An Application of Fixed Point Theory to a Nonlinear Differential Equation

Abstract

We introduce a new family of mappings on by relaxing the nondecreasing condition on the mappings and by using the properties of this new family we present some fixed point theorems for --contractive-type mappings in the setting of complete metric spaces. By applying our obtained results, we also assure the fixed point theorems in partially ordered complete metric spaces and as an application of the main results we provide an existence theorem for a nonlinear differential equation.

1. Introduction and Preliminaries

Fixed point theory has fascinated many researchers since 1922 with the celebrated Banach fixed point theorem. There exists a vast literature on the topic field and this is very active field of research at present. Fixed point theorems are very important tools for proving the existence and uniqueness of the solutions to various mathematical models (integral and partial equations, variational inequalities, etc). It can be applied to, for example, variational inequalities, optimization, and approximation theory. The fixed point theory has been continually studied by many researchers (see, e.g., [1–5] and references contained therein). It is well known that the contractive-type conditions are very indispensable in the study of fixed point theory. The first important result on fixed points for contractive-type mappings was the well-known Banach-Caccioppoli theorem which was published in 1922 in [6] and it also appeared in [7]. Later in 1968, Kannan [8] studied a new type of contractive mappings. Since then, there have been many results related to mappings satisfying various types of contractive inequalities; we refer to ([9–12] etc.) and references contained therein.

Recently, Samet et al. [5] introduced a new category of contractive-type mappings known as - contractive-type mappings. The results obtained by Samet et al. [5] extended and generalized the existing fixed point results in the literature, in particular the Banach contraction principle. Salimi et al. [4] and Karapinar and Samet [3] generalized the - contractive-type mappings and obtained various fixed point theorems for this generalized class of contractive mappings [3, 4].

Most of papers (see, for instance, [3–5] and references contained therein) have considered the - contractive-type mapping for a nondecreasing mapping with for all . The convergence of and nondecreasing condition for are restrictive and it is a fact that such a mapping is differentiable almost everywhere and hence continuous why was one of our aims to write this paper in order to consider a family of mappings by relaxing nondecreasing condition and the convergence of the series . This paper is inspired and motivated by research works [4, 5]; we will introduce a new family of mappings on and prove the fixed point theorems for mappings using properties of this new family in complete metric spaces. By applying our obtained results, we also assure the fixed point theorems in partially ordered complete metric spaces and give the applications to ordinary differential equations.

In the rest of the paper, we introduce some notations and definitions that will be used in the sequel.

Lemma 1 (see [5]). Suppose that . If is nondecreasing, then for each , implies that .

Remark 2. It is easily seen that if is nondecreasing and , for all , then .

Definition 3 (see [5]). Let and let . We say that is -admissible if, for all , implies .

In 2012, Samet et al. [5] introduced the concept of --contractive-type mappings, where and

Definition 4 (see [5]). Let be a metric space and let be a mapping. We say that is an --contractive mapping if there exist two functions and where such that for all .

In [5], the authors assured the existence of the fixed point theorems for the mentioned mappings satisfying -admissibility in the complete metric spaces.

Recently, Salimi et al. [4] modified the concept of -admissibility.

Definition 5 (see [4]). Let and . We say that is -admissible with respect to if, for all , implies .

Remark 6. If we suppose that , for all , Definition 5 is reduced to Definition 3.

Salimi et al. [4] proved the existence of fixed point theorems for generalized --contractive-type mappings where . They also assure the fixed point theorems generalized --contractive-type mappings where is a nondecreasing continuous mapping and .

In this work, we will introduce a new family of mappings on without assuming the nondecreasing condition for and prove the fixed point theorems for --contractive-type mappings using properties of this new family in complete metric spaces. We will use our result to obtain fixed point results in partially ordered complete metric spaces and to give an application to nonlinear differential equations.

2. Main Results

We now introduce a new family of mappings and prove the existence of fixed point results for --contractive-type mappings where .

Denote by the family of mappings such that(i) is an upper semicontinuous mapping from the right;(ii) for all ;(iii).

Remark 7. By Lemma 1, for each , we have for all and by Remark 2 we obtain that .

Remark 8. Since every nondecreasing mapping is differentiable almost everywhere (see [13]), we observe that nondecreasing condition is closed to continuity and it is restrictive.

Example 9. The floor function is upper semicontinuous function from the right and nondecreasing but is not continuous.

Example 10. Let be a mapping defined by
We have that is upper semicontinuous from the right and for all . Furthermore, is not nondecreasing.

Example 11. Let be a mapping defined by
Thus, is upper semicontinuous from the right, for all and . Moreover, is not nondecreasing.

We now the prove the existence of the fixed point theorem for -admissible mappings with respect to where .

Theorem 12. Let be a complete metric space and . Suppose that is a mapping satisfying the following conditions: (i) is -admissible with respect to ;(ii)if and , then ;(iii)there exists such that ;(iv) is continuous or if is a sequence in such that for all and as , and then for all .Then, has a fixed point.

Proof. Since , there exists such that . Therefore, we can construct the sequence in such that If , for some , then has a fixed point. Assume that for all . Since and is -admissible with respect to , we obtain that By continuing the process as above, we have Applying (ii), we obtain that for all . Since for all , we have for all . Therefore, is a nonincreasing sequence. It follows that there exists such that We will prove that . Suppose that . Since is upper semicontinuous from the right using (9), we have which is a contradiction. Therefore, This implies that for each , there exists such that We obtain that Therefore, is a Cauchy sequence and so converges to some . By continuity of , we have This implies that is a fixed point of . On the other hand, since and converges to , we obtain that Using (ii), for each , we have Since is upper semicontinuous from the right, we obtain that By taking the limit as , this yields and hence .

Theorem 13. Suppose all hypotheses of Theorem 12 hold. Assume that, for all , there exists such that and . Then, has a unique fixed point.

Proof. Assume that and are two fixed points of . This implies that there exists such that Since is -admissible with respect to , for each , we obtain that It follows that Therefore, is a nonincreasing sequence and then converges to some . We will show that . Suppose that . Since is upper semicontinuous from the right, we have which is a contradiction. It follows that Similarly, by the same argument, we can prove that Since the limit of the sequence is unique, we have .

Applying Theorems 12 and 13, we immediately obtain the following result.

Corollary 14. Let be a complete metric space and . Suppose that is an --contractive mapping satisfying the following conditions: (i) is -admissible;(ii)there exists such that ;(iii) is continuous or if is a sequence in such that for all and as , and then for all ;(iv)for all , there exists such that and . Then, has a unique fixed point.

Bhaskar and Lakshmikantham [9] introduced the definition of coupled fixed points.

Definition 15 (see [9]). Let be a given mapping. We say that is a coupled fixed point of if

Remark 16. Let be a given mapping. Define the mapping by Therefore, is a coupled fixed point of if and only if is a fixed point of .

By using the analogous proof appeared in [5], we obtain the coupled fixed point results assuming .

Theorem 17. Let be a complete metric space and be a given mapping. Suppose that there exist and a function such that for all . Suppose that, (i)for all , one has (ii)there exists such that (iii) is continuous. Then, has a coupled fixed point.

Theorem 18. Let be a complete metric space and be a given mapping. Suppose that there exist and a function such that for all . Suppose that, (i)for all , we have (ii)there exists such that (iii)if and are sequences in such that then Then, has a coupled fixed point.

Theorem 19. Suppose that all hypotheses of Theorem 17 (resp., Theorem 18) hold. Assume that, for all , there exists such that Then, has a unique coupled fixed point.

3. Consequences

We now prove the fixed point theorems in complete metric spaces and partially ordered complete metric spaces using our obtained results.

Theorem 20 (Banach [6]). Let be a complete metric space and be a mapping satisfying for all , where . Then, has a unique fixed point.

Proof. Let be mappings defined by It follows that is -admissible with respect to . Suppose that defined by for all . This implies that is upper semicontinuous from the right, for all and . Furthermore, we can see that all assumptions in Theorem 13 are now satisfied. This completes the proof.

Theorem 21 (Ran and Reurings [14]). Let be a partially ordered set and suppose that there exists a metric in such that the metric space is complete. Let be a continuous and nondecreasing mapping with respect to . Assume that the following conditions hold: (i)there exists such that for all with ;(ii)there exists such that ;(iii) is continuous.Then, has a fixed point.

Proof. Suppose that are mappings defined by Let such that . This implies that . Since is nondecreasing with respect to , we obtain that . Therefore, . It follows that is -admissible with respect to . Define a mapping defined by for all . We can see that . For each with , we obtain that and this yields By using (ii), we have . Hence, all assumptions in Theorem 12 are now satisfied. Thus, we obtain the desired result.

Theorem 22 (Nieto and Rodríguez-López [12]). Let be a partially ordered set and suppose that there exists a metric in such that the metric space is complete. Let be a nondecreasing mapping with respect to . Assume that the following conditions hold: (i)there exists such that for all with ;(ii)there exists such that ;(iii)if is a nondecreasing sequence in such that as , then for all .Then, has a fixed point.

Proof. Suppose that and are mappings defined as in the proof of Theorem 21. Assume that is a sequence in such that for all and as . This implies that for all . Using (iii), this yield for all . Therefore, for all . Hence, all assumptions in Theorem 12 are now satisfied. Thus, we obtain the desired result.

Theorem 23. Suppose that all hypotheses of Theorem 21 (resp., Theorem 22) hold. Assume that, for all , there exists such that and . Then has a unique fixed point.

Proof. Suppose that and are mappings defined as in the proof of Theorem 21. Let . It follows that there exists such that and . Therefore, and . Hence, all assumptions in Theorem 13 are now satisfied. So, the proof is complete.

4. Applications to Ordinary Differential Equations

The following ordinary differential equation is taken from Samet et al. [5].

Denote by the set of all continuous functions defined on and let be defined by It is well known that is a complete metric space. Let us consider the two-point boundary value problem of the second-order differential equation: where is continuous. The Green function associated to (42) is defined by Assume that the following conditions hold: (i)there exists a function such that, for all , for all with , we have where ;(ii)there exists such that, for all , we have (iii)for all , for all , (iv)if is a sequence in such that and , for all , then for all .We now prove that existence of a solution of the mentioned second-order differential equation. The idea of proving the following theorem is taken from [5] but is slightly different.

Theorem 24. Under assumptions (i)–(iv), (42) has a solution in .

Proof. It is well known that is a solution of (42) is equivalent to is a solution of the integral equation (see [5]) Let be a mapping defined by Suppose that such that for all . By applying (i), we obtain that Since , for all , we have . It follows that for each , such that for all .
Let be mappings defined by Let such that . This implies that for all . Therefore, Furthermore, if such that , then by using (iii) we have It follows that is -admissible with respect to . By (ii), there exists such that Applying Theorem 12, we obtain that has a fixed point in ; say . Hence, is a solution of (42).

Corollary 25. Assume that the following conditions hold: (i) is continuous and nondecreasing;(ii)for all , for all with , one has where ;(iii)there exists such that, for all , one has Then, (42) has a unique solution in .

Proof. Define a mapping by By the analogous proof, as in Theorem 24, we obtain that (42) has at least one solution. Since, for each , there exists a mapping such that and . This implies that the solution of (42) is unique by Theorem 13.

Conflict of Interests

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

Acknowledgments

This work is supported by Naresuan University under Grant R2557B055. The authors would like to express their deep thanks to Naresuan University.