Upper and Lower Solution Method for Boundary Value Problems at Resonance

We consider two simple boundary value problems at resonance for an ordinary differential equation. Employing a shift argument, a regular fixed point operator is constructed. We employ the monotone method coupled with a method of upper and lower solutions and obtain sufficient conditions for the existence of solutions of boundary value problems at resonance for nonlinear boundary value problems. Three applications are presented in which explicit upper solutions and lower solutions are exhibited for the first boundary value problem. Two applications are presented for the second boundary value problem. Of interest, the upper and lower solutions are easily and explicitly constructed. Of primary interest, the upper and lower solutions are elements of the kernel of the linear problem at resonance.


Introduction
We consider two boundary value problems at resonance for second order ordinary differential equations.Specifically, we shall consider where f : [0, 1] × R → R is continuous and y(0) = 0, y (0) = y (1), (1.4) where f : [0, 1] × R 2 → R is continuous.We shall employ the method of upper and lower solutions coupled with monotone methods.
Boundary value problems at resonance have been investigated for many years; coincidence degree theory, credited to Mawhin [24,25], has been employed by many researchers and we cite, for example, [3, 6-8, 10, 13, 18, 19, 21, 22].More recently, beginning with interest to obtain sufficient conditions for the existence of solutions in a cone, researchers have been developing a variety of new methods.As examples, the following methods have been developed: (i) a coincidence theorem of Schauder type [30], (ii) a Lyapunov-Schmidt procedure [23], (iii) topological degree [5,12,27,29], (iv) a Leggett-Williams type theorem for coincidences [9,15,28], (v) a fixed point index theorem [2,4,19,20], and (vi) fixed point index theory [32].More in line with the approach employed in this work, Han [14] modified the problem at resonance and considered a regular boundary value problem (a method referred as the shift argument by Infante, Pietramala and Tojo [16]) in order to apply the Krasnosel skiȋ-Guo fixed point theorem [11].
Szyma ńska-D ębowska [31] generalized Miranda's theorem [26] and provided applications to boundary value problems at resonance for second order ordinary differential equations.Yang et al. [33] recently extended the work in [31] to nth order ordinary differential equations.
Infante, Pietramala and Tojo [16] provided a thorough study of boundary value problems related to the Neumann boundary conditions, (1.2), using the shift argument.Motivated by [16], Almansour and Eloe [1] applied the shift argument and presented three applications, one using the Krasnosel skiȋ-Guo fixed point theorem, motivated by Han [14], one using the Schauder fixed point theorem and one using the Leray-Schauder nonlinear alternative.
In this work, we develop the monotone method, coupled with the method of upper and lower solutions, for the shifted boundary value problem.We revisit the applications of Almansour and Eloe [1].We also present some new applications; in particular, we develop the monotone method, coupled with the method of upper and lower solutions for the more complicated problem, (1.3)- (1.4).The boundary value problem (1.3)-(1.4) is more complicated because nonlinear dependence on velocity is assumed.In each application, explicit upper and lower solutions are exhibited and thus, a numerical algorithm to estimate solutions is implied.However, the primary contribution of this work is that the upper and lower solutions, in each application, are nontrivial solutions of the homogeneous problem at resonance.
For boundary value problems not at resonance, the method of upper and lower solutions provides a stand alone method for studying existence of solutions of boundary value problems [17].In this case, one employs the upper solution and the lower solution to truncate the problem and then applies the Schauder fixed point theorem to a bounded nonlinearity.We are unsuccessful to employ the method of upper and lower solutions as a stand alone method for boundary value problems at resonance and we shall address this observation in a remark in Section 2.

The monotone method coupled with the method of upper and lower solutions
Since we couple the monotone method with the method of upper and lower solutions, the analysis is simple.Hence, as this is not the primary contribution of this work, we present the method briefly.Throughout, C[0, 1] will denote the space of continuous real valued functions defined on [0, 1], where for y ∈ C[0, 1] the norm is the usual supremum norm, C 1 [0, 1] will denote the space of continuously differentiable real valued functions defined on [0, 1], where for y ∈ C 1 [0, 1] the norm is the standard y = max{ y 0 , y 0 }.
First, consider the boundary value problem (1.1)-(1.2).Assume β ∈ R and define g(t, y) = f (t, y) + β 2 y.To employ the monotone methods, we shall assume that g is increasing in y.Consider an equivalent boundary value problem, with the boundary conditions (1.2).Assume throughout that f : [0, 1] × R → R is continuous and when considering the boundary value problem (1.1)-(1.2),we shall assume The Green's function, G 1 (β; t, s), for the boundary value problem (2.1)-(1.2) exists and has the form in particular, y is a solution of the boundary value problem (2.1)-(1.2) if, and only if, y ∈ C[0, 1] and Then y is a solution of the boundary value problem (1.1)-(1.2) if, and only if, y ∈ C[0, 1] and Note that under the assumption (2.2), it follows that G 1 (β; t, s) > 0, (t, s) ∈ (0, 1) × (0, 1); so under an additional assumption that g(t, y) = f (t, y) + β 2 y is increasing in y, it is the case that K 1 is a monotone operator; that is, if In particular, 2) and assume g(t, y) = f (t, y) + β 2 y increasing in y.Assume there exist lower and upper solutions, w 0 , Then there exists a solution y of the boundary value problem (2.6) Then if y is a solution of (1.1)-(1.2) satisfying (2.6), then, for each n = 0, 1, . . ., (2.8) and each of w and v are solutions of the boundary value problem (1.1)-(1.2).
Proof.Define the operator K 1 by (2.4).Since g(t, y) = f (t, y) + β 2 y is increasing in y, and G 1 (β; t, s) > 0 on (0, 1) × (0, 1), then K 1 is monotone as stated in (2.5).Define sequences by {w n (t)} and {v n (t)} by (2.7).Since K 1 is monotone, and w 0 (t) ≤ v 0 (t), for 0 ≤ t ≤ 1, it follows inductively that for each n = 0, 1, . . . .Moreover, it is the case that To see this, note that w 0 is the solution of Thus, Since w 0 is a lower solution and in particular, satisfies the differential inequality In particular, and now inductively, n = 0, 1, . . ., follows by the monotonicity of K 1 .Similarly, it is shown that n = 0, 1, . . . .And so, it follows from (2.10), (2.11) and (2.12) that (2.8) is valid.From (2.8), it follows that {w n } is monotone increasing and bounded above by v 0 .By Dini's theorem, there exists w ∈ C[0, 1] such that {w n } converges uniformly to w.Similarly, {v n } is a monotone decreasing sequence and bounded below by w 0 .Thus there exists v ∈ C[0, 1] such that {v n } converges uniformly to v. Thus, From the continuity of g and K 1 (not shown here), and from the uniform convergence of w n+1 (t) = K 1 w n (t) and v n+1 (t) = K 1 v n (t), it follows that w(t) = K 1 w(t) or v(t) = K 2 v(t) and the proof is complete.Remark 2.2.It is interesting to note that we are unable to develop a stand alone method of upper and lower solutions for the boundary value problem at resonance (1.1)-(1.2).For the regular boundary value problem (1.1) with Dirichlet boundary conditions, y(0) = 0, y(1) = 0, the corresponding Green's function for this boundary value problem is negative on (0, 1) × (0, 1) and in the definition of upper solution, one assumes, One then shows that the solution y of the truncated problem (obtained as an application of the Schauder fixed point theorem) satisfies by showing that sign of the differential inequality contradicts the second derivative test for local maximum values.For the problem considered in Theorem 2.1, the Green's function, G 1 , is positive on (0, 1) × (0, 1).This implies that in the definition of upper solution, the differential inequality is reversed; in particular, There is no contradiction to the second derivative test.

Construction of upper and lower solutions
The method of upper and lower solutions is of value in the case when explicit upper and lower solutions can be constructed.In this section we exhibit explicit upper and lower solutions for five applications.Each application can be obtained using standard fixed point theorems (following the shift argument).In each application, the explicit upper and lower solutions are nontrivial solutions of the original linear problem at resonance.The first three applications illustrate the usage of Theorem 2.1.The fourth and fifth applications will illustrate the usage of Theorem 2.3.2) in the case that g is bounded.
Proof.Since g is bounded, assume M > 0 such that Construct constant upper and lower solutions, The hypotheses of Theorem 2.1 are satisfied.
is increasing in y.Assume, moreover, that f (t, y) ≥ −β 2 y holds.Assume f satisfies the asymptotic properties (1) lim sup (2) lim inf Then there is at least one positive solution for the boundary value problem (1.1)-(1.2).
is increasing in y.Assume there exist σ ∈ C[0, 1] and a nondecreasing function ψ : Moreover, assume there exists M > 0 such that Then the boundary value problem Proof.To exhibit v 0 , an upper solution, set To exhibit w 0 a lower solution, set w 0 = −M. Then, In particular, and the hypotheses of Theorem 2.1 are satisfied.
is increasing in y, and there exist σ ∈ C[0, 1] and some 0 < α < is bounded on [0, 1] × R 2 .Moreover, assume that g is increasing in each of y 1 and y 2 .Then there exists a solution of the boundary value problem (1.3)-(1.4).
Remark 3.9.Analogous to Remark 3.2, remove the hypotheses that g is increasing in each of y 1 and y 2 and the Schauder fixed point theorem implies the existence of solutions in the case that g is bounded.
Set w 0 (t) = −v 0 (t) and the remainder of the verification is clear.

Remark 3 . 2 .
and g is increasing in y.Then there exists a solution of the boundary value problem (1.1)-(1.2).Remove the hypothesis that g is increasing in y and the Schauder fixed point theorem implies the existence of a solution of the shifted boundary value problem (1.1)-(1.

( 1 . 1 )-( 1 . 2 )Remark 3 . 6 .
has a solution.Remove the hypothesis that g is increasing in y and the Leray-Schauder alternative theorem implies the existence of a solution of the shifted boundary value problem (1.1)-(1.2).
implies the existence of a positive solution of the shifted boundary value problem (1.1)-(1.2) in the case that f (t, y) ≥ −β 2 y and f satisfies (1) and (2).