#### Abstract

We introduce some generalizations of Prešić type contractions and establish some fixed point theorems for mappings satisfying Prešić-Hardy-Rogers type contractive conditions in metric spaces. Our results generalize and extend several known results in metric spaces. Some examples are included which illustrate the cases when new results can be applied while old ones cannot.

#### 1. Introduction

The well-known Banach contraction mapping principle states that if is a complete metric space and is a self-mapping such that for all , where , then there exists a unique such that . This point is called the fixed point of mapping .

On the other hand, for mappings , Kannan [1] introduced the contractive condition: for all , where is a constant and proved a fixed point theorem using (2) instead of (1). The conditions (1) and (2) are independent, as it was shown by two examples in [2].

Reich [3], for mappings , generalized Banach and Kannan fixed point theorems, using contractive condition: for all , where are nonnegative constants with . An example in [3] shows that the condition (3) is a proper generalization of (1) and (2).

For mapping Chatterjea [4] introduced the contractive condition: for all , where is a constant and proved a fixed point result using (4).

Ćirić [5], for mappings , generalized all above mappings, using contractive condition: for all , where are nonnegative constants with . A mapping satisfying (5) is called Generalized contraction.

Hardy and Rogers [6], for mappings , used the contractive condition: for all , where are nonnegative constants with and proved fixed point result. Note that condition (6) generalizes all the previous conditions.

In 1965, Prešić [7, 8] extended Banach contraction mapping principle to mappings defined on product spaces and proved the following theorem.

Theorem 1. Let be a complete metric space, a positive integer, and a mapping satisfying the following contractive type condition: for every , where are nonnegative constants such that . Then there exists a unique point such that . Moreover if are arbitrary points in and for , then the sequence is convergent and .

Note that condition (7) in the case reduces to the well-known Banach contraction mapping principle. So, Theorem 1 is a generalization of the Banach fixed point theorem. Some generalizations and applications of Prešić theorem can be seen in [918].

The -step iterative sequence given by (8) represents a nonlinear difference equation and the solution of this equation can be assumed to be a fixed point of ; that is, solution of (8) is a point such that . The Prešić theorem insures the convergence of the sequence defined by (8) and provides a sufficient condition for the existence of solution of (8) in the case when mapping satisfies the condition (7). A condition, independent from (7); namely, the Prešić-Kannan condition, is considered in [11] (for the proof of independency of these conditions in case , we refer [1, 2]). In this paper, we introduce some generalizations of Prešić type contractions in metric spaces and use a more general condition; namely, the Prešić-Hardy-Rogers type condition, to prove the existence of fixed point of in metric spaces. We note that this condition generalizes the result of Prešić [7, 8], Păcurar [11], Hardy and Rogers [6], and several known results in metric spaces. Some examples are included which illustrate the cases when new results can be applied while old ones cannot.

#### 2. Some Generalizations of Presić Type Contractions

In this section, we introduced some Prešić type contractions in metric spaces.

Let be a metric space, a positive integer, and be a mapping. (i) is said to be a Prešić contraction if satisfies the condition (7). (ii) is said to be a Prešić-Kannan contraction (see [11] for detail) if satisfies following condition: for all , where (iii) is said to be a Prešić-Reich contraction if satisfies following condition: for all , where are nonnegative constants such that (iv) is said to be a Prešić-Chatterjea contraction if satisfies following condition: for all , where (v) is said to be a Generalized-Prešić contraction if satisfies following condition: for all , where are nonnegative constants such that (vi) is said to be a Prešić-Hardy-Rogers contraction if satisfies following condition: for all , where are nonnegative constants such that

Remark 2. Note that for for all with and for all , the Prešić-Hardy-Rogers contraction reduces into the Generalized-Prešić contraction. With , the Generalized-Prešić contraction reduces into the Prešić-Reich contraction and with for all ,   for all , and , the Generalized-Prešić contraction reduces into the Prešić-Chatterjea contraction. With for all , the Prešić-Reich contraction reduces into the Prešić-Kannan contraction and with for all , the Prešić-Reich contraction reduces into the Prešić contraction. Therefore among all above definitions, the Prešić-Hardy-Rogers contraction is the most general contraction.

Remark 3. It is easy to see that for , Prešić-Hardy-Rogers contraction reduces into Hardy-Rogers contraction and for , Generalized-Prešić contraction reduces into Generalized contraction and so forth; therefore, the comparison as considered in [19] shows that the above generalization is proper.

Now, we shall prove some fixed point results for Prešić-Hardy-Rogers type contractions in metric spaces.

#### 3. Main Results

The following theorem is the fixed point result for Prešić-Hardy-Rogers type contractions and the main result of this paper.

Theorem 4. Let be any complete metric space, a positive integer. Let be a Prešić-Hardy-Rogers contraction, then has a unique fixed point in .

Proof. Let be arbitrary. Define a sequence in by If for any then is a fixed point of . Therefore we assume for all .
We shall show that this sequence is a Cauchy sequence in .
For simplicity, set For any , we obtain using (17), it follows from above inequality that that is, that is, where , and are the coefficients of , and , respectively, in the above inequality.
By definition, , therefore that is, Again, as , interchanging the role of and , and repeating above process, we obtain It follows from (26) and (27) that where .
Using (18), we obtain So . By (28), we obtain Suppose with . Then as , it follows from the above inequality that . Therefore is a Cauchy sequence. By completeness of , there exists such that .
We shall show that is the fixed point of . Note that using a similar process as used in the calculation of , we obtain that is, Using the fact that , it follows from the above inequality that Thus is a fixed point of . For uniqueness, let be another fixed point of , that is, . Again using a similar process as used in the calculation of , we obtain as , we obtain , that is, . Thus fixed point is unique.

Remark 5. For in the above theorem, we obtain the result of Hardy and Rogers [6]. For for all , we obtain the fixed point result of Prešić. Therefore, above theorem is a generalization of the results of Hardy and Rogers and Prešić.

With Remark 2, the following corollaries are obtained.

Corollary 6. Let be any complete metric space, a positive integer, and a Generalized Prešić contraction. Then has a unique fixed point in .

For in above corollary, we obtain the fixed point result of Ćirić [5].

Corollary 7. Let be any complete metric space, a positive integer, and a Prešić-Reich contraction. Then has a unique fixed point in .

For in the above corollary, we obtain the fixed point result of Reich [3].

Corollary 8. Let be any complete metric space, a positive integer, and a Prešić-Kannan contraction. Then has a unique fixed point in .

For in above the corollary, we obtain the fixed point result of Kannan [2].

Corollary 9. Let be any complete metric space, a positive integer, and a Prešić-Chatterjea contraction. Then has a unique fixed point in .

For in above corollary, we obtain the fixed point result of Chatterjea [4].

The following are some examples which illustrate the cases when known results are not applicable, while our new results can be used to conclude the existence of fixed point of mapping.

Example 10. Let with usual metric. For define by Then (i) is a Prešić-Reich contraction with ; (ii) is not a Prešić contraction; (iii) is not a Prešić-Kannan contraction.

Proof. (i) Note that for with , Therefore conditions (11) and (12) are satisfied for and with .
If any one of is then proof is similar. If any two of are , for example, if and , then As , so conditions (11) and (12) are satisfied for and with .
Similarly in all possible cases conditions (11) and (12) are satisfied with , . Therefore is a Prešić-Reich contraction. All other conditions of Corollary 7 are satisfied and is the unique fixed point of .
(ii) Note that for and Therefore, we cannot find nonnegative constants such that condition (7) is satisfied with . So is not a Prešić contraction.
(iii) Again for , Therefore, we cannot find nonnegative constant such that conditions (9) and (10) are satisfied. So is not a Prešić-Kannan contraction.

Remark 11. In the above example, we cannot apply the result of Prešić [7, 8] and Păcurar [11] to conclude the existence of fixed point of . But Corollary 7 is applicable which insures the existence of unique fixed point of .

Example 12. Let with usual metric. For , define by Then (i) is a Prešić-Chatterjea contraction with ; (ii) is not a Prešić contraction; (iii) is not a Prešić-Kannan contraction.

Proof. (i) Note that if or any one of is , then conditions (17) and (18) are satisfied trivially.
If any two of are , for example, if , , then Therefore conditions (13) and (14) are satisfied with . Also all other conditions of Corollary 9 are satisfied and has a unique fixed point .
(ii) For , , , we have Therefore we cannot find nonnegative constants such that condition (7) is satisfied with . So is not a Prešić contraction.
(iii) For , , we have Therefore we cannot find nonnegative constant such that conditions (9) and (10) are satisfied. So is not a Prešić-Kannan contraction.

Remark 13. In the above example, we cannot apply the result of Prešić [7, 8] and Păcurar [11] to conclude the existence of fixed point of . But Corollary 9 is applicable which insures the existence of unique fixed point of .

The following theorem is a consequence of Theorem 4 and the recent result of Aydi et al. [20].

Theorem 14. Let be any complete metric space and a positive integer. Let and be two mappings such that the following condition holds: for all , where are nonnegative constants such that and is continuous, injective, and sequentially convergent. Then has a unique fixed point in .

Proof. Define a mapping by Then is a complete metric space (see [20]). Note that condition (46) reduces to the condition (17); that is, mapping reduces to Prešić-Hardy-Rogers contraction with respect to metric . So the rest of the proof followed Theorem 4.

#### Acknowledgment

The first author is thankful to Mr. Rajpal Tomar for his help in typing the manuscript.