Boundary Value Problems from Higher Order Differential Equations

The following sections are included:

• PREFACE

• CONTENTS

Boundary value problems (hereafter to be abbreviated as BVPs) manifest themselves in almost all branches of Science, Engineering and Technology, for instance, boundary layer theory in fluid mechanics, heat power transmission theory, space technology and also control and optimization theory, to cite only a few. The following examples provide a variety of situations of occurrence of BVPs and the motivation for some of the problems we shall consider in these notes.

For the nth order linear differential equation

where p_i(t), 1 ≤ i ≤ n and f(t) are continuous on I = [a,b], we shall consider the following separated boundary conditions

where a ≤ a₁ ≤ a₂ ≤ … ≤ a_n ≤ b…

Throughout, we shall assume that the condition (2.15) is satisfied. Thus, the existence of the fundamental system of solutions {x_i(t)}, 1 ≤ i ≤ n of (2.12) satisfying

is assured. We shall denote by D_i(t), 1 ≤ i ≤ n the cofactor of the element in the Wronskian W(t). For convenience we shall write a = a₀, b = a_n+1, x₀(t) = x_n+1(t) = D₀(t) = D_n+1(t) = 0. The square a ≤ t, s ≤ b will be represented by K; the same square with straight lines of the form s = a_i rejected from it we shall denote by K₀; K₀ with rejected diagonal t = s by K₁…

Shooting methods, in which the numerical solution of a BVP is found by integrating an appropriate initial value problem, have been the subject of several papers [1-5, 9-11 and references therein] and books [8, 13, 14], and a large part of the monographs [7, 12]. The attraction of these methods is in the availability on most computers of reasonably adequate subroutines of the numerical solution of initial value problems. Goodman and Lance [6] gave two such methods for the numerical solution of linear two point BVPs, the method of complementary functions and the method of adjoints, which are now in common use. Here, we shall formulate the method of complementary functions for the BVP (2.1), (2.2) whereas the method of adjoints is the subject of our next section…

As the name suggests we use the adjoint equation of (2.1) to obtain the solution of the BVP (2.1), (2.2). However, to avoid unnecessary differentiability conditions on the functions P_i(t), 1 ≤ i ≤ n appearing in (2.1) we shall formulate this method for the first order differential systems…

This is another practical shooting method originally developed by Gel’fand and Lokutsiyevskii, however first appeared in English literature only recently [4]. Na [6] has briefly described the method and given different formulations for the different particular cases of (2.1), (2.2). Here, we shall follow [1-3] to provide the general formulation of the method for the BVP (2.1), (2.2). The power of the method is illustrated by solving known Holt’s problem…

The BVP

where f(t,x₀,x₁) is continuous and satisfies a uniform Lipschitz condition

on [a₁,a₂] × R² has a long history going back to Picard [11] 1893 (from the literature it appears that before 1893 the main attack was to construct the solution to a given problem tacitly assuming existence and uniqueness). He proved that if (a₂-a₁) is sufficiently small then the sequence {x_m(t)} of functions generated by the iterative scheme

In polynomial interpolation theory, the following result is well known :

Theorem 8.1 Let x(t) ∈ C⁽ⁿ⁾[a,bl, satisfying (3.8). Then,

where and …

The necessary and sufficient condition (2.15) for the existence of a unique solution of the BVP (2.1), (2.2) is only of theoretical value because it can be verified only for some trivial problems. Further, the method used in Theorem 7.2 to obtain explicit bounds for the existence of a solution of the BVP (7.1), (7.7), (7.8) leads to an insurmountable task for (2.1), (2.2). However, using the inequalities obtained in the previous section, it is possible to provide explicit bounds for the existence and uniqueness of the solutions even for the nonlinear nth order differential equation

together with each of the boundary conditions (2.3)-(2.9). In the differential equation (9.1), x stands for (x,x′,…,x^(q)), 0 ⪯ q ⪯ n-1. Throughout, we shall assume that the function f is continuous at least in the interior of the domain of interest.

The Picard method of successive approximations mentioned in Section 7 for the BVP (7.1), (7.2) has an important characteristic, that it is constructive; moreover, bounds of the difference between iterates and the solution are easily available. In this section we shall discuss this method only for the BVP (9.1), (2.4). For other BVPs analogous results can be stated without much difficulty. For this, we need

Theorem 10.1 (Contraction mapping principle) [3] Let B be a Banach space, and let r > 0; S(x₀,r) = {x ∈ B : ||x −x₀||≤r}. Let T map into B and

(i) for all x,y ε , ||Tx − Ty|| ≤ α ||x - y|| where 0 ≤ α < 1

(ii) r₀ = (1−α)⁻¹ ||Tx₀ − x₀|| ≤ r.

It is well recognised that quasilinearization is a fruitful practical method to construct the solution of nonlinear problems in an iterative way. Although the technique as originally developed by Bellman and Kalaba [5] was motivated by dynamic programming, it is not necessary to know or to employ dynamic programming to use quasilinear method…

Here, we shall employ weight function technique of Weissinger [15] and Collatz [10] to show that the region of existence and uniqueness of solutions of BVPs can be enlarged as compared to what can be deduced from the results of sections 9 and 10. More so, in some special cases where the explicit form of the Green’s function and some of its properties are known, it is possible to find best existence and uniqueness intervals. For this, we need

Definition 12.1 A system of nonnegative continuous functions (weight functions) w_j(t), 0 ≤ j ≤ q in [a₁, a_r] is called admissible with respect to the Green’s function g₂(t,s) if

(i) there exist smallest positive constants k_j., 0 ≤ j ≤ q such that

(ii) there exist finite smallest positive constants , 0 ≤ i, j ≤ q such that

In example 12.6, we employed weight function technique for the BVP (9.1), (2.7) when q ≤ p and obtained best possible existence and uniqueness interval in terms of the Lipschitz constants L₀, …, L_q. Here, we shall again assume that q ≤ p, but the function f satisfies the condition

whenever x_j ≥ y_j, 0 ≤ j ≤ q to obtain best possible existence and uniqueness interval for each of the BVPs (9.1), (2.7); (9.1), (2.8) in terms of M_j, K_j, 0 ≤ j ≤ q. Obviously, the condition (13.1) is equivalent to the Lipschitz condition (9.12), but more informative particularly since there are no sign restrictions on the constants.

In [3], a fixed point theorem for isotone operators in partially ordered spaces has been proved which is distinguished for its constructive character. Its slightly strengthened version is given by the following :

Theorem 14.1 Let (E,≤) be a partially ordered space and x₀ ≤ y₀ be two elements of E. [x₀,y₀] denotes the interval {x ∈ E : x ≤ x ≤ y₀}. Let T : [x₀, y₀] → E be isotone (T(x) ≤ T(y), whenever x ≤ y) and let it possess the properties

(i) x₀ ≤ T(x₀)

(ii) the (nondecreasing) sequence {T^m(x₀)} where T⁰(x₀) = x₀, T^m+1(x₀) = T[T^m(x₀)] for each m = 0,1,2,… is well defined, i.e., T^m(x₀) ≤ y₀ for each natural m

(iii) the sequence {T^m(x₀)} has sup x ∈ E, i.e., T^m(x₀) ↑ x

(iv) T^m+1(x₀) ↑ T(x)…

In Section 2, we have noticed that the uniqueness of solutions of the linear BVP (2.1), (2.2) implies the existence of solutions. The argument employed in proving this assertion is algebraic and is based on the linear structure of the fundamental system of solutions of (2.12) and the linearity of the boundary conditions (2.2). The question of whether or not nonlinear equation (9.1) could have this property will be discussed here. For this, we need Kamke’s convergence theorem [2].

The proposition mentioned in Remark 15.4 is that “conditions (A)-(D) of Theorem 15.5 are sufficient to imply the conclusion of Theorem 15.7”. This is indeed the case when the differential equation (9.1) is of second or third order. In fact for these particular cases conditions (A)-(D) imply the compactness condition (E). To show this we note that Theorem 15.1 can be stated as follows :

Theorem 16.1 Assume that for the differential equation (9.1) the conditions (A), (B) and (C) of Theorem 15.5 are satisfied. Then, if {x_m(t)} is a sequence of solutions of (9.1) and [c,d] is a compact sub-interval of (a,b), either there is a subsequence {x_m(j)(t)} such that converges uniformly on [c,d] for each 0 ≤ i ≤ n-1, or

uniformly on [c,d] as m → ∞…

Let the BVP (7.1), (7.2) have two solutions x_l(t) and x₂(t). Since x₁(a_i) - x₂(a_i) = 0 for i = 1,2 there exists some t₀ ∈ (a₁,a₂) at which . Thus, x₁(t) and x₂(t) are both solutions of the two BVPs : (7.1) together with

and (7.1) together with

where . Thus, if a₁ < t₁ < a₂ and uniqueness holds for all BVPs (7.1), (17.1) whenever t₀ ∈ (a₁,t₁], and if uniqueness holds for all BVPs (7.1), (17.2) then uniqueness holds for all BVPs (7.1), (7.2) also…

The concept of boundary value functions for the third order linear differential equations appears to have been used first by Azbelev and Tsalyuk [1], whereas for the nth order linear differential equations by Sherman [11]. Peterson [3] was the first to extend this notion to nonlinear differential equation (9.1)…

A number of authors have employed various topological principles to study second order BVPs. For example, in [1-5,13,16] variations and refinements of Wazewski’s topological method have been obtained for the first order differential system u′ = F(t,u) and subsequently these are used to prove the existence and continuous dependence of solutions on boundary data for a class of second order BVPs more general than (7.1), (7.2). In [6] Wazewski’s topological method together with the compactness properties of solution funnels have been used to prove the existence of solutions. In [15] Kelley has employed a method similar to Wazewski’s to study the existence of the solutions of the BVP (9.1), (2.7) with q = n-1 and p = n-2. In this section, we shall use a newly developed topological transversality method due to Granas [10,11] which is a generalization of the continuation theorem of Leray and Schauder found in [9], to study the existence of solutions of a more general class of BVP

where P_i(t), 1 ≤ i ≤ n are continuous on I = [a,b], f is continuous on I × R^q+1, U : C^(n-1) (I) → Rⁿ is a continuous linear operator and l is a given vector in Rⁿ.

In Section 10, we have used Contraction Mapping Principle and obtained the conclusion that a given BVP has a unique solution provided certain inequality over the length of the interval is satisfied. In general this inequality does not provide the best possible length estimates in the sense that unique solutions may exist on longer intervals. The results obtained in Sections 12 and 13 are some examples where the weight function technique or shooting methods indeed give best possible intervals. In this section, we shall apply techniques from optimal control theory to obtain some more best possible results…

This technique is useful in proving the existence of solutions of a given BVP with the help of solutions of several other related BVPs. The main idea of this method is contained in the following : Let x₁(t) be a solution of a BVP on [a,b] and x₂(t) be a solution of a BVP on [b,c], where b ∈ (a,c) and fixed, then the function x(t) defined by

is a solution of the given BVP on [a,c]…

In this section, we shall consider the question of existence of a maximal solution for the initial value problem : (9.1) together with

where t₀ ∈ (a,b) and f is assumed to be continuous on (a,b) × R^q+1…

‘Everyone knows’ that if x ∈ C⁽²⁾ [a₁,a₂] and x^″(t) ≥ 0 on (a₁,a₂), then x satisfies the maximum principle, i.e., if x attains its maximum at an interior point of [a₁,a₂], then x is identically constant on [a₁,a₂]. This principle, however, is not true for functions satisfying higher order inequalities. For example, let x = −t², it follows that x satisfies the inequality x⁽⁴⁾(t) ≥ 0 on [−1,1], and yet x assumes its maximum at t = 0…

In this section, we are concerned with the existence of solutions of BVPs on semi-infinite and infinite intervals. In Theorems 24.2 and 24.3, we shall provide sufficient conditions for the BVP : (9.1) with q = 0 and

where to have a solution on [a₁,∞] and respectively.…

In this final section, we shall first consider the following differential equation with deviating arguments

where <x⁽ⁱ⁾> stands for . The function f is assumed to be continuous on [a_l,a_r] × R^N, where . The functions W_ij(t); 1 ≤ j ≤ p(i); 0 ≤ i ≤ q are continuous on [a₁, a₁, a_r]…

The following sections are included:

• NAME INDEX