Positive solutions of second order boundary value problems

In this paper we investigate the existence of positive solutions of two-point boundary value problems for nonlinear second order differential equations of the


Introduction
We are interested in the existence of positive solutions of the two-point boundary value problem, (py ) (t) + q(t)y(t) = f (t, y(t), y (t)), 0 < t < 1, y(0) = y(1) = 0. (1.1) Problems of this type arise naturally in the description of physical phenomena, where only positive solutions, that is, solutions y satisfying y(t) > 0 for all t ∈ (0, 1), are meaningful.It is well known that Krasnoselskii's fixed point theorem in a cone has been instrumental in proving existence of positive solutions of problem (1.1).Most of the previous works deal with the case p(t) = 1, q(t) = 0, for all t ∈ [0, 1], and assume that f is nonnegative, that f does not depend on y , and that f : [0, 1] × [0, +∞) → [0, +∞) is continuous and satisfies, either lim inf See for instance [1], [7], [12], [13], [16] and the references therein.The above conditions have been relaxed in [17] and [18], where the authors remove the condition f is nonnegative, and they consider the behavior of f relative to π 2 .Notice that π 2 is the first eigenvalue of the operator u → −u , subject to the boundary conditions in (1.1).The arguments in [17] and [18] are based on fixed point index theory in a cone.When the nonlinear term depends also on the first derivative of y, we refer the interested reader to [2], [3], [19].The authors in [2], [8] and [20] deal with a singular problem.Several papers are concerned with the problem of the existence of multiple solutions.See for instance [3], [13], [14], [15] and [21].However, our assumptions are simple and more general.In fact, we obtain a multiplicity result as a byproduct of our main result, with no extra assumptions.We exploit the fact that the nonlinearity changes sign with respect to its second argument.We do not rely on cone preserving mappings.Also, the sign of the Green's function of the corresponding linear homogeneous problem plays no role in our study.However, we assume the existence of positive lower and upper solutions.We provide an example to motivate our assumptions.Our results complement and generalize those obtained in [21].

Topological Transversality Theory
In this section, we recall the most important notions and results related to the topological transversality theory due to Granas.See Granas-Dugundji [10] for the details of the theory.Let X be a Banach space, C a convex subset of X and U an open set in C.

A Linear Problem
Consider the following linear boundary value problem where the coefficient functions p and q satisfy (H0) p ∈ C 1 (I), q ∈ C(I), p(t) ≥ p 0 > 0 for all t ∈ I, q(t) ≤ p 0 π 2 with strict inequality on a subset of I with positive measure.Lemma 3.1 Assume that (H0) is satisfied.Then for any nontrivial y ∈ C 1 0 (I), we have Consider the functional χ : C 1 0 (I) → R defined by Results from the classical calculus of variations (see [6]) shows that χ(y) ≥ 0 for all y ∈ C 1 0 (I).Hence, the conclusion of the Lemma holds.
Proof.Assume on the contrary that the problem has a nontrivial solution y 0 .Then, we have This contradiction shows that y 0 = 0, and the proof is complete.
It follows from the above that L is one-to-one and onto, and By the compactness of the embedding W 2,1 (I) → C 1 (I), the operator L −1 maps L 1 (I) into C 1 0 (I) and is compact.

Carathéodory function and satisfies
(H1) There exist positive functions α ≤ β in C 1 0 (I) such that for all t ∈ I, Remarks.
We have two possibilities.(i) I 1 ⊂ I. Then z(a) = 0, z(b) = 0 and z(t) > 0 for all t ∈ (a, b) .Hence, by (H1), for all t ∈ (a, b) On the other hand, Lemma 3.2 implies that b a Lz(t)z(t)dt < 0. This is a contradiction.
Therefore, we conclude y(t) ≤ β(t) for all t ∈ I.
Step.2.A priori bound on the derivative y for solutions y of (1.2 λ ) satisfying the inequality (1.4).
Define K 1 > 0 by the formula We want to show that |y (t)| ≤ K 1 for all t ∈ I. Suppose, on the contrary that there exists τ 1 such that |y (τ 1 )| > K 1 .Then, there exists an interval [µ, ξ] ⊂ [0, 1] EJQTDE, 2006 No. 7, p. 7 such that the following situations occur: We study the first case.The others can be handled in a similar way.For (i), the differential equation in (1.2 λ ) and condition (H2) imply Since we have It follows from (1.5) that Notice that Ψ(z) ≥ 1 for all z ≥ 0, so that This implies Integration from µ to ξ, and a change of variables (see [8,Lemma A.10]) lead to EJQTDE, 2006 No. 7, p. 8 This clearly contradicts the definition of K 1 .
Taking into consideration all the four cases above, we see that Then, any solution y of (1.2 λ ) satisfying the inequality (1.4), is such that its first derivative y will satisfy the a priori bound As a consequence of Step 1 and Step 2 above, we deduce that any solution y of (1.2 λ ) satisfies Since f is an L 1 -Caratheodory function, it follows from (1.6) that there exists h K ∈ L 1 (I : R + ) such that |f (t, y(t), y (t))| ≤ h K (t), for almost every t ∈ I. Now, the differential equation in (1.1) implies there exists φ ∈ L 1 (I : R + ), depending on only p, ||p || 0 , ||q|| 0 , h K such that y (t) ≤ φ(t) for almost every t ∈ I.In particular, y ∈ L 1 (I : R + ).
If G(t, s) is the Green's function corresponding to the linear homogeneous problem (py ) (t) + q(t)y(t) = 0, y(0) = 0 = y (1) , then problem (1.2 λ ) is equivalent to Let be defined by, for all t ∈ I, Therefore, H(λ, •) : U → C 1 0 (I) is an admissible homotopy between the constant map H(0, •) = 0 and the compact map H(1, •).Since 0 ∈ U, we have that H(0, •) is essential.By the topological transversality theorem of Granas, H(1, •) is essential.This implies that it has a fixed point in U, and this fixed point is a solution of (1.2 1 ).Since solutions of (1.2 1 ) are solutions of (1.1) we conclude that (1.1) has at least one solution, which is necessarily positive because of (1.4).
This completes the proof of the main result.
Remark.It is possible to obtain a uniqueness result if we assume, in addition to (H1) and (H2), the following condition: (H3) There exists a constant M, such that q(t) + M ≤ q 0 π 2 , with strict inequality on a subset of I with positive measure, and f (t, y 1 , z) − f (t, y 2 , z) ≥ −M(y 1 − y 2 ) for all t ∈ I, z ∈ R and α ≤ y 2 ≤ y 1 ≤ β.Assume first that u(t) ≤ v(t).Then w(t) := v(t) − u(t) ≥ 0 for all t ∈ I.
We know of no applicable previous published works.However, f satisfies condition (H4) of Theorem 5.1, hence Problem (1.9) has infinitely many positive solutions.
Let I denote the real interval [0, 1], and let R + denote the set of all nonnegative real numbers.For k = 0, 1, . . ., C k (I) denotes the space of all functions u : I → R, whose derivatives up to order k are continuous.For u ∈ C k (I) we define its norm by u = k i=0 {max u (i) (t) : t ∈ I}.Equipped with this norm C k (I) is a Banach space.When k = 0, we shall use the notation u 0 for the norm of u ∈ C(I).Also C 1 0 (I) shall denote the space {y ∈ C 1 (I) : y(0) = y(1) = 0}.It can be easily shown that (C 1 0 (I), • ) is a Banach space.A real valued function f defined on I × R 2 is said to be an L 1 −Caratheodory function if it satisfies the following conditions EJQTDE, 2006 No. 7, p. 3
called admissible if g is compact and has no fixed points on Γ = ∂U.Γ (U , C) is called inessential if there is a fixed point free compact map h : U → C such that g| Γ = h| Γ .Otherwise, g is called essential Lemma 2.1 Let d be an arbitrary point in U and g ∈ M Γ (U , C) be the constant map g(x) ≡ d.Then g is essential.
Since L −1 is compact andF 1 is continuous it follows that H(λ, •) is a compact operator.It is easily seen that H(•, •) is uniformly continuous in λ.It is clear from Steps 1 and 2 above and the choice of U that there is no y ∈ ∂U such that H(λ, y) = y for λ ∈ [0, 1].This shows that H(λ, •) : U → C 1 0 (I) is an admissible homotopy; i.e. a compact homotopy without fixed points on ∂U, the boundary of U.