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Some New Results of Interpolative Hardy–Rogers and Ćirić–Reich–Rus Type Contraction
In this paper, we present new concepts on completeness of Hardy–Rogers type contraction mappings in metric space to prove the existence of fixed points. Furthermore, we introduce the concept of -interpolative Hardy–Rogers type contractions in -metric spaces to prove the existence of the coincidence point. Lastly, we add a third concept, interpolative weakly contractive mapping type, Ćirić–Reich–Rus, to show the existence of fixed points. These results are a generalization of previous results, which we have reinforced with examples.
1. Introduction and Preliminaries
The theory of fixed points has known a lot of evolution. It has been given great merit and concern, thanks to its many uses in several fields of mathematics, such as differential equations, graph theory, and nonlinear analysis [1–8]. Besides, the emergence of the fixed theorem with Banach  in 1922 on complete normed space was followed by several improvements and generalizations of this theorem on two levels: the first level is related to the applications used, and the second to the spaces used in them. It first knew improvements with Kannan  in 1968, and later with other researchers such as Rus, Ćirić, Reich, Hardy, and Rogers. Afterwards, it took another turning with Karapinar  in 2018 in a new version, which has made several researchers pursue this field (see [11–19]). Thus, the concept has been applied in various spaces: metric space, -metric space, rectangular -metric spaces, and the Branciari distance. More recently, Errai et al.  have inserted -interpolation over Ćirić–Reich–Rus type contraction. They have also introduced the concept of interpolative weakly contractive mapping, which makes us use these two concepts in this paper: the first concept on Hardy–Rogers type contraction and the second on Ćirić–Reich–Rus type contraction as a generalization of the previous findings, reinforced by various examples. This leads us to come up with some remarks. Before starting, we will take some basic concepts that we will use in this article.
Definition 1. (see [20, 21]). Let be a given real number and be a nonempty set. A function is a -metric if the following conditions are met for all if and only if The pair is called a -metric space.
It is worth mentioning that -metric spaces are a broader category than metric spaces.
The definitions of -convergent and -Cauchy sequences, as well as -complete -metric spaces, are defined in the same way as usual metric spaces (see, e.g., ).
Remark 1. (see ). The following assertions apply in a -metric space :(1)Each -convergent sequence is a -Cauchy sequence.(2)A -convergent sequence has a unique limit.(3)In general, a -metric is not continuous.
To prove our results, the fact in the previous remark necessitates the following lemma regarding -convergent sequences:
Lemma 1. (see ). Let be a -metric space with , and assume that and are -convergent to , respectively, so we haveIn particular, if , then we have . In addition, for each , we have
Lemma 2. Let be a sequence defined on a -metric space and meets the conditions:(i) is -convergent sequence in (ii), where Then, is a -Cauchy sequence in .
Proof. Let ; using the triangle inequality of the -metric space and condition (ii), we haveSince for each , then , which implies for any finite integer :Then, is a -Cauchy sequence in . □
The set of functions which satisfies for all is denoted by .
Definition 4. Consider the metric space . If there exist with , the self mapping is named a -interpolative Hardy–Rogers type contraction, such thatfor all , where and .
The following is our key finding:
Theorem 1. In a complete metric space , a -interpolative Hardy–Rogers type contraction , we assume there exists such that . Then, has a fixed point in .
Proof. Let be the sequence defined by and for all integer . If there exists such that , then is a fixed point of . The proof is complete. Suppose that for all .
By substituting the values and in (5), we haveUsing the fact for each , we obtainIf for some , thenFrom (7), we obtainThus,which implieswhich contradicts with and . Then, for all , and the sequence . From (7), we deduceSince , so there exists a real such that and .
By (12), we obtainAssume that there exists a real such thatFrom (12), we deducewhich giveswhere for all with .
Since ; we have .
Consequently,which is convergent, so is Cauchy sequence in , and then it converges to some . Suppose that , we find by (5):Passing the limit as , we get . So which is a contradiction. Then, .
Example 1. Let be endowed with metric , defined byLet be defined asand the function for all .
Choose ; ; ; and .
We have for all .
The following issues are discussed:
First case: if or for all , we have for all , and for all or for all .
Consequently, in this case, inequality (5) is satisfied.
Second case: if and , we haveThird case: if and with , we haveFourth case: if and with , we haveHence, in all cases, we havefor all .
As a result, all the conditions of Theorem 1 are fulfilled, and has a fixed point, .
Example 2. Let be endowed with the metric given in the following chart (Table 1).
Consider the self mapping on asTake , for all ; ; ; and .
We havefor all .
Then, has two fixed points, which are and .
If we replace with in Theorem 1, we get the following corollary.
Corollary 1. Let be a complete metric space and is self mapping on such thatis satisfied for all , where and such that .
If there exists such that , then has a fixed point in .
Definition 5. Let be a -metric space and be self mappings on . We say that is a -interpolative Hardy–Rogers type contraction if there exist and such thatis satisfied for all such that , and .
Theorem 2. In a -complete -metric space , if is a -interpolative Hardy–Rogers type contraction such that(1)(2) is closedThen, and have a coincidence point in .
Proof. Let , since , we can inductively define a sequence such thatIf there exists such that , then is a coincidence point of and . Assume that , for all . By (28), we obtainUsing the fact for each , we obtainSuppose that for some . Then,Thus, from inequality (31), we haveThis impliesSo, we getwhich is a contradiction. Thus,This means that the positive sequence is monotone decreasing, and consequently, there exists such that .
From (36), we haveSo, (31) together with the nondecreasing character of , we obtainSimilar to the previous method, we findLetting in (39) and using the fact for each , we deduce that , that is,By Lemma 2, we deduce that is a -Cauchy sequence, and consequently, is also a -Cauchy sequence. Let such that,And, since , there exists such that . We claim that is a coincidence point of and . Thus, if we assume that , we obtainConsequently,which is a contradiction. This implies thatThen, is a coincidence point in of and .
Example 3. Let the set and a function defined as follows (Table 2)
One can check that the function is a -metric for s = 2 by doing a simple calculation. Self-mappings on are defined asChoose ; and for all .
For all , it is obvious that fulfills (28). Furthermore, and are two coincidence points of and .
Example 4. Let and be defined byThen, is a complete -metric space.
Define two self-mappings and on by , for all and is a -interpolative Hardy–Rogers type contraction for ; ; and . Let , for all .
Hence, the following issues are discussed.
First case: if or for all , this is obvious.
Second case: if and .
ConsiderWe haveandTherefore,Third case: if and .
We haveandTherefore,Fourth case: if and , we haveUsing the property of , we getTherefore,For all , it is obvious that fulfills (28). Furthermore, one is a coincidence point of and .
The two previous examples lead us to the following remark.
Remark 2. In the Theorem 2, and do not need a fixed point, just as and accept a coincidence point and are not necessarily unique.
Definition 6. In a metric space , we say that a self mapping is an interpolative weakly contractive mapping type Ćirić–Reich–Rus, if there exists a constant such thatfor all , where is a lower semicontinuous function with if and only if , is a continuous monotone nondecreasing function with if and only if .
Theorem 3. In a complete metric space , if is a interpolative weakly contractive mapping type Ćirić–Reich–Rus, then has a fixed point.
Proof. For any , we consider a sequence by and ,
If there exists such that , then is clearly a fixed point in . Otherwise, for each .
Substituting and in (58), we obtain thatUsing property of function , we getWe deriveTherefore,It follows that the positive sequence is decreasing. Eventually, there exists such that .
Taking in inequality (60), we obtainWe deduce that . Hence,Therefore, is a Cauchy sequence. Suppose not, then there exists a real number , for any such thatPutting and in (58) and using (66), we obtainwhereandLetting and using (65), we concludeThen,which is contradiction with . Thus, is a Cauchy sequence, and since is complete, we obtain such that and assume that , we havewhereUsing (65), we getLetting in (72), we getwhich is a contradiction. Thus, . □
Example 5. Consider the space