Application and Theory of Functional Analytical Methods in SummabilityView this Special Issue
A Multiple Fixed Point Result for -Type Contractions in the Partially Ordered -Distance Spaces with an Application
In this manuscript, the aim is to prove a multiple fixed point (FP) result for partially ordered -distance spaces under -type weak contractive condition. The result will generalize some well-known results in literature such as coupled FP (Guo and Lakshmikantham, 1987), triple fixed point (Berinde and Borcut, 2011), and quadruple FP results (Karapinar, 2011). Moreover, to validate the result, an application for the existence of solution of a system of integral equations is also provided.
In pure mathematics, the Banach fixed-point theorem  (contractive mapping theorem or also known as the contraction mapping theorem) is main result in the study of metric spaces; it assurances the uniqueness and existence of FP of certain self-maps of metric spaces and requires a constructive technique to discover those FP. It can be understood as an abstract formulation of Picard’s method of successive approximations. The theorem is named after Stefan Banach (1892-1945) who first stated it in 1922. It has numerous applications in different fields such as computer science, physics, engineering, and various branches of mathematics itself. FP theory as a whole got an upward flight after this celebrated result.
Many authors started working in this field, and soon it became a hot field of research. A number of authors have extended this fundamental result to nonlinear analysis . Following this streak, Guo and Lakshmikantham  established the idea of doubled FP. This is considered to be the first of its nature and was extended to triple FP by Berinde and Borcut . Continuing in this direction, Karapinar  used four variable to strengthen the idea of quadruple FP in partially ordered metric spaces. In 2012, Berzig and Samet  discussed the existence of the fixed point of -order for -mixed monotone mappings in complete ordered metric spaces. In the same year, Roldán et al.  extended the notion of the FP of -order to the -fixed point and obtained some -fixed point theorems for a mixed monotone mapping in partially ordered complete metric spaces. In , Karapinar et al. studied the existence and uniqueness of a FP of the multidimensional operators which satisfy Meir-Keeler type contraction condition. Soon after, a number of articles were published to discussed the concept of a “ multidimensional FP” or “an -tuples fixed point.” For applications of such results, we refer the reader to  and the references cited therein.
In 2016, Choban and Berinde  generalized metric spaces to distance spaces. They established multidimensional FP results for ordered spaces with distance under certain contractive conditions [4, 11]. They pointed out that the concept allowed them to reduce the multidimensional case of FP and coincidence points to the one dimensional case. Recently, Rashid et al.  established some multiple FP findings for the -distance spaces in the existence of various contractive mapping.
The abovementioned ideas serve as motivation of the work in the present paper. We have extended the results of  for partially ordered -distance space, in terms of a significant multiple FP result under -type contractive condition . In support of this result, an example is also given.
Since, in Section 1, introduction and historical background of generalized metric spaces is given. In Section 2, preliminaries and some basic definitions are stated. In Section 3, main result is stated, and in Section 2, some consequences and examples of our main result will be described. In Section 3, an application is stated to support our main result. In the last section, article is concluded.
Definition 1 . Consider the , as a nonempty set and a function is called -distance on if for all , satisfies the following axioms: (1)(2) if and only if (3)For a positive real number An -distance space is said to be a symmetric -distance space if . Now, some remarks and examples as given below.
Remark 2. (1)Every -metric space is an -distance space but not conversely(2)In -distance space, distance is not necessarily a continuous function, i.e., if and then (3)In an -distance space, the limit of a convergent sequence may not be unique
Example 3. Let , and for all Hence, is an -distance space with . However, it is not a -metric space.
Fix and is said to be a collection of mappings such that
Let be a distance space and also be a mapping. The mapping which is a composition of and is gvien as for any point and . A point is considered as a -multiple FP of if it is a FP of , i.e., and
Let be a distance space, . On , consider the distance
Obviously, is a distance space, too.
The following are some basic concepts from :
Consider a partially ordered distance space , be a positive integer and be a partition of , i.e., , , and . Define … the Cartesian product of the set . Define a partial order over as follows:
For any , if and only if for all , where
The function given by defines a distance on , where and . Obviously, is a partially ordered distance space, so that it inherits the properties of and if and only if as for all
Definition 4. . Let be a self mapping on . A mapping has the mixed -monotone property with respect to the partition , if is -monotone nondecreasing in arguments of and -monotone nonincreasing in arguments of , i.e., , , , and
If is the identity mapping on , then the mapping has the mixed monotone property with respect to the partition . Define a set of mappings by such that if and if
Definition 5. If a function is continuous, nondecreasing, and , then it is called an altering distance function.
Definition 6. The metric space is called regular if it satisfies the following properties: (1)If is a nondecreasing sequence and then for all (2)If is a nonincreasing sequence and then for all
3. Main Result
Berinde and Borcut  extended the concept of multidimensional FP to ordered distance spaces by utilizing the properties of contractive type mappings. Keeping ourselves in touch with all these concepts, we are extending these results to symmetric -distance spaces by using a combination of altering distance functions. This result will generalize the main theorems of , in which the space considered is a metric space. It is also valid for -metric spaces.
Theorem 7. Consider a complete partially ordered symmetric -distance space.
and be collection of mappings verifying if and if If the mapping satisfies the following conditions: (a)For ()
where is an altering distance function and and are upper semicontinuous and increasing functions such that satisfying for all with (b)There is such that for all (c) has mixed monotone property with respect to (d) is continuous, or is regular; then, has -multiple FP(e)Moreover, if for and in , there is such that and ; then, has a unique -multiple FP
Proof. Step 1. Let be the th Picard iterate of under , i.e., , where
By condition and definition of , it follows that Since, has mixed monotone property so is monotone nondecreasing ; therefore
Step 2. We need to show that . Set and If for some then which means has -multiple FP which completes the proof. Assume for all . Since and , it follows that
for any and . Now, using condition (12)
Since, and both are increasing and so that
for any , since is a finite set, there is an index such that From above inequality, it follow that
for all Therefore, from the inequality we get
Combining (19) and (20), we have
for all . Since is an altering distance function, it follows that
Hence, the sequence and are monotone decreasing and bounded below. So, we have such that
We need to show that . Suppose , then by applying limit as in (19) and utilizing the properties of and , we have , which contradicts . Hence
Step 3. Now to prove that the sequence is Cauchy. Suppose on contrary that it is not Cauchy, then there exists for which there are subsequences and of with such that
Let be the smallest integer satisfying above, then
Assume that , then consider
On letting and using condition (24), we get
which leads to a contradiction. Now consider , then
Again letting and using condition (24), we get
From , it follows that
Since and , we have
for any and By condition , it follows that
for any . Since is finite, there will be an index in , so
From inequality (29)
Since and are increasing functions and , so
Applying over above inequality and then using (29), we get
which is a contradiction to the condition Hence, is a Cauchy sequence.
Step 4. Since the space is complete so is complete. Therefore, we have such that, i.e., Next, we show that is a -multiple FP of operator . If condition holds and is continuous, then Above shows that which means the point is a -FP of . Next, suppose is regular. The above relation implies . On the other hand Since, for any . By using Taking into account above and letting in the last inequality, it implies Hence This completes the existence of -multiple FP.
Step 5. In this step, we will show that the FP of is unique.
Suppose be another FP of . By condition , there exists such that and .
Put and define By the induction method, we have for all . By condition , . Assume that above condition holds for .
Using the procedure of Step 1, it can be shown that for all ; that is, . Similarly, we can prove the second inequality. Further, we prove that For this, we first show that if for some then , for all . Indeed, from above condition, it follows that for all and . From , it follows that for all and . Recall that . Hence for all . Taking into account , it follows for all and . Combining (50) and (52), we get for . Since, is an altering distance function Now, it is obvious that if , then for all
Next, assume that for all Using the same manner adopted in Step 1, it can be proved that Hence, there exists such that In similar way, it can be shown On the other hand, Applying over the above inequality, it follows This implies Hence, has a unique multidimensional FP.☐
4. Some Consequences and Example of Theorem 7
In the following section, some important concept regarding the consequences of Theorem 7 are discussed which are in terms of the main results represented in the articles [4, 16]. An illustrative example is also added in this section which will be helpful for the readers to understand the structure of multidimensional FP under the weak contractions for partially ordered -distance spaces.
Corollary 8. Let is known as complete partially ordered -distance space and be a collection of mappings verifying if and if Assume that the mapping satisfies the following conditions: (1)If there exists such thatfor all and with satisfies conditions. (2)To of the above theorem then has a unique multidimensional FP
Proof. We can prove this corollary easily by taking , , and in the above theorem.☐
Remark 9. Theorems 2.1 and 2.2 of , in -distance spaces, follow from Corollary 8. In , contractive condition is and for such that , it implies Applying Corollary 8, we get the desired result in -distance space.
Remark 10. Corollary 8 also generalizes the main triple FP result of , in which the space under consideration is a metric space. We generalize it for -metric space and for an -distance space. In , , is chosen as , collection of mappings is defined as the contractive condition in  is for any with and and Obviously Applying Corollary 8, we get the desired result in -distance space.
Corollary 11. Consider as a complete partially ordered -metric space and be collection of mappings verifying if and if Assume that the mapping satisfies conditions to , then has a unique multidimensional FP.
Example 12. Let , Define and , for all . Now, define , as .
Since “” is a partial order on , therefore where and partitions such that and
A function defined as