Abstract
In this paper, we introduce a new concept of -admissible almost type -contraction and prove some fixed point results for this new class of contractions in the context of complete metric spaces. The presented results generalize and unify several existing results in the literature.
1. Introduction and Preliminaries
Fixed point theory is one of the fundamental research areas in nonlinear functional analysis, and it plays a powerful role in the resolution of several mathematical problems with potential applicability in various fields such economics and game theory (see, e.g., [1, 2]) and computer sciences (see, e.g., [3, 4]).
The well-known Banach contraction principle (BCP) is the most impressive theoretical development in the evolution of the metric fixed point theory, and this principle has been extended, improved, and generalized in different approaches (see [5–18]). In particular, in 2015, Khojasteh et al. [5] coined the concept of simulation functions and defined a new class of nonlinear contractions, namely, -contractions, which generalize the Banach contraction principle and unify several known types of contractions. For other results related to this interesting approach, refer [8, 17–21].
In [6], Samet et al. introduced the notion of --contractive-type mappings by defining the concept of -admissibility and using Bianchini–Grandolfi gauge functions, and the authors inspected for the existence and the uniqueness of fixed points for such mappings. Later on, Karapinar and Samet [7] generalized the results derived in [6] by proposing the concept of generalized --contractive type. In 2016, Karapinar [8] originated the concept of -admissible -contraction by combining the ideas in [5, 6] to obtain some interesting fixed point results in the context of complete metric spaces. In fact, Karapinar [8] proved, among other results, that several existing fixed point theorems can be expressed easily from the main results of [8].
In the present paper, we introduce a new concept of -admissible almost type -contraction with respect to a simulation function , and we prove some results about existence and uniqueness of fixed points for such mappings; the presented results unify several well-known types of contraction and generalize several existing results in the literature.
Let be the family of nondecreasing functions satisfying the following condition.
There exist and and a convergent series of nonnegative terms such that
The class of such functions is called as Bianchini–Grandolfi gauge functions [10, 22] or c-comparison functions [11].
Lemma 1 (see [11]). If , then the following holds:(i) converges to 0 as for all .(ii), for any .(iii) is continuous at 0.(iv)The series converges for all .
In 2012, Karapinar and Samet [7] introduced a new class of contractive mappings via the following concept of -admissibility presented in [6].
Definition 1 (see [6]). Let be a self-mapping and be a function. is said to be -admissible if
Example 1 (see [6]). Let , and we define the mappings and as follows. , andThen, is -admissible.
Definition 2 (see [7]. Let be a metric space and be a given mapping. We say that T is a generalized --contractive mapping if there exist two functions and such thatwhere
In 2014, Popescu [12] suggested the concept of triangular -orbital admissible as an improvement of the triangular -admissible notion proposed in [9].
Definition 3 (see [12]). Let be a mapping and be a function. Then, is said to be -orbital admissible ifMoreover, is called a triangular -orbital admissible if it satisfies the following conditions:(i): is -orbital admissible.(ii): and .On the contrary, Khojasteh et al. [5] defined a new family of contractions by using the following notion of simulation functions.
Definition 4 (see [5]). The function is said to be a simulation function, if it satisfies the following conditions:(i): .(ii): for all .(iii): if are sequences in such that , then .The set of all simulation functions is denoted by .
Example 2 (see [5]). Let be defined as follows:(1) for all , where are two continuous functions such that if and only if and for all .(2) for all , where is a continuous function such that if and only if .(3) for all , where are continuous functions with respect to each variable such that for all .(4) for are simulation functions. For other interesting examples of simulation functions, readers are referred to [5, 19, 20].
Definition 5 (see [5]). Let be a metric space, a self-mapping, and . We say that is a -contraction with respect to , if the following condition is satisfied:Now, we state the result proved in [5] as follows.
Theorem 1 (see [5]). Let be a complete metric space and be a -contraction with respect to a simulation function . Then, has unique fixed point in , and for every , the Picard sequence , where for all , converges to the fixed point of .
2. Main Results
First, we present the following concept.
Definition 6. Let be a metric space and . We say that is an -admissible almost -contraction if there exists and a constant such thatwhere
Remark 1. If is an -admissible almost -contraction with respect to , thenOur first result is the following theorem.
Theorem 2. Let be a complete metric space, and let be an -admissible almost -contraction with respect to and satisfying the following conditions:(i) is triangular -orbital admissible.(ii)There exists such that .(iii) is continuous.Then, there exists such that .
Proof. Using condition (ii), there exists such that , and let be the iterative sequence in defined by
If for some , then is a fixed point of . Therefore, to continue our proof, we suppose that for all . Since is an -admissible mapping, we have
By induction, we obtain that
Applying the condition (7) with and and using (12), we getwhere
By (13) and taking in account (12), (14), and (15), we derive thatfor all . Now, if for some , then from the above inequality, we getwhich is a contradiction. Therefore,
Hence,
Consequently, we deduce that the sequence is a decreasing positive real number. Thus, there exists such that . We claim that
On the contrary, assume that . It follows from the inequality (19) that
Now, if we take the sequences and and considering (21), then ; therefore, by , we get thatis a contradiction, we deduce that , and equation (20) holds.
Next, we show that is Cauchy sequence in , reasoning by the method of reductio ad absurdum. Suppose to the contrary that is not a Cauchy sequence. So, there exists ; for every , there exist such that and . In account of (20), there exists such that
We can find two subsequences and of such that andwhere is the smallest index satisfying (24). Then,
Now, using (24), (25), and the triangular inequality, we obtain
Letting and using equation (20), we derive that
Again, using the triangular inequality, we get
Also, we have
By taking the limit as on both sides of (28), (29), and using (20), we deduce that
By the same reasoning as above, we get that
As is triangular -orbital admissible, we have
Moreover, since is an -admissible almost -contraction with respect to , we obtain
Hence,for all , where
Taking the limit as in (35) and (36), using (20), (27), (30), and (31), we get
From (34), (37), and (38), we derive that as ; therefore by , we getwhich is a contradiction. It follows that is a Cauchy sequence in the complete metric space . Therefore, there exists such that
Furthermore, by the continuity of , we obtain that
Taking into account (40), (41), and the uniqueness of the limit, we deduce that is a fixed point of and .
Theorem 3. Let be a complete metric space and be an -admissible almost -contraction with respect to satisfying the following conditions:(i) is triangular -orbital admissible.(ii)There exists such that .(iii)If is a sequence in such that for all and as , then there exists a subsequence of such that for all k.Then, there exists such that .
Proof. Following the lines of the proof of Theorem 2, we obtain that the sequence defined by for all is a Cauchy sequence in . Since is complete, there exists such that as . By (12) and the condition (iii), there exists a subsequence of such that for all . Using (7), we obtain that
Hence,where
Letting in the above equalities, we get that
Suppose that . From (43), we derive that
Now, letting in the above inequality, taking into account (45) and (46), we obtain thatis a contradiction, and hence, , that is, .
To ensure the uniqueness of a fixed point of a -admissible almost -contraction with respect to , we shall consider the following condition: For all Fix (T), we have , where Fix (T) denotes the set of fixed points of .
Theorem 4. Adding condition to the hypotheses of Theorem 2 (resp., Theorem 3), we obtain the uniqueness of the fixed point of .
Proof. We argue by contradiction, suppose that there exist such that and with . From assumption , we have
Therefore, it follows from equation (7) and thatwhere
From (50), together with (51) and (52), we deduce that
Using (49), it follows thatwhich is a contradiction. Hence, .
3. Consequences
In this section, we will show that several known fixed point results can be easily concluded from our obtained results.
Corollary 1 (see Karapinar and Samet [7]). Let be a complete metric space. Suppose that is a generalized --contractive mapping and satisfies the following conditions:(i) is -admissible.(ii)There exists such that .(iii)Either is continuous or(iv)If is a sequence in such that for all and as , then there exists a subsequence of such that for all k.Then, there exists such that .
Proof. Taking and a simulation function defined by for all where , in Theorem 4, we obtain that
The mapping is an -admissible almost -contraction with respect to , and the conclusion follows.
Corollary 2 (see [7]). Let be a complete metric space and be a given mapping. Suppose that there exists a function such thatThen, has a unique fixed point.
Proof. It suffices to choose the mapping such that , for all and with in Theorem 4.
Corollary 3 (see Berinde [23]). Let be a complete metric space and be a given mapping. Suppose that there exists a function such that
Then, has a unique fixed point.
Corollary 4 (see Ćirić [13]). Let be a complete metric space and be a given mapping. Suppose that there exists a constant such thatfor all . Then, has a unique fixed point.
Proof. It is enough to choose the mapping such that , for all and with in Theorem 4.
Corollary 5 (see Hardy and Rogers [14]). Let be a complete metric space and be a given mapping. Suppose that there exists a constant with such thatfor all . Then, has a unique fixed point.
Corollary 6 (see Banach contraction principle). Let be a complete metric space and be a given mapping. Suppose that there exists a constant such thatThen, has a unique fixed point.
Corollary 7 (see Kannan [15]). Let be a complete metric space and be a given mapping. Suppose that there exists a constant such that
Then, has a unique fixed point.
Corollary 8 (see Chatterjea [16]). Let be a complete metric space and be a given mapping. Suppose that there exists a constant such that
Then, has a unique fixed point.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.