Electronic Journal of Qualitative Theory of Differential Equations

This article investigates nonlinear, second-order ordinary differential equations subject to various two-point boundary conditions. A condition is introduced that ensures a priori bounds on the derivatives of solutions to the problem. In particular, quadratic growth conditions on the right-hand side of the differential equation are not employed. The ideas are then applied to ensure the existence of at least one solution. The main tools involve differential inequalities and fixed-point methods.


Introduction
This paper considers the nonlinear, second-order differential equation coupled with any of the following boundary conditions: x (0) = 0 = x (T); (1.3) x(0) = x(T), x (0) = x (T). Our motivation for studying the above problems naturally arises in the following ways. Consider the following nonlinear partial differential equations ∆u = g(t, x, u t , u x ), (t, x) ∈ D, (nonlinear Laplace equation); (1.5) u xx − u t = g(t, x, u t , u x ), (t, x) ∈ D, (nonlinear heat equation); (1.6) where each equation is subjected to appropriate boundary conditions. The applications of (1.5), (1.6) are well known. If stationary solutions u(x, t) = u(x) to the above equations are sought then both (1.5) and (1.6) become the nonlinear, second-order ordinary differential equation As a particular example to illustrate the above point, consider (1.1), (1.2) with: a 1 = 0 = b 1 = b 2 ; a 2 = 1, a 3 arbitrary, a 4 = 1 and T = 1. These then are models for a thermostat. Solutions of these ODEs are stationary solutions for a (nonlinear) one-dimensional heat equation, corresponding to a heated bar, with a controller at t = 1 adding or removing heat dependent on the temperature detected by a sensor at t = 1. The particular boundary conditions (1.2) correspond to the end of the bar at t = 0 being insulated, see [26, pp. 672-3].
A further example of steady-state temperature distribution in rods that naturally involves (1.1), (1.2) with each a i > 0 can be found in [10, p. 79].
Recently, [23] presented a firm mathematical foundation for the boundary value problem associated with the nonrelativistic Thomas-Fermi equation for heavy atoms in intense magnetic fields. The analysis involved the BVP Another area of motivation for studying (1.1) is the appearance of ordinary differential equations in their own right, as opposed to simplifications of PDE. The reader is referred to [4, Chapter 1], [2, Chapter 1] for some nice examples, including applications of boundary value problems involving ordinary differential equations to physics, engineering and science.
These types of above applications naturally motivate a deeper theoretical study of the subject of BVPs involving ordinary differential equations.
In this work, the interest is in the existence of solutions to the BVP (1.1) subject to (1.2), (1.3) or (1.4). In particular, we are concerned with those f (t, p, q) that do not satisfy the standard growth conditions in q.
The tools used in this paper involve new differential inequalities, topological degree and related fixed-point methods of integral operators. One of the useful building blocks for using topological degree theory on operator equations, is the obtention of a priori bounds on solutions to a certain family of equations that are related to the equation that is under consideration. In the field of second-order, nonlinear BVPs this is equivalent to obtaining conditions under which certain a priori bounds are guaranteed on solutions x and its first derivative x .
The classical method of upper and lower solutions has been used to bound solutions x a priori where the ideas involve certain differential inequalities on the right-hand side of the differential equation and a simple maximum principle [7].
In this paper an alternate method for bounding x is introduced. The ideas do not follow any of the above approaches, in particular, no growth conditions of | f (t, p, q)| in |q| are used.
The new results compliment and extend previous works in the literature and an example is provided to which the new results apply but the classical growth conditions do not.
The article is organised as follows. Section 2 contains the new a priori bound results for first derivatives of solutions to (1.1) subject to (1.2), (1.3) or (1.4). The conditions of the theorems feature simple, wide-ranging differential inequalities that are easily verifiable in practice.
In Section 3 the ideas of Section 2 are applied, in conjunction with topological degree and fixed-point theory, to gain new theorems that ensure the existence of solutions to (1.1) subject to (1.2), (1.3) or (1.4). Section 4 contains an example that demonstrates how to apply the new results and to clearly demonstrate the advancement made over current literature. In particular, an example is constructed so that the classical Bernstein-Nagumo growth conditions do not apply.

A priori bounds
In this section, some new a priori bound results are presented involving the first derivative of solutions to (1.  Proof. If α = 0 then −K ≤ x (t) ≤ K for all t ∈ [0, T] and an integration on [0, t] leads to |x (t)| ≤ KT + N 1 , for all t ∈ [0, T].
If α > 0 then we have 0 ≤ αx (t) + K for all t ∈ [0, T] with an integration over [0, t] and [t, T] giving, respectively Thus the desired bound on x follows by defining N as In a similar fashion to Lemma 2.1 we have the following result.
Then the conclusion of Lemma 2.1 holds.
continuous, let R be a non-negative constant and assume each a i > 0. If there exist non-negative constants α and K such that where N is a constant involving: α, K, T, R, each a i and each |b i |.
so that (2.2) holds. Hence the desired a priori bound on x follows from Lemma 2.1.
The following a priori bound result may be obtained for a different class of f than those dealt with in Theorem 2.3.
(2.5) then the conclusion of Theorem 2.3 holds.
Proof. The proof is similar to that of Theorem 2.3 and so is omitted.
Example 2.5. Comparing Theorems 2.3 and 2.4, we see that satisfies (2.5) for α = 1, K = 8, R = 1; but f 1 cannot satisfy (2.4) for any choice of non-negative α, K and positive R. Conversely, see that satisfies (2.4) for α = 1, K = 5, R = 1; but f 2 cannot satisfy (2.5) for any choice of non-negative α, K and positive R. Similarly to Theorem 2.6, the following result may be obtained. Theorem 2.7. If the conditions of Theorem 2.6 hold with "(2.4)" replaced with "(2.5)" then the conclusion of Theorem 2.6 holds.
Our focus is now on a priori bounds on derivatives of solutions to the periodic BVP (1.1), (1.4). It is difficult to immediately verify that (1.4) implies (2.2) for this case so we adopt an alternative approach.
Consider the following BVP that is equivalent to (1.1), (1.4): We can equivalently rewrite the BVP (2.6), (2.7) in the following integral form ∂G ∂t (t, s) . We will need this constant in the theorems that follow. Proof. Let x be a solution to the BVP (1.1), (1.4) that satisfies |x(t)| ≤ R for all t ∈ [0, T]. Note that x also satisfies the BVP (2.6), (2.7) and the integral equation (2.8). If we differentiate both sides of (2.8) then x must satisfy, for each t ∈ [0, T] and |x(t)| ≤ R we have Hence the desired a priori bound on x follows.
Similarly, the following result may be obtained. Theorem 2.9. Let the conditions of Theorem 2.8 hold with "(2.4)" replaced with "(2.5)". Then the conclusion of Theorem 2.8 holds.
Some important corollaries to the theorems in this section now follow, where we assume, respectively, that f (t, p, q) is bounded below or bounded above for all t ∈ [0, T], |p| ≤ R, q ∈ R. Proof. We want to show that there exists non-negative constants α and K such that (2.4) holds. If f (t, p, q) is bounded below for all t ∈ [0, T], |p| ≤ R, q ∈ R then there exists a constant C such that Thus (2.4) holds with α = 1 and K = −2C.
Similarly, the following result can be obtained. Proof. The proof is similar to that of Corollary 2.10 and thus is omitted.
The following lemmas will be a useful tool in gaining a priori bounds on solutions x to our BVPs. In particular, they will be needed in our existence proofs in Section 3. The proofs of the following results are well known and use standard maximum principle techinques from the theory of lower and upper solutions [7,9]. Thus the proofs are omitted for brevity.
For convenience, it has been assumed that each a i > 0 in (1.2). This can naturally be relaxed to a 2 1 + a 2 2 > 0 and a 2 3 + a 2 4 > 0 with one of a 1 or a 3 being zero. In particular, this allows the treatment of BVPs involving, respectively, Nicoletti and Corduneanu boundary conditions by suitably modifying the conditions of the theorems in this section and their associated proofs.

Solvability
In this section the results of Section 2 are applied, in conjunction with topological degree and fixed-point theory, to obtain some new existence theorems for solutions to (1. Proof. Since each a i > 0 and since f is continuous, we can equivalently rewrite the BVP (1.1), (1.2) in the following integral form and φ : [0, T] → R is the unique, continuously differentiable solution to the BVP In view of (3.1) and its context, we define H : It is well known that H is a compact map. If we can show that H has at least one fixed-point, that is, Hx = x for at least one x, then (3.1) will have at least one solution and so will the BVP (1.1), (1.2).
With this in mind, consider the family of equations where N is defined in Lemma 2.1 (with N 1 defined in the proof of Theorem 2.3). Let x be a solution to (3.3) and see that x is also a solution to the family of BVPs where λ ∈ [0, 1]. We will show that all possible solutions to (3.3) that satisfy x ∈ Ω also satisfy x / ∈ ∂Ω. This is equivalent to showing that these solutions to the family of BVPs (3.4)-(3.6) satisfy a particular a priori bound, with the bound being independent of λ.
If λ = 0 then we have the zero solution to (3.4)-(3.6), so assume λ ∈ (0, 1] from now on. If (2.9)-(2.11) and (2.4) hold then for λ ∈ (0, 1] it is not difficult to show that the family of BVPs (3.4)-(3.6) satisfy the conditions of Lemma 2.12. Thus, all solutions to (3.4)-(3.6) that In addition, (3.4)-(3.6) satisfies the conditions of Theorem 2.3 with Hence |x (t)| ≤ N for all t ∈ [0, T]. Above, note that R 1 , R 2 and N are all independent of λ. Thus, if I denotes the identity, the following Leray-Schauder degrees are defined and a homotopy principle applies [17,Chap.4]   Proof. Consider the following BVP that is equivalent to (1.1), (1.3): x (0) = x (T). Above, G : [0, T] × [0, T] → R is the unique, continuously differentiable Green's function for the following BVP In view of (3.9) and its context, we define H : It is well known that H is a compact map. If we can show that H has at least one fixed-point, that is, Hx = x for at least one x, then (3.9) will have at least one solution and so will the BVP (1.1), (1.3).
With this in mind, consider the family of equations Let x be a solution to (3.11) and observe that x is also a solution to the family of BVPs x (0) = x (T); (3.13) where λ ∈ [0, 1]. We show that these solutions to the family of BVPs (3.12), (3.13) satisfy a particular a priori bound, with the bound being independent of λ.
If λ = 0 then we have the zero solution to (3.12), (3.13), so assume λ ∈ (0, 1] from now on. If (2.9), (2.10) and (2.4) hold then for λ ∈ (0, 1] it is not difficult to show that the family of BVPs (3.12), (3.13) satisfy the conditions of Lemma 2.13. Thus, all solutions to (3.12), (3.13) that satisfy −R 1 ≤ x(t) ≤ R 2 for all t ∈ [0, T] must also satisfy −R 1 < x(t) < R 2 for all t ∈ [0, T]. Note that R 1 and R 2 are independent of λ. Now, if (2.4) holds, then for all λ ∈ [0, 1], t ∈ [0, T] and |p| ≤ R we have and so |λ f (t, p, q) and hence Therefore it follows |g λ (t, p, q)| ≤ αg λ (t, p, q) + K + 2R for all t ∈ [0, T], |p| ≤ R, q ∈ R and all λ ∈ [0, 1] so that Theorem 2.6 applies to the family (3.12), (3.13) with N 1 = 0, N being independent of λ and specifically given by We have shown that all possible solutions to (3.11) that satisfy x ∈ Ω also satisfy x / ∈ ∂Ω. Thus, if I denotes the identity, then the following Leray-Schauder degrees are defined and a homotopy principle applies [17,Chapter 4]   Proof. The proof is similar to that of Theorem 3.2 and so is only sketched. Consider the following BVP that is equivalent to (1.1), (1.4): 14) x(0) = x(T), Since f is continuous, we can equivalently rewrite the BVP (3.14), (3.15) in the following integral form It is well known that H is a compact map. If we can show that H has at least one fixed-point, that is, Hx = x for at least one x, then (3.16) will have at least one solution and so will the BVP (1.1), (1.4). Apply Theorem 2.8 and Lemma 2.14 to the family of BVPs x(0) = x(T), x (0) = x (T); (3.19) where λ ∈ [0, 1] and then use degree theory in the same way as proofs of our previous results.
An simple but useful corollary to Theorems 3.1, 3.2, and 3.3 now follows. The proof follows that of the respective theorems by respectively applying Corollary 2.10 to an appropriate family of BVPs. The proof is omitted for brevity.
A more abstract generalization of Theorems 3.1 now follows in which we replace the assumptions involving upper and lower solutions with the a priori knowledge of a certain bound on solutions x to appropriate families of BVPs.  Similarly, the next two abstract results follow as natural corollaries to Theorem 3.5.  It is clear that the above results for the Sturm-Liouville BVP may be also modified to treat the Neumann and periodic BVPs, but for brevity the statements of these results are omitted. In addition, it is possible to gain existence of solutions for the Nicoletti BVP (1.1), (2.12) and the Corduneanu BVP (1.1), (2.13) under suitably modified assumptions connected with the ideas of this section and Section 2.

An example
In this final section, a simple example is considered to which our new theorems are applicable. In particular, the classical Bernstein-Nagumo theory is inapplicable to what follows. For the above f , the conditions Theorem 3.1 hold.
Thus (2.4) holds for any choice of R > 0. We will choose R = 3. Note that (2.9) and (2.10) hold for the choices R 1 = 1/2 and R 2 = 3. Thus, for suitable a i and b i , the BVP (1.1), (1.2) with the above f admits at least one solution from Theorem 3.1.
Thus the classical Bernstein-Nagumo quadratic growth condition does not apply to the above example.