ON Modified (𝛼−𝜂)-Contractive Mappings

Abstract

Hussain et al. (2013) established new fixed point results in complete metric space. In this paper, we prove fixed point results of α-admissible mappings with respect to η, for modified contractive condition in complete metric space. An example is given to show the validity of our work. Our results generalize/improve several recent and classical results existing in the literature.

1. Preliminaries and Scope

The study of fixed point problems in nonlinear analysis has emerged as a powerful and very important tool in the last 60 years. Particularly, the technique of fixed point theory has been applicable to many diverse fields of sciences such as engineering, chemistry, biology, physics, and game theory. Over the years, fixed point theory has been generlized in many directions by several mathematicians (see [1–36]).

In 1973, Geraghty [12] studied different contractive conditions and established some useful fixed point theorems.

In 2012, Samet et al. [33] introduced a concept of -contractive type mappings and established various fixed point theorems for mappings in complete metric spaces. Afterwards Karapinar and Samet [10] refined the notions and obtained various fixed point results. Hussain et al. [17] extended the concept of -admissible mappings and obtained useful fixed point theorems. Subsequently, Abdeljawad [4] introduced pairs of -admissible mappings satisfying new sufficient contractive conditions different from those in [17, 33] and proved fixed point and common fixed point theorems. Lately, Salimi et al. [32] modified the concept of -contractive mappings and established fixed point results.

We define the family of nondecreasing functions such that , and for each where is the th term of .

Lemma 1 (see [32]). If , then for all .

Definition 2 (see [33]). Let be a metric space and let be a given mapping. We say that is an -contractive mapping if there exist two functions and such that for all .

Definition 3 (see [33]). Let and . One says that is -admissible if , .

Example 4. Consider . Define and by , for all and Then is -admissible.

Definition 5 (see [32]). Let and let be two functions. One says that is -admissible mapping with respect to if , . Note that if one takes , then this definition reduces to definition [33]. Also if we take , then one says that is an -subadmissible mapping.

2. Main Results

In this section, we prove fixed point theorems for -admissible mappings with respect to , satisfying modified ()-contractive condition in complete metric space.

Theorem 6. Let be a complete metric space and let is -admissible mappings with respect to . Assume that there exists a function such that, for any bounded sequence of positive reals, implies such that for all where ; then suppose that one of the following holds: (i)is continuous;(ii)if is a sequence in such that for all and as , then If there exists such that , then has a unique fixed point.

Proof. Let and define We will assume that for each . Otherwise, there exists an such that . Then and is a fixed point of . Since and is -admissible mapping with respect to , we have By continuing in this way, we have for all . From (7), we have Thus applying the inequality (3), with and , we obtain which implies that We suppose that Then we prove that . It is clear that is a decreasing sequence. Therefore, there exists some positive number such that . Now we will prove that . From (10), we have Now by taking limit , we have By using property of function, we have . Thus Now we prove that sequence is Cauchy sequence. Suppose on contrary that is not a Cauchy sequence. Then there exists and sequences and such that, for all positive integers , we have , By the triangle inequality, we have for all . Now taking limit as in (16) and using (14), we have Again using triangle inequality, we have Taking limit as and using (14) and (17), we obtain By using (3), (17), and (19), we have which implies that Therefore, we have Now taking limit as in (22), we get Hence , which is a contradiction. Hence is a Cauchy sequence. Since is complete so there exists such that . Now we prove that . Suppose (i) holds; that is, is continuous, so we get Thus . Now we suppose that (ii) holds. Since for all . By the hypotheses of (ii), we have Using the triangle inequality and (3), we have which implies that Letting then we have . Thus . Let there exists to be another fixed point of , s.t ; which implies that By the property of function, , implies ; then we have . Hence has a unique fixed point.

If in Theorem 6,   we get the following corollary.

Corollary 7 (see [17]). Let be a complete metric space and let be -admissible mapping. Assume that there exists a function such that, for any bounded sequence of positive reals, implies such that for all , where . Suppose that either(i) is continuous, or(ii)if is a sequence in such that for all and as , then If there exists such that ; then has a fixed point.

If in Theorem 6, we get the following corollary.

Corollary 8. Let be a complete metric space and let be -subadmissible mapping. Assume that there exists a function such that, for any bounded sequence of positive reals, implies such that for all where ; then suppose that one of the following holds:(i) is continuous;(ii)if is a sequence in such that for all and as , then If there exists such that , then has a fixed point.

Example 9. Let with usual metric for all and , and for all be defined by We prove that Corollary 7 can be applied to . Let ; clearly and , then of -admissible mapping , and , , and imply that If , then we have Let and ; then

Theorem 10. Let be a complete metric space and let be -admissible mappings with respect to . Assume that there exists a function such that, for any bounded sequence of positive reals, implies such that for all ; then suppose that one of the following holds:(i) is continuous;(ii)if is a sequence in such that for all and as , then If there exists such that , then has a fixed point.

Proof. Let and define We will assume that for each . Otherwise, there exists an such that . Then and is a fixed point of . Since and is -admissible mapping with respect to , we have By continuing in this way, we have for all . From (43), we have Thus applying the inequality (39), with and , we obtain which implies that We suppose that Then we prove that . It is clear that is a decreasing sequence. Therefore, there exists some positive number such that . Now we will prove that . From (47), we have Now by taking limit , we have By using property of function, we have . Thus Now we prove that sequence is Cauchy sequence. Suppose on contrary that is not a Cauchy sequence. Then there exists and sequences and such that, for all positive integers , we have , By the triangle inequality, we have for all . Now taking limit as in (52) and using (50), we have Again using triangle inequality, we have Taking limit as and using (50) and (53), we obtain By using (39), (53), and (55), we have which implies that Therfore, we have
Now taking limit as in (58), we get Hence , which is a contradiction. Hence is a Cauchy sequence. Since is complete so there exists such that . Now we prove that . Suppose (i) holds; that is, is continuous, so we get Thus . Now we suppose that (ii) holds. Since for all . By the hypotheses of (ii), we have Using the triangle inequality and (39), we have which implies that Letting , we have . Thus . Let there exists to be another fixed point of , s.t ; implies By the property of function, implies ; then we have . Hence has a unique fixed point.

If in Theorem 10,   we get the following corollary.

Corollary 11 (see [17]). Let be a complete metric space and let be -admissible mapping. Assume that there exists a function such that, for any bounded sequence of positive reals, implies such that for all . Suppose that either(i) is continuous, or(ii)if is a sequence in such that for all and as , then If there exists such that , then has a fixed point. Our results are more general than those in [17, 32, 33] and improve several results existing in the literature.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

All authors contributed equally and significantly to writing this paper. All authors read and approved the final paper.

Acknowledgments

Marwan Amin Kutbi gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. The authors thank the editor and the referees for their valuable comments and suggestions which improved greatly the quality of this paper.