Existence of solutions of nonlinear third-order two-point boundary value problems

We study various two-point boundary value problems for the equation x′′′ = f (t, x, x′, x′′). Using barrier strips type conditions, we give sufficient conditions guaranteeing positive or non-negative, monotone, convex or concave C3[0, 1]-solutions.


Introduction
In this paper, we are concerned with boundary value problems (BVPs) for the differential equation x = f (t, x, x , x ), t ∈ (0, 1), ( with boundary conditions either or where f : [0, 1] × D x × D p × D q → R, and D x , D p , D q ⊆ R.
We study the existence of C 3 [0, 1]-solutions to the above problems which do not change their sign, are monotone and do not change their curvature.
Third-order differential equations arise in a large number of physical and technological processes, see, for example, M. Aïboudi and B. Brighi [1], J. R. Graef et al. [9], Z. Zhang [33] for facts and references.Recently, various third-order BVPs have received much attention and a lot of research has been done in this area.Here, we cite sources devoted to two-point BVPs.
We do not use the above tools.The imposed condition in this paper allows the main nonlinearity to be defined on a bounded set, to be continuous on a suitable subset of its domain and to change its sign.So, our results rely on the following hypotheses.
(H 1 ) There are constants F i , L i , i = 1, 2, and a sufficiently small σ > 0 such that Besides, we will say that for some of the BVPs (1.1),(1.k),k = 2, 3, 4, 5, 6 (k = 2, 6 for short), the condition (H 2 ) holds for constants m i ≤ M i , i = 0, 2, (these constants will be specified later for each problem) if: where σ is as in (H 1 ), and f (t, x, p, q) is continuous on Such type of conditions have been used for studying the solvability of various problems for first and second order differential equations, see P. Kelevedjiev and N. Popivanov [14] and R. Ma et al. [16] for results and references.Here we adapt this approach for the considered problems developing ideas partially announced in P. Kelevedjiev et al. [15] on the BVP (1.1), (1.8).(H 1 ) ensures priori bounds for x (t), x (t) and x(t), in this order, for each eventual solution x(t) ∈ C 3 [0, 1] to the families of BVPs for

Global existence theorem
Let E be a Banach space, Y be its convex subset, and with the property G/∂U = F/∂U has a fixed point in U. Clearly, every essential map has a fixed point in U.
, has at least one fixed point in U and in particular there is a x 0 ∈ U such that x 0 = F(x 0 ).
Besides, for λ ∈ [0, 1] consider the family of BVPs for , and a, b, c are as above.
Finally, let BC be the set of functions satisfying boundary conditions (2.2), ∩ BC, BC 0 be the set of functions satisfying the homogeneous boundary conditions We are now ready to state our basic existence result which is a variant of [10, Chapter I, Theorem 5.1 and Chapter V, Theorem 1.2].
Proof.For a start, introduce the set and define the maps and for λ ∈ [0, 1] Our first task is to establish that exists and is continuous.Therefore, we use (iii) which implies that for each y ∈ C[0, 1] the BVP where x i (t), i = 1, 2, 3, are linearly independent solutions to the homogeneous equation η(t) is a solution to the inhomogeneous equation, and The last means that det[V i (x j )] = 0 and so the system is the unique C 3 [0, 1]-solution to the homogeneous equation (2.3) satisfying the inhomogeneous boundary conditions As a result, conclude that L −1 exists and where h is continuous and so L −1 is also continuous.Now, introduce the homotopy The map j is a completely continuous embedding and U is a bounded set, hence the set j(U) is compact.The set Φ λ (j(U)), λ ∈ [0, 1], is also compact since the map Φ λ is continuous on j(U) in view of (v).Finally, because of the continuity of L −1 proved above, the set

Auxiliary results
The results stated in this part guarantee the bounds from (iv) of Theorem 2.3.
On the other hand, since x(t) is a C 3 [0, 1]-solution to (1.1) λ , we have in particular and (1.9) it follows x (γ) ≤ 0, a contradiction.Thus, In an analogous way, using (1.10), we can prove that Proof.Let firstly the solution satisfies x (0) = B.Then, by the mean value theorem, for each t ∈ (0, 1] there is a ξ ∈ (0, t) such that from where, using Lemma 3.1, derive (3.1).If x (1) = B, we obtain similarly that for each t ∈ [0, 1) there is a η ∈ (t, 1) with the property which implies (3.1).Using again the mean value theorem and (3.1), we get the bound for |x(t)| in both cases x(1) = C and x(0 Proof.From Lemma 3.1 we know that Then, for t ∈ (0, 1] we get and from where (3.2) follows.Similarly, integrating (3.2) from t ∈ [0, 1) to 1 we get which implies the bounds for x(t).
Using similar arguments to those in the proof of Lemma 3.3, we can also show that the following three auxiliary results are held.
Proof.It is clear, there is a µ ∈ (0, 1) with the property x (µ) = C − B. Then, for each t ∈ [0, µ) there is a ξ ∈ (t, µ) such that Similarly establish that the same bound is valid for t ∈ [µ, 1].Using again the mean value theorem, we obtain that for each t ∈ (0, 1] and some η ∈ (0, t) we have and As a result, . This fact together with B, C ≥ 0 means that x(t) ≥ min{B, C} on [0, 1], which completes the proof.
Proof.We will show that each BVP for (1.1) λ , λ ∈ [0, 1], with one of the boundary conditions (1.k), k = 2, 5, satisfies all hypotheses of Theorem 2.3.It is not hard to check that (i) holds for each BVP for (1.1) 0 with one of the boundary conditions (1.k), k = 2, 5. Obviously, each BVP for (1.1) is equivalent to the BVP for (1.1) 1 with the same boundary conditions, that is, (ii) is satisfied.Because now L h = x , (iii) also holds.Further, for each solution x(t) ∈ C 3 [0, 1] to a BVP for (1.1) λ , λ ∈ [0, 1], with one of the boundary conditions (1.k), k = 2, 5, we have Because of the continuity of f on [0, 1] × J there are constants m 3 and M 3 such that Hence, (iv) also holds.Finally, (v) follows from the continuity of f on the set J. So, we can apply Theorem 2.3 to conclude that the assertion is true.
The following results guarantee C 3 [0, 1]-solutions with important properties.
We will illustrate the application of the obtained results.
Consider the case P n (q) < 0 for q ∈ (q 1 , q 1 + θ] and P n (q) > 0 for q ∈ [q 2 − θ, q 2 ); the other cases for the sign of P n (q) around the zeros can be studied by analogy.In this case, if we choose, for example, and (H 2 ) hold and so each BVP for (4.1) with one of the boundary conditions ( It is not hard to see that if, for example, The assumptions of Theorem 4.5 are satisfied for F 2 = −7, F 1 = −6, L 1 = −2, L 2 = −1 and σ = 0.1, for example.Thus, the considered problem has a positive, increasing, concave solution in C 3 [0, 1] by Theorem 4. Then Then BVP (1.1), (1.6) has at least one positive (non-negative), concave solution in C 3 [0, 1].

Theorem 2 . 1 (
[10,  Chapter I, Theorem 2.2]).Let p ∈ U be fixed and F ∈ L ∂U (U, Y) be the constant map F(x) = p for x ∈ U. Then F is essential.
the homotopy is compact.For its fixed points we have x = L −1 Φλjx and Lx = Φλjx which means that the fixed points of H λ are precisely the solutions of family (2.1) λ , (2.2) and in view of (iv) we conclude that the homotopy is fixed point free on the boundary of U. Using (i), we see that H 0 = x 0 , x 0 ∈ U, is essential by Theorem 2.1.Then, H 1 is also essential by Theorem 2.2 and so it has a fixed point, that is, (2.1) λ , (2.2) has a solution in C 3 [0, 1] when λ = 1, and, by (ii), problem (2.1), (2.2) has a solution in C 3 [0, 1].