The Shooting Method and Nonhomogeneous Multipoint Bvps of Second-order Ode

Recommended by Kanishka Perera In a recent paper, Sun et al. (2007) studied the existence of positive solutions of nonhomo-geneous multipoint boundary value problems for a second-order differential equation. It is the purpose of this paper to show that the shooting method approach proposed in the recent paper by the first author can be extended to treat this more general problem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Introduction
In a previous paper [1], the first author demonstrated that the classical shooting method could be effectively used to establish existence and multiplicity results for boundary value problems of second-order ordinary differential equations. This approach has an advantage over the traditional method of using fixed point theorems on cones by Krasnosel'skiȋ [2]. It has come to our attention after the publication of [1] that Baxley and Haywood [3] had also used similar ideas to study Dirichlet boundary value problems.
In this article, we continue our exposition by further extending this shooting method approach to treat multipoint boundary value problems with a nonhomogeneous boundary condition at the right endpoint, and homogeneous boundary condition at the left endpoint of the most general type, that is, the Robin boundary condition which includes both Dirichlet and Neumann boundary conditions as special cases.
The study of multipoint boundary value problems for linear second-order differential equations was initiated by Il'in and Moiseev [4,5]. Nonlinear second-order boundary value problems with three-point boundary conditions were first studied by Gupta [6,7] followed by many others, notably Marano [8]. Please consult the articles cited in the References Section.

Boundary Value Problems
Symmetric positive solutions for Dirichlet boundary value problems, which are related to second-order elliptic partial differential equations, were studied by Constantian [9], Avery [10], and Henderson and Thompson [11]. We defer a discussion of these results in relation to ours to the last section of this paper.
We will first establish two existence results (Theorems 3.1 and 3.2) on multipoint problems for the second-order differential equation u (t) + a(t) f u(t) = 0, t ∈ (0,1), (1.1) where the nonlinear term is in a separable format, and a and f are continuous functions satisfying Note that the assumption that f (u) does not vanish for u > 0 is a technical assumption imposed for convenience. Without this assumption, the second inequality sign in (1.8) and (1.9) below may not be strict. Analogous results (Theorems 3.3 and 3.4) are then formulated and extended to nonlinear equations of the more general form y (t) + F t, y(t) = 0, t ∈ (0,1), (1.3) where the nonlinear term may not be in a separable format. In both [12,13], the Neumann boundary condition is imposed on the left endpoint. Some other authors use the Dirichlet condition The results in this paper are applicable to the most general Robin boundary condition of the form where θ is a given number in [0,3π/4). The choices θ = 0 and π/2 correspond, respectively, to the Neumann and Dirichlet conditions (1.4) and (1.5). We leave out those θ in [3π/4,π] as solutions satisfying the corresponding Robin's condition cannot furnish a positive solution for our boundary value problem. To see this, note that if θ ∈ [3π/4,π], then u (0) = u(0)tanθ ≤ −u(0). Since u(t) is concave, u(t) must lie below the line joining the points (0,u(0)) and (1,0), so u(t) cannot be positive in [0,1]. The second boundary condition we impose involves m − 2 given points ξ i ∈ (0,1), i = 1,...,m − 2, together with t = 1. Let k i , i = 1,...,m − 2 be another set of m − 2 given positive numbers, and b ≥ 0. We require the solution to satisfy (1.7) The boundary value problem for the differential equation (1.1) with boundary conditions (1.6) and (1.7) is often referred to as the m-point problem. When b = 0, the multipoint boundary condition is said to be homogeneous. Otherwise, it is called nonhomogeneous.
In the special case when m = 3, only one interior point ξ = ξ 1 is used and the boundary value problem is called a three-point problem.
In the case of left Neumann problem, it is known that a necessary condition for the existence of a positive solution is To see this, we put b = 0 in (1.7) and use the fact that In the case of the left Dirichlet problem, the corresponding necessary condition is To see this, we use the fact that u(t) is a concave function, and so u(t) lies strictly above the straight line joining the origin (0,0) with the point (1,u(1)). Therefore, u(ξ i ) > ξ i u (1) for all i. Plugging these inequalities and b = 0 into (1.7) gives (1.9). We will state and prove the corresponding necessary condition for the general Robin condition in the next Section, see Lemma 2.2.
In [12], Ma proved the following existence result for the homogeneous three-point problem. Define (1.10) For the nonhomogeneous problem, Ma [14] has the following result for the superlinear case.
has the solution u(t) = −t 2 /2 + 2b + 7/8 for all b > 0, showing that the result as stated in [13, Theorem 1.2] is false. Since our technique of proof uses the shooting method, the issues of continuability and uniqueness of initial value problems for the differential equations (1.1) or (1.3) arise naturally. In fact, these issues have already been discussed in [1]. The readers can be referred to that paper for more details. We only give a brief summary below. It is well known that continuability and uniqueness may not always hold for initial value problems of general nonlinear equations. In particular, it is known, see, for example, Coffman and Wong [15], that solutions of superlinear equation may not be continuable to a solution defined on the entire interval [0,1]. This is not a problem for our study because in our technique, we only need to be able to extend the solution up to its first zero. Since the solution is concave, this poses no problem at all. We also know that solutions of initial value problems may not be unique if f (u) is not Lipschitz continuous. In such a situation, we can approximate f (u) by Lipschitz continuous functions, obtain existence for the smoothed equation, and then use a compactness (passing to limit) argument to derive solutions for the original equation.

Auxiliary lemmas
Our first Lemma has already been presented in [1]. It is repeated here for the sake of easy reference. It is a simple consequence of a well-known fact in the Sturm Comparison theory of linear differential equations.
M. K. Kwong and J. S. W. Wong 5 Proof. The classical Sturm comparison theorem has a strong form that yields the inequality where strict inequality will hold if we know, in addition, that b ≡ B in [0,t]. One way to prove this is to note that the function r(t) = Y (t)/Y(t) satisfies a Riccati equation of the form The function s(t) = Z (t)/Z(t) satisfies an analogous Riccati equation. The inequality r(t) ≥ s(t) follows by applying results in differential inequalities to compare the two Riccati equations.
Let ξ be any point in (0,τ). By integrating over [ξ,τ], we see that In particular, the inequality is true for ξ = ξ i , and the conclusion of the lemma follows by taking the appropriate linear combination of the various fractions.
Lemma 2.2. A necessary condition for the homogeneous Robin multipoint boundary value problem, with θ = π/2, to have a positive solution is Proof. Let S be the tangent line to the solution curve u(t) at the initial point (0,u(0)). Let Y (t) be the function that is represented by S. Then Y satisfies the simple differential equation Substituting these inequalities into the homogeneous Robin boundary condition gives (2.6).
The next lemma is reminiscent of the eigenvalue problem of a linear equation.

Lemma 2.3. Consider the homogeneous linear multipoint boundary value problem
where λ is a positive parameter, and a(t) ≥ 0, a(t) ≡ 0. Furthermore, assume that (2.6) holds. Then there exists a unique constant L θ > 0 for which the problem, with λ = L θ , has a positive nontrivial solution.
In the proof of Lemma 2.3, we see that if y(1,λ) > 0, then φ(λ) is defined and finite. Later in Section 3, we have occasions to make use of the inverse of this simple fact, namely, that if φ(λ) (or a similar function) is defined and finite, then y(1;λ) (or the value at t = 1 of a similar function) is positive.

Main results
To study the multipoint problem, we use the shooting solution u(t;h), which satisfies the initial condition for θ = π/2, and for the Dirichlet case. The function u(t;h) concaves downwards. It can happen that u(t;h) intersects the taxis at some point t = τ ≤ 1. Such a function cannot be a positive solution to our boundary value problem. In the contrary case, suppose that u(t;h) remains positive in [0,1].
We define two functions It is obvious from the definition that for b > 0, u(t;κ) furnishes a solution to our multipoint problem if and only if ψ(κ) = b. For b = 0, u(t;κ) is a nontrivial solution if and only if κ = 0 and is a boundary point of the set of points {h > 0 | ψ(h) = 0} (in other words, ψ(κ) = 0, and every neighborhood of κ contains points for which ψ(h) > 0).
We can now state our first result.  Proof. The first condition means that when u is sufficiently small, the nonlinear term a(t) f (u(t)) is dominated by the linear function La(t)u(t). More precisely, let L 1 be any number such that Then there exists a u 1 > 0 such that for all u ∈ [0,u 1 ], Let us shoot a solution u(t;h) with a sufficiently small h. Since u(t;h) concaves downwards, its curve lies below the straight line that is tangent to the curve at the point (0,h). By choosing h sufficiently small, say for h < h 1 for some h 1 > 0, we can guarantee that u(t;h) ≤ u 1 for all t ∈ [0,1]. The inequality (3.7), therefore, holds for all t close to t = 0, up to the first zero of u(t,h) if there is one before t = 1. This allows us to compare u(t;h) with solutions of It follows that u(1,h) − k i u(ξ i ;h) > 1 and consequently ψ(h) > 0. Recall that this fact is proved for all h ∈ (0,h 1 ). Next, let us study the function ψ(h) when h is large. The second condition of (3.5) is similar to the first one, and suggests an analogous situation. Let L 2 be any number between L θ and liminf u→∞ f (u)/u. By hypothesis, we can find a u 2 large enough such that for all u ≥ u 2 . This allows us to compare solutions of (1.1) with solutions of (note that w(t) is simply hy(t;L 2 )) as long as u(t;h) remains above u 2 . This last requirement complicates our arguments because we have no guarantee that u(t;h) ≥ u 2 when t is near 1. The Dirichlet case has an additional complication because u(t;0) = 0, and we have to deal with those points that are near t = 0. In the following, we present the detailed proof for the Neumann case. The proof for the general case is similar, with an appropriate modification of the value of τ. We leave the Dirichlet case to the readers.
We now assume only the Neumann case with u(0;h) = h and u (0,h) = 0. Suppose that u(τ;h) = u 2 for some t = τ. We claim that if τ ≤ 1 − u 2 /h, then u(t;h) must vanish somewhere in [τ,1]. In such cases, by definition, ψ(h) = 0. To prove the claim, in the tuplane, we draw a straight line S joining the points (0,h) and (1,0). The point (1 − u 2 /h,u 2 ) lies on S. The solution curve u(t;h) intersects the straight line S at the initial point (0, h) but stays above S at least for a neighborhood near t = 0. If τ ≤ 1 − u 2 /h, then the point on the curve at t = τ is below S. Therefore, the solution curve intersects S at a second point somewhere before τ. Since u(t;h) is concave, it cannot intersect S at a third point, so u(t;h) must lie strictly below S in [t,1], forcing it to vanish somewhere before reaching t = 1.
It, therefore, remains to find what ψ(h) is when τ > 1 − u 2 /h. By choosing h sufficiently large, we can make τ as close to 1 as we please. Let us determine how close it should be in order to work for us. We know that φ(L 2 ) > φ(L θ ) = 1. By continuity, we can pick a point τ 1 close to, but distinct from, 1 such that k i y ξ i ;L 2 y τ 1 ;L 2 ≥ 1. We claim that ψ(h) = 0 for all h > h 2 . Let us consider all shooting solutions u(t;h) with initial height h ≥ h 2 . If u(τ 1 ;h) ≤ u 2 , then u(t;h) must have reached u 2 before τ 1 < 1 − u 2 /h. By the above claim, we know that ψ(h) = 0. So we assume that u(τ 1 ;h) > u 2 . In the interval [0, τ 1 ], comparing (1.1) with w is valid and Lemma 2.1 gives (3.14) It then follows that ψ(h) = 0. To summarize, the continuous function ψ(h) is positive in a right neighborhood of h = 0, and 0 for all h > h 2 . It, therefore, is bounded above. Let the least upper bound be b * , which is obviously positive, and suppose that it is attained at a point κ * > 0, ψ(κ * ) = b * . Figure 3.1 illustrates a concrete example.
Let b ∈ (0,b * ). Then, by continuity, ψ(h) must assume the value b at least twice: once at a point κ 1 in (0,κ * ) and once at a point κ 2 in (κ * ,∞). Each of these furnishes a solution to the multipoint problem. It is, of course, possible that there may be other solutions, in particular when the function ψ(h) has multiple local maxima and local minima. If b = b * , then h = κ * gives a solution to the multipoint problem. If b = 0, then the first value κ in (κ * ,∞), for which ψ(κ) = 0, gives a solution to the multipoint problem. There may or may not be a solution for h in (0,κ * ) because it can happen that the only value h that solves ψ(h) = 0 is h = 0, which corresponds to the trivial solution. For b > b * , ψ(h) = b has no solution and neither does the multipoint boundary value problem. Proof. The arguments are the same as those used to prove Theorem 3.1, except that we interchange the parts regarding large and small h, respectively, and the conclusions are different.
Assume first that only (3.15) holds. Let L 1 be a number between limsup u→∞ f (u)/u and L θ , and let u 1 be so large that f (u)/u ≤ L 1 < L θ , for u ≥ u 1 .
We can compare the shooting solution u(t;h) with z(t) = hy(t;L 1 ) as we do in the proof of Theorem 3.1. Since z(1) > 0, we can take h > u 1 /z (1). This ensures that u(t;h) remains greater than u 1 for all t ∈ [0,1] so that the comparison is valid in the entire interval [0,1]. In particular, we have u(1;h) ≥ z(1) = hy 1;L θ . (3.17) We see that as h → ∞. We thus see that ψ(h) is a continuous function on [0,∞), with the properties that ψ(0) = 0 and ψ(h) → ∞. This is depicted in Figure 3.2. Note also that the function ψ(h) may vanish on a subinterval of [0, ∞), and this situation is also illustrated in Figure 3.2.
On account of this, any b > 0 is in the range of ψ(h) and the corresponding boundary value problem has at least one positive solution.
To prove the part concerning b = 0, we have to show that ψ(h) = 0 for all h that are sufficiently small. This is done by using the second condition (3.16) to compare u(t;h) with w(t) = hy(t;L 2 ) as in the proof of Theorem 3.1. In fact, the argument is easier this time since the comparison condition is now satisfied for all t ∈ [0,1], and we do not need to find special treatments for a set of t such as those near t = 1 in the proof of Theorem 3.1.
We remark that Theorem 3.2 does not assert the uniqueness of the positive solution when b = 0. In fact, in the example shown in Figure 3.2, there are three positive solutions, corresponding to the points κ 1 , κ 2 , and κ 3 .
It is interesting to note the asymmetry between Theorems 3.1 and 3.2. In Theorem 3.1, we need both asymptotic conditions to prove the existence of positive solutions to the boundary value problem, while in Theorem 3.2, we need only one asymptotic condition to get the existence of the solutions. In addition, Theorem 3.1 gives at least two positive solutions to the boundary value problem for b < b * , while Theorem 3.2 can only guarantee at least one positive solution for all b > 0.
By examining the proof of Theorem 3.1 more closely, it is not difficult to see that the only places where the separable format of the nonlinear term is needed are to enable us to compare the nonlinear term of (1.1) with the linear terms of the comparing equations (3.8) and (3.11). If we include that as part of the hypotheses, we can obtain the following analogous results concerning the more general nonlinear equivalent (1.3).
Theorem 3.3. Suppose that (1.2) hold, and there exist four constants L 1 , L 2 , u 1 , and u 2 such that Likewise, we have the following.
Theorem 3.4. If we replace condition (3.15) by the existence of two constants L 1 and u 1 such that

21)
and condition (3.16) by the existence of two constants L 2 and u 2 such that Let us assume, in addition, that a(t) > 0 for all t ∈ [0,1], and b(t) ≥ 0 is a given function on [0,1]. Theorem 3.4 obviously implies the following extension of Theorem 3.2 to the "forced" equation  (3.23).

Proof. Let F(t,u) = a(t) f (u) + b(t).
Then the hypotheses of Theorem 3.5 imply the hypotheses of Theorem 3.4, and the same conclusions of Theorem 3.2 hold.
Note again the asymmetry, Theorem 3.1 does not appear to have a similar extension to the forced equation.

Discussion and examples
(1) Theorem 1.1 has been extended by Raffoul [16] by showing that it still holds if f 0 and f ∞ are positive finite constants satisfying certain bounds. The constants given by Raffoul are not the best possible. Our Theorems 3.1 and 3.2 include Raffoul's results and give the best possible constant L θ . A discussion of Raffoul's results can be found in [1]. In [17], Liu and Yu proved results similar to that of Theorem 1.1 for the homogeneous threepoint problem. Liu [18] further improved Theorem 1.1 by considering cases when both of f 0 and f ∞ are finite or zero. He also proved existence theorems when both f 0 and f ∞ are finite. In all these cases, he also assumed that f (u) is either bounded above or below by a constant multiple of |u| in certain specified intervals. The techniques used in this paper can easily be extended to generalize these results.
(2) With the exception of a few, for example, [1,3], most of the results on the existence of solutions to multipoint boundary value problems are based upon fixed-point theorems of Krasnoselskii's type; see Krasnoselskii [2], Guo and Lakshmikantham [19], and recent articles of Ma [12,14,20], Liu [18], and Sun et al. [13]. Other methods in nonlinear functional analysis such as Leray-Schauder continuation theorem, nonlinear alternative of Leray-Schauder fixed point theorem, and coincidence degree theory have also been used, see Mawhin [21,22]. In most cases, the conditions imposed are stronger than those required by using the shooting method and Sturm comparison theorem as discussed in the previous work [1] and in this paper. Nevertheless, the abstract methods in Banach spaces have the important advantage over the shooting method; in that, it can be applied to higher-order equations, in higher dimensions, and to equations with deviating arguments.
(3) The shooting method and Sturm's comparison theorem are effective when the nonlinear function is bounded by a linear growth. In the case of (1.1), the corresponding linear differential operator is invertible on a suitable function space. This is known as the nonresonance case and Leray-Schauder degree theory has been applied to study m-point problems. For results on three-point boundary conditions, see Gupta et al. [23,24] and Gupta and Trofimchuk [25]. Gupta [6,7] further employed a nonlinear alternative of Leray-Schauder fixed point theorem to obtain sharper conditions where the coefficient function in (1.1) is not required to be continuous but only integrable in [0,1]. For results related to multipoint boundary conditions, see Guo et al. [26]. For the resonance case with nonlinear function still subject to linear growth, see Gupta [27,28] and Feng and Webb [29].
(4) In Feng and Webb [29,30] and Feng [31], more general nonlinear functions subject to certain sign conditions and quadratic growth are studied. It is interesting to see if the shooting method and Sturm comparison theorem can be extended to such cases.
(5) Multipoint problems have been studied for p-Laplacian equations; see, for example, [32]. The shooting method can be easily adapted to get similar results for such equations because the Sturm comparison theorem remains valid for p-Laplacian equations.
(6) Another generalization of the multipoint problem can be obtained by replacing (1.7) by an integral condition (sometimes called a nonlocal condition) such as (4.1) and the corresponding nonhomogeneous form. This can be considered as the continuous analogue of the discrete m-point condition. The shooting method can also be easily adapted to treat such generalizations. (7) Numerous authors have derived results guaranteeing multiple solutions to various boundary value problems. Typically, assumptions are imposed on the nonlinear function so that it is alternatively large and then small in successive subintervals of [0,∞). See [1] for a discussion of such results.

Example 4.2.
Consider the second-order nonlinear differential equation subject to the Robin boundary condition at the left end-point Here, f 0 = 0 and f ∞ = 1 as defined by (1.10).
Theorem 3.1 shows that there exists a positive number b * such that the boundary value problem (4.3) and (4.4) has at least two positive solutions for all b, 0 ≤ b < b * , at least one positive solution for b = b * , and has no solution for b * > 0. Here, Theorem 1.2 and other similar results such as Raffoul [16], Liu [18], and Liu and Yu [17] are not applicable. 1 + u(t) = 0, t ∈ (0,1), 0 < γ < 1, (4.5) subject to the Dirichlet boundary condition at the left end-point Here, f 0 = ∞ and f ∞ = 0, which is typical in the sublinear case. Theorem 3.2 is applicable (by choosing any L θ > 0) and we conclude that the boundary value problem (4.5) and (4.6) always has a positive solution for all b ≥ 0. This example is not covered by either Theorem 1.1 or any other previously known results applicable to the sublinear case.