Approximate Coincidence Point of Two Nonlinear Mappings
We study the approximate coincidence point of two nonlinear functions introduced by Geraghty in 1973 and Mizoguchi and Takahashi (-function) in 1989.
Fixed point theory has been an important tool for solving various problems in nonlinear functional analysis as well as a useful tool for proving the existence theorems for nonlinear differential and integral equations. However, in many practical situations, the conditions in the fixed point theorems are too strong and so the existence of a fixed point is not guaranteed. In that situation, one can consider nearly fixed points what we call as approximate fixed points. By an approximate fixed point of a function we mean in a sense that is “near to” . The study of approximate fixed point of a function we mean in a sense that is “near to” . The study of approximate fixed point theorems is equally interesting to that of fixed point theorems. Motivated by the article of Tijs et al. , Berinde  established some fundamental approximate fixed point theorems in metric space. In a recent paper, Dey and Saha  studied the existence of approximate fixed point for the Reich operator  which in turn generalizes the results of Berinde . Coincidence point theory has a vast literature, and many generalizations have been emerged so far (see [5–11]). The aim of this paper is to define approximate coincidence point for a pair of single valued self-mappings to obtain some important results on approximate coincidence point using two nonlinear functions by Geraghty  in 1973 and Mizoguchi and Takahashi  (-function) in 1989.
2. Approximate Coincidence Point
Definition 1. Let be a metric space and , , . Then is an -fixed point (approximate fixed point) of if .
The set of all -fixed points of , for a given , is denoted by
Definition 2. Let . Then has the approximate fixed point property if
Definition 3. Let be a metric space, and let be two single valued maps. The maps and are said to have coincidence point if say, is called a point of coincidence of and . If , then is called a common fixed point of and .
3. Approximate Coincidence Point Results for Two Nonlinear Maps
In this section, we establish existence of some results concerning approximate coincidence point for various types of nonlinear contractive maps in the setting of general metric spaces. For this purpose, we first define approximate coincidence point for two self-maps in metric space and prove results on approximate coincidence point using the idea of the Geraghty-type contractive condition .
Let denote the class of real functions satisfying condition An example of a function in may be given by for and . We now prove our result using this -function.
Definition 4. Let be a metric space, and let be two single-valued self-maps. The maps and are said to have approximate coincidence point property provided or, equivalently, for any , there exists such that The set of approximate coincidence point of and is denoted by .
Theorem 5. Let be a metric space and let be two self-mappings such that satisfying for all and . Then the following statements hold. and have the approximate coincidence point property on , that is, . There exists a sequence in such that is a Cauchy sequence and .
Proof. Let be arbitrary. Since , we can choose such that . Continuing this process, we obtain a sequence in as follows:
We will suppose for all , since if for some , then for every , implying that have approximate coincidence point and thus completes the proof. So we suppose that . Then by (6),
and so . Hence, is a strictly decreasing and bounded below, thus converging to some . Without loss of generality, let . Then using (8), we get
Now passing limit as on (9), we get
Now, using property of the function , we conclude that . Assume for , so . It implies that for all . Since as , it follows that . So and have approximate coincidence point in . So follows.
Now it suffices to prove that is a Cauchy sequence in . Let . Then using the property of and from (9), we have . Again using (9), we get . In this process, we obtain Thus, for with , it follows from (11) that Since , as and so is a Cauchy sequence. Hence, follows.
Example 6. Let with be usual metric and be defined by Also take for and . Then one can check the inequality (6) satisfied with , . Also it is easy to see that all the conditions of Theorem 5 are satisfied having the approximate coincidence point property. In fact , if we select such that with . On the contrary, it is clear that and have no coincidence point in .
In 1989, Mizoguchi and Takahashi  introduced -function as follows.
A function is said to be an -function, if
It is obvious that if is a monotone, then is an -function.
Definition 7. Let be a metric space and is an -function. Then is said to be an -type mapping if
Using this function, Mizoguchi and Takahashi  proved a fixed point theorem for multivalued mapping, which is a generalization of Nadler’s fixed point theorem which extends the Banach contraction principle for multivalued maps, but its primitive proof is different (see ). But we only restrict ourselves in proving results for single valued mapping. In this aspect, we formulate our next results using -function. For the properties and characterizations of -function one can see [16, 17] for details.
Now, we establish the following approximate coincidence point property using the concept of Mizoguchi and Takahashi ()-type mappings.
Theorem 8. Let be a metric space and let be two single valued self-maps such that satisfying
for all where is an -function.
Then the following statements hold.() and have the approximate coincidence point property on , that is, .() There exists a sequence in such that is a Cauchy sequence and .
Proof. Proof is similar to that of Theorem 5 and is left out. The only difference lies in the character of the -function and -function which are supplementary to each other.
Example 9. Let with be usual metric and be defined by If we take defined by , then all the conditions of the Theorem 8 are satisfied.
It is easy to arrive at the following corollaries.
Corollary 10. Let be a metric space and let be a single valued self-map satisfying
for all where is an -function.
Then the following statements hold.(B1) has the approximate fixed point on , that is, .(B2)There exists a sequence in such that is a Cauchy sequence and .
Corollary 11. Let be a metric space and let be two single valued self-mapping satisfying
for all and . Then the following statements hold.
has the approximate fixed point on , that is, .
There exists a sequence in such that is a Cauchy sequence and .
The authors are thankful to the reviewers for their valuable suggestions.