Some Notes on the Existence of Solution for Ordinary Differential Equations via Fixed Point Theory

Abstract

We establish a fixed point theorem with w-distance for nonlinear contractive mappings in complete metric spaces. As applications of our results, we derive the existence and uniqueness of solution for a first-order ordinary differential equation with periodic boundary conditions. Here, we need not assume that the equation has a lower solution.

1. Introduction and Preliminaries

The main purpose of this paper is to obtain the existence and uniqueness of solution for a periodic boundary value problem. This topic has been considered in [1–11] and the references therein. A powerful tool to solve the problem is the fixed point theorem in partially ordered metric spaces. On the existing research results, admitting the existence of a lower solution is necessary. In this paper, we establish a fixed point theorem for w-distance contraction type maps in complete metric spaces without partially ordered structure. From this, we obtain some results on the existence and uniqueness for ordinary differential equations. We will see that the assumption of the existence of a lower solution can be removed. In particular, we also show that under assumptions of a recent result in [10] the equation has a unique zero solution.

In this note, we consider the following periodic boundary value problem: where and is a continuous function.

Definition 1. A lower solution for (1) is a function such that
Let be the set of all real continuous functions on a closed interval . We endow with the norm for all . Obviously, this space is a Banach space and the norm induces a complete metric on as follows: for all .
Let denote the class of those functions which satisfy the condition
Let denote the class of the functions which have the following properties:(i) is increasing;(ii)for each , ;(iii).
For example, , where , , and are in .
Recently, Amini-Harandi and Emami [4] proved the following existence theorem.

Theorem 2 (see [4, Theorem 3.1]). Consider problem (1) with continuous and suppose that there exists such that for with , where . Then the existence of a lower solution of (1) provides the existence of a unique solution of (1).

Then, Caballero et al. [6] also give an existence theorem.

Theorem 3 (see [6, Theorem 3.2]). Consider problem (1) with continuous and suppose that there exist with such that for with where . Then the existence of a lower solution of (1) provides the existence of a solution of (1).

Very recently, Hussain et al. [10] obtain the following result.

Theorem 4 (see [10, Section 3]). Consider problem (1) with continuous and suppose that there exists such that for where and . Then the existence of a lower solution of (1) provides the existence of a solution of (1).

Remark 5. If we take for all in the condition (9), then we deduce that
This means that is a solution of (1) (of course, a lower solution of (1)). In Section 3, we will show this fact again; see Remark 17.

Now, let us recall the concept of w-distance, which was introduced by Kada et al. [12].

Definition 6. Let be a metric space with metric . Then a function is called a w-distance on if the following are satisfied:(w1), for any ;(w2)for any , is lower semicontinuous; that is, , whenever and ;(w3)for any , there exists such that and imply .

Let us give some basic examples of w-distances (see [12]).

Example 7. Let be a metric space. Then the metric is a w-distance on .

Example 8. Let be a normed space with norm . Then the function defined by for every is a w-distance on .

Example 9. Let be a normed space with norm . Then the function defined by for every is a w-distance on .

We also need the following example in Section 3.

Example 10. Let be endowed with the norm for all . We easily deduce that the function defined by for every is a w-distance on .

The following lemma has been proved in [12].

Lemma 11. Let be a metric space with metric and let be a w-distance on . Let and be sequences in , let and be sequences in converging to 0, and let . Then the following hold.(i)If and for any , then . In particular, if and , then .(ii)If and for any , then converges to .(iii)If for any with , then is a Cauchy sequence.

2. Fixed Point Results with -Distance

We are now ready to state and prove our main theorem.

Theorem 12. Let be a complete metric space and let be a w-distance on . Let be a map and suppose that there exists such that for all . Then has a unique fixed point in and .

Proof. Starting with an arbitrary , we construct by induction a sequence in . Put .
Applying the condition (15) with and , we have
Therefore is a decreasing sequence and bounded below and hence .
Let us prove that . Assume that . From (15), we obtain Taking limit when , the above inequality yields , and since this implies . Thus . Analogously, it can be proved that .
Now, we show that
Suppose that this is not true. Then there exists for which we can find subsequences and of with such that
By the triangle inequality and the condition (15), for every ,
From this, we obtain
Letting in the above inequality, using and , we deduce that
But since we get . This is a contradiction and hence we have shown that (18) holds.
From (18) and Lemma 11(iii), it follows that is a Cauchy sequence in . Since is a complete metric space, there exists a such that .
Next, we prove that is a fixed point of . To the end, we prove that and . Let be given. Using (18), there exists such that for all . Letting , by (w2) we have for all . This means that . Applying the condition (15) with and , we have
Due to , it follows that . According to Lemma 11(i), we obtain ; that is, is a fixed point of .
To prove the uniqueness of the fixed point of , let us suppose that is another fixed point of . Using the condition (15), we get
Therefore is a decreasing sequence and bounded below and hence . Suppose that . Using the condition (15), we have
Taking limit when , the above inequality yields , and since this implies . Then . Combining this and , we get by Lemma 11(i) that .
Finally, we prove when is a fixed point of . Applying the condition (15) with and , we obtain and hence . Now that , it follows that .

The following corollary is immediate result from Theorem 12 and Example 7.

Corollary 13. Let be a complete metric space. Let be a map and suppose that there exists such that for all . Then has a unique fixed point in .

3. Application to Ordinary Differential Equations

Now, we prove some results on the existence of solution for problem (1).

Theorem 14. Consider problem (1) with continuous and suppose that there exists such that for any where . Then there exists a unique solution for (1).

Proof. Problem (1) can be rewritten as Using variation of parameters formula, we can easily deduce that problem (1) is equivalent to the integral equation where Define by Note that if is a fixed point of then is a solution of (1).
Now, we check that hypotheses of Corollary 13. Consider the space with the metric Then by (28) for any , As the function is increasing, we obtain From Corollary 13, we see that has a unique fixed point.

Remark 15. Clearly, we see that the condition (6) of Theorem 2 implies the condition (28) of Theorem 14. In addition, Theorem 14 need not assume the existence of a lower solution for (1). Hence this result is an improvement of Theorem 2.

Theorem 16. Consider problem (1) with continuous and suppose that there exists such that for any where . Then there exists a unique solution for (1).

Proof. Problem (1) is equivalent to the integral equation where is the same as (31). Define by Notice that if is a fixed point of then is a solution of (1).
Now, we check that hypotheses of Theorem 12. Consider the space with the metric We take a w-distance on by ; see Example 10. Using the monotene property of the function and condition (36), for any , From Theorem 12, we deduce that has a unique fixed point.

Remark 17. Let and . If we denote then and . Thus Theorem 4 is a sepical case of Theorem 16. However, from Theorem 12 we get if is a solution of (1). This means for all . In other words, the unique solution for (1) from Theorem 16 (in particular, from Theorem 4) can only be a zero function.

Theorem 18. Consider problem (1) with continuous and suppose that there exist and with such that for where . Then there exists a unique solution for (1).

Proof. Problem (1) is equivalent to the integral equation where is the same as (31). Define by Notice that if is a fixed point of then is a solution of (1).
Now, we check that hypotheses of Corollary 13. By (43), for any ,
Using the Hölder inequality in the last integral, we get where with . The first integral gives us As the function is increasing and , from (47) we obtain From Corollary 13, we deduce that has a unique fixed point.

From Theorem 18, we obtain the following result when we take .

Corollary 19. Consider problem (1) with continuous and suppose that there exist with such that for where . Then there exists a unique solution for (1).

Remark 20. From [6, Remark 3.3], we see that if , then . The condition (8) of Theorem 3 implies the condition (51) of Corollary 19. In addition, Corollary 19 need not assume the existence of a lower solution for (1). Thus Corollary 19 is an improvement of Theorem 3.

Remark 21. We easily check that and are in . Therefore the condition of [8, Theorem 3.1], [11, Theorem 25], or [7, Theorem 3.2] implies the condition (51) of Corollary 19. In addition, Corollary 19 need not assume the existence of a lower solution for (1). Thus Corollary 19 is an improvement of [8, Theorem 3.1], [11, Theorem 25], and [7, Theorem 3.2].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported in part by Scientific Studies of Higher Education Institution of Inner Mongolia (NJZZ13019) and in part by Program of Higher-level talents of Inner Mongolia University (30105-125150, 30105-135117).