A Priori Bounds and Existence Results for Singular Boundary Value Problems

This article furnishes qualitative properties for solutions to two-point boundary value problems (BVPs) which are systems of singular, second-order, nonlinear ordinary differential equations. The right-hand side of the differential equation is allowed to be unrestricted in the growth of its variables and may depend on the derivative of the solution, which incurs additional difficultly in the mathematical proofs. A new approach is introduced by using a singular differential inequality that ensures that all possible solutions satisfy certain a priori bounds, including their " derivatives " , to the singular BVP under consideration. Topological methods, in particular Schauder's fixed point theorem are then applied to generate new existence results for solutions to the singular boundary value problems. Many of the results are novel for both the singular and the nonsingular cases.


Introduction
This work focuses on achieving novel a priori bounds and existence results to systems of nonlinear, singular, second order boundary value problems (BVPs) given by: 1 p(t) (p(t)y (t)) = q(t)f(t, y(t), p(t)y (t)), 0 < t < T; (1.1) with various forms of the boundary conditions −αy(0) + β lim t→0 + p(t)y (t) = c; (1.2) 3) The study of these singular BVPs are partially motivated by their application in modelling a variety of physical phenomena, some examples can be found in [11,22].The strategy used to achieve a priori bounds herein is to consider the BVP as an integral representation and use novel differential inequalities to yield a priori bounds on solutions.To prove the existence of solutions, the application of Schauder's fixed point theorem [24] is utilised once these bounds are known.
One of the contributions of this work is that it removes the widely used, Nagumo condition [20] in obtaining a priori bounds of solutions to derivative dependent nonlinear systems of singular BVPs.The use of the Nagumo condition has been studied extensively by [4,6,7,12,18,26].In a recent paper by Fewster-Young [9], the singular vector-valued version of the Nagumo condition is introduced, that is: there is a positive continuous function φ However, it is not too hard to produce an example where (1.6) does not hold like the next problem.
Example 1.1.Consider the singular BVP for 0 < t < 1, In this BVP, the functions p(t) = t 1/2 , q(t) = 1/t and Now, by considering (1.6) for this example, the function φ does not exist when u ≤ R, t ∈ [0, 1] since Also, Fewster-Young & Tisdell [10] used a Hartman inequality [15], namely where V, W are non-negative constants to produce a priori bounds and existence results for singular BVPs.They showed for singular BVPs that the Nagumo condition could be replaced by the assumption introduced by Hartman [15]: 2VR < 1 where R is a non-negative constant and all solutions satisfy max t∈[0,T] y(t) ≤ R.However, the condition 2VR < 1 is a difficult assumption to satisfy since the constant R depends on V and can be very large.The first result builds off this idea and relaxes this condition to 2V d < 1, a very manageable assumption for applications.In addition, Hartman [15] only deals with Dirichlet boundary conditions where herein a Sturm-Liouville and a Dirichlet boundary condition are dealt with.
The final new result increases the freedom of the differential inequality by stating it in a general form using Lyapunov functions.This means the growth of the function f can be unrestricted and leads to many more examples able to be discussed.In the example below, the inequality (1.9) is not satisfied.In addition, in the scalar case, Bobisud [4] introduced a sufficient condition on the relationship between the functions p, q to be p 2 q ≤ 1 on [0, T].This new result removes this condition, thus expanding the possible scenarios.For instance, the following scalar example fails both conditions and will be used to illustrate the new result.
Example 1.2.Consider the singular BVP: In this BVP, the functions p(t) = t 1/4 , q(t) = 1/t and f (t, u, v) = t 1/2 v 2 + u 3 .Notice that the condition p 2 q ≤ 1 on [0, T] does not hold.In addition, suppose that g is a positive function such that g(u) ≥ −u for all u ∈ R. See that the inequality (1.9) does not necessarily hold because 2 and the term: g(u)v 2 can not be bounded for all u, v ∈ R.

Main results
In this section, the main results are presented and the approach is to provide the a priori bounds on all possible solutions to (1.1)-(1.3)first then to follow with the existence of at least one solution to (1.1)-(1.3).The first result was proved by Fewster-Young & Tisdell [10] and provides an a priori bound on solutions to (1.1)-(1.3).It can be stated as follows.(1.4), (1.5) hold and let then all solutions y = y(t) to the singular BVP (1.1), (1.2), (1.3) satisfy where Proof.Let y = y(t) be any solution to the singular BVP (1.1)-(1.3).From [22, p. 14], the BVP (1.1)-( 1.3) has the equivalent integral equation given by for all t ∈ [0, T] and where 2 where y is a solution to (1.1) then for all t ∈ (0, T): we have The condition p 2 q ≤ 1 on [0, T], the integral conditions (2.1) and (1.1) yields If we apply the Schwarz inequality [14] to the last term then Consider the inequality: We now claim that The essence of the procedure is to show lim t→T − p(t)y (t) is bounded.Consider the integral equation: Next, consider the equation (2.10) and apply (2.2) to see that Since r(t) = y(t) 2 and r (t) = 2 y(t), y (t) , this gives lim By applying the Schwarz inequality, we have lim By using the condition, 2V d < 1, we can rearrange this inequality to yield an a priori bound: Substituting (2.6) and integrating yields By employing the boundary condition (1.2), this results in If we apply the Schwarz inequality and employ our bounds on the terms then this produces: So far, we have achieved bounds on η and κ.Now, if we estimate (2.9) then the bounds on κ and η imply sup By combining the two previous results, the following result now proves the existence of solutions to the BVP (1.1)-(1.3).

Theorem 2.3. If the conditions of Theorem 2.2 are satisfied then exists at least one solution to the singular BVP (1.1)-(1.3).
Proof.Define the norm Consider the Banach space: p(x)q(x)f(x, y(x), p(x)y (x)) dx ds, for t ∈ [0, T] and where A is given in (2.4).Consequently the solutions to (1.1)-(1.3)are the fixed points of the operator T. The aim now is to use the Schauder fixed point theorem [24] to prove the existence of fixed points of T. This requires T to be a continuous compact mapping from U to U. Now, the integral conditions (2.1) and f is continuous on X implies that T is a continuous mapping.To prove that T is compact, see that for any bounded set V ⊂ X, T maps V to a bounded set in X by the assumptions (2.1) and f is continuous.Furthermore, this means there is a positive constant M such that max Furthermore, consider y ∈ U, r, t ∈ [0, T] where t ≥ r; so this gives Also, this gives p(t)(Ty) (t) − p(t)(Ty) (r) ≤ t r p(x)q(x) f(x, y(x), p(x)y (x)) dx ds.
The condition (2.1) implies that the functions where > 0. Thus, it follows that Ty(t) − Ty(r) 1 ≤ , whenever |t − r| ≤ δ and so T is equicontinuous.Consequently, the Arzelà-Ascoli theorem [3] implies that T : U → X is a continuous compact mapping.We now show that for any y ∈ U, Ty ∈ U, that is T(U) ⊂ U.The assumptions of Theorem 2.2 hold, this means by the proof of that Theorem 2.2 that By estimating Ty 1 , we obtain sup By using the bounds on κ, η, it follows that This means for any y ∈ U, Ty ∈ U.By applying Schauder's fixed point theorem, the operator T has at least one fixed point.Moreover, the integral representation implies that y ∈ C 2 ((0, T); R n ), py ∈ C([0, T]; R n ) and y satisfies the boundary conditions (1.2), (1.3).This proves that there is at least one solution to the BVP (1.1)-(1.3).
The Example 1.1 is next examined by applying the previous results to show the existence of at least one solution to the singular BVP (1.7), (1.8).
Example 2.4.Consider the singular BVP: (1.7), (1.8).Notice that the functions here are p(t) = t 1/2 , q(t) = 1/t and See that the conditions relating to the functions p, q are satisfied since In next part of the proof, the following inequalities are used: The next condition to check is the inequality (2.2), if we choose V = 1/2 and W = 3/4 then In this example, d = 0, so the condition 2V d < 1 is satisfied and finally by applying Theorem 2.3 there exists at least one solution to the singular BVP (1.7), (1.8) The next result removes the condition, p 2 q ≤ 1 on [0, T] and generalises the differential inequality (2.2) in the previous result by using a general Lyapunov function.The Lyapunov function is of two variables t and u := y(t) 2  2 )) for all t ∈ (0, T). ( where L := lim t→T − p(t)r t (t, y(t) 2 ).
Proof.Let y = y(t) be any solution to (1.1)-(1.3),which has the integral representation given by (2.3).We now wish to estimate the terms κ and η as in Theorem 2.
The condition (2.12) is satisfied with B = 0 since by substituting the boundary condition, Thus, all the conditions of Theorem 2.5 are satisfied and so Theorem 2.7 implies there exists at least one solution to the singular BVP 1.

u 1 and
the convex set in X, U := {y ∈ X : y 1 ≤ max{R, S}}.Define the operator T : U → X by Ty := d − A