Positive solutions of nonlinear Dirichlet BVPs in ODEs with time and space singularities

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 a\in (-1,0). The nonlinearity f(t,x,y) may be singular for the space variables x=0 and/or y=0. Moreover, since a\ne 0, the differential operator on the left-hand side of the differential equation is singular at t=0. 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.

In the present work, we analyze the existence of positive solutions to the singular Dirichlet BVP,

(1a)

(1b)

Here, we assume that T>0, a\in (-1,0) and f satisfies the local Carathéodory conditions on [0,T]\times \mathcal{D}, where \mathcal{D}:={\mathbb{R}}_{+}\times {\mathbb{R}}_0 and {\mathbb{R}}_{+}:=(0,\mathrm{\infty }), {\mathbb{R}}_0:=\mathbb{R}\setminus \{0\}. Let us recall that a function h:[0,T]\times \mathcal{A}\to \mathbb{R}, \mathcal{A}\subset \mathbb{R}\times \mathbb{R}, satisfies the local Carathéodory conditions on [0,T]\times \mathcal{A} if

1. (i)

   h(\cdot ,x,y):[0,T]\to \mathbb{R} is measurable for all (x,y)\in \mathcal{A},

2. (ii)

   h(t,\cdot ,\cdot ):\mathcal{A}\to \mathbb{R} is continuous for a.e. t\in [0,T],

3. (iii)

   for each compact set \mathcal{U}\subset \mathcal{A}, there exists a function m_{\mathcal{U}}\in L^1[0,T] such that

   |h(t,x,y)|\le m_{\mathcal{U}}(t)\text{for a.e. }t\in [0,T]\text{ and all }(x,y)\in \mathcal{U}.

For such functions, we use the notation h\in Car([0,T]\times \mathcal{A}). Moreover, f(t,x,y) may become singular when the space variables x and/or y vanish, which means that f(t,x,y) may become unbounded for x=0 and a.e. t\in [0,T] and all y\in {\mathbb{R}}_0, and/or it may be unbounded for y=0 and a.e. t\in [0,T] and all x\in {\mathbb{R}}_{+}. Finally, since a\ne 0, Eq. (1a) has a singularity of the first kind at the time variable t=0 because

The differential operator on the left-hand side of Eq. (1a) can be equivalently written as {(t^{-a}{(t^{a}u)}^{\prime })}^{\prime } and, after the substitution x(t)=t^{a}u(t), it takes the form {(t^{-a}{x}^{\prime })}^{\prime }, 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 [7], in relativistic cosmology for description of particles which can be treated as domains in the universe [8], and in the nonlinear field theory [9], in particular, when describing bubbles generated by scalar fields of Higgs type in the Minkowski spaces [10].

The aim of this paper is to study the case a\in (-1,0) which is fundamentally different from the case a\in (-\mathrm{\infty },-1). 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 a\in (-1,0) 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 y=0. This means that we have to deal with the following additional difficulties.

Let u be a positive solution of problem (1a) and (1b), where f(t,x,y) has a singularity at y=0. Then there exists t_0\in (0,T) such that u(t_0)>0, u^{\prime }(t_0)=0 and hence f is unbounded in a neighborhood of the point (t_0,u(t_0),u^{\prime }(t_0)). Unfortunately, we do not know the exact position of t_0 and therefore, it is not possible to construct a universal Lebesgue integrable majorant for all functions f(t,u_n(t),u_n^{\prime }(t)), where u_n 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) f\in Car([0,T]\times \mathcal{D}), where \mathcal{D}={\mathbb{R}}_{+}\times {\mathbb{R}}_0.

(H₂) There exists an \epsilon >0 such that

(H₃) For a.e. t\in [0,T] and all (x,y)\in \mathcal{D}, the estimate

holds, where \phi \in L^1[0,T], h\in C([0,\mathrm{\infty })\times [0,\mathrm{\infty })), g,r\in C({\mathbb{R}}_{+}) are positive, h is nondecreasing in both its arguments, g and r are nonincreasing, and

By

we denote the norms in C[0,T] and L^1[0,T], respectively. A{C}^1[0,T] denotes the set of functions whose first derivative is absolutely continuous on [0,T], while A{C}_{\mathrm{loc}}^1(0,T] is the set of functions having absolutely continuous first derivative on each compact subinterval of (0,T]. We use the symbol meas(\mathcal{M}) to denote the Lebesgue measure of ℳ.

Definition 1 We say that a function u\in A{C}^1[0,T] is a positive solution of problem (1a) and (1b) if u>0 on (0,T), u satisfies the boundary conditions (1b) and (1a) holds for a.e. t\in [0,T].

Remark 1 Let a function g have the properties specified in (H₃). Then for each b,c\in {\mathbb{R}}_{+}, , and it follows from the inequality

that

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 n\in \mathbb{N} we define functions f_n^{\ast }:[0,T]\times (\mathbb{R}\times {\mathbb{R}}_0)\to \mathbb{R} and f_n:[0,T]\times {\mathbb{R}}^2\to \mathbb{R} by

and

respectively. Then it follows from (H₁) that f_n\in Car([0,T]\times {\mathbb{R}}^2) and (H₂) and (H₃) yield

(3)

(4)

Hence,

and

As a first step in the analysis, we investigate auxiliary regular BVPs of the form

(7a)

(7b)

To show the solvability of problem (7a) and (7b), we use the following alternative of Leray-Schauder type which follows from [[16], Theorem 5.1].

Lemma 1 Let E be a Banach space, U be an open subset of E and \ell \in U. Assume that \mathcal{F}:\overline{U}\to E is a compact operator. Then either

A1: ℱ has a fixed point in \overline{U}, or

A2: there exists an element u\in \partial U and \lambda \in (0,1) with u=\lambda \mathcal{F}(u)+(1-\lambda )\ell.

In limit processes, we apply the following Vitali convergence theorem; cf. [17–19].

Lemma 2 Let \{{\rho }_n\}\subset L^1[0,T] and let {lim}_{n\to \mathrm{\infty }}{\rho }_n(t)=\rho (t) for a.e. t\in [0,T]. Then the following statements are equivalent:

1. (i)

   \rho \in L^1[0,T] and {lim}_{n\to \mathrm{\infty }}{\parallel {\rho }_n-\rho \parallel }_1=0,

2. (ii)

   the sequence \{{\rho }_n\} is uniformly integrable on [0,T].

We recall that a sequence \{{\rho }_n\}\subset L^1[0,T] is called uniformly integrable on [0,T] if for any \epsilon >0 there exists \delta >0 such that if \mathcal{M}\subset [0,T] 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 a\in (-1,0).

In this section, auxiliary statements necessary for the subsequent analysis are formulated.

Lemma 3 Let \rho \in L^1[0,T] and

Then

1. (i)

   r can be extended on [0,T] with r\in C[0,T] and r(0)=0,

2. (ii)

   H can be extended on [0,T] with H\in A{C}^1[0,T], and the equality

   H^{″}(t)+\frac{a}{t}{H}^{\prime }(t)-\frac{a}{t^2}H(t)=-\rho (t)

   (8)

holds for a.e. t\in [0,T].

Proof (i) It is clear that r\in C(0,T]. Since

we have {lim}_{t\to 0^{+}}r(t)=0. Setting r(0):=0, r\in C[0,T] follows.

1. (ii)

   Let

   p(t)={\int }_t^T\frac{1}{s^{a+2}}({\int }_0^s{\xi }^{a+1}\rho (\xi )\mathrm{d}\xi )\mathrm{d}s\text{for }t\in (0,T].

Then p\in C(0,T] and H(t)=tp(t) for t\in (0,T]. We now show that p can be extended on [0,T] in such a way that p\in C[0,T]. Integrating by parts yields

(10)

for t\in (0,T]. Hence {lim}_{t\to 0^{+}}p(t)=A, where

Let p(0):=A. Then p\in C[0,T]. Since H^{\prime }(t)=p(t)+t{p}^{\prime }(t)=p(t)-r(t) for t\in (0,T], we see that H can be extended on [0,T] with H\in C^1[0,T]. Moreover,

In particular,

Hence, cf. (10),

and therefore, H^{″}\in L^1[0,T]. Consequently, H\in A{C}^1[0,T]. Finally, it follows from H^{\prime }(t)=p(t)-r(t) and p(t)=\frac{H(t)}{t} that r(t)=\frac{H(t)}{t}-H^{\prime }(t). Since, by (11), \frac{a}{t}r(t)=H^{″}(t)+\rho (t), we see that equality (8) is satisfied for a.e. t\in [0,T] which completes the proof. □

Lemma 4 Let \{{\rho }_n\}\subset L^1[0,T] be a uniformly integrable sequence on [0,T] and let {lim}_{n\to \mathrm{\infty }}{\rho }_n(t)=\rho (t) for a.e. t\in [0,T]. Then the sequence

Proof It follows from Lemma 2 that {\parallel {\rho }_n\parallel }_1\le L for n\in \mathbb{N}, where L is a positive constant. Recall that by Lemma 3(i), \{\frac{1}{t^{a+1}}{\int }_0^t{s}^{a+1}{\rho }_n(s)\mathrm{d}s\}\subset C[0,T]. Let us assume that (12) does not hold. Then there exist \epsilon >0, \{k_n\}\subset \mathbb{N} and \{{\xi }_n\},\{{\eta }_n\}\subset [0,T] such that {lim}_{n\to \mathrm{\infty }}{k}_n=\mathrm{\infty }, {lim}_{n\to \mathrm{\infty }}({\xi }_n-{\eta }_n)=0 and

Since \{{\xi }_n\} and \{{\eta }_n\} are bounded sequences, we may assume that they are convergent, and {lim}_{n\to \mathrm{\infty }}{\xi }_n=\tau ={lim}_{n\to \mathrm{\infty }}{\eta }_n. If \tau =0, then (cf. (9))

which contradicts (13). Let \tau \in (0,T]. Since {lim}_{n\to \mathrm{\infty }}(\frac{1}{{\xi }_n^{a+1}}-\frac{1}{{\eta }_n^{a+1}})=0 and since the uniform integrability of the sequence \{{\rho }_n\} on [0,T] results in

we conclude from the relation

that

The last equality contradicts (13). Consequently, (12) holds and the result follows. □

Lemma 5 Let \rho \in L^1[0,T]. Then

Proof Since (cf. (10))

for t\in [0,T], estimate (14) holds. □

Here, we investigate asymptotic properties of solutions of (7a). We also provide a related integral equation this solution satisfies.

Lemma 6 Let (H₁) hold. Let u\in A{C}_{\mathrm{loc}}^1(0,T] satisfy Eq. (7a) for a.e. t\in [0,T] and . Then u can be extended on [0,T] with u\in A{C}^1[0,T], and there exists c\in \mathbb{R} such that the integral equation

holds for t\in [0,T].

Proof Choose n\in \mathbb{N} and denote by \rho (t)=f_n(t,u(t),u^{\prime }(t)) for a.e. t\in [0,T]. In order to prove that \rho \in L^1[0,T], define for m\in \mathbb{N}, ,

and

Then {\rho }_m\in L^1[0,T] and {lim}_{m\to \mathrm{\infty }}{\rho }_m(t)=\rho (t) for a.e. t\in [0,T]. Moreover, |{\rho }_m(t)|\le \mu (t) for a.e. t\in [0,T] and all m\in \mathbb{N}, where \mu (t)=sup\{|f_n(t,x,y)|:|x|\le L,|y|\le L\}\in L^1[0,T]. Consequently, by the Lebesgue dominated convergence theorem, \rho \in L^1[0,T].

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 [0,T] with H\in A{C}^1[0,T] and −H satisfies (16) for a.e. t\in [0,T]. Therefore, each function v\in A{C}_{\mathrm{loc}}^1(0,T] which satisfies Eq. (16) a.e. on [0,T] has the form v(t)=c^{\ast }t+d^{\ast }{t}^{-a}-H(t) for t\in (0,T], with some c^{\ast },d^{\ast }\in \mathbb{R}. By assumption we know that u\in A{C}_{\mathrm{loc}}^1(0,T] satisfies (16) a.e. on [0,T], and therefore there exist c,d\in \mathbb{R} such that u(t)=ct+d{t}^{-a}-H(t), t\in (0,T]. Since by assumption |u^{\prime }|\le L on (0,T], we have d=0. Consequently, the function u can be extended on the interval [0,T] in the class A{C}^1[0,T] and (15) holds on [0,T]. □

Corollary 1 Let (H₁) hold. Let u\in A{C}^1[0,T] be a solution of Eq. (7a). Then there exists a constant c\in \mathbb{R} such that equality (15) is satisfied for t\in [0,T].

Proof The result holds by Lemma 6 with . □

Remark 2 Corollary 1 says that the set of all solutions u\in A{C}^1[0,T] of Eq. (7a) depends on one parameter c\in \mathbb{R} and u(0)=0.

In order to prove the solvability of problem (7a) and (7b), we first have to investigate the problem

(17a)

(17b)

depending on the parameter λ. Here, \epsilon >0 is from (H₂) and n\in \mathbb{N}.

The following result shows that the solvability of problem (17a) and (17b) is equivalent to the solvability of an integral equation in the set C^1[0,T].

Lemma 7 Let (H₁) hold. Then u is a solution of problem (17a) and (17b) if and only if u is a solution of the integral equation

in the set C^1[0,T].

Proof Let u be a solution of Eq. (17a). Then u\in A{C}^1[0,T], and by Corollary 1 (with f_n replaced by \lambda f_n-(1-\lambda )\epsilon), there exists c\in \mathbb{R} such that the equation

holds for t\in [0,T]. Hence, u(0)=0 and u(T)=0 if and only if c=0. Consequently, if u is a solution of problem (17a) and (17b), then u is a solution of Eq. (18) in C^1[0,T].

Let u be a solution of Eq. (18) in C^1[0,T]. Then f_n(t,u(t),u^{\prime }(t))\in L^1[0,T]. Hence, Lemma 3(ii) (with ρ replaced by -\lambda f_n+(1-\lambda )\epsilon) guarantees that u\in A{C}^1[0,T] and u is a solution of Eq. (17a). Moreover, u(0)=u(T)=0. 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 8 Let (H₁)-(H₃) hold. Then there exists a positive constant S (independent of n\in \mathbb{N} and \lambda \in [0,1]) such that for all solutions u of problem (17a) and (17b), the estimates

(19)

(20)

hold. Moreover, for any solution u of problem (17a) and (17b), there exists \xi \in (0,T) such that

Proof Let u be a solution of problem (17a) and (17b). Then by Lemma 7, equality (18) holds for t\in [0,T]. Since by (5), \lambda f_n(t,u(t),u^{\prime }(t))-(1-\lambda )\epsilon \le -\epsilon for a.e. t\in [0,T], the relation

follows from (18). Hence, g(u(t))\le g(\frac{\epsilon }{a+2}t(T-t)) for t\in (0,T) because g is nonincreasing on {\mathbb{R}}_{+}. 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 n\in \mathbb{N}, \lambda \in [0,1].

We now show that inequality (21) holds for some \xi \in (0,T). Differentiation of (18) gives

Since , it follows from (5) and (23) that

Hence, u^{\prime } is decreasing on [0,T], and therefore u^{\prime } vanishes at a unique point \xi \in (0,T) due to u(0)=u(T)=0. The inequality (21) now follows from the relations

Hence, r(|u^{\prime }(t)|)\le r(\frac{2\epsilon }{a+2}|t-\xi |) on [0,T]\setminus \{\xi \}, and

In particular,

Let W=\frac{a+3}{a+1}. Taking into account (6), (9), (18), (22), (24), and Lemma 5, we obtain

It follows from u(t)={\int }_0^t{u}^{\prime }(s)\mathrm{d}s for t\in [0,T],

and therefore, we have

where K=W{\parallel \phi \parallel }_1 and M=W(L+V). By (H₃),

Consequently, there exists S>0 such that for z\ge S. 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 9 Let (H₁)-(H₃) hold. Then for each n\in \mathbb{N}, problem (7a) and (7b) has a solution u satisfying inequalities (19)-(21), where S is a positive constant independent of n.

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 C^1[0,T]. Keeping in mind Lemma 3, define an operator \mathcal{K}:[0,T]\times \overline{\mathrm{\Omega }}\to C^1[0,T] by the formula

By Lemma 7, any fixed point of the operator \mathcal{K}(1,\cdot ) is a solution of problem (7a) and (7b). In order to show the existence of a fixed point of \mathcal{K}(1,\cdot ), we apply Lemma 1 with E=C^1[0,T], U=\mathrm{\Omega }, \mathcal{F}=\mathcal{K}(1,\cdot ) and \ell =\frac{\epsilon }{a+2}t(T-t). Especially, we show that

1. (i)

   \mathcal{K}(1,\cdot ):\overline{\mathrm{\Omega }}\to C^1[0,T] is a compact operator, and

2. (ii)

   \mathcal{K}(\lambda ,x)\ne x for each \lambda \in (0,1] and x\in \partial \mathrm{\Omega }.

We first verify that \mathcal{K}(1,\cdot ) is a continuous operator. To this end, let \{x_m\}\subset \overline{\mathrm{\Omega }} be a convergent sequence, and let {lim}_{m\to \mathrm{\infty }}{x}_m=x in C^1[0,T]. Let

It follows from Lemma 5 and (9) that

for t\in [0,T]. Here \mathcal{K}{(1,x)}^{\prime }=\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{K}(1,x). In particular, for m\in \mathbb{N},

Since {lim}_{m\to \mathrm{\infty }}{f}_n(t,x_m(t),x_m^{\prime }(t))=f_n(t,x(t),x^{\prime }(t)) for a.e. t\in [0,T] and there exists \rho \in L^1[0,T] such that

we have {lim}_{m\to \mathrm{\infty }}{\parallel r_m\parallel }_1=0 by the Lebesgue dominated convergence theorem. Hence, by (28), \mathcal{K}(1,\cdot ) is a continuous operator. We now show that the set \mathcal{K}(1,\overline{\mathrm{\Omega }}) is relatively compact in C^1[0,T]. It follows from f_n\in Car([0,T]\times {\mathbb{R}}^2) and \overline{\mathrm{\Omega }} bounded in C^1[0,T] that there exists \mu \in L^1[0,T] such that

Then by Lemma 5 and (9), the inequalities

are satisfied for t\in [0,T] and x\in \overline{\mathrm{\Omega }}, and therefore, the set \mathcal{K}(1,\overline{\mathrm{\Omega }}) is bounded in C^1[0,T]. Moreover, the relation

holds for a.e. t\in [0,T] and all x\in \overline{\mathrm{\Omega }} (cf. (9)). Consequently, the set \{\mathcal{K}{(1,x)}^{\prime }:x\in \overline{\mathrm{\Omega }}\} is equicontinuous on [0,T]. Hence, the set \mathcal{K}(1,\overline{\mathrm{\Omega }}) is relatively compact in C^1[0,T] by the Arzelà-Ascoli theorem. As a result, \mathcal{K}(1,\cdot ) is a compact operator and the condition (i) follows.

Due to the fact that by Lemma 7 any fixed point u of the operator \mathcal{K}(\lambda ,\cdot ) is a solution of problem (17a) and (17b), Lemma 8 guarantees that u satisfies inequality (20). Therefore, \mathcal{K} has property (ii). Consequently, by Lemmas 1 and 8, for each n\in \mathbb{N}, problem (7a) and (7b) has a solution u satisfying estimates (19)-(21). □

Let u_n be a solution of problem (7a) and (7b) for n\in \mathbb{N}. The following property of the sequence \{|f_n(t,u_n(t),u_n^{\prime }(t))|\} is an important prerequisite for solving problem (1a) and (1b).

Lemma 10 Let (H₁)-(H₃) hold. Let u_n be a solution of problem (7a) and (7b) for n\in \mathbb{N}. Then the sequence \{|f_n(t,u_n(t),u_n^{\prime }(t))|\} is uniformly integrable on [0,T].

Proof

By Lemma 9, the inequalities

(29)

(30)

(31)

hold, where S is a positive constant and {\xi }_n\in (0,T). Hence, by (3) and (4),

for a.e. t\in [0,T], where g_{\ast }(t)=g(\frac{\epsilon }{a+2}t(T-t))\in L^1[0,T], see Remark 1. Since the sequence \{r(\frac{2\epsilon }{a+2}|t-{\xi }_n|)\} is uniformly integrable on [0,T] (cf. [[20], criterion A.4], [21, 22]), it follows from (32) that \{|f_n(t,u_n(t),u_n^{\prime }(t))|\} is uniformly integrable on [0,T] and the result follows. □

This section is devoted to the main result on the existence of positive solutions to the original BVP (1a) and (1b).

Theorem 1 Let (H₁)-(H₃) hold. Then problem (1a) and (1b) has at least one positive solution.

Proof By Lemma 9, for each n\in \mathbb{N}, problem (7a) and (7b) has a solution u_n satisfying inequalities (29)-(31), where S is a positive constant and {\xi }_n\in (0,T). Moreover, by Lemma 10, the sequence \{|f_n(t,u_n(t),u_n^{\prime }(t))|\} is uniformly integrable on [0,T]. We now prove that \{u_n^{\prime }\} is equicontinuous on [0,T]. Since u_n is a fixed point of the operator \mathcal{K}(1,\cdot ) given in (27), the equality

holds for a.e. t\in [0,T] and all n\in \mathbb{N}. Let . Then

Let r_n(t)=\frac{1}{t^{a+1}}{\int }_0^t{s}^{a+1}{f}_n(s,u_n(s),u_n^{\prime }(s))\mathrm{d}s. By Lemma 3(i), \{r_n\}\subset C[0,T] and r_n(0)=0. Integrating by parts yields

and

follows. By Lemma 4 (for {\rho }_n(t)=f_n(t,u_n(t),u_n^{\prime }(t))), the sequence \{r_n\} is equicontinuous on [0,T]. Since the sequence \{|f_n(t,u_n(t),u_n^{\prime }(t))|\} is uniformly integrable on [0,T], the sequence \{{\int }_0^t{f}_n(s,u(s),u^{\prime }(s))\mathrm{d}s\} is equicontinuous on [0,T]. Hence, it follows from (33), that \{u_n^{\prime }\} is equicontinuous on [0,T]. We summarize: \{u_n\} is bounded in C^1[0,T] and \{u_n^{\prime }\} is equicontinuous on [0,T]. Also, \{{\xi }_n\}\subset (0,T). Using appropriate subsequences, if necessary, we can assume, by the Arzelà-Ascoli theorem and the Bolzano-Weierstrass theorem, that \{u_n\} is convergent in C^1[0,T] and \{{\xi }_n\} is convergent in ℝ. Let {lim}_{n\to \mathrm{\infty }}{u}_n=:u and {lim}_{n\to \mathrm{\infty }}{\xi }_n=:\xi. With n\to \mathrm{\infty } in (29)-(31), we conclude

In addition, u(0)=0, u(T)=0. Since

it follows from Lemma 2 that

and f(t,u(t),u^{\prime }(t))\in L^1[0,T]. We now deduce from the inequality (cf. Lemma 5)

that

Taking the limit n\to \mathrm{\infty } in