Existence for Nonlinear Fourth-Order Two-Point Boundary Value Problems

: The present paper is devoted to the solvability of various two-point boundary value problems for the equation y ( 4 ) = f ( t , y , y (cid:48) , y (cid:48)(cid:48) , y (cid:48)(cid:48)(cid:48) ) , where the nonlinearity f may be deﬁned on a bounded set and is needed to be continuous on a suitable subset of its domain. The established existence results guarantee not just a solution to the considered boundary value problems but also guarantee the existence of monotone solutions with suitable signs and curvature. The obtained results rely on a basic existence theorem, which is a variant of a theorem due to A. Granas, R. Guenther and J. Lee. The a priori bounds necessary for the application of the basic theorem are provided by the barrier strip technique. The existence results are illustrated with examples.


Introduction
This paper studies the solvability in C 4 [0, 1] of boundary value problems (BVPs) for the equation y (4) = f (t, y, y , y , y ), t ∈ (0, 1), where f (t, y, u, v, w) is a scalar function defined on [0, 1] × D y × D u × D v × D w , and D y , D u , D v , D w ⊆ R.
Results guaranteeing positive solutions can be found in [2,3,[5][6][7][8][9][10]12,14,18,20,21]. In [9], the most recent of these articles, the following nonlinear fourth-order two-point boundary value problem y (4) = f (t, y ), t ∈ (0, 1), y(0) = y (0) = y (1) = y (1) + g(y(1)) = 0, is considered under the assumption that there exist two suitable real numbers b > a ≥ 0 and a non-negative function l ∈ C(0, 1) ∩ L 1 [0, 1] such that the function f : (0, 1) × [0, b] → R is continuous and | f (t, y)| ≤ l(t) for (t, x) ∈ (0, 1) × [0, b], and the function g : [0, b] → (0, ∞) is continuous and increasing. The authors establish that this problem has two nontrivial solutions x * , y * ∈ C[0, 1] with at 2 ≤ x * ≤ y * ≤ bt 2 , t ∈ [0, 1], which are limits of sequences with first terms x 0 (t) = at 2 and y 0 (t) = bt 2 , respectively. A classic tool for studying the solvability of initial and boundary value problems is the lower and upper solutions technique. It was probably E. Picard [36], in 1893, who first used an initial version of this technique to study a first-order initial boundary value problem. This idea was further developed later by G. Scorca Dragoni [37]. The lower and upper solutions technique is often used together with so-called growth conditions imposed on the main nonlinearity of the differential equation. S. Bernstein [38], in 1912, first used such a condition to establish the solvability of a second-order boundary value problem with Dirichlet boundary conditions. Subsequently, his idea was further developed by a number of mathematicians, with M. Nagumo [39] being the first to do this in 1937. In a series of papers of recent decades, R. Agarwal and D. O'Regan, see for example [40], study the solvability of various nonsingular and singular initial and boundary value problems under the assumption that the main nonlinearity does not change its sign.
We use other tools. In [45], under a barrier strips condition, we study the solvability of BVPs for (1) with BCs, including y (1) = C. In the present paper, we extend the list of BVPs considered in [45], imposing a different barrier strips condition, which is adapted to the new BC for the third derivative. Barrier strip conditions have also been used by W. Qin [42] for studying the solvability of a three-point BVP for Equation (1).
Our results rely on the following assumptions: Hypothesis 1 (H1). There exist constants F i , L i , i = 1, 2, with the following properties: Hypothesis 2 (H2). There exist constants m k ≤ M k , k = 0, 3, with the properties: In Lemma 1, we will see that the strips [0, 1] × [L 1 , L 2 ] and [0, 1] × [F 2 , F 1 ] from (H 1 ) control the behavior of y (t) on [0, 1] and, in this way, guarantee a priori bounds for y (t). These strips are called barrier ones-see P. Kelevedjiev [46]; more details on the nature of the barrier technique are presented in Section 5. Note also that (H 1 ) and (H 2 ) allow the sets D y , D u , D v and D w to be bounded and the nonlinearity f to be continuous only on a bounded subset of its domain.
The barrier idea can be used in various variants. One of the possibilities is to replace the constants F i , L i , i = 1, 2, from (H 1 ) by continuous functions having suitable monotonicity. Such curvilinear strips have been used in P. Kelevedjiev [47] [47]. A disadvantage of barrier segments is that the right side of the equation must not become zero on them. Barrier segments have also been used by I. Rachůnková and S. Staněk [48] and R. Ma [49] for studying the solvability of various BVPs. Discontinuous barrier strips, curvilinear strips and barrier segments can also be useful; see [47].
Our basic existence result is stated in Section 2. There, we also give auxiliary results, which guarantee a priori bounds for each eventual solution y(t) ∈ C 4 [0, 1] to the families of BVPs for y (4) = λ f (t, y, y , y , y ), λ ∈ [0, 1], t ∈ (0, 1), (1) λ with BCs either (2)-(4) or (5); these a priori bounds are necessary for the application of the basic existence theorem. Moreover, the barrier condition (H 1 ) first provides the a priori bound for y (t), and those for y(t), y (t) and y (t) are a consequence of it. The existence results are stated in Section 3. They are based on the simultaneous use of (H 1 ) and (H 2 ) and guarantee not just a solution to the considered boundary value problems but also solutions with important properties such as an invariant sign, increasing, decreasing, convexity, and concavity. In Section 4, we illustrate the application of the obtained existence theorems with examples.

Basic Existence Results, Auxiliary Results
Following [45], we first introduce the notation needed to formulate the basic existence theorem.
is one-to-one.
(v) There is a sufficiently small where the set J is as in (H 2 ), and the constants m k , M k , k = 0, 3, are as in (iv). Then BVP (10) has at least one solution in C 4 [0, 1].
We skip the proof, it can be found in [45]. Our first auxiliary result guarantees a priori bounds for the third derivatives of all eventual C 4 [0, 1]-solutions to the families of BVPs (1.1) λ , (2)-(5).

Proof. Suppose that
is a non-empty set. Then, bearing in mind that y (0) ≤ L 1 and y (t) is continuous on [0, 1], we conclude that there exists an α ∈ T − such that For α, we have In view of (8), this means that The obtained contradiction shows that the set T − is empty and so Similarly, assuming, on the contrary, that is a non-empty set and using (9), we arrive at a contradiction, which implies

Lemma 3. Assume that (H 1 ) holds. Then the bounds
Proof. Following the proof of Lemma 2, we obtain the bound for |y (t)|. Next, consider that there exists a ν ∈ (0, 1) with the property from which, using the established estimate for |y (γ)|, we obtain We can proceed analogously to see that this bound is also valid in the interval [ν, 1]. For each t ∈ (0, 1], again by the mean value theorem, there exists a δ ∈ (0, t) such that |y(t)| ≤ |y(0)| + |y (δ)|t, t ∈ (0, 1], and using (14), we establish the a priori bound for |y(t)|.
from where, using |y (γ)| ≤ max{|F 1 |, |L 1 |}, which Lemma 1 gives, we obtain This estimate is also valid in the interval [ν, 1], and it is established with similar reasoning. Following the proof of Lemma 2, establish the assertion for |y (t)|. Finally, for each t ∈ [0, 1), there exists a δ ∈ (t, 1) for which which gives the assertion for |y(t)|.
Proof. In view of Lemma 1, we have which yields (17). Further, use the fact that there exists a ν ∈ (0, 1) such that y (ν) = D − C to establish consecutively Similarly from As a result, keeping in mind that B, F 1 ≤ 0, we obtain Now, by the mean value theorem, for any t ∈ (0, 1] there exists a γ ∈ (0, t) such that from where it follows Since B ≤ 0, y(t) is concave on [0, 1] in view of (17). In addition, y(0) = C ≥ 0 and y(1) = D ≥ 0, which means y(t) ≥ min{C, D} ≥ 0 on [0, 1], from where the assertion for y(t) follows.
Proof. From the proof of Lemma 5, we know that there exists a ν ∈ (0, 1) such that y (ν) = D − C − B. Now the integration (18) of the estimates for y (t) that Lemma 1 guarantees gives us and On the other hand, the integration (19) gives Bearing in mind that F 1 ≤ 0, on the whole interval [0, 1], we obtain which means y (t) ≥ 0 for t ∈ [0, 1] because of the assumption D − C ≥ B − F 1 and so y (t) is non-decreasing on the interval [0, 1]. Then, in view of the boundary condition y (0) = B, we have Thus, the bound for y (t) from Lemma 5 takes the form Further, from y (t) ≥ B ≥ 0, it follows that y(t) is non-decreasing. This fact, together with the bound for |y(t)| from Lemma 5, gives C ≤ y(t) ≤ |B| + |C| + |D − C − B| + max{|F 1 |, |L 1 |}, t ∈ [0, 1], from where the assertion for y(t) follows.

Proof. Lemma 6 implies
and Lemma 1 yields Further, as in the proof of Theorem 2, we establish that problem (1), (2) has a solution 1], this solution has the specified properties.
Clearly, there is a τ > 0 such that Let, for concreteness, the other cases can be considered in a similar way. It is clear that for the considered case, we have Q n (w) < 0 on (w 1 , w 1 + τ) and Q n (w) > 0 on (w 2 − τ, w 2 ).
The nonlinearity holds for the above constants m k , M k , k = 0, 1, 2, 3, and ε = 0.1, for example. Therefore, we can apply Theorem 3 to conclude that this problem has a positive, increasing, convex solution in C 4 [0, 1].
This problem is of the type (1), (5) with boundary values A = −2, B = 1, C = 1 and D = 10, which satisfy the condition of Theorem 9.

Discussion
The barrier strips technique used in this paper was introduced in 1994 in [46]. This technique does not use the classical tools mentioned in the introduction. It is based on the assumption that the right-hand side of the equation has suitable different signs on suitable subsets of its domain. Subsequently, barrier strips are used by a number of authors investigating the solvability of various boundary value problems for differential, difference and fractional differential equations, as well as of functional boundary value problems for differential equations.
This paper shows how the barrier strips technique (based here on assumption (H 1 )) can be used not only to establish the solvability of the boundary value problems under consideration but also to establish the existence of solutions that have important properties, namely, solutions that are monotonous, convex or concave and do not change their sign.
In principle, the barrier strips technique provides an a priori estimate for the (n − 1)th derivative of initial and boundary value problems for nth-order equations. As a consequence, it provides a priori estimates for both the unknown function and its remaining derivatives if at least one value for all of them is known. Moreover, the type of barrier condition depends on what value of the variable is the known value of the (n − 1)th derivative-at the end of the set interval or at its interior point. All this makes the barrier strips technique applicable to a wide class of initial and boundary value problems.
Author Contributions: Investigation, R.A., G.M., and P.K. All authors have read and agreed to the published version of the manuscript.