A Note on the Paper “Multipoint BVPs for Second-Order Differential Equations with Impulses” by Xuxin Yang, Zhimin He, and Jianhua Shen

Abstract

We give a counter example to the comparison principle for the multipoint BVPs (by Xuxin Yang, Zhimin He, and Jianhua Shen, in Mathematical Problems in Engineering, Volume 2009, Article ID 258090, doi:10.1155/2009/258090). Then we suggest and prove a corrected version of the comparison principle.

1. Introduction and Preliminaries

Consider the following multipoint BVPs [1]: −𝑢^′′(𝑡)=𝑓(𝑡,𝑢(𝑡),𝑢(𝜃(𝑡))),𝑡≠𝑡_𝑘,𝑡∈𝐽=[0,1],Δ𝑢^′(𝑡_𝑘)=𝐼_𝑘(𝑢(𝑡_𝑘)),𝑘=1,2,…,𝑚,𝑢(0)−𝑎𝑢^′(0)=𝑐𝑢(𝜂),𝑢(1)+𝑏𝑢^′(1)=𝑑𝑢(𝜉), (1.1) where 0≤𝜃(𝑡)≤𝑡,𝜃∈𝐶(𝐽),0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯<𝑡_𝑚<𝑡_𝑚+1=1, 𝑓 is continuous everywhere except at {𝑡_𝑘}×𝑅²;𝑓(𝑡⁺_𝑘,⋅,⋅) and 𝑓(𝑡⁻_𝑘,⋅,⋅) exist with 𝑓(𝑡⁻_𝑘,⋅,⋅)=𝑓(𝑡_𝑘,⋅,⋅);𝐼_𝑘∈𝐶(𝑅,𝑅), and Δ𝑢^′(𝑡_𝑘)=𝑢^′(𝑡⁺_𝑘)−𝑢^′(𝑡⁻_𝑘), 𝑎≥0, 𝑏≥0, 0≤𝑐≤1, 0≤𝑑≤1, 𝑎+𝑐>0,𝑏+𝑑>0,0<𝜂,𝜉<1.

Let 𝑃𝐶(𝐽)={𝑥∶𝐽→𝑅;𝑥(𝑡) be continuous everywhere expect for some 𝑡_𝑘 at which 𝑥(𝑡⁺_𝑘) and 𝑥(𝑡⁻_𝑘) exist and 𝑥(𝑡_𝑘)=𝑥(𝑡⁻_𝑘),𝑘=1,2,…,𝑚}; 𝑃𝐶¹(𝐽)={𝑥∈𝑃𝐶(𝐽)∶𝑥^′(𝑡) is continuous everywhere expect for some 𝑡_𝑘 at which 𝑥^′(𝑡⁺_𝑘) and 𝑥^′(𝑡⁻_𝑘) exist and 𝑥^′(𝑡_𝑘)=𝑥^′(𝑡⁻_𝑘),𝑘=1,2,…,𝑚}. Let 𝐽⁻=𝐽⧵{𝑡_𝑘,𝑘=1,2,…,𝑚}, and 𝐸=𝑃𝐶¹(𝐽,𝑅)∩𝐶²(𝐽⁻,𝑅). a function 𝑥∈𝐸 is called a solution of BVPS (1.1) if it satisfies (1.1).

The purpose of this note is to point out that the results basing on the comparison principle [1, Theorem 2.1] are not true. Then we give a new comparison principle.

2. Problem and Statement

The authors [1] proved some existence results for multipoint BVPs (1.1) by use of the following comparison principle [1, Theorem 2.1].

Assume that 𝑢∈𝐸 satisfies −𝑢^′′(𝑡)+𝑀𝑢(𝑡)+𝑁𝑢(𝜃(𝑡))≤0,𝑡≠𝑡_𝑘,𝑡∈𝐽=[0,1],Δ𝑢^′(𝑡_𝑘)≥𝐿_𝑘𝑢(𝑡_𝑘),𝑘=1,2,…,𝑚,𝑢(0)−𝑎𝑢^′(0)≤𝑐𝑢(𝜂),𝑢(1)+𝑏𝑢^′(1)≤𝑑𝑢(𝜉), (2.1) where 𝑎≥0, 𝑏≥0, 0≤𝑐≤1, 0≤𝑑≤1, 𝑎+𝑐>0,𝑏+𝑑>0,0<𝜂,𝜉<1, 𝐿_𝑘≥0, and constants 𝑀,𝑁 satisfy 𝑀>0,𝑁≥0,𝑀+𝑁2+^𝑚_𝑘=1𝐿_𝑘≤1. (2.2) Then 𝑢(𝑡)≤0 for 𝑡∈𝐽.

However, the comparison principle above is not true.

A Counter Example
Let 3𝑢(𝑡)=2𝑡²1+20,𝑡∈[0,25],2𝑡²1+3,𝑡∈(2,1]. (2.3) Then 𝑢^′1(𝑡)=3𝑡,𝑡∈[0,21],5𝑡,𝑡∈(2𝑢,1],_′′1(𝑡)=3,𝑡∈[0,21],5,𝑡∈(2,1]. (2.4) And let 𝑀=𝑁=1/1000,𝑎=𝑏=𝑐=𝑑=1,𝑚=1,𝑡₁=1/2,𝐿₁=1/1000,𝜃(𝑡)=(1/2)𝑡,𝜂=1/3,and𝜉=1/6. When 𝑡∈[0,1/2), then 1(310002𝑡²1+20)+(310002×𝑡²4+20)≤3. (2.5) When 𝑡∈(1/2,1], then 1(510002𝑡²1+3)+(510002×𝑡²4+3)≤5. (2.6) Hence −𝑢^′′(𝑡)+𝑀𝑢(𝑡)+𝑁𝑢(𝜃(𝑡))≤0.Δ𝑢^′(12)=𝑢^′(12+)−𝑢^′(121)=5×21−(3×2)=1, (2.7)111000𝑢(21)=(310002×141+20)=×10001638. (2.8) Hence Δ𝑢^′(𝑡₁)≥𝐿₁𝑢(𝑡₁).𝑢(0)−𝑢^′1(0)=20,𝑢(33)=2×19+20. (2.9) Hence 𝑢(0)−𝑎𝑢^′(0)≤𝑐𝑢(1/3).𝑢(1)+𝑢^′5(1)=2+3+5=2121,𝑢(63)=2×136+20. (2.10) Hence 𝑢(1)+𝑏𝑢^′(1)≤𝑑𝑢(1/6).𝑀+𝑁2+^𝑚_𝑘=1𝐿_𝑘=21000<1. (2.11) But we easily show that 𝑢(𝑡)>0,forall𝑡∈[0,1], which is a contradiction with (Theorem 2.1) in [1]. In fact, we can correct Theorem 2.1 in [1] as follows.

Theorem 2.1. Suppose 𝑢∈𝐸∩𝐶(𝐽) such that −𝑢^′′(𝑡)+𝑀𝑢(𝑡)+𝑁𝑢(𝜃(𝑡))≤0𝑡≠𝑡_𝑘,𝑡∈𝐽=[0,1],Δ𝑢^′(𝑡_𝑘)≥𝐿_𝑘𝑢(𝑡_𝑘),𝑘=1,2,…,𝑚,𝑢(0)−𝑎𝑢^′(0)≤𝑐𝑢(𝜂),𝑢(1)+𝑏𝑢^′(1)≤𝑑𝑢(𝜉), (2.12) where 𝑎≥0, 𝑏≥0, 0≤𝑐≤1, 0≤𝑑≤1, 0<𝜂,𝜉<1,𝑎+𝑐>0,𝑏+𝑑>0,𝐿_𝑘>0, and constants 𝑀,𝑁 satisfy 𝑀>0,𝑁≥0,𝑀+𝑁2+^𝑚_𝑘=1𝐿_𝑘≤1. (2.13) Then 𝑢(𝑡)≤0 for 𝑡∈𝐽.

Remark 2.2. In this Theorem, we have to add 𝑢∈𝐶(𝐽).

Proof. Suppose to contrary that there exist some 𝑡∈𝐽, such that 𝑢(𝑡)>0.
If 𝑢(1)=max_𝑡∈𝐽𝑢(𝑡)>0, we have 𝑢^′(1)≥0, 𝑢(1)≥𝑢(𝜉), and 𝑑𝑢(𝜉)≤𝑢(1)≤𝑢(1)+𝑏𝑢^′(1)≤𝑑𝑢(𝜉). (2.14) Therefore, 𝑑=1 and 𝑢(𝜉) is maximum value.
If 𝑢(0)=max_𝑡∈𝐽𝑢(𝑡)>0, we have 𝑢^′(0)≤0, 𝑢(0)≥𝑢(𝜂), and 𝑐𝑢(𝜂)≤𝑢(0)≤𝑢(0)−𝑎𝑢^′(0)≤𝑐𝑢(𝜂). (2.15) Therefore, 𝑐=1 and 𝑢(𝜂) is maximum value.
So there is a 𝛿∈(0,1) such that 𝑢(𝛿)=max_𝑡∈𝐽𝑢(𝑡)>0,byΔ𝑢=0,then𝑢^′(𝛿⁺)≤0,𝑢^′(𝛿⁻)≥0. (2.16) It is obvious to see that 𝛿∉{𝑡_𝑘,𝑘=1,2,…,𝑚} by Δ𝑢^′(𝛿)=𝑢^′(𝛿⁺)−𝑢^′(𝛿)≥𝐿_𝑘𝑢(𝛿)>0 (2.17) which is a contradiction because of (2.16).
(i)Suppose that 𝑢(𝑡)≥0 for 𝑡∈[0,𝛿].
By 𝑢(𝛿)=max_𝑡∈𝐽𝑢(𝑡)>0, we get 𝛿∈𝐽⁻, 𝑢^′′(𝛿)≤0. On the other hand, by (2.12), we have 0<𝑀𝑢(𝛿)+𝑁𝑢(𝜃(𝑡))≤𝑢^′′(𝛿) (2.18) which is a contradiction.
(ii)Suppose there exists 𝑡_∗∈[0,𝛿] such that 𝑢(𝑡_∗)=min_{𝑡∈[0,𝛿)}𝑢(𝑡)<0. By (2.12), we get 𝑢^′′(𝑡)≥(𝑀+𝑁)𝑢(𝑡_∗),𝑡∈[0,𝛿),𝑡≠𝑡_𝑘,Δ𝑢(𝑡_𝑘)=0,Δ𝑢^′(𝑡_𝑘)≥𝐿_𝑘𝑢(𝑡_𝑘),𝑘=1,2,…,𝑚. (2.19)Integrating from 𝑠(𝑡_∗≤𝑠≤𝛿) to 𝛿, we get 𝑢^′(𝛿)−𝑢^′(𝑠)≥^𝛿_𝑠(𝑀+𝑁)𝑢(𝑡_∗)𝑑𝑠+_{𝑠<𝑡𝑘<𝛿}𝐿_𝑘𝑢(𝑡_𝑘)=(𝛿−𝑠)(𝑀+𝑁)𝑢(𝑡_∗)+_{𝑠<𝑡𝑘<𝛿}𝐿_𝑘𝑢(𝑡_𝑘)≥(𝛿−𝑠)(𝑀+𝑁)𝑢(𝑡_∗)+^𝑚_𝑘=1𝐿_𝑘𝑢(𝑡_∗). (2.20) Hence −𝑢^′(𝑠)≥(𝛿−𝑠)(𝑀+𝑁)𝑢(𝑡_∗)+^𝑚_𝑘=1𝐿_𝑘𝑢(𝑡_∗),𝑡_∗≤𝑠≤𝛿. (2.21) Then integrate from 𝑡_∗ to 𝛿 to obtain −𝑢(𝑡_∗)<𝑢(𝛿)−𝑢(𝑡_∗)≤^𝛿_𝑡∗(𝑀+𝑁)𝑢(𝑡_∗)(𝑠−𝛿)𝑑𝑠−^𝑚_𝑘=1𝐿_𝑘𝑢(𝑡_∗)(𝛿−𝑡_∗)=(𝑀+𝑁)𝑢(𝑡_∗)[−(𝑡_∗−𝛿)²2]−^𝑚_𝑘=1𝐿_𝑘𝑢(𝑡_∗)(𝛿−𝑡_∗)≤−[𝑀+𝑁2(𝛿−𝑡_∗)²+^𝑚_𝑘=1𝐿_𝑘]𝑢(𝑡_∗)≤−(𝑀+𝑁2+^𝑚_𝑘=1𝐿_𝑘)𝑢(𝑡_∗). (2.22) By (2.13), we get 𝑢(𝑡_∗)>0 which is a contradiction. We complete the proof.

This implies that in order to get the existence results of the multipoint BVPs [1], we have to require an additional continuity hypotheses on the function space.

Acknowledgments

This project is supported by NNSF of China, Grant no. 10971019 and NSF of Guangxi, Grant no. 2010GXNSFA013114.