A Fixed Point Theorem for Contraction Type Maps in Partially Ordered Metric Spaces and Application to Ordinary Differential Equations

Abstract

We present a fixed point theorem for generalized contraction in partially ordered complete metric spaces. As an application, we give an existence and uniqueness for the solution of a periodic boundary value problem.

1. Introduction

The contraction mapping theorem and the abstract monotone iterative technique are well known and are applicable to a variety of situations. Recently, there has been a trend to weaken the requirement on the contraction by considering metric spaces endowed with partial order (see [1–7]). It is of interest to determine if it is still possible to establish the existence of a unique fixed point assuming that the operator considered is monotone in such a setting. Such a fixed point theorem is useful, for example, in establishing the existence of a unique solution to periodic boundary value problems, besides many others.

That approach was initiated by Ran and Reurings in [8], where some applications to matrix equations were studied. This fixed point theorem was refined and extended in [7, 9] and applied to the periodic boundary value problem in the monotone case. In this paper, we consider a special case of the following periodic boundary value problem 𝑢^′(𝑡)((𝑡[],𝑢(0)(𝑇),=𝑓𝑡,𝑢))if𝑡∈𝐼=0,𝑇=𝑢 (1.1) where 𝑇>0 and 𝑓∶𝐼×ℝ→ℝ is continuous function.

Definition 1.1. A lower solution for (1.1) is a function 𝑢∈𝐶¹(𝐼,ℝ) such that 𝑢^′(𝑡)((𝑡[],𝑢(0)(𝑇).≤𝑓𝑡,𝑢))for𝑡∈𝐼=0,𝑇≤𝑢 (1.2)

Very recently, Harjani and Sadarangani [4] proved the following existence theorem.

Theorem 1.2. Consider problem (1.1) with 𝑓∶𝐼×ℝ→ℝ is continuous and suppose that there exists 𝜆>0 such that for 𝑥,𝑦∈ℝ with 𝑦≥𝑥0≤𝑓𝑡,𝑦+𝜆𝑦−𝑓(𝑡,𝑥)+𝜆𝑥≤𝜆𝜙𝑦−𝑥, (1.3) where 𝜙∶[0,∞)→[0,∞) can be written by 𝜙(𝑥)=𝑥−𝜓(𝑥) with 𝜓∶[0,∞)→[0,∞) continuous increasing positive in (0,∞), 𝜓(0)=0 and lim_𝑡→∞𝜓(𝑡)=∞. Then the existence of a lower solution of (1.1) provides the existence of a unique solution of (1.1).

In Section 2, we prove a new fixed point theorem in partially ordered complete metric spaces. In Section 3, existence of a unique solution for problem (1.1) is obtained under suitable conditions.

2. Fixed Point Theorem

Let 𝑆 denote the class of those functions 𝛼∶[0,∞)→[0,1) which satisfies the condition limsup_𝑠→𝑡+𝛼(𝑠)<1,∀t∈[0,∞). (2.1) We prove the main theorem of the paper as follows.

Theorem 2.1. Let (𝑋,≼) be a partially order metric space that there exists a metric 𝑑 in 𝑋 such that (𝑋,𝑑) is a complete metric space. Let 𝑓∶𝑋→𝑋 be an increasing mapping such that there exists 𝑥₀∈𝑋 with 𝑥₀≼𝑓(𝑥₀). Suppose that there exists 𝛼∈𝑆 such that 𝑑𝑓(𝑥),𝑓𝑦≤𝛼𝑑𝑥,𝑦𝑑𝑥,𝑦, (2.2) for all comparable 𝑥,𝑦∈𝑋. If 𝑓is𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 (2.3) or 𝑖𝑓𝑎𝑛𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑥_𝑛→𝑥𝑖𝑛𝑋,𝑡ℎ𝑒𝑛𝑥_𝑛≼𝑥,∀𝑛∈ℕ. (2.4) Besides, if 𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑥,𝑦∈𝑋,𝑡ℎ𝑒𝑟𝑒𝑒𝑥𝑖𝑠𝑡𝑠𝑧∈𝑋𝑤ℎ𝑖𝑐ℎ𝑖𝑠𝑐𝑜𝑚𝑝𝑎𝑟𝑎𝑏𝑙𝑒𝑡𝑜𝑥𝑎𝑛𝑑𝑦, (2.5) then 𝑓 have a unique fixed point.

Proof. We first show that 𝑓 has a fixed point. Since 𝑥₀≼𝑓(𝑥₀) and 𝑓 is increasing function, we obtain by induction that 𝑥₀≼𝑓𝑥₀≼𝑓²𝑥₀≼𝑓³𝑥₀≼⋯≼𝑓^𝑛𝑥₀≼𝑓^𝑛+1𝑥₀⋯. (2.6) Put 𝑥_𝑛=𝑓^𝑛(𝑥₀), 𝑛=1,2,…. Since 𝑥_𝑛≼𝑥_𝑛+1 for each 𝑛∈ℕ then by (2.2) 𝑑𝑥_𝑛+1,𝑥_𝑛+2=𝑑𝑓^𝑛+1𝑥₀,𝑓^𝑛+2𝑥₀≤𝛼𝑑𝑥_𝑛,𝑥_𝑛+1𝑑𝑥_𝑛,𝑥_𝑛+1≤𝑑𝑥_𝑛,𝑥_𝑛+1. (2.7) And so the sequence {𝑑(𝑥_𝑛+1,𝑥_𝑛)} is nonincreasing and bounded below. Thus there exists 𝜏≥0 such that lim_𝑛→∞𝑑(𝑥_𝑛+1,𝑥_𝑛)=𝜏. Since limsup_𝑠→𝜏+𝛼(𝑠)<1 and 𝛼(𝜏)<1 then there exist 𝑟∈[0,1) and 𝜖>0 such that 𝛼(𝑠)<𝑟 for all 𝑠∈[𝜏,𝜏+𝜖]. We can take 𝜈∈ℕ such that 𝜏≤𝑑(𝑥_𝑛+1,𝑥_𝑛)≤𝜏+𝜖 for all 𝑛∈ℕ with 𝑛≥𝜈. Then since 𝑑𝑥_𝑛+1,𝑥_𝑛+2≤𝛼𝑑𝑥_𝑛,𝑥_𝑛+1𝑑𝑥_𝑛,𝑥_𝑛+1≤𝑟𝑑𝑥_𝑛,𝑥_𝑛+1, (2.8) for all 𝑛∈ℕ with 𝑛≥𝜈 we have ^∞_𝑛=1𝑑𝑥_𝑛,𝑥_𝑛+1≤^𝜈_𝑛=1𝑑𝑥_𝑛,𝑥_𝑛+1+^∞_𝑛=1𝑟^𝑛𝑑𝑥_𝜈,𝑥_𝜈+1<∞, (2.9) and hence {𝑥_𝑛} is a Cauchy sequence. Since 𝑋 is complete, {𝑥_𝑛} converges to some point 𝑧∈𝑋. To prove that 𝑧 is a fixed point of 𝑓, if 𝑓 is a continuous, then 𝑧=lim_𝑛→∞𝑥_𝑛=lim_𝑛→∞𝑓^𝑛𝑥₀=lim_𝑛→∞𝑓^𝑛+1𝑥₀=𝑓lim_𝑛→∞𝑓^𝑛𝑥₀=𝑓(𝑧); (2.10) hence 𝑧=𝑓(𝑧). If case (2.4) holds then 𝑑𝑓(𝑧),𝑧≤𝑑𝑓(𝑧),𝑓𝑥_𝑛+𝑑𝑓𝑥_𝑛𝑑𝑥,𝑧≤𝛼_𝑛,𝑧𝑑𝑥_𝑛,𝑧+𝑑𝑥_𝑛+1𝑥,𝑧≤𝑑_𝑛,𝑧+𝑑𝑥_𝑛+1,𝑧. (2.11) Since 𝑑(𝑥_𝑛,𝑧)→0 then we get 𝑓(𝑧)=𝑧. To prove the uniqueness of the fixed point, let 𝑦 be another fixed point of 𝑓. From (2.5) there exists 𝑥∈𝑋 which is comparable to 𝑦 and 𝑧. Monotonicity implies that 𝑓^𝑛(𝑥) is comparable to 𝑓^𝑛(𝑦)=𝑦 and 𝑓^𝑛(𝑧)=𝑧 for 𝑛=0,1,…. Moreover, 𝑑𝑧,𝑓^𝑛(𝑥)=𝑑𝑓^𝑛(𝑧),𝑓^𝑛(𝑥𝑑𝑓)≤𝛼^𝑛−1(𝑧),𝑓^𝑛−1(𝑥)𝑑𝑓^𝑛−1(𝑧),𝑓^𝑛−1(𝑥)≤𝑑𝑓^𝑛−1(𝑧),𝑓^𝑛−1(𝑥)=𝑑𝑧,𝑓^𝑛−1(𝑥). (2.12) Consequently, the sequence 𝜁^𝑧_𝑛∶=𝑑(𝑧,𝑓^𝑛(𝑥)) is nonnegative and decreasing and so lim_𝑛→∞𝑑(𝑧,𝑓^𝑛(𝑥))=𝜁_𝑧∈ℝ. Similarly we can show that the sequence 𝜁^𝑦_𝑛∶=𝑑(𝑦,𝑓^𝑛(𝑥)) is nonnegative and decreasing and so lim_𝑛→∞𝑑(𝑦,𝑓^𝑛(𝑥))=𝜁_𝑦∈ℝ. Now similarly the above method we can choose 𝑟₁,𝑟₂ in [0,1) and 𝜏₁∈ℕ such that 𝑑𝑧,𝑓^𝑛(𝑥)≤𝛼𝑑𝑧,𝑓^𝑛−1(𝑥)𝑑𝑧,𝑓^𝑛−1(𝑥)≤𝑟₁𝑑𝑧,𝑓^𝑛−1(𝑥,𝑑)𝑦,𝑓^𝑛(𝑥)≤𝛼𝑑𝑦,𝑓^𝑛−1(𝑥)𝑑𝑦,𝑓^𝑛−1(𝑥)≤𝑟₂𝑑𝑦,𝑓^𝑛−1(𝑥), (2.13) for all 𝑛∈ℕ with 𝑛>𝜏₁. Finally 𝑑𝑧,𝑦≤𝑑𝑧,𝑓^𝑛(𝑥)+𝑑𝑓^𝑛(𝑥),𝑦≤𝑟^𝑛−𝜏1₁𝑑𝑧,𝑓^𝜏1𝑥₀+𝑟^𝑛−𝜏1₂𝑑𝑦,𝑓^𝜏1𝑥₀, (2.14) for all 𝑛∈ℕ with 𝑛>𝜏₁. Therefor if in (2.14) taking 𝑛→∞ yields 𝑑(𝑧,𝑦)=0.

3. Application to Ordinary Differential Equation

In this section we present an example where Theorem 2.1 can be applied. This example is inspired in [2, 4, 7].

Definition 3.1. Let 𝔅 denote the class of those functions 𝜙∶[0,∞)→[0,∞) which satisfies the following condition:(i)𝜙 is increasing,(ii)for each 𝑥>0, 𝜙(𝑥)<𝑥,(iii)𝛽(𝑥)=𝜙(𝑥)/𝑥∈𝑆.

For example, 𝜙(𝑥)=𝑎𝑥, where 0≤𝑎<1, 𝜙(𝑥)=𝑥/(𝑥+1), and 𝜙(𝑥)=ln(1+𝑥) are in 𝔅.

Theorem 3.2. Consider problem (1.1) with 𝑓∶𝐼×ℝ→ℝ is continuous and suppose that there exists 𝜆>0 such that for 𝑥,𝑦∈ℝ with 𝑦≥𝑥0≤𝑓𝑡,𝑦+𝜆𝑦−𝑓(𝑡,𝑥)+𝜆𝑥≤𝜆𝜙𝑦−𝑥, (3.1) where 𝜙∈𝔅. Then the existence of a lower solution of (1.1) provides the existence of a unique solution of (1.1).

Proof. Problem (1.1) is equivalent to the integral equation 𝑢(𝑡)=^𝑇₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))+𝜆𝑢(𝑠)𝑑𝑠, (3.2) where 𝐺()=𝑒𝑡,𝑠𝑒^{𝜆(𝑇+𝑠−𝑡)𝜆𝑇}𝑒−1,0≤𝑠<𝑡≤𝑇,_{𝜆(𝑠−𝑡)}𝑒^𝜆𝑇−1,0≤𝑡<𝑠≤𝑇. (3.3) Define 𝐹∶𝐶(𝐼,ℝ)→𝐶(𝐼,ℝ) by (𝐹𝑢)(𝑡)=^𝑇₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))+𝜆𝑢(𝑠)𝑑𝑠. (3.4) Note that if 𝑢∈𝐶(𝐼,ℝ) is a fixed point of 𝐹 then 𝑢∈𝐶¹(𝐼,ℝ) is a solution of (1.1). Now, we check that hypotheses in Theorem 2.1 are satisfied. Indeed, 𝑋=𝐶(𝐼,ℝ) is a partially ordered set if we define the following order relation in 𝑋: 𝑥,𝑦∈𝐶(𝐼,ℝ),𝑥≤𝑦iff𝑥(𝑡)≤𝑦(𝑡),∀𝑡∈𝐼. (3.5) The mapping 𝐹 is increasing since, by hypotheses, for𝑢≥𝑣𝑓(𝑡,𝑢)+𝜆𝑢≥𝑓(𝑡,𝑣)+𝜆𝑣 (3.6) which implies for 𝑡∈𝐼, using that 𝐺(𝑡,𝑠)>0 for (𝑡,𝑠)∈𝐼×𝐼, that (𝐹𝑢)(𝑡)=^𝑇₀𝐺(𝑓((𝑠(𝑠≥𝑡,𝑠)𝑠,𝑢))+𝜆𝑢)𝑑𝑠_𝑇0𝐺(𝑡,𝑠)𝑓(𝑠,𝑣(𝑠))+𝜆𝑣(𝑠)𝑑𝑠=(𝐹𝑣)(𝑡). (3.7) Beside, for 𝑢≥𝑣𝑑(𝐹𝑢,𝐹𝑣)=sup_𝑡∈𝐼|(𝐹𝑢)(𝑡)−(𝐹𝑣)(𝑡)|≤sup_𝑡∈𝐼^𝑇₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))+𝜆𝑢(s)−𝑓(𝑠,𝑣(𝑠))+𝜆𝑣(𝑠)𝑑𝑠≤sup_𝑡∈𝐼^𝑇₀𝐺(𝑡,𝑠)⋅𝜆𝜙(𝑢(𝑠)−𝑣(𝑠))𝑑𝑠. (3.8) As the function 𝜙(𝑥) is increasing and 𝑢≥𝑣 then 𝜙(𝑢(𝑠)−𝑣(𝑠))≤𝜙(𝑑(𝑢,𝑣)) we obtain 𝑑(𝐹𝑢,𝐹𝑣)≤sup_𝑡∈𝐼^𝑇₀𝐺()(𝑢(𝑠)(𝑠(𝑑(𝑡,𝑠⋅𝜆𝜙−𝑣))𝑑𝑠≤𝜆𝜙𝑢,𝑣))sup_𝑡∈𝐼^𝑇₀𝐺()(𝑑(𝑡,𝑠𝑑𝑠=𝜆𝜙𝑢,𝑣))sup_𝑡∈𝐼1𝑒^𝜆𝑇1−1𝜆𝑒^{𝜆(𝑇+𝑠−𝑡)}]^𝑡₀+1𝜆𝑒^{𝜆(𝑠−𝑡)}]^𝑇_𝑡(𝑑(⋅1=𝜆𝜙𝑢,𝑣))𝜆𝑒^𝜆𝑇𝑒−1^𝜆𝑇−1=𝜙(𝑑(𝑢,𝑣)). (3.9) Then for 𝑢≥𝑣𝑑(𝐹𝑢,𝐹𝑣)≤𝛼(𝑑(𝑢,𝑣))𝑑(𝑢,𝑣). (3.10) Finally, let 𝛽(𝑡) be a lower solution of (1.1), and we will show that 𝛽≤𝐹(𝛽).
Indeed, 𝛽^′(𝑡)+𝜆𝛽(𝑡)≤𝑓𝑡,𝛽(𝑡)+𝜆𝛽(𝑡),for𝑡∈𝐼. (3.11) Multiplying by 𝑒^𝜆𝑡 we get 𝛽(𝑡)𝑒^𝜆𝑡′≤𝑓𝑡,𝛽(𝑡)+𝜆𝛽(𝑡)𝑒^𝜆𝑡,for𝑡∈𝐼, (3.12) and this gives us 𝛽(𝑡)𝑒^𝜆𝑡≤𝛽(0)+^𝑡₀𝑓𝑠,𝛽(𝑠)+𝜆𝛽(𝑠)𝑒^𝜆𝑠𝑑𝑠,for𝑡∈𝐼 (3.13) which implies that 𝛽(0)𝑒^𝜆𝑇≤𝛽(𝑇)𝑒^𝜆𝑇≤𝛽(0)+^𝑇₀𝑓𝑠,𝛽(𝑠)+𝜆𝛽(𝑠)𝑒^𝜆𝑠𝑑𝑠, (3.14) and so 𝛽(0)≤^𝑇₀𝑒^𝜆𝑠𝑒^𝜆𝑇𝑓(𝑠(𝑠−1𝑠,𝛽)+𝜆𝛽)𝑑𝑠. (3.15) From this equality and (3.13) we obtain 𝛽(𝑡)𝑒^𝜆𝑡≤^𝑡₀𝑒^{𝜆(𝑇+𝑠)}𝑒^𝜆𝑇𝑓(𝑠(𝑠−1𝑠,𝛽)+𝜆𝛽)𝑑𝑠+^𝑇_𝑡𝑒^𝜆𝑠𝑒^𝜆𝑇𝑓(𝑠(𝑠−1𝑠,𝛽)+𝜆𝛽)𝑑𝑠, (3.16) and, consequently, 𝛽(𝑡)≤^𝑡₀𝑒^{𝜆(𝑇+𝑠−𝑡)}𝑒^𝜆𝑇𝑓(𝑠(𝑠−1𝑠,𝛽)+𝜆𝛽)𝑑𝑠+^𝑇_𝑡𝑒^{𝜆(𝑠−𝑡)}𝑒^𝜆𝑇𝑓(𝑠(𝑠−1𝑠,𝛽)+𝜆𝛽)𝑑𝑠. (3.17) Hence 𝛽(𝑡)≤^𝑇₀𝐺(𝑡,𝑠)𝑓𝑠,𝛽(𝑠)+𝜆𝛽(𝑠)𝑑𝑠=𝐹𝛽(𝑡),for𝑡∈𝐼. (3.18) Finally, Theorem 2.1 gives that 𝐹 has a unique fixed point.

Example 3.3. Let 𝜙₀∶[0,∞)→[0,∞) be a defined 𝜙₀(𝑡)=30,0≤𝑡≤3,3𝑡−9,3<𝑡≤4,4𝑡,4<𝑡. (3.19) Let 𝑓∶𝐼×ℝ→ℝ be continuous and suppose that there exists 𝜆>0 such that for 𝑥,𝑦∈ℝ with𝑦≥𝑥0≤𝑓𝑡,𝑦+𝜆𝑦−𝑓(𝑡,𝑥)+𝜆𝑥≤𝜆𝜙₀𝑦−𝑥. (3.20) Then be Theorem 2.1, the existence of a lower solution for (1.1) provides the existence of a unique solution of (1.1).
The example discussed above cannot be the result of Harjani and Sadarangani noted Theorem  1.2, because 𝜓(𝑥)=𝑥−𝜙₀(𝑥) is not increasing.

Acknowledgment

The third author of this work was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no.: 2010-0010243).