In this paper, we discuss the existence of positive solutions to the singular Dirichlet boundary value problems (BVPs) for ordinary differential equations (ODEs) of the form
where . The nonlinearity may be singular for the space variables and/or . Moreover, since , the differential operator on the left-hand side of the differential equation is singular at . Sufficient conditions for the existence of positive solutions of the above BVPs are formulated and asymptotic properties of solutions are specified. The theory is illustrated by numerical experiments computed using the open domain MATLAB code bvpsuite, based on polynomial collocation.
MSC: 34B18, 34B16, 34A12.
Keywords:singular ordinary differential equation of the second order; time singularities; space singularities; positive solutions; existence of solutions; polynomial collocation
In the present work, we analyze the existence of positive solutions to the singular Dirichlet BVP,
Here, we assume that , and f satisfies the local Carathéodory conditions on , where and , . Let us recall that a function , , satisfies the local Carathéodory conditions on if
(i) is measurable for all ,
(ii) is continuous for a.e. ,
(iii) for each compact set , there exists a function such that
For such functions, we use the notation . Moreover, may become singular when the space variables x and/or y vanish, which means that may become unbounded for and a.e. and all , and/or it may be unbounded for and a.e. and all . Finally, since , Eq. (1a) has a singularity of the first kind at the time variable because
The differential operator on the left-hand side of Eq. (1a) can be equivalently written as and, after the substitution , it takes the form , which arises in numerous important applications. Operators of such type were studied in phase transitions of Van der Waals fluids [1-4], in population genetics, especially in models for the spatial distribution of the genetic composition of a population [5,6], in the homogeneous nucleation theory , in relativistic cosmology for description of particles which can be treated as domains in the universe , and in the nonlinear field theory , in particular, when describing bubbles generated by scalar fields of Higgs type in the Minkowski spaces .
The aim of this paper is to study the case which is fundamentally different from the case . The latter setting was studied in [11,12], where the structure and properties of the set of all positive solutions to (1a) and (1b) were investigated (the cardinality of this set is a continuum).
In the sequel, we introduce the basic notation and state the preliminary results required in the analysis of problem (1a) and (1b). Here, we focus our attention on the case and prove the existence of at least one positive solution of (1a) and (1b). In contrast to [11,12], we consider the more general situation in which f may be also singular at . This means that we have to deal with the following additional difficulties.
Let u be a positive solution of problem (1a) and (1b), where has a singularity at . Then there exists such that , and hence f is unbounded in a neighborhood of the point . Unfortunately, we do not know the exact position of and therefore, it is not possible to construct a universal Lebesgue integrable majorant for all functions , where are positive solutions of a sequence of auxiliary regular problems. Consequently, the Lebesgue dominated convergence theorem is not applicable and we have to use arguments based on the Vitali convergence theorem instead; see Lemma 2. Another tool used in the proofs is a combination of regularization and sequential techniques with the Leray-Schauder nonlinear alternative.
The investigation of singular Dirichlet BVPs has a long history and a lot of methods for their analysis are available. One of the most important ones is the topological degree method providing various fixed point theorems and existence alternative theorems; see, e.g., Lemma 1. For more information on the topological degree method and its application to numerous BVPs, including Dirichlet problems, we refer the reader to the monographs by Mawhin [13-15].
Throughout this paper, we work with the following conditions on the function f in (1a):
(H1) , where .
(H2) There exists an such that
(H3) For a.e. and all , the estimate
holds, where , , are positive, h is nondecreasing in both its arguments, g and r are nonincreasing, and
we denote the norms in and , respectively. denotes the set of functions whose first derivative is absolutely continuous on , while is the set of functions having absolutely continuous first derivative on each compact subinterval of . We use the symbol to denote the Lebesgue measure of ℳ.
Definition 1 We say that a function is a positive solution of problem (1a) and (1b) if on , u satisfies the boundary conditions (1b) and (1a) holds for a.e. .
Remark 1 Let a function g have the properties specified in (H3). Then for each , , and it follows from the inequality
In order to prove that the singular problem (1a) and (1b) has a positive solution, we use regularization and sequential techniques. To this end, for we define functions and by
respectively. Then it follows from (H1) that and (H2) and (H3) yield
As a first step in the analysis, we investigate auxiliary regular BVPs of the form
To show the solvability of problem (7a) and (7b), we use the following alternative of Leray-Schauder type which follows from [, Theorem 5.1].
Lemma 1LetEbe a Banach space, Ube an open subset ofEand . Assume that is a compact operator. Then either
A1: ℱ has a fixed point in , or
A2: there exists an element and with .
Lemma 2Let and let for a.e. . Then the following statements are equivalent:
(i) and ,
(ii) the sequence is uniformly integrable on .
We recall that a sequence is called uniformly integrable on if for any there exists such that if and , then
The paper is organized as follows. In Section 2, we collect auxiliary results used in the subsequent analysis. Section 3 is devoted to the study of limit properties of solutions to Eq. (7a). In Section 4, we investigate auxiliary regular problems associated with the singular problem (1a) and (1b). We show their solvability and describe properties of their solutions. An existence result for the singular problem (1a) and (1b) is given in Section 5. Finally, in Section 6, we illustrate the theoretical findings by means of numerical experiments.
Throughout the paper .
In this section, auxiliary statements necessary for the subsequent analysis are formulated.
Lemma 3Let and
(i) rcan be extended on with and ,
(ii) Hcan be extended on with , and the equality
holds for a.e. .
Proof (i) It is clear that . Since
we have . Setting , follows.
Then and for . We now show that p can be extended on in such a way that . Integrating by parts yields
for . Hence , where
Let . Then . Since for , we see that H can be extended on with . Moreover,
Hence, cf. (10),
and therefore, . Consequently, . Finally, it follows from and that . Since, by (11), , we see that equality (8) is satisfied for a.e. which completes the proof. □
Lemma 4Let be a uniformly integrable sequence on and let for a.e. . Then the sequence
Proof It follows from Lemma 2 that for , where L is a positive constant. Recall that by Lemma 3(i), . Let us assume that (12) does not hold. Then there exist , and such that , and
Since and are bounded sequences, we may assume that they are convergent, and . If , then (cf. (9))
which contradicts (13). Let . Since and since the uniform integrability of the sequence on results in
we conclude from the relation
The last equality contradicts (13). Consequently, (12) holds and the result follows. □
Lemma 5Let . Then
Proof Since (cf. (10))
for , estimate (14) holds. □
3 Limit properties of solutions to Eq. (7a)
Here, we investigate asymptotic properties of solutions of (7a). We also provide a related integral equation this solution satisfies.
Lemma 6Let (H1) hold. Let satisfy Eq. (7a) for a.e. and . Thenucan be extended on with , and there exists such that the integral equation
holds for .
Proof Choose and denote by for a.e. . In order to prove that , define for , ,
Then and for a.e. . Moreover, for a.e. and all , where . Consequently, by the Lebesgue dominated convergence theorem, .
We now discuss the linear Euler differential equation
Let H be the function given in Lemma 3. By Lemma 3(ii), H can be extended on with and −H satisfies (16) for a.e. . Therefore, each function which satisfies Eq. (16) a.e. on has the form for , with some . By assumption we know that satisfies (16) a.e. on , and therefore there exist such that , . Since by assumption on , we have . Consequently, the function u can be extended on the interval in the class and (15) holds on . □
Corollary 1Let (H1) hold. Let be a solution of Eq. (7a). Then there exists a constant such that equality (15) is satisfied for .
Proof The result holds by Lemma 6 with . □
Remark 2 Corollary 1 says that the set of all solutions of Eq. (7a) depends on one parameter and .
4 Auxiliary regular problems
In order to prove the solvability of problem (7a) and (7b), we first have to investigate the problem
depending on the parameter λ. Here, is from (H2) and .
The following result shows that the solvability of problem (17a) and (17b) is equivalent to the solvability of an integral equation in the set .
Lemma 7Let (H1) hold. Thenuis a solution of problem (17a) and (17b) if and only ifuis a solution of the integral equation
in the set .
Proof Let u be a solution of Eq. (17a). Then , and by Corollary 1 (with replaced by ), there exists such that the equation
holds for . Hence, and if and only if . Consequently, if u is a solution of problem (17a) and (17b), then u is a solution of Eq. (18) in .
Let u be a solution of Eq. (18) in . Then . Hence, Lemma 3(ii) (with ρ replaced by ) guarantees that and u is a solution of Eq. (17a). Moreover, . Consequently, u is a solution of problem (17a) and (17b) which completes the proof. □
The following results provide bounds for solutions of problem (17a) and (17b).
Lemma 8Let (H1)-(H3) hold. Then there exists a positive constantS (independent of and ) such that for all solutionsuof problem (17a) and (17b), the estimates
hold. Moreover, for any solutionuof problem (17a) and (17b), there exists such that
Proof Let u be a solution of problem (17a) and (17b). Then by Lemma 7, equality (18) holds for . Since by (5), for a.e. , the relation
follows from (18). Hence, for because g is nonincreasing on . Due to Remark 1, , which means that
It is clear that L is independent of the choice of solution u to problem (17a) and (17b) and independent of , .
We now show that inequality (21) holds for some . Differentiation of (18) gives
Since , it follows from (5) and (23) that
Hence, is decreasing on , and therefore vanishes at a unique point due to . The inequality (21) now follows from the relations
Hence, on , and
Let . Taking into account (6), (9), (18), (22), (24), and Lemma 5, we obtain
It follows from for ,
and therefore, we have
where and . By (H3),
Consequently, there exists such that for . Now, due to (26), , and therefore, by (25), . □
We are now in the position to prove the existence result for problem (7a) and (7b).
Lemma 9Let (H1)-(H3) hold. Then for each , problem (7a) and (7b) has a solutionusatisfying inequalities (19)-(21), whereSis a positive constant independent ofn.
Proof Let S be a positive constant in Lemma 8 and let us define
Then Ω is an open and bounded subset of the Banach space . Keeping in mind Lemma 3, define an operator by the formula
By Lemma 7, any fixed point of the operator is a solution of problem (7a) and (7b). In order to show the existence of a fixed point of , we apply Lemma 1 with , , and . Especially, we show that
(i) is a compact operator, and
(ii) for each and .
We first verify that is a continuous operator. To this end, let be a convergent sequence, and let in . Let
It follows from Lemma 5 and (9) that
for . Here . In particular, for ,
Since for a.e. and there exists such that
we have by the Lebesgue dominated convergence theorem. Hence, by (28), is a continuous operator. We now show that the set is relatively compact in . It follows from and bounded in that there exists such that
Then by Lemma 5 and (9), the inequalities
are satisfied for and , and therefore, the set is bounded in . Moreover, the relation
holds for a.e. and all (cf. (9)). Consequently, the set is equicontinuous on . Hence, the set is relatively compact in by the Arzelà-Ascoli theorem. As a result, is a compact operator and the condition (i) follows.
Due to the fact that by Lemma 7 any fixed point u of the operator is a solution of problem (17a) and (17b), Lemma 8 guarantees that u satisfies inequality (20). Therefore, has property (ii). Consequently, by Lemmas 1 and 8, for each , problem (7a) and (7b) has a solution u satisfying estimates (19)-(21). □
Let be a solution of problem (7a) and (7b) for . The following property of the sequence is an important prerequisite for solving problem (1a) and (1b).
Lemma 10Let (H1)-(H3) hold. Let be a solution of problem (7a) and (7b) for . Then the sequence is uniformly integrable on .
By Lemma 9, the inequalities
hold, where S is a positive constant and . Hence, by (3) and (4),
5 The existence result for BVP (1a) and (1b)
This section is devoted to the main result on the existence of positive solutions to the original BVP (1a) and (1b).
Theorem 1Let (H1)-(H3) hold. Then problem (1a) and (1b) has at least one positive solution.
Proof By Lemma 9, for each , problem (7a) and (7b) has a solution satisfying inequalities (29)-(31), where S is a positive constant and . Moreover, by Lemma 10, the sequence is uniformly integrable on . We now prove that is equicontinuous on . Since is a fixed point of the operator given in (27), the equality
holds for a.e. and all . Let . Then
Let . By Lemma 3(i), and . Integrating by parts yields
follows. By Lemma 4 (for ), the sequence is equicontinuous on . Since the sequence is uniformly integrable on , the sequence is equicontinuous on . Hence, it follows from (33), that is equicontinuous on . We summarize: is bounded in and is equicontinuous on . Also, . Using appropriate subsequences, if necessary, we can assume, by the Arzelà-Ascoli theorem and the Bolzano-Weierstrass theorem, that is convergent in and is convergent in ℝ. Let and . With in (29)-(31), we conclude
In addition, , . Since
it follows from Lemma 2 that
and . We now deduce from the inequality (cf. Lemma 5)
Taking the limit in
and by Lemma 3(ii). This means that u is a positive solution of problem (1a) and (1b) and the result follows. □
6 Numerical simulations
For the numerical simulation, we choose and use an alternative formulation of problem (1a) and (1b),
where is a parameter. We can use the above formulation because problem (1a) and (1b) is solvable for f satisfying the assumptions of Theorem 1 and, therefore, solutions of problem (1a) and (1b) can be computed as solutions of problem (36a) and (36b) using the proper value depending on f. The values are provided for given f in Examples 1 and 2, below.
The reason for changing the boundary conditions from (1b) to (36b) is that the differential equation (36a) subject to (1b) is not well posed; see . However, to enable successful numerical treatment, well-posedness of the model is crucial. This property means that Eq. (36a) subject to proper boundary conditions has at least a locally unique solution,a and this solution depends continuously on the problem data. The well-posedness of the problem is important for two reasons. First of all, it allows to express errors in the solution of the analytical problem in terms of modeling errors and data errors (all measured via appropriate norms). Therefore, when the errors in the data become smaller due to more precise modeling or smaller measurement inaccuracies, the errors in the solution will decrease. The second reason is that the well-posedness decides if the numerical simulation will be at all successful. If the analytical problem is ill-posed, then the inevitable round-off errors can become extremely magnified and fully spoil the accuracy of the approximation.
In what follows, we work with for a.e. and all , and, according to the next numerical approach (see Section 6.2), we consider Eq. (36a), where , that is,
By , problem (37), (36b) is well posed and therefore it is suitable for the numerical treatment. To see this, we need to look at a general solution of the homogeneous equation
If we set , we arrive at the characteristic polynomial of (38),
whose roots and are positive. Therefore, conditions for u and can be prescribed at as it is done in (36b).
6.1 MATLAB Code bvpsuite
To illustrate the analytical results discussed in the previous section, we solved numerically examples of the form (36a) and (36b) using a MATLAB™ software package bvpsuite designed to solve BVPs in ODEs and differential algebraic equations. The solver routine is based on a class of collocation methods whose orders may vary from two to eight. Collocation has been investigated in the context of singular differential equations of first and second order in [24,25], respectively. This method could be shown to be robust with respect to singularities in time and retains its high convergence order in the case that the analytical solution is appropriately smooth. The code also provides an asymptotically correct estimate for the global error of the numerical approximation. To enhance the efficiency of the method, a mesh adaptation strategy is implemented, which attempts to choose grids related to the solution behavior in such a way that the tolerance is satisfied with the least possible effort. Error estimate procedure and the mesh adaptation work dependably provided that the solution of the problem and its global error are appropriately smooth.b The code and the manual can be downloaded from http://www.math.tuwien.ac.at/~ewa webcite. For further information, see . This software proved useful for the approximation of numerous singular BVPs important for applications; see, e.g., [3,9,27,28].
Before dealing with two nonlinear models specified in Sections 6.3 and 6.4, we have to compute numerical solutions for a simpler linearc model of the form
where a was chosen as . Since in this case the exact solution is given, , the value is available, , respectively. In Figure 1, the numerical solutions of BVPs (39a) and (39b) are shown. They will be used as starting values for the numerical solution of Examples 1 and 2; see Sections 6.3 and 6.4, respectively. All numerical results have been obtained using collocation with five Gaussian collocation points on an equidistant grid (justified by a very simple solution structure) with the step size 0.01.
Figure 1. Problem (39a) and (39b): Numerical solutions for different values ofa.
6.3 Example 1
We first investigate the following problem:
The nonlinearity f in (40a) has the form
and it satisfies (H1)-(H3) with , for and
It follows from Theorem 1 that there exists at least one value of such that the related solution u of problem (40a) and (40b) with is positive on with . Using formula (35), we now determine an interval containing all admissible values of c.
Let u be a solution of problem (1a) and (1b) with f from (41). Then by (35), we obtain
Then (42) implies and due to (35) and (41),
In order to solve the nonlinear problem (40a) and (40b), we first have to solve a series of auxiliary problems for parameter-dependent differential equations
We begin the calculations with and increase its value gradually until we arrive at ; cf. (40a). In each step we use the solution of the previous problem to solve the next one. The aim is to find a good starting value for both the solution u and the value before solving the BVP (40a) and (40b), i.e., find the final value of such that .
In the case of Example 1 and , this chain has the following structure:
1. Numerical approximation of BVP (39a) and (39b) is used as an initial guess for ODE (45) with subject to terminal conditions , .
2. Use the above approximation as an initial guess for ODE (45) with subject to terminal conditions , .
3. Use the above approximation as an initial guess for ODE (45) with subject to terminal conditions , .
4. Use the above approximation as an initial guess for ODE (45) with subject to terminal conditions , .
After the last step, we have solved problem (40a) and (40b) subject to boundary conditions , . In this case, the value of was not small enough to consider it a reasonable approximation for . Therefore, we use a shooting idea combined with a bisection strategy to find a better value for . The complete numerical results for Example 1 can be found in Table 1 and Figure 2.
Figure 2. Problem (40a) and (40b): Numerical solutions for different values ofa. Values of .
Table 1. Problem (40a) and (40b): Complete data of the numerical simulation for different values ofa
6.4 Example 2
The above approach has been also accordingly applied for Example 2. Here, we consider the problem
The right-hand side f in Eq. (46a) now reads
and has a singularity at . The function f satisfies conditions (H1)-(H3) with , for and
Theorem 1 guarantees the existence of at least one such that a solution u of problem (46a) and (46b) is positive on and holds. We now again determine an interval containing all such values of c. Let u be a solution of problem (1a) and (1b) with f given in (47). Inequality (19) yields
and hence by (35),
For Example 2, the auxiliary ODE is constructed using ODE (39a),
For all values of a, we choose and analogously carry out the path-following in δ first. The related chain for is as follows.
1. Numerical approximation of BVP (39a) and (39b) is used as an initial guess for ODE (49) with subject to terminal conditions , .
2. Use the above approximation as an initial guess for ODE (49) with subject to terminal conditions , .
3. Use the above approximation as an initial guess for ODE (49) with subject to terminal conditions , .
4. Use the above approximation as an initial guess for ODE (49) with subject to terminal conditions , .
5. Use the above approximation as an initial guess for ODE (49) with subject to terminal conditions , .
After the last step, we have solved BVP (46a) and (46b) subject to boundary conditions , , but also, in this case, the value of is too large and we have to find a better value for . The complete numerical results for Example 2 can be found in Table 2 and Figure 3.
In the present article, we deal with the existence of positive solutions to the singular Dirichlet problem of the form
where , and the nonlinearity may be singular at the space variables and/or . The main result for the existence of positive solutions of the above BVP is Theorem 1. It is illustrated by numerical simulations using the MATLAB code bvpsuite, based on polynomial collocation. For the successful numerical treatment, the above problem has to be reformulated to obtain its well-posed form
Here, it is only known that , where can be specified depending on functions f arising in Examples 1 and 2. Now, a simple shooting method combined with the bisection idea is used to find c in such a way that .
The authors declare that they have no competing interests.
IR and SS contributed to the analytical part of the work and AS and EBW contributed to its numerical part. All authors read and approved the final version of the manuscript.
Dedicated to Jean Mawhin on the occasion of his 70th birthday.
This research was supported by the grant Matematické modely a struktury, PrF-2012-017. The authors thank the referees for suggestions which improved the paper.
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