On the Solvability of Nonlinear Third-Order Two-Point Boundary Value Problems

: Under barrier strips type assumptions we study the existence of C 3 [ 0, 1 ] —solutions to various two-point boundary value problems for the equation x (cid:48)(cid:48)(cid:48) = f ( t , x , x (cid:48) , x (cid:48)(cid:48) ) . We give also some results guaranteeing positive or non-negative, monotone, convex or concave solutions.

As a rule, the main nonlinearity is defined and continuous on a set such that each dependent variable changes in a left-and/or a right-unbounded set; in Reference [13] it is a Carathéodory function on an unbounded set. Besides, the main nonlinearity is monotone with respect to some of the variables in References [1,5], does not change its sign in References [2][3][4]14] and satisfies Nagumo type growth conditions in Reference [14]. Maximum principles have been used in References [8,10], Green's functions in References [1,2,4,5], and upper and lower solutions in References [1,[7][8][9][10][11].
Here, we use a different tool-barrier strips which allow the right side of the equation to be defined and continuous on a bounded subset of its domain and to change its sign.
To prove our existence results we apply a basic existence theorem whose formulation requires the introduction of the BVP V i (x) = r i , i = 1, 2, 3(i = 1, 3 for short), with constants a ij and b ij such that ∑ 2 j=0 (a 2 ij + b 2 ij ) > 0, i = 1, 3, and r i ∈ R, i = 1, 3. Next, consider the family of BVPs for 1], and a, b, c are as above. Finally, BC denotes the set of functions satisfying boundary conditions (8), and BC 0 denotes the set of functions satisfying the homogeneous boundary conditions V i (x) = 0, i = 1, 3.
Besides, let C 3 The proofs of our existence results are based on the following theorem. It is a variant of Reference [12] (Chapter I, Theorem 5.1 and Chapter V, Theorem 1.2). Its proof can be found in Reference [15]; see also the similar result in Reference [16] (Theorem 4).
For us, the equation from (7) λ has the form Preparing the application of Lemma 1, we impose conditions which ensure the a priori bounds from (iv) for the eventual C 3 [0, 1] -solutions of the families of BVPs for (7) λ , λ ∈ [0, 1], with one of the boundary conditions (k), k = 2, 6.
So, we will say that for some of the BVPs (1), (k), k = 2, 6, the conditions (H 1 ) and (H 2 ) hold for a K ∈ R (it will be specified later for each problem) if: (H 2 ) There are constants F i , L i , i = 1, 2, such that Besides, we will say that for some of the BVPs (1), (k), k = 2, 6, the condition (H 3 ) holds for constants m i ≤ M i , i = 0, 2, (they also will be specified later for each problem) if: where J is as in (v) of Lemma 1, and σ > 0 is sufficiently small.
The auxiliary results which guarantee a priori bounds are given in Section 2, and the existence theorems are in Section 3. The ability to use (H 1 ) and (H 2 ) for studying the existence of solutions with important properties is shown in Appendix A. Examples are given in Section 4.
Proof. By contradiction, assume that x (t) > L 1 for some t ∈ [a, b). This means that the set (9) imply Along similar lines, assuming on the contrary that the set is not empty and using (10), we achieve a contradiction which implies that The proof of the next assertion is virtually the same as that of Lemma 2 and is omitted; it can be found in [15].
Let us recall, conditions of type (H 1 ) and (H 2 ) are called barrier strips, see P. Kelevedjiev [17]. As can we see from Lemmas Proof. Let first x(t) be a solution to (1) λ , (2). Using Lemma 2 we conclude that (11) is true. Then, according to the mean value theorem, for each t ∈ [0, 1) there is a ξ ∈ (t, 1) such that which together with (11) gives the bound for |x (t)|. Again from the mean value theorem for each t ∈ (0, 1] there is an η ∈ (0, t) with the property which yields the bound for |x(t)|. The assertion follows similarly for (1) λ , (3).

Existence Results
Theorem 1. Let (H 1 ) hold for K = C and (H 3 ) hold for Then each of BVPs (1), (2) and (1) Proof. We will establish that the assertion is true for problem (1), (2) after checking that the hypotheses of Lemma 1 are fulfilled; it follows similarly and for (1), (3). We easily check that (i) holds for (1) 0 , (2). Clearly, BVP (1), (2) is equivalent to BVP (1) 1 , (2) and so (ii) is satisfied. Since now L h = x , (iii) also holds. Next, according to Lemma 4, for each solution x ∈ C 3 [0, 1] to (1) λ , (2) we have Now use that f is continuous on [0, 1] × J to conclude that there are constants m 3 and M 3 such that which together with (x(t), x (t), x (t)) ∈ J for t ∈ [0, 1] and Equation (1) λ implies These observations imply that (iv) holds, too. Finally, the continuity of f on the set J gives (v) and so the assertion is true by Lemma 1.

Proof. It follows the lines of the proof of Theorem 1. Now the bounds
for each solution x ∈ C 3 [0, 1] to a (1) λ , (4) follow from Lemma 5.

Examples
Through several examples we will illustrate the application of the obtained results. Example 1. Consider the BVPs for the equation with boundary conditions (2) or (3).

Example 2. Consider the BVP
where ϕ : [0, 1] × R 2 → R is continuous and does not change its sign.
Author Contributions: All authors contributed equally. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
In this part we show how the barrier strips can be used for studying the existence of positive or non-negative, monotone, convex or concave C 3 [0, 1] -solutions. Here, we demonstrate this on problem (1), (4) but it can be done for the rest of the BVPs considered in this paper. Similar results for various other two-point boundary conditions can be found in R. Agarwal and P. Kelevedjiev [16] and P. Kelevedjiev and T. Todorov [15].