1Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India 2Department of Mathematics, BITS Pilani, Rajasthan, Pilani 333031, India
Received 20 May 2011; Revised 24 August 2011; Accepted 13 September 2011
We present a constructive approach to establish existence and uniqueness of solution of singular boundary value problem for Here on allowing . Further may be allowed to have integrable discontinuity at , so the problem may be doubly singular.
1. Introduction
Consider the following singular boundary value problem:
where , , is any finite constant. We assume that and satisfy the following conditions:(A1) in , and ,(A2) in and .
In this work we establish existence and uniqueness of solution of the singular problem (1.1)-(1.2). We use monotone iterative method. For this we require an appropriate iterative scheme. In this regard Cherpion et al. [1] suggest the following approximation scheme:
for the regular boundary value problem . They also suggest that (1.3) with or with constant and does not work for the Dirichlet boundary condition.
Thus, for our problem we consider the following iterative scheme:
where
We assume that and satisfy the following conditions.(A3), , such that .(A4), that is,.(A5)Further, we assume that the homogeneous boundary value problem , , and has only trivial solution.
For , several researchers ([2–5]) suggest to reduce the singular problem to regular problem by a change of variable. But in [6] it is suggested that a direct consideration of singular problems provide better results.
Further, the following sign restrictions are imposed by several researchers ([4, 5, 7–9]): (i), , where is a constant ([7–9]) or(ii) ([4, 5]).
But such sign restrictions are quite restrictive as the simple differential equation fails to satisfy the sign restrictions (i) and (ii) ([7]).
In the present work we consider the singular boundary value problem (SBVP) directly and do not impose any sign restriction. Further, we do not assume that the point is a regular singular point as assumed in [6, 9]. We use iterative scheme (1.4) to establish existence and uniqueness of the solution of the problem. With the help of nonnegativity of Green's function, existence uniqueness of linear singular boundary value problem (LSBVP) is established.
This paper is divided in four sections. In Section 2, we show that singular point is of limit circle type; hence, spectrum is pure point spectrum with complete set of orthonormal eigenfunctions. In Section 3, we prove the existence uniqueness of the corresponding LSBVP. Finally, in Section 4, using the results of Section 3, we establish the existence uniqueness of solutions of the nonlinear problem (1.1)-(1.2).
2. Eigenfunction Expansion
Let be a Hilbert space with weight and the inner product defined as
Conditions on , , , and guarantee that the singular point is of limit circle type (Weyl's Theorem, [10, page 438]. Thus, we have pure point spectrum ([11, page 125]. Next, from the Lagrange's identity, it is easy to see that all the eigenvalues are real, simple, positive, and eigenfunctions are orthogonal. Let the eigenvalues be , and let the corresponding eigenfunctions be , respectively. Next, we transform by changing variable , where , to
where . Now following the analysis of Theorem 2.7, (i), (ii) and Theorem 2.17 of [11] for the operator , where is defined by (1.1), the following results can be established.
Theorem 2.1. Let be the primitive of an absolutely continuous function, and let
for every nonreal , where is a solution of (2.2) and is the wronskian of and . Then
being the series absolutely and uniformly convergent on .
Theorem 2.2. Let . Then
Theorem 2.3. Let , and let be the solution of
satisfying , where . Then for not equal to any of the values of , one has
where the series is absolutely convergent.
Remark 2.4. Since on is equivalent to on , we can apply Theorems 2.1–2.3 in also.
3. Linear Singular Sturm-Liouville’s Problem
In this section we apply Theorem 1.1 of [12] to the differential operator and generate two linearly independent solutions of the linear problem. Further, with the help of these solutions, Green's function is constructed, and nonnegativity of the Green's function is established.
Theorem 3.1. Let , , , and satisfy (A1), (A2), (A3), and (A4), respectively. Then the initial value problems (IVPs)
have a solution in or equivalently in (Remark 2.4).
3.1. Green's Function
Green's function for the differential operator can be defined as
where , , , , , , and is a nontrivial solution of IVP (3.1) with , . From (A5), it is easy to conclude that ; thus, and are linearly independent.
Next we establish nonnegativity of Green's function. For this we need to establish following results.
Lemma 3.2. If satisfies , for , , and , where , , , and satisfy (A1), (A2), (A3), and (A4), respectively, then provided that .
Proof. We divide the proof in two cases as follows. Case i. When . On contrary assume that there exists a point such that . Then from the continuity of the solutions there exists a point such that , , and . Now at the point , we have
which is a contradiction. Hence, when . Case ii. When . Using the same notations as in the previous case, we have for and for . Now, we consider the interval where , in , and in . Then
Integrating the first term by parts, we get
which is again a contradiction. Thus, when .
Lemma 3.3. Consider the following differential equation:
where , , , and satisfy (A1), (A2), (A3), and (A4), respectively, with the boundary conditions:
Then LSBVP (3.7)-(3.8) has a unique solution given by
provided that is none of the eigenvalues of the corresponding eigenvalue problem and satisfies (3.1). Moreover, if and , where is the first positive zero of .
Proof. From Theorem 3.1, it is easy to see that the unique solution of (3.7)-(3.8) can be written as
provided that ; that is, is none of the eigenvalue of the corresponding eigenvalue problem (Section 2). Since , , and does not change sign for , we get that for , provided that .
Lemma 3.4. For the linear differential operator associated with
with , the generalized Green's function for the corresponding homogeneous boundary value problem is given by
where are the normalized eigenfunctions corresponding to the eigenvalue . satisfies the homogeneous boundary condition provided that . Solution of the non-homogeneous LSBVP (3.11)-(3.12) is
The series on the right is absolutely convergent.
Proof. The solution of (3.11)-(3.12) can be written as sum of the solution of (3.11) with boundary condition and solution of (3.7) with boundary condition , ,
where is Green's function defined by (3.3). Now using the analysis of ([11, page 38], it is easy to show that the generalized Green's function is given by
and absolute convergence of the series on the right-hand side follows from the analysis of ([11, page 38]. This completes the proof.
Lemma 3.5. If , and , then solution of (3.11)-(3.12) is nonnegative provided that .
Proof. We first show that for all if . Fixing , satisfies , where . Since , for , from Lemma 3.3 for , provided that . By the symmetry, continuity, and for , it follows that for , provided that . The result follows.
Corollary 3.6. If satisfies for and , , then , provided that .
In this section, we establish the existence uniqueness of solution of the nonlinear problem (1.1)-(1.2). For this, first we prove that the sequences generated by (1.4) are monotonic sequences (Lemmas 4.2 and 4.3). Then using the bound for (Lemmas 4.9 and 4.10), the uniform convergence of these sequences to a solution of the nonlinear problem is established (Theorem 4.11). Finally the uniqueness of the solution is established in Theorem 4.14.
The nonlinear boundary value problem
can be transformed to
with . Further, the functions and satisfy the same Lipschitz condition, so we may work with the boundary value problem
Next, we define upper solution and lower solution such that , which work as initial iterates for our constructive approach.
Definition 4.1. A function is an upper solution if
and a function is a lower solution if
Lemma 4.2. If , , , for , , and , then
provided that
hold. Here,
Proof. The solution of the equation , , and is given by (3.15) where is defined by (3.3). Substituting from (3.15) into (4.6), it is easy to see that we require the following inequalities in order to complete the proof
Here satisfies the IVP at ; that is, , and , and satisfies the IVP at ; that is, , and . The solutions and cannot have either point of maxima (at the point of maxima the or will be contradicted) or point of minima (since to have minima, maxima is bound to occur). So, finally we have and on . As , it is enough to prove the following inequalities:
Next, we prove the inequality (4.12), and the other one can be proved in a similar manner. By the mean value theorem, there exist such that . Writing in the following form:
and integrating it first from to and then to , we get that
Here is given by (4.8). Now, the result follows from (4.7), (4.12), and (4.15).
Lemma 4.3. If , , , for , , and , then
provided that
or
hold.
Proof. Similar to the proof of Lemma 4.2, we need to establish two inequalities (4.10)-(4.11) for . Here and cannot have the point of minima in , because at the point of minima, the differential equation or will be contradicted. So either and or and both are concave downwards on . Thus we can divide the proof in two cases: Case i. and both are concave downwards. We prove for as similar analysis provides result for . Let the point of maxima be . Then for and for . On both sides of , the inequality (4.10) will be reduced into the following two inequalities:
For a point on the left side of , we integrate (4.14) from to twice and get
Similarly for any point on the right side of , we get
Now, the result follows from the fact that and from (4.18) to (4.21). Case ii. When and . To establish the inequality (4.16), we require to establish the inequalities (4.12)-(4.13). We prove the inequality (4.12), and the proof for (4.13) is quite similar. By the mean value theorem, there exists such that . Integrating (4.14) first from to and then from to , we get
and the result follows from (4.12), (4.17), and (4.22). This completes the proof.
Lemma 4.4. If is an upper solution of (4.3) and is defined by (1.4)–(1.5), then for .
Proof. Let . satisfies , and the result follows from Corollary 3.6.
Proposition 4.5. Let be an upper solution of (4.3), and let satisfy the following(F1) is continuous on(F2) such that for all ,(F3) such that for all ,
and (4.7), (4.17), or (4.18) hold. Then the functions defined by (1.4)–(1.5) are such that, for all , (i) is upper solution of (4.3) and (ii) .
Proof. Since is an upper solution from Lemma 4.4, we have . Assume that the claim is true for ; that is, is an upper solution and . Let . We have
and from Lemmas 4.2 and 4.3 we get . Thus, is an upper solution for all . From Lemma 4.4 we have . Hence, the result follows. Similar results (Lemma 4.6, Proposition 4.7) follow for lower solutions.
Lemma 4.6. If is a lower solution of (4.3) and is defined by (1.4)–(1.5) then for .
Proposition 4.7. Let be a lower solution of (4.3), let satisfies (F1)–(F3) and (4.7), (4.17), or (4.18) hold. Then the functions defined by (1.4)–(1.5) are such that, for all , (i) is lower solution of (4.3) and (ii) .
Proposition 4.8. If satisfies(F4) for such that and ,and in addition let (F1)–(F3) and (4.7), (4.17), or (4.18) hold, then for all the functions and defined by (1.4)–(1.5) satisfy .
Proof. Let , then satisfies for all such that
Since , we prove that . Since is solution of , and , from Corollary 3.6 we have . Let , let , and , then we prove that and . Consider
Since is a solution of , , and ; hence, from Lemmas 4.2 and 4.3, we have . Thus, from Corollary 3.6 on , and , we have , that is, . This completes the proof.
Lemma 4.9. If satisfies(F5) for all , where is continuous and satisfies
where , then there exists such that any solution of
with for all , satisfies .
Proof. We divide the proof in three parts. Case i. If solution is not monotone throughout the interval, then we consider the interval such that and for . Integrating (4.30) from to we get
From (F5) we can choose such that
which gives
Now we consider the case in which for , , and proceeding in the similar way we get
and the result follows.Case ii. If is monotonically increasing in , that is, in , then by the mean value theorem there exists a point such that
Now, integrating (4.30) from to , we get
Further, from (F5) we can choose such that
which gives .Case iii. If is monotonically decreasing in ; that is, in , then argument similar to Case ii yields
and we get
and the result follows.
Lemma 4.10. If satisfies (F5), then there exists such that any solution of
with for all , satisfies .
Proof. Proof follows from the analysis of Lemma 4.9.
Theorem 4.11. Let and be upper and lower solutions. Let satisfy (F1) to (F5) and (4.7), (4.17), or (4.18) hold. Then, boundary value problem (4.3) has at least one solution in the region . If is chosen such that and , where is the first positive eigenvalue of the corresponding eigenvalue problem, then the sequences and generated by (1.4)–(1.5) with initial iterate and converge monotonically and uniformly towards solutions and of (4.3). Any solution in must satisfy .
Proof. From Lemmas 4.2–4.10, Propositions 4.5–4.8, and we get two monotonic sequences and which are bounded by and ; respectively, and by Dini's Theorem their uniform convergence is assured. Let and converge uniformly to and . By Lemmas 4.9 and 4.10, it is easy to see that the sequences and are uniformly bounded. Now, from
uniform convergence of , properties (A1)–(A4), and (F1), it is easy to prove that is equicontinuous. Hence, by Arzela-Ascoli's Theorem there exist a uniform convergent subsequence of . Since limit is unique so original sequence will also converge uniformly to the same limit say . It is easy to see that, if , then . Therefore sequences and converge uniformly to and , respectively. Let be Green's function for the linear boundary value problem , , and . Then solution of (1.4)–(1.5) can be written as
where and . Now, uniform convergence of , and continuity of imply that converges uniformly in . Hence, converges in the sense of mean in . Taking limit as and using Lemma 2.4 ([11, page 27]), we get
which is the solution of the boundary value problem (4.3). Any solution in plays the role of . Hence . Similarly, . This completes the proof.
Remark 4.12. The case when corresponds to the case when . In such cases the boundary value problem (4.3) can be reduced to two initial value problems , and , . From the assumptions on , , and , one can easily conclude existence uniqueness of solutions of the nonlinear boundary value problem.
Remark 4.13. Suppose, in addition to the hypothesis of Theorem 4.11, in . Then lower solution and upper solution may be obtained as solution of the following linear boundary value problems:
Theorem 4.14. Suppose that satisfies (F1), (F3), and constants such that
Then the boundary value problem (4.3) has unique solution.
Proof. Let and be two solutions of (4.3), then we get
Since , from Corollary 3.6 we get or . Similarly . Therefore, the solution of (4.3) is unique.
Acknowledgments
The authors are grateful to the reviewers for their critical comments and valuable suggestions. This work is supported by Council of Scientific and Industrial Research (CSIR) and DST, New Delhi, India.
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