Fixed-Point Theorems Involving Lipschitz in the Small

Abstract

Lipschitz in the small is a generalization of the Lipschitz condition. The Lipschitz condition guarantees the uniqueness of the solution of the initial value problems. A special Lipschitz condition in the small is a contraction in the small. Based on the Lipschitz in the small in this paper, fixed-point theorems involving contraction in the small will be presented. The results will be applied to develop Picard’s theorem.

1. Introduction

A Lipschitz condition is an interesting condition, both in analysis and in its applications. In Picard’s theorem, the Lipschitz condition guarantees the uniqueness of the solution of an initial value problem (IVP). Picard proved the guarantee of the uniqueness by using the equivalency of the solution of the IVP with the solution of its associated integral problem.

Picard’s theorem can be proven by using the Banach contraction theorem or by constructing a sequence of functions that converges uniformly to the solution [1]. Contraction is a specific condition of the Lipschitz condition, where its Lipschitz constant is in the interval Many researchers generalized the concept of the Lipschitz function to have fixed-point theorems, such as Hussain et al., Liu and Xu, Pata, Xu and Radenović [2–5].

The concept of the Lipschitz condition has been developed as the Lipschitz in the small [6]. We recall the definition of Lipschitz in the small in Definition 1.

Definition 1. Let and be metric spaces. A function is said to be Lipschitz in the small if there exist an and a , such that for every , , we have

The constant is called the constant of Lipschitz in the small of on .

From the definition, a Lipschitz function in the small is uniformly continuous, but the converse is not always true. Garrido and Jaramillo state that a uniformly continuous function can be approached by the Lipschitz function in the small or the local Lipschitz function.

A Lipschitz function is Lipschitz in the small but the converse is not always true [6]. Based on that fact, Garrido and Jaramillo define a small-determined metric space. A small-determined metric space is a metric space with the set of all the Lipschitz functions equal to the set of all Lipschitz functions in the small. One of the examples of a small-determined metric space is a quasiconvex metric space.

A metric space is called quasiconvex if there exists a positive constant , such that for every two points in can be joined by a continuous path with its length not greater than times the distance between those two points and .

Based on Garrido's results, Leung and Tang [7] give necessary and sufficient conditions on a subset of such that is the Lipschitz for every function that is Lipschitz in the small on [7].

In this paper, the influence of the Lipschitz in the small condition in Picard’s theorem will be seen. After defining the contraction in the small, its characteristics due to fixed-point theorems will be presented. The results will be applied to develop Picard’s theorem using contraction in the small. The proof of the uniqueness in Picard’s theorem will be provided by using Gronwall’s inequality.

We recall Gronwall’s inequality as in Theorem 2.

Theorem 2. (Gronwall’s inequality) [1]. Let and be two continuous positive real-valued functions on an interval and If satisfies for every , then for every

2. Results

Based on the definition of Lipschitz in the small it will be defined as the contraction in the small. A fixed-point theorem using contraction will be developed by using contraction in the small. Furthermore, it will be applied to develop a new version of Picard’s theorem.

Definition 3. A function is said to be a contraction in the small if there exist and , such that for every , , we have

Contraction in the small is a special case of the Lipschitz condition with its constant less than 1. It is clear that every contraction function is a contraction in the small function. The converse is not always true. For example, let us consider the function , where , . The function is a contraction in the small on , but it is not a contraction on .

Lemma 4. If the function is bounded and the Lipschitz in the small with in Definition 3 is less than 1, then is a contraction.

Proof. Since is bounded and Lipschitz in the small then is Lipschitz.

Lemma 5. Let and be metric spaces. If is compact and if the function is Lipschitz in the small then is Lipschitz.

Proof. The function is Lipschitz in the small then is uniformly continuous. Since is compact, then is bounded. As a corollary, is Lipschitz.

Corollary 6. Let and be metric spaces. If is compact and if the function is a contraction in the small, then is a contraction.

Proof. It is proven by Lemma 4.

The relationships between a contraction in the small function and its fixed points will be discussed in Lemma 7 and Theorem 8.

Lemma 7. Let be a metric space and . If is a contraction in the small and , then there exists a positive such that

Proof. Since is a contraction in the small, there exist a positive less than and a positive such that for every, Put Let be an arbitrary point. There is a point with As a corollary, we have This means that .

The converse of Lemma 7 is not always true. For example, let us consider the metric space with its metric defined by for every . The function , where is a contraction in the small. For and However,

Based on Lemma 7, it will be given a fixed-point theorem involving the contraction in the small in Theorem 8.

Theorem 8. Let be a complete metric space and . If is a contraction in the small with constant such that , then there exists as a fixed point of . Moreover, if , then is a unique fixed point of on .

Proof. Since is a contraction in the small, there exist and such that for every , , Let us define and , for every , From the hypothesis, for every . Let us consider that for every , As a corollary, for , there is , with . Therefore, Since the sequence is a Cauchy sequence in . The completeness of implies the existence of such that converges to in . Since for every , then . From the hypothesis, . Therefore, This means that is a fixed point of the function .
Furthermore, if with , then Since we have . This means that .

Before we develop Picard’s theorem using Lipschitz in the small let us consider that for every , the standard/usual metric in will bring that is equal with .

Definition 9. Let . A function is said to be Lipschitz in the small in the second component on , if there exists a positive and such that for every , we have

Let . A function is said to be a contraction in the small in the second component on , if there exist a positive and such that for every , , we have

In the next discussion, the word domain refers to an open connected set in .

Let consider the initial value problem (IVP) with is a continuous real-valued function on a domain.

Picard’s Theorem still holds if the condition of Lipschitz is replaced by a contraction in the small. The important thing in the proof is in choosing the value of such that the constructed sequence satisfies for every and for every . In the discussion, the function is a nonzero function on the domain.

Theorem 10. Let be a domain, be an interior point of , and . If the function is continuous on and satisfies the contraction in the small in the second component on , then there exist a positive and a unique function such that is a solution of the initial value problem in (15) on .

Proof. Since is an interior point of , there exist positive numbers such that Since is continuous on and is compact, then attains its maximum. Put . Since is a contraction in the small in the second component on , there exist an and such that for every , , we have Put .
Let us define and By using the induction method, it can be proven that for every , we have From (20) and by considering that and , by using the induction method, we can deduce that for every and for every , Let us consider that Let assume the formula is held for , i.e., Therefore, The induction method gives the formula which holds for every .
Moreover, for every and for every , Therefore, Since the series is convergent, via the Weierstrass -test, then the sequence converges uniformly to a function, say , on .
Since converges uniformly to on , and for every , is continuous on , and the function is continuous on . Furthermore, Therefore, for every , This means that for every .
Since is continuous, then is differentiable on .
Moreover, . This means that is a solution of the IVP (15) on .
Uniqueness. Let and be two solutions of the IVP (15) on , for every By using Gronwall’s inequality with and for every , we get This means that for every .

Similarly, for we get for every . Therefore, we have on .

Remark 11. Theorem 10 still holds if the condition contraction in the small in the second component on is replaced by the contraction in the small.

The in Theorem 10 may be extended as stated in Theorem 12.

Theorem 12. Let and . If is an interior point of and is a continuous function on and satisfies the contraction in the small in the second component on , then there exists a unique function as a solution of the initial value problem in (15) on . Moreover, if and , then the unique solution of the initial value problem in (15) exists on .

Proof. From Theorem 10, there exist an and a unique solution of the initial value problem in (15) on . (i)Put and . It is clear that is an interior point of . By Theorem 10, there exist an and a unique solution on of the initial value problem (ii)Put and . It is clear that is an interior point of . Based on Theorem 10, there exist an and a unique solution on of the initial value problem Continuing the process, we have , , , , , and (i)a unique solution on (ii)unique solutions on , (iii) on , As a corollary, we have a unique solution, where For every Let and , . We have

3. Concluding Remarks

Theorem 8 has given a fixed-point theorem involving a contraction in the small. The proof of the developed Picard’s Theorem has been provided in Theorem 10 by constructing a sequence of functions as in Picard approximation. The key is to have an such that the contraction condition still holds for every , , for every .

Data Availability

No data were used in the research.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

The author would like to thank the Faculty of Mathematics and Natural Sciences, Universitas Gadjah Mada, for the support of the research.