

Remark 2. A-metric may be reduced to a -generalized metric [4] for a rectangular metric [4] for and , a rectangular -metric [6] for a -metric [3] for , and a usual metric [1] for

In 2018, Karahan and Isik [9] introduced the -partial metric.

Definition 3. A partial -metric on a nonempty set with is a map satisfying and are distinct. A pair is a partial -metric space.

Remark 4. A partial -metric may be reduced to a rectangular partial metric [15] for and , a rectangular partial -metric for a partial -metric [16] for , and a partial metric [17] for
In 2019, Asim et al. [10] have familiarized the notion of an -metric space.

Definition 5. An -metric on a nonempty set is a map satisfying and are distinct. A pair is an -metric space.

Example 6 (see [10]). Let Define by, thenis an-metric space.

Remark 7. In the above example, if , will be an -metric [5]. In this case, for

If it will be a rectangular -metric [18]. Again, in this case, for ,

This suggests that if in an -metric, we subtract the term , then the function will no longer be a metric. However, the terms like are indispensable in the analogous generalizations of a partial metric as a self-distance of a point in this space is not essentially zero.

For instance, if , then for a finite and is not equal to zero.

Definition 8 (see [19]). A function is a subadditive altering distance function if (i) is an altering distance function (i.e., is continuous, strictly increasing, and iff )(ii)

3. Main Results

We use notations and . First, we introduce an -metric space.

Definition 9. An -metric on a nonempty set with is a map satisfying and are distinct. A pair is called an -metric space.

Example 10. Let and be defined by . By routine calculation, we verify that is an -metric space with . But is not an -metric space. Since, for and , we attain Therefore,

Also, if , is an improvement and extension of an -metric space [10]. Inclusion of the terms containing nonzero self-distances demonstrate that it is a proper generalization of a notion of an -metric space and consequently partial metric space as well. In particular, is an -metric [18] for and rectangular -metric space [20] for .

Theorem 11. Let be an -metric on a set . Let then is a generalized partial -metric on .

Proof. Let :

Let
So, i.e.,
Hence,
For and
Now, .
Symmetric condition is trivially satisfied.
For , Thus, is a generalized partial -metric space.

Theorem 12. Let be an -metric on a set . Let then is a -metric on .

Proof. For Since, i.e.,
Symmetric condition is trivially satisfied.
For , Thus, is a -metric.

Remark 13. Let be a partial -metric on a set , then is -metric space.
To discuss the topology corresponding to -metric, the open ball with centre at and radius is defined as
Similarly, the closed ball with centre at and radius is defined as

Lemma 14. In an -metric space every open ball is an open set.

Proof. Let then Choose
Again, let so and choose
Proceeding as above, let so , choose
Now, for Hence,

Theorem 15. If is an -metric space and a topology generated by the open balls , then is a -space.

Proof. Let be an -metric space and are two distinct points. Then, i.e., or
Firstly, assume that For if we chose such that so
Next, assume that then : Again, for if we chose such that so
Similarly, for one may easily find an open ball so that and , i.e., for two distinct points there is a ball including the point but not including the other point . Thus, is a -space.

Now, we discuss the convergence of the sequence and introduce definitions related to it.

Definition 16. (i)A sequence in is -convergent to iff In other words, a sequence in a topological space converges to a point in if for each open ball containing there exists so that for each . (ii)A sequence in is -Cauchy sequence iff and exist and are finite(iii)An -metric space is -complete if each -Cauchy sequence converges to a point so that and

Lemma 17. Let be an -metric space and a self-map on so that there exists , satisfying Consider the sequence defined by If as then as

Proof. If , i.e., as
Now, if , then by (15),

There are two cases.

Case 1. If , then as
Since we have Again, implies that
Since, so
Now, from equation (16),

Case 2. If , then as .
So, as above as
Now, we prove the first main result for a Kannan type contraction.

Theorem 18. Let be an -complete metric space with coefficient . Suppose, a self-map satisfies Then, has a unique fixed point so that and the sequence of iterates converges to .

Proof. Starting from the given element , form the sequence , where . If , then and , and this completes the proof.
Further, take , . For , utilizing condition (18), Let, then
On repeating these steps, we get
Now, First, we show that for Suppose, for then
Now, using inequality (18), for and , a contradiction. Thus, for

Now, we assert that is a Cauchy sequence in We discuss two cases.

Case 1. First, let be odd, i.e., , for Now, by using for i.e.,

Case 2. Now, let is even, i.e., for Now, by using for i.e.,
So,
Let Now, as
Hence,
Therefore, the sequence is -Cauchy in
Since, is -complete, there exists so that Now, we assert that : i.e., or .
Hence, i.e., is a fixed point of .
Now, we assert that : a contradiction. Hence,
Suppose, and are two different fixed points of , so Hence,

Now, we furnish two examples (one of continuous and another of discontinuous self-map) to validate Theorem 18 and demonstrate the fact that the continuity of a self-map is not indispensable for the survival of a unique fixed point in an -metric space for the Kannan type contraction. Also, the self-distance of a fixed point is zero, and the sequence of iterates converges to a fixed point.

Example 19. Let and an -metric be defined by Then, is a complete -metric space for . Define a self-map on as Observe that, for all we obtain i.e.,
Thus, all the postulates of Theorem 18 are verified and has a unique fixed point in and clearly, Further, there exists a sequence , which -converges to 0.

Example 20. Consider and an -metric defined by Then, is a complete -metric space for . Define a self-map on as

Observe that, for all we obtain the following.

Case 21. When

Case 22. When

Case 23. When i.e.,
Thus, all the postulates of Theorem 18 are verified and has a unique fixed point in , and clearly, Further, there exists a sequence , which -converges to .

Remark 24. Examples 19 and 20 demonstrate that Theorem 18 is an extension, generalization, and improvement of Asim et al. [10], Banach [21], Branciari [4], Kannan [22], Karahan and Isik [9], Mitrović and Radenović [7], Özgür et al. [8], Shukla [15, 16], and references therein to an -metric space. It is fascinating to see that the continuity of a self-map is not an indispensable requirement for the survival of a unique fixed point satisfying (18) in this novel space.

Our next result extends the result of Reich [23] to -metric space.

Theorem 25. Let be an -complete metric space with coefficient . Suppose a self-map satisfies Then, has a unique fixed point so that and the sequence of iterates converges to .

Proof. It follows the pattern of Theorem 18.
Theorem 25 is an extension, improvement, and generalization of Banach [21] (), Kannan [22] (), Reich [23], and Asim et al. [10] () to an -metric space.
Now, we prove the result for a Hardy and Rogers type contraction [24], which includes all the results stated above as a special case.

Theorem 26. Let be an -complete metric space. Suppose, a self-map satisfies Then, has a unique fixed point so that and the sequence of iterates converges to .

Proof. It follows the pattern of Theorem 18.

Theorem 26 is an extension, improvement, and generalization of Banach [21] (), Kannan [22] (), Chatterjee [25] (), Reich [23] (), Hardy and Rogers [24] (), Asim et al. [10] (), and references therein to an -metric space.

The following result is more fascinating as it is proved by altering the distances between the points, exploiting subadditive altering distance function [19].

Theorem 27. Let be a -complete metric space. Suppose, we have a self-map satisfying Then, has a unique fixed point so that and the sequence of iterates converges to and for

Proof. For an arbitrary let Then, Now, starting from the given element , form the sequence , where . If , then and , and this completes the proof.

Further, take , . For , utilizing condition (36),

Also,

So, following a similar pattern as that of Theorem 18, we may easily conclude that has a unique fixed point where

Finally, by taking , from (36),

Now, we furnish two examples of a discontinuous self-map to validate Theorem 27.

Example 1. Let and an -metric be defined as with .
Define a self-map by and as then observe that, for all , we obtain Thus, all the postulates of Theorem 27 are verified and has a unique fixed point in , and clearly, Further, there exists a sequence , which -converges to .

Example 2. Let and an -metric be defined as Define a self-map by and as then observe that, for all , we obtain Thus, all the postulates of Theorem 27 are verified and has a unique fixed point in , and clearly, Further, there exists a sequence , which -converges to 0.

Remark 28. Examples 1 and 2 demonstrate that Theorem 27 is an extension, generalization, and improvement of Asim et al. [10], Banach [21], Kannan [22], Hardy and Rogers [24], Karahan and Isik [9], Khan et al. [19], Mlaiki et al. [18], Mitrović and Radenović [7], Reich [23], and references therein to -metric space. It is fascinating to see that continuity of a self-map is not indispensable for the existence of a fixed point satisfying (34) in this novel space.

Corollary 29. The conclusion of Theorem 27 is true even if we replace (34) by

Proof. The proof follows, if we take, in (34).

Corollary 30. The conclusion of Theorem 27 is true even if we replace (34) by

Proof. Applying Theorem 27 to the self-map we get has a unique fixed point, say i.e., .
Now, so is a unique fixed point of .

Corollary 31. The conclusion of Theorem 27 remains true even if we replace (34) by

Proof. The proof follows the pattern of Theorem 27 if we take in (34).

Theorem 32. Let be an-complete metric space with coefficient Suppose satisfies Then, has a unique fixed point so that and the sequence of iterates converges to and for

Proof. The proof follows the pattern of Theorem 27.

Theorem 33. The conclusion of Theorem 27 remains true even if we replace inequality (34) by

Proof. The proof follows the pattern of Theorem 27.
We note that when , the above theorem is the same as Theorem 18.

4. Existence of a Fixed Circle/Disc

Following Özgür and Tas [26], we first familiarize fixed circle in an -metric space to establish fixed circle theorems. Next, we exploit -metric version (-Caristi map) of the classical Caristi map [27] to establish that the set of nonunique fixed points of a map includes a circle. Also, following Aydi et al. [28], we familiarize fixed disc to establish the greatest fixed disc in an -metric space.

Definition 34. We define a circle centred at and radius in an -metric space as

Definition 35. We define a disc centred at and radius in an -metric space as

Geometrically, it is not necessary that a circle/disc defined in a -metric space is the same as the circle/disc in a Euclidean space.

Definition 36. Let / be a circle/disc centred at and radius in an -metric space . For a self-map in an -metric space , if then / is called the fixed circle/fixed disc of

Theorem 37. Let be a circle in an -metric space . Define as If there exists a self-map so that (iv)If , then is a unique fixed circle of

Proof. Let be any arbitrary point. Using (i) and equation (54),
But by the definition of an -metric space, . So, Using (iii) . Again, by the definition of an -metric space, Hence, i.e., is a fixed point of So, the self-map fixes the circle , i.e., the set of nonunique fixed points of a map includes a circle.
Let there exist two fixed circles, and of i.e., satisfies all the conditions (i) to (iii) for each of the circles and . Let and Using (iv), a contradiction.

Hence, is a unique fixed circle of

The following example illustrates Theorem 37.

Example 38. Let and an -metric be defined as with The circle Now, we discuss circles for different values of : (i)When , then(ii)When , thenNow, define a self-map as Then, the self-map verifies all the postulates of Theorem 37 except (iv) and fixes the circle Clearly, the set of fixed points of contains a circle . However, one may notice that there are more than one circle corresponding to each value of For instance, is also a fixed circle of .
It is obvious that geometrically (i) implies that is in the exterior of a circle and (ii) implies that is in the interior of a circle. It means (see Example 38).

Theorem 39. The conclusion of Theorem 37 is true even if we substitute (i) by (i) and (ii) by (ii):

Proof. Let be any arbitrary point. Using (i) and equation (54) But by the definition of an -metric space, .
So, Using (63) and (iii), . Also, by definition of an -metric So, i.e., is a fixed point of i.e., the self-map fixes the circle , i.e., the set of fixed points of a map includes a circle.
Uniqueness of a fixed circle may be proved as in Theorem 37.

Example 40. Let and an -metric be defined as with There arises two cases: (i) ,(ii),,Define a self-map as Then, map verifies all the postulates of Theorem 39 and fixes the unique circle i.e., the set of nonunique fixed points of contains a unique circle
It is clear that geometrically (i) implies that is in the interior of a circle and (ii) implies that is in the exterior of a circle. It means (see Example 40).
In the above theorems, we have assumed Banach contraction to demonstrate the uniqueness of fixed circle. So, it is significant to establish the uniqueness of the fixed circle using different contractive conditions. In the following, we establish uniqueness using a more general contractive condition.

Theorem 41. Letbe an-metric space andbe a circle onLetbe a self-map satisfying (i)-(iii), (i)and (ii)of Theorem37or39along with the contraction condition where is a continuous nondecreasing function and then is a unique fixed circle of

Proof. Let and be two fixed circles of i.e., satisfies the conditions (i)-(iii), (i) and (ii) (Theorem 37 and 39) for both the circles and . Let and Using inequality (67), a contradiction. Hence, is a unique fixed circle of
Next, we establish the existence of a greatest fixed disc.

Theorem 42. Let be a disc in an -metric space. Define as in (54). If there exists a self-map so that (i) then is a fixed disc of (ii)If then is a fixed disc of maximum radius , i.e., there is no fixed disc of having a radius greater than

Proof. Let be any arbitrary point: But by the definition of an -metric space, .
So, Using (70) and (iii), . Also, by definition of an -metric So, i.e., is a fixed point of , i.e., the self-map fixes the disc i.e., the set of fixed points of a map includes a disc.
Let there exist two fixed discs and of i.e., satisfies the conditions (i) to (iii) for each of the discs and
Let and such that .
Using (iv), a contradiction. Hence, is a fixed disc of having maximum a radius

Remark 43. Following the pattern of Theorem 41, we may prove the existence of a greatest fixed disc of having a maximum radius.

Remark 44. For more work on the set of nonunique fixed points forming a circle or a disc or an ellipse, one may refer to [26, 28–31] and references therein.

5. Application to Cantilever Beam

As an application of Theorem 18, we solve fourth-order differential equation arising in two point boundary value problem of a bending of an elastic beam. This problem is employed in the distortion of an elastic beam in equilibrium, of which one end is free whereas another is fixed. Let and be the set of all continuous functions on Define -metric, as and .

Theorem 45. We consider the Cantilever beam problem where is a continuous function. If then Problem (72) has a unique solution.

Proof. The Cantilever beam problem (72) may be rewritten as where the Green function Define a map by Now, is a solution of (72) iff it is a fixed point of Equations (77) and (78) imply that i.e., satisfies Theorem 18, and hence, has a unique fixed point, i.e., a Cantilever beam problem has a unique solution.

6. Conclusion

We have introduced -metric as an improvement and generalization of an -metric which need not be continuous and defined topological notions like open ball, closed ball, convergence of a sequence, Cauchy sequence, and completeness of a space to discuss the topology of -metric and to create an environment for the survival of a unique fixed point in an -metric space. Further, we have demonstrated that the collection of open ball, which forms a basis on -metric space and generates a topology on it. Also, we have introduced a notion of a fixed circle and a fixed disc to study the geometry of the set of nonunique fixed points of a discontinuous self-map. Our results are sharpened versions of the well-known results, wherein continuity of a self-map is not indispensable for the survival of a unique fixed point. Examples and applications to solve a Cantilever beam problem employed in distortion of an elastic beam in equilibrium substantiate the utility of these improvements and extensions. It is interesting to mention that the Cantilever structures permit overhanging constructions deprived of peripheral bracing.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

All authors contributed equally to this research.

Acknowledgments

The authors are thankful to the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University, Al-Kharj, Kingdom of Saudi Arabia, for supporting this research.

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