Uniqueness of solutions of boundary value problems at resonance

In this paper, the method of upper and lower solutions is employed to obtain uniqueness of solutions for a boundary value problem at resonance. The shift method is applied to show the existence of solutions. A monotone iteration scheme is developed and sequences of approximate solutions are constructed that converge monotonically to the unique solution of the boundary value problem at resonance. Two examples are provided in which explicit upper and lower solutions are exhibited.


Introduction
The method of upper and lower solutions and monotone methods have been useful in the study of boundary value problems for nonlinear ordinary differential equations.For many problems, the associated Green's function has fixed sign that agrees with a maximum principle or an anti-maximum principle [7].Then monotonicity of iterates can occur naturally by assuming the nonlinearity is monotone with respect to the unknown function or the monotonicity of iterates can be forced by various methods.We refer the reader to [7,8,9,10] for discussions and applications of maximum or anti-maximum principles or to [4,11,12,14,25] for discussion and applications of the so-called isotone operators.
The method of quasilinearization, introduced by Bellman [5,6] in the 1960s, offers a numerical method to approximate solutions of nonlinear problems with sequences of solutions of linear problems.Under suitable hypotheses, the sequences of approximate solutions converge monotonically and quadratically.In many applications, the iterates converge to a unique solution of the boundary value problem.The quasilinearization method has been particularly useful in the study of boundary value problems for ordinary differential equations and we cite a number of those applications here [1,3,13,15,19,20,21,22,23,26].In these works, the monotonicity is obtained rather delicately and the uniqueness of solutions plays a key role in obtaining the monotonicity.
To obtain the monotonicity of the iterates in, for example, [1,3,13,15,19,23], a standard hypothesis is that the nonlinear term is increasing as a function of the unknown function.This hypothesis is used only to show the uniqueness of solutions.In this work, we shall assume the standard hypothesis and we shall assume in addition that the nonlinear term is increasing as a function of the derivative of the unknown function.The new hypothesis is only employed in the analysis at a boundary point.It is of interest to note that the new hypothesis is used to show the uniqueness of solutions and to show the existence of solutions.
Recently [2], the method of quasilinearization was applied to a two-point boundary value problem for an ordinary differential equation at resonance.In this article we shall consider a new two-point boundary value problem at resonance and we shall construct the monotone iteration scheme associated with the method quasilinearization.One key contribution of this work is that the nonlinear term depends on the unknown function and the derivative of the unknown function.In [2], a shift argument [17] is employed to obtain existence of solution.The shift that is employed depends on the unknown function.In this work, we shall apply the shift argument with a shift that depends on the derivative of the unknown function.In doing so, we shall successfully construct the monotone method; however, we currently cannot verify quadratic convergence.We shall leave for future work the application of the shift argument employed in [2] with the intention to obtain quadratic convergence.
The paper is organized as follows.In Section 2 we shall first employ the method of upper and lower solutions and under suitable hypotheses obtain the uniqueness of solutions of a two-point boundary value problem at resonance for a second order ordinary differential equation.In Section 3, we shall apply the shift argument and obtain the existence of that unique solution.In Section 4, we shall construct the monotone method.Sections 2, 3 and 4 apply to a problem where the nonlinear term depends on the unknown function and the derivative of the unknown function.In Section 5, we shall show how the methods of Sections 2, 3 and 4 apply to a problem where the nonlinear term only depends on the unknown function.We close in Section 6 with two examples in which upper and lower solutions are explicitly exhibited.
The application of the method of quasilinearization to boundary value problems at resonance is not new; see [27,28].The motivation and development here is different than that in [27] or [28], since uniqueness of solutions is a key feature in this work and multiplicity of solutions is key in [27] or [28].

Uniqueness of solutions
Assume f : [0, 1] × R 2 → R is continuous.We shall consider the boundary value problem 2) The boundary value problem (2.1), (2.2) is at resonance since the linear functions, y = ct, c ∈ R, are solutions of the homogeneous problem y = 0 and satisfy the homogeneous boundary conditions (2.2).With the notation f (t, y 1 , y 2 ), we begin with the assumption that f is increasing in y 1 for each (t, y 2 ) ∈ [0, 1] × R and f is increasing in y 2 for each (t, y 1 ) ∈ [0, 1] × R to obtain results for the uniqueness of solutions of the boundary value problem (2.1), (2.2).
Then solutions of the boundary value problem (2.1), (2.2) are unique, if they exist.
Proof.Assume for the sake of contradiction that y(t) and z(t) denote two distinct solutions of the boundary value problem (2.1), (2.2).Assume without loss of generality that y −z has a positive maximum at t 0 ∈ [0, 1].
(If this is not the case, then z − y has a positive maximum at some t 0 ∈ [0, 1].) First, assume, t 0 ∈ (0, 1).Then (y − z) (t 0 ) ≤ 0. However, y and z each satisfy (2.1), and so, . This is a contradiction, and we shall refer to this contradiction as the usual contradiction.Thus, y − z does not have a positive maximum at t 0 ∈ (0, 1).Second, assume t 0 = 0 and recall the boundary condition y(0) = z(0) = 0. Thus, y − z does not have a positive maximum at t 0 = 0.
Remark 2.4.We point out here that Theorem 2.1 follows as an immediate corollary of Theorem 2.3 since a solution of boundary value problem (2.1), (2.2) is both an upper solution of the boundary value problem (2.1), (2.2) and a lower solution of the boundary value problem (2.1), (2.2).

Existence of solutions
To obtain existence of solutions, we shall apply the shift argument [17].Assume λ > 0 and consider the shifted equation The boundary value problem, (3.1), (2.2) is not at resonance for any λ > 0 since Theorem 2.1 (modified with the hypothesis f y 1 ≥ 0 on [0, 1] × R) applies to the homogeneous problem with boundary conditions (2.2), if we rewrite y (t) − λy (t) = 0 in the form y (t) = λy (t).Thus, the Green's function for the boundary value problem (3.1), (2.2) can be constructed and has the form We observe the following properties of G(λ; t, s).
Theorem 3.1.Let G(λ; t, s) denote the Green's function of the boundary value problem (3.1), (2.2).Then Proof.The term e λt e λ(1−s) −e λ(1−s) is decreasing in t for 0 ≤ s ≤ t ≤ 1 and negative at t = s.And so, G(λ; t, s) < 0, on [0, 1] × [0, 1].The sign of G t (λ; t, s) is determined similarly.Note that 1 0 G(λ; t, s)ds is the solution of the boundary value problem (2.1), (2.2) for f ≡ 1.Thus, then G(λ; t, s) is represented as the convolution of Green's functions for lower order problems.G t (λ; t, s) denotes the Green's function for the periodic boundary value problem and the function denotes the Cauchy function for the initial value problem Periodic boundary value problems have been studied extensively and we refer the reader to the recent monograph [7] for an authoritative account.The proof of Theorem 3.1 can be obtained as a corollary of the observation of (3.4) as a convolution of Green's functions for lower order problems.
Prior to obtaining existence of solutions of the boundary value problem (2.1), (2.2) we state without proof versions of the Kamke convergence criterion for solutions of initial value problems for ordinary differential equations [16,18].We also state a version of the Schauder fixed point theorem [24].
continuous.We say g satisfies a Nagumo condition with respect to y 2 if, for each M > 0, there exists h M defined on (0, ∞) such that and such that An important consequence of the Nagumo condition is that if g satisfies a Nagumo condition, then solutions of y (t) = g(t, y(t), y (t)) either extend to the interval [0, 1] or the functional value y(t) becomes unbounded on its maximal interval of existence.We refer the reader to [16] or [18] for further details.The next theorem is a version of the Kamke convergence criterion and again we refer the reader to [16] or [18] for further details.Theorem 3.4.For k ∈ {0, 1, . . .} assume the functions f k (t, y 1 , y 2 ) are continuous on [0, 1]×R 2 and assume that there exists f (t, y 1 , y 2 ) such that Then there exists a subsequence {y k j } of {y k } and a solution y of We shall employ the Kamke criterion in the form of the following corollary which is proved in [15].Corollary 3.5.For k ∈ {0, 1, . . .} assume the functions f k (t, y 1 , y 2 ) are continuous on [0, 1]×R 2 and assume that there exists f (t, y 1 , y 2 ) such that uniformly on compact subsets of [0, 1] × R 2 .Assume f satisfies a Nagumo condition in y 2 .Assume that, for k = 0, 1, . . ., y k (t) is a solution of y (t) = f k (t, y(t), y (t)) and assume y k (t) satisfies the boundary conditions (2.2).Assume that {y k } is monotone decreasing and bounded below by a continuously differentiable function α(t).Then there exists a solution y of Our final preliminary result is the following version of the Schauder fixed point theorem.
Theorem 3.6.If U is a closed convex subset of a Banach space B, if T : U → U is continuous on U, and if T (U) is a compact subset of B, then T has a fixed point in U.
We are now in a position to provide sufficient conditions for the existence of a solution of the boundary value problem (2.1), (2.2).It is of interest to note that the hypothesis f y 1 > 0 on [0, 1] × R 2 is not employed and the hypothesis f y 2 > 0 on [0, 1] × R 2 will be employed at the boundary point t 0 = 1.In particular, the following theorem makes no claims to uniqueness of solutions.
Then there exists a solution y of (2.1), (2.2) satisfying Proof.Let λ > 0 and first define a truncation, F, of g(t, y(t), y (t)) = f (t, y(t), y (t)) − λy (t) by Define a further truncation F k , for k = 0, 1, . . .by uniformly on compacts subsets of [0, 1] × R 2 .More importantly, in order to apply Corollary 3.5 note that For each integer k ≥ 0, define an operator where G(λ; t, s) is given by (3.2).The purpose of the Green's function is to provide the following equivalent statements.A function y ∈ C 2 [0, 1] is a solution of the boundary value problem with boundary conditions (2.2) if, and only if, y ∈ C 1 [0, 1] and That is, y ∈ C 2 [0, 1] is a solution of the boundary value problem (3.5), (2.2) if, and only if, y ∈ C [0, 1] and Since G(λ; t, s) and G t (λ; t, s) are continuous on Since the further truncation, F k , is bounded and continuous on [0, 1] × R 2 it is a straightforward application of the Schauder fixed point theorem to show that the boundary value problem (3.5), (2.2) has a solution.To see this, let Then U is a closed convex subset of C 1 [0, 1] and T k : U → U. To show that T (U) is compact, we apply the Arzela-Ascoli Theorem to each set {(T k y) : y ∈ U} and {(T k y) : y ∈ U}.Each of the sets {(T k y) : y ∈ U} and {(T k y) : y ∈ U} is uniformly bounded by M λ .Since {(T k y) : y ∈ U} is uniformly bounded, an application of the mean value theorem implies that {(T k y) : y ∈ U} is equicontinuous.Moreover, note that if y ∈ U then .
We now argue that Details are similar to the proof of Theorem 2.1 and we highlight the differences in the details due to the truncation, F k .
producing the usual contradiction.So, t 0 / ∈ (0, 1).It is important to note that the hypothesis f y1 > 0 on [0, 1] × R 2 is not required to obtain the usual contradiction.
Second, t 0 = 0 by the boundary condition (2.2).Finally assume y k − β has a positive maximum at t 0 = 1 and so (y Then the calculation in (3.6) is valid at t 0 = 1 and (y k − β) (1) > 0, which produces the usual contradiction.

The monotone method
In this section we develop the monotone method.The construction is modeled after the construction in [15].
Proof.The existence of a unique solution y satisfying follows from Theorems 2.1, 2.3 and 3.7.
To complete the proof, {α n } and {β n } are monotone sequences of continuous functions bounded above or below, respectively, on a compact domain.So by Dini's theorem, each converges uniformly to α(t), β(t) respectively on [0, 1].We can not apply Corollary 3.5 directly since neither h(t, y 1 , y 2 ; α n , β n , β n ) nor k(t, y 1 , y 2 ; β n , β n ) converge uniformly to f (t, y 1 , y 2 ) uniformly on compact domains of [0, 1] × R 2 .To see this, write h(t, y 1 , y Corollary 3.5 applies to the boundary value problem

A simplified problem
We have proved the main results in this paper for the case where f depends on the derivative of the unknown function.The results are also valid if f is independent of the derivative of the unknown function.
Assume f : [0, 1] × R → R is continuous and consider the boundary value problem with the boundary conditions (2.2).We state the following three theorems without proof as the proofs follow in a straightforward way from the proofs of Theorems 2.1, 2.3 and 3.7 very closely and are more simple.and there exists M > 0 such that λM M 1 ψ(M ) > 1.