ABSTRACT
In this paper we are considering a third-order three-point equation with nonhomogeneous conditions in the boundary. Using Krasnoselskii’s Theorem and Leray-Schauder Alternative we provide existence results of positive solutions for this problem. Nontrivials examples are given and a numerical method is introduced.
Key words
numerical solutions; third-order; boundary value problem; Krasnoselskii’s Theorem
RESUMO
Neste artigo, consideramos uma equação com três pontos de fronteira de terceira ordem com condições de contorno não homogêneas. Com uso do Teorema de Krasnoselskii e da Alternativa de Leray-Schauder, apresentamos resultados de existência para soluções positivas. Exemplos não triviais são fornecidos e um método numérico é introduzido.
Palavras-chave
soluções numéricas; terceira-ordem; problema de valor de contorno; Teorema de Krasnoselskii
1 INTRODUCTION
Multi-point boundary value problems there has been attention of several studies mainly focused on the existence of solutions with qualitative and quantitative aspects, we recommend[2[2] D. Anderson. Multiple Positive Solutions for a Three-Point Boundary Value Problem. Mathematical and Computer Modelling, 27(6) (1998), 49–57.,3[3] D.R. Anderson. Green’s function for a third-order generalized right focal problem. Journal of Mathematical Analysis and Applications, 288(1) (2003), 1–14. doi:10.1016/S0022-247X(03) 00132-X.
https://doi.org/10.1016/S0022-247X(03)00...
,4[4] D.R. Anderson & J.M. Davis. Multiple Solutions and Eigenvalues for Third-Order Right Focal Boundary Value Problems. Journal of Mathematical Analysis and Applications, 267(1) (2002), 135– 157. doi:10.1006/jmaa.2001.7756. URL http://linkinghub.elsevier.com/retrieve/pii/S0022247X0197756X.
http://linkinghub.elsevier.com/retrieve/...
,5[5] A. Boucherif & N. Al-Malki. Nonlinear three-point third-order boundary value problems. Applied Mathematics and Computation, 190(2) (2007), 1168–1177. doi:10.1016/j.amc.2007.02.039. URL http://linkinghub.elsevier.com/retrieve/pii/S0096300307001476.
http://linkinghub.elsevier.com/retrieve/...
,6[6] H. Chen. Positive solutions for the nonhomogeneous three-point boundary value problem of secondorder differential equations. Math. Comput. Modelling, 45(7-8) (2007), 844–852. doi:10.1016/j.mcm. 2006.08.004. URL http://dx.doi.org/10.1016/j.mcm.2006.08.004.
http://dx.doi.org/10.1016/j.mcm.2006.08....
,7[7] Z.B. Fei & Xiangli. Existence of triple positive solutions for a third order generalized right focal problem. Math. Inequal. Appl, 9(3) (2006), 2006.,8[8] J. Graef & B. Yang. Multiple Positive Solutions to a Three Point Third Order Boundary Value Problem. Discrete and Continuous Dynamical Systems, 2005 (2005), 337–344.,9[9] Y. Lin & M. Cui. A numerical solution to nonlinear multi-point boundary value problems in the reproducing kernel space. Mathematical Methods in the Applied Sciences, 34(1) (2011), 44–47. doi: 10.1002/mma.1327.
https://doi.org/10.1002/mma.1327...
,10[10] Z. Liu & F. Li. On the existence of positive solutions of an elliptic boundary value problem. Chinese Ann. Math. Ser. B, 21(4) (2000), 499–510. doi:10.1142/S0252959900000492. URL http://dx.doi.org/10.1142/S0252959900000492.
http://dx.doi.org/10.1142/S0252959900000...
,11[11] R. Ma. Existence Theorems for a Second Order m-Point Boundary Value Problem. Journal of Mathematical Analysis and Applications, 211(2) (1997), 545–555. doi:10.1006/jmaa.1997.5416. URL http://www.sciencedirect.com/science/article/pii/S0022247X97954160.
http://www.sciencedirect.com/science/art...
,12[12] Y.H. Ma & R.Y. Ma. Positive solutions of a singular nonlinear three-point boundary value problem. Acta Math. Sci. Ser. A Chin. Ed., 23(5) (2003), 583–588. doi:10.1016/j.jmaa.2004.10.029.
https://doi.org/10.1016/j.jmaa.2004.10.0...
,14[14] Q.L. Yao. The existence and multiplicity of positive solutions for a third-order three-point boundary value problem. Acta Mathematicae Applicatae Sinica, 19(1) (2003), 117–122. doi:10.1007/ s10255-003-0087-1.
https://doi.org/10.1007/s10255-003-0087-...
,15][15] H. Yu, L. Haiyan & Y. Liu. Multiple positive solutions to third-order three-point singular semipositone boundary value problem. Proceedings Mathematical Sciences, 114(4) (2004), 409–422. and the references therein. It is well known that the Krasnoselskii’s fixed point theorem, Avery-Peterson and Leggett-Williams theorems are massively used in this line.
In this paper, motived by[13][13] Y. Sun. Positive solutions for third-order three-point nonhomogeneous boundary value problems. Applied Mathematics Letters, 22(1) (2009), 45–51. doi:10.1016/j.aml.2008.02.002. URL http://linkinghub.elsevier.com/retrieve/pii/S0893965908000827.
http://linkinghub.elsevier.com/retrieve/...
, we discuss the existence of a positive solution for the third-order boundary value problem:
2 BACKGROUND MATERIAL
We begin this section by stating the following results.
Theorem 1. Let be a Banach space, a closed and convex set, an open set in and . Then each completely continuous mapping has at least one of the following properties:
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(A1) T has a fixed point in .
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(A2) There are and such that
Theorem 2. Let be a Banach space and let be a cone in . Assume , are bounded open subsets of with , and let be a completely continuous operator such that, either
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(B1) , and , or
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(B2) , and .
Then has a fixed point in .
The first theorem is a well-known Leray-Schauder alternative and the second theorem is due to Krasnoselskii, see[1][1] R.P. Agarwal, M. Meehan & D. O’Regan. “Fixed point theory and applications”, volume 141. Cambridge university press (2001)..
Let us set an auxiliary problem that will be useful in our context.
Related to this problem we have an important lemma.Lemma 3.Let , then we have a unique solution for (2.1)-(2.2). Moreover, this solution is expressed by
where is the Green’s function: andProof. If is solution of (2.1) , we can suppose that
From condition (2.2), we have and Thus (2.1)-(2.2) has a unique solution. Furthermore
Defining in Lemma 3 is easy to see that the solution of (1.1)-(1.2) can be expressed as fixed point of the operator defined by:
Remark 4. Related to and we have useful properties that will be used in the next section.
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For all:
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For all:
3 POSITIVE SOLUTIONS
Let, where be the Banach space of continuously differentiable functions in equipped with
Remark 1. If then satisfies . Besides .
In order to prove the existence we need to consider some basic assumptions.
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(H1) There exist positive constants , and such that
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-
-
.
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Lemma 2. Suppose that holds. Thus the problem (1.1)-(1.2) has a solution with .
Proof. Let us consider the Theorem 1 with and .
We claim that is continuous and completely continuous. In fact, the continuity follows immediately from the Lebesgue dominated convergence theorem and noting that
with . To show complete continuity we will use the Arzela-Ascoli’s theorem. Let be bounded, in other words, there exists with for each . Now if we have Then is a bounded equicontinuous family on . Consequently the Arzela-Ascoli theorem implies is completely continuous.In addition, suppose there are and with . According we have:
Therefore, and in Theorem 1cannot occur. Thus holds and there is such that .
Theorem 3. Suppose that holds and . Then (1.1)-(1.2) has at least one positive solution .
Proof. We start the proof defining the cone by
From and the definition of and , we have that applies in . As seen in the last result, is completely continuous.We shall apply Theorem 2. Thus, we will define , and we will show that the following conditions are true for all :
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if then ;
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if then .
In fact, the demonstration of (a) is similar to the proof of the Lemma 2. To prove (b) is necessary to verify that there is with
where .Let us assume that the inequality is false, that is, for every such that there exists with and . Thus for all with , we can find such that
Then and , when . Being continuous, we have . On the other hand, using and the definition of and we have which is a contradiction. Therefore we have the result.Remark 4. Note that the most important step in the proof of Theorem 3 is to impose conditions to conclude that is not fixed point of .
Example 3.1. Let us consider (1.1)-(1.2) with
Choosing the constants we can easily verify that in these conditions the hypotheses are satisfied.Example 3.2.Let us define
As before, choosing the constantswe can verify that is satisfied.4 NUMERICAL SOLUTIONS
In this section we show the existence and uniqueness for (1.1)-(1.2) using Banach Fixed Point Theorem. This approach is classical but very important to define numerical methods for our problem. Let us consider the iterative sequence
and the basic assumptionsTheorem 1. Suppose that , and are satisfied. Then (1.1)- (1.2) has a unique solution with . Moreover, .
Proof. Let us consider with and . Then
Using we obtain
Motivated by the last result we can define Algorithm 1.
Example 4.1. In this example, we consider
The the analytical solution is . The Table 1 contains results of application in Example 4.1.
We can make additional tests. From Theorem 3 we have a solution for Examples3.1 and 3.2 but in both case, we do not know which they are. Let us apply Algorithm 1 in these problems. For this purpose, we can consider the condition
as stopping criterion for the algorithm. The results for these examples are presented in Table 2 and 3, respectively. The illustrations of these results are given in Figure 1 and 2.REFERENCES
-
[1]R.P. Agarwal, M. Meehan & D. O’Regan. “Fixed point theory and applications”, volume 141. Cambridge university press (2001).
-
[2]D. Anderson. Multiple Positive Solutions for a Three-Point Boundary Value Problem. Mathematical and Computer Modelling, 27(6) (1998), 49–57.
-
[3]D.R. Anderson. Green’s function for a third-order generalized right focal problem. Journal of Mathematical Analysis and Applications, 288(1) (2003), 1–14. doi:10.1016/S0022-247X(03) 00132-X.
-
[4]D.R. Anderson & J.M. Davis. Multiple Solutions and Eigenvalues for Third-Order Right Focal Boundary Value Problems. Journal of Mathematical Analysis and Applications, 267(1) (2002), 135– 157. doi:10.1006/jmaa.2001.7756. URL http://linkinghub.elsevier.com/retrieve/pii/S0022247X0197756X
» http://linkinghub.elsevier.com/retrieve/pii/S0022247X0197756X -
[5]A. Boucherif & N. Al-Malki. Nonlinear three-point third-order boundary value problems. Applied Mathematics and Computation, 190(2) (2007), 1168–1177. doi:10.1016/j.amc.2007.02.039. URL http://linkinghub.elsevier.com/retrieve/pii/S0096300307001476
» http://linkinghub.elsevier.com/retrieve/pii/S0096300307001476 -
[6]H. Chen. Positive solutions for the nonhomogeneous three-point boundary value problem of secondorder differential equations. Math. Comput. Modelling, 45(7-8) (2007), 844–852. doi:10.1016/j.mcm. 2006.08.004. URL http://dx.doi.org/10.1016/j.mcm.2006.08.004
» http://dx.doi.org/10.1016/j.mcm.2006.08.004 -
[7]Z.B. Fei & Xiangli. Existence of triple positive solutions for a third order generalized right focal problem. Math. Inequal. Appl, 9(3) (2006), 2006.
-
[8]J. Graef & B. Yang. Multiple Positive Solutions to a Three Point Third Order Boundary Value Problem. Discrete and Continuous Dynamical Systems, 2005 (2005), 337–344.
-
[9]Y. Lin & M. Cui. A numerical solution to nonlinear multi-point boundary value problems in the reproducing kernel space. Mathematical Methods in the Applied Sciences, 34(1) (2011), 44–47. doi: 10.1002/mma.1327.
-
[10]Z. Liu & F. Li. On the existence of positive solutions of an elliptic boundary value problem. Chinese Ann. Math. Ser. B, 21(4) (2000), 499–510. doi:10.1142/S0252959900000492. URL http://dx.doi.org/10.1142/S0252959900000492
» http://dx.doi.org/10.1142/S0252959900000492 -
[11]R. Ma. Existence Theorems for a Second Order m-Point Boundary Value Problem. Journal of Mathematical Analysis and Applications, 211(2) (1997), 545–555. doi:10.1006/jmaa.1997.5416. URL http://www.sciencedirect.com/science/article/pii/S0022247X97954160
» http://www.sciencedirect.com/science/article/pii/S0022247X97954160 -
[12]Y.H. Ma & R.Y. Ma. Positive solutions of a singular nonlinear three-point boundary value problem. Acta Math. Sci. Ser. A Chin. Ed., 23(5) (2003), 583–588. doi:10.1016/j.jmaa.2004.10.029.
-
[13]Y. Sun. Positive solutions for third-order three-point nonhomogeneous boundary value problems. Applied Mathematics Letters, 22(1) (2009), 45–51. doi:10.1016/j.aml.2008.02.002. URL http://linkinghub.elsevier.com/retrieve/pii/S0893965908000827
» http://linkinghub.elsevier.com/retrieve/pii/S0893965908000827 -
[14]Q.L. Yao. The existence and multiplicity of positive solutions for a third-order three-point boundary value problem. Acta Mathematicae Applicatae Sinica, 19(1) (2003), 117–122. doi:10.1007/ s10255-003-0087-1.
-
[15]H. Yu, L. Haiyan & Y. Liu. Multiple positive solutions to third-order three-point singular semipositone boundary value problem. Proceedings Mathematical Sciences, 114(4) (2004), 409–422.
Publication Dates
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Publication in this collection
13 Dec 2019 -
Date of issue
Sep-Dec 2019
History
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Received
3 Dec 2018 -
Accepted
21 Feb 2019