The Scientific World Journal

Volume 2014 (2014), Article ID 585964, 14 pages

http://dx.doi.org/10.1155/2014/585964

Research Article

Fixed Point Results for --Contractive Maps with Application to Boundary Value Problems

Nawab Hussain,¹ Vahid Parvaneh,² and Jamal Rezaei Roshan³

¹Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

²Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran

³Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran

Received 9 March 2014; Accepted 29 March 2014; Published 7 May 2014

Academic Editor: M. Mursaleen

Copyright © 2014 Nawab Hussain et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We unify the concepts of G-metric, metric-like, and b-metric to define new notion of generalized b-metric-like space and discuss its topological and structural properties. In addition, certain fixed point theorems for two classes of G-α-admissible contractive mappings in such spaces are obtained and some new fixed point results are derived in corresponding partially ordered space. Moreover, some examples and an application to the existence of a solution for the first-order periodic boundary value problem are provided here to illustrate the usability of the obtained results.

1. Introduction and Mathematical Preliminaries

The concept of a -metric space was introduced by Czerwik [1]. After that, several interesting results about the existence of fixed point for single-valued and multivalued operators in (ordered) -metric spaces have been obtained (see, e.g., [2–11]).

Definition 1 (see [1]). Let be a (nonempty) set and a given real number. A function is a -metric on if, for all , the following conditions hold:

    if and only if ,   ,   .

In this case, the pair is called a -metric space.

The concept of a generalized metric space, or a -metric space, was introduced by Mustafa and Sims [12].

Definition 2 (see [12]). Let be a nonempty set and a function satisfying the following properties:

if and only if ;, for all with ;, for all with ;, where is any permutation of (symmetry in all three variables);, for all (rectangle inequality).

Then, the function is called a -metric on and the pair is called a -metric space.

Definition 3 (see [13]). A metric-like on a nonempty set is a mapping such that, for all , the following hold:

    implies ;   ;   .

The pair is called a metric-like space.

Below, we give some examples of metric-like spaces.

Example 4 (see [14]). Let . Then, the mapping defined by is a metric-like on .

Example 5 (see [14]). Let ; then the mappings defined by

are metric-likes on , where and .

Definition 6 (see [15]). Let be a nonempty set and a given real number. A function is a -metric-like if, for all , the following conditions are satisfied:

    implies ;   ;   .

A -metric-like space is a pair such that is a nonempty set and is a -metric-like on . The number is called the coefficient of .

In a -metric-like space if and , then , but the converse may not be true and may be positive for all . It is clear that every -metric space is a -metric-like space with the same coefficient but not conversely in general.

Example 7 (see [8]). Let , let be a constant, and let be defined by

Then, is a -metric-like space with coefficient .

The following propositions help us to construct some more examples of -metric-like spaces.

Proposition 8 (see [8]). Let be a metric-like space and , where is a real number. Then, is a -metric-like with coefficient .

From the above proposition and Examples 4 and 5, we have the following examples of -metric-like spaces.

Example 9 (see [8]). Let . Then, the mapping defined by , where is a real number, is a -metric-like on with coefficient .

Example 10 (see [8]). Let . Then, the mappings defined by

are -metric-like on , where , , and .

Each -metric-like on generates a topology on whose base is the family of all open -balls , where for all and .

Now, we introduce the concept of generalized -metric-like space, or -metric space, as a proper generalization of both of the concepts of -metric-like spaces and -metric spaces.

Definition 11. Let be a nonempty set. Suppose that a mapping satisfies the following:

implies ;, where is any permutation of (symmetry in all three variables); for all (rectangle inequality).

Then, is called a -metric and is called a generalized -metric-like space.

The following proposition will be useful in constructing examples of a generalized -metric-like space.

Proposition 12. Let be a -metric-like space with coefficient . Then,

are two generalized -metric-like functions on .

Proof. It is clear that and satisfy conditions and of Definition 11. So, we only show that is satisfied by and . Let . Then, using the triangular inequality in -metric-like spaces, we have

Also,

According to the above proposition, we provide some examples of generalized -metric-like spaces.

Example 13. Let , let be a constant, and let be defined by

for all . Then, and are generalized -metric-like spaces with coefficient . Note that, for ,  and .

Example 14. Let . Then, the mappings defined by

where is a real number, are generalized -metric-like spaces with coefficient .

By some straight forward calculations, we can establish the following.

Proposition 15. Let be a -metric space. Then, for each , it follows that:

(1) for ;(2);(3);(4).

Definition 16. Let be a -metric space. Then, for any and , the -ball with center and radius is

The family of all -balls

is a base of a topology on , which we call it -metric topology.

Definition 17. Let be a -metric space. Let be a sequence in . Consider the following.

(1)A point is said to be a limit of the sequence , denoted by , if .(2) is said to be a -Cauchy sequence, if exists (and is finite).(3) is said to be -complete if every -Cauchy sequence in is -convergent to an such that

Using the above definitions, one can easily prove the following proposition.

Proposition 18. Let be a -metric space. Then, for any sequence in X and a point , the following are equivalent:

(1) is -convergent to ;(2), as ;(3), as ;

Definition 19. Let and be two generalized -metric like spaces and let be a mapping. Then, is said to be -continuous at a point if, for a given , there exists such that and imply that . The mapping is -continuous on if it is -continuous at all . For simplicity, we say that is continuous.

Proposition 20. Let and be two generalized -metric like spaces. Then, a mapping is -continuous at a point if and only if it is -sequentially continuous at ; that is, whenever is -convergent to , is -convergent to .

We need the following simple lemma about the -convergent sequences in the proof of our main results.

Lemma 21. Let be a -metric space and suppose that , , and are -convergent to , , and , respectively. Then, we have

In particular, if are constant, then

Proof. Using the rectangle inequality, we obtain

Taking the lower limit as in the first inequality and the upper limit as in the second inequality, we obtain the desired result.

If , then

Again taking the lower limit as in the first inequality and the upper limit as in the second inequality, we obtain the desired result.

2. Main Results

Samet et al. [16] defined the notion of -admissible mappings and proved the following result.

Definition 22. Let be a self-mapping on and a function. We say that is an -admissible mapping if

Denote with the family of all nondecreasing functions such that for all , where is the th iterate of .

Theorem 23. Let be a complete metric space and an -admissible mapping. Assume that

where . Also, suppose that the following assertions hold:

(i)there exists such that ; (ii)either is continuous or, for any sequence in with for all such that as , we have for all .

Then, has a fixed point.

For more details on -admissible mappings, we refer the reader to [17–20].

Definition 24 (see [21]). Let    be a -metric space, let be a self-mapping on , and let be a function. We say that is a--admissible mapping if

Motivated by [22], let denote the class of all functions satisfying the following condition:

Definition 25. Let and . We say that is a rectangular --admissible mapping if

implies ; implies .

From now on, let be a function and

Theorem 26. Let be a -complete generalized -metric-like space and let be a rectangular --admissible mapping. Suppose that

for all .

Also, suppose that the following assertions hold:

(i)there exists such that ; (ii) is continuous and, for any sequence in with for all such that as , we have for all .

Then, has a fixed point.

Proof. Let   be such that . Define a sequence by for all . Since is a --admissible mapping and , we deduce that . Continuing this process, we get for all .

Step I. We will show that . If for some , then . Thus, is a fixed point of . Therefore, we assume that for all .

Since for each , then we can apply (21) which yields

Therefore, is a decreasing and bounded sequence of nonnegative real numbers. Then, there exists such that . Letting in (22), we have

Since , we deduce that , that is

By Proposition 15(2), we conclude that

Step II. Now, we prove that the sequence is a -Cauchy sequence. For this purpose, we will show that

Using the rectangular inequality with (21) (as , since is a rectangular --admissible mapping), we have

Taking limit as in the above inequality and applying (25) and (26), we have

Here,

Letting in the above inequality, we get

Hence, from (29) and (31), we obtain

If , then we get

Since , we deduce that

which is a contradiction. Consequently, is a -Cauchy sequence in . Since is -complete, there exists such that , as . Now, from (34) and -completeness of ,

Step III. Now, we show that is a fixed point of .

Using the rectangle inequality, we get

Letting and using the continuity of and (35), we obtain

Note that, from (21), as , we have

where, by (37),

Hence, as for all , we have . Thus, by (37), we obtain that . But then, using (38), we get that

which is a contradiction. Hence, we have . Thus, is a fixed point of .

We replace condition (ii) in Theorem 26 by regularity of the space .

Theorem 27. Under the same hypotheses of Theorem 26, instead of condition , assume that whenever in is a sequence such that for all and as , one has for all . Then, has a fixed point.

Proof. Repeating the proof of Theorem 26, we can construct a sequence in such that for all and for some . Using the assumption on , we have for all . Now, we show that . By Lemma 21 and (35),

where

Therefore, we deduce that . Hence, we have .

A mapping is called a comparison function if it is increasing and , as for any (see, e.g., [23, 24] for more details and examples).

Definition 28. A function is said to be a -comparison function if

is increasing,there exists , , and a convergent series of nonnegative terms such that for and any .

Later, Berinde [5] introduced the notion of -comparison function as a generalization of -comparison function.

Definition 29 (see [5]). Let be a real number. A mapping is called a -comparison function if the following conditions are fulfilled:

(1) is increasing;(2)there exist , , and a convergent series of nonnegative terms such that for and any .

Let be the class of all -comparison functions . It is clear that the notion of -comparison function coincides with -comparison function for .

Lemma 30 (see [25]). If is a -comparison function, then we have the following:

(1)the series converges for any ;(2)the function defined by , , is increasing and continuous at .

Remark 31. It is easy to see that if , then we have and for each and is continuous at .

In the next example, we present a class of -comparison functions.

Example 32. Any function of the form for all where is a -comparison function.

Proof. From the part of Lemma 30, the necessary condition is that the series converges for any . But, for each and , we have

So, according to the comparison test of the series, we should have . On the other hand, we have

Therefore, for any convergent series of nonnegative terms and each , we have

For example, for and , the function is a -comparison function.

Theorem 33. Let be a -complete generalized -metric-like space and let be a --admissible mapping. Suppose that

for all where and

Also, suppose that the following assertions hold:

(i)there exists such that ; (ii)   

(a) is continuous and, for any sequence in with for all such that as , one has for all ;(b)assume that whenever in is a sequence such that for all and as , one has for all .

Then, has a fixed point.

Proof. Let   be such that  . Define a sequence by for all . Since is a --admissible mapping and , we deduce that . Continuing this process, we get for all .

If there exists such that , then and so we have nothing to prove. Hence, for all , we assume that .

Step I (Cauchyness of ). As for all , using condition (46), we obtain

Using Proposition 15(2) as , we get

Hence,

By induction, since , we get that

Let be arbitrary. Then, there exists a natural number such that

Let . Then, by the rectangular inequality and Proposition 15(2) as , we get

Consequently, is a -Cauchy sequence in . Since is -complete, so there exists such that

Step II. Now, we show that is a fixed point of . Suppose to the contrary, that is, , then, we have .

Let the part (a) of (ii) holds.

Using the rectangle inequality, we get

Letting and using the continuity of , we get

From (46) and part (a) of condition (ii), we have

where, by using (56), we have

Hence, from properties of , . Thus, by (56), we obtain that

Moreover, (57) yields that . This is impossible, according to our assumptions on . Hence, we have . Thus, is a fixed point of .

Now, let part (b) of (ii) holds.

As is a sequence such that for all and as , we have for all .

Now, we show that . By (46), we have

where

Letting in the above inequality and using (54) and Lemma 21, we get

Again, taking the upper limit as in (60) and using (62) and Lemma 21, we obtain

So, we get . That is, .

Let be a partially ordered -metric-like space. We say that is an increasing mapping on if [26]. Fixed point theorems for monotone operators in ordered metric spaces are widely investigated and have found various applications in differential and integral equations (see [27–30] and references therein). From the results proved above, we derive the following new results in partially ordered -metric-like space.

Theorem 34. Let be a partially ordered -complete generalized -metric-like space and let be an increasing mapping. Suppose that

for all with , where

Also, suppose that the following assertions hold:

(i)there exists such that ; (ii) is continuous or assume that whenever in is an increasing sequence such that as , one has for all .

Then, has a fixed point.

Proof. Define by

First, we prove that is a triangular -admissible mapping. Hence, we assume that . Therefore, we have . Since is increasing, we get ; that is, . Also, let and ; then and . Consequently, we deduce that ; that is, . Thus, is a triangular -admissible mapping. Since satisfies (64) so, by the definition of , we have

for all . Therefore, satisfies the contractive condition (21). From (i), there exists such that ; that is, . According to (ii), we conclude that all the conditions of Theorems 26 and 27 are satisfied and so has a fixed point.

Similarly, using Theorem 33, we can prove following result.

Theorem 35. Let be an ordered -complete generalized -metric-like space and let be an increasing mapping. Suppose that

for all with where and

Also, suppose that the following assertions hold:

(i)there exists such that ; (ii)   

  is continuous;  assume that whenever in is an increasing sequence such that as , one has for all .

Then, has a fixed point.

We conclude this section by presenting some examples that illustrate our results.

Example 36. Let be endowed with the usual ordering on . Define the generalized -metric-like function given by

with . Consider the mapping defined by and the function given by ,  , and . It is easy to see that is an increasing function on . We show that is -continuous on . By Proposition 20, it is sufficient to show that is -sequentially continuous on . Let be a sequence in such that , so we have   and, equivalently, . On the other hand, we have

So, is -sequentially continuous on .

For all elements , and the fact that is an increasing function on , we have

Hence, satisfies all the assumptions of Theorem 34 and thus it has a fixed point (which is .

Example 37. Let with the usual ordering on . Define the generalized -metric-like function given by with . Consider the mapping defined by and the function given by . It is easy to see that is increasing function. Now, we show that is a -continuous function on .

Let be a sequence in such that , so we have   and, equally, . On the other hand, we have

So, is -sequentially continuous on .

For all elements , we have

Hence, satisfies all the assumptions of Theorem 35 and thus it has a fixed point (which is .

3. Application

In this section, we present an application of our results to establish the existence of a solution to a periodic boundary value problem (see [30, 31]).

Let be the set of all real continuous functions on [0,T]. We first endow with the -metric-like

for all where and then we endow it with the generalized -metric-like defined by

Clearly, is a complete generalized -metric-like space with parameter . We equip with a partial order given by

Moreover, as in [30], it is proved that is regular; that is, whenever in is an increasing sequence such that as , we have for all .

Consider the first-order periodic boundary value problem

where , with , and is a continuous function.

A lower solution for (78) is a function such that

where .

Assume that there exists such that, for all , we have

where . Then, the existence of a lower solution for (78) provides the existence of a solution of (78).

Problem (78) can be rewritten as

Consider

where .

Using variation of parameters formula, we get

which yields

Since , we get

or

Substituting the value of in (83), we arrive at

where

Now, define the operator as

The mapping is increasing [31]. Note that if is a fixed point of , then is a solution of (78).

Let . Then, we have

Similarly,

where

Equivalently,

which yields that

Finally, it is easy to obtain that

Finally, since is a lower solution for (78), so it is easy to show that [31].

Hence, the hypotheses of Theorem 35 are satisfied, with where . Hence, there exists a fixed point such that which is a solution to periodic boundary value problem (78).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU, for financial support.

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