Some New Existence Results of Positive Solutions to an Even-Order Boundary Value Problem on Time Scales

We consider a high-order three-point boundary value problem. Firstly, some new existence results of at least one positive solution for a noneigenvalue problem and an eigenvalue problem are established. Our approach is based on the application of three different fixed point theorems, which have extended and improved the famous Guo-Krasnosel’skii fixed point theorem at different aspects. Secondly, some examples are included to illustrate our results.

In recent years, the theory of time scales, which has received a lot of attention, was introduced by Hilger in his Ph.D. thesis [5] in 1988 in order to unify continuous and discrete analysis. In particular, the theory is also widely applied to stock market, biology, heat transfer, and epidemic models; for details, see [6][7][8][9][10] and the references therein. An important class of time scales are boundary value problems, and such investigations can provide accurate information of phenomena that manifest themselves partly in continuous time and partly in discrete time. There is much attention paid to the existence of positive solution for higher-order two-point boundary value problems on time scales [11][12][13][14][15]. However, to the best of our knowledge, there are not many results concerning multipoint boundary value problems of higher-order on time scales; we refer the reader to [16][17][18][19][20][21][22][23][24] for some recent results.
We would like to mention some results of Anderson and Avery [16], Anderson and Karaca [17], Han and Liu [20], and Hu [21]. In [16], Anderson and Avery studied the following even-order BVP: 2 Abstract and Applied Analysis They have studied the existence of at least one positive solution to the BVP (3) using the functional-type cone expansion-compression fixed point theorem.
In [17], Anderson and Karaca were concerned with the dynamic three-point boundary value problem Existence results of bounded solutions of a noneigenvalue problem are first established as a result of the Schauder fixed point theorem. Second, the monotone method is discussed to ensure the existence of solutions of the BVP (4). Third, they established criteria for the existence of at least one positive solution of the eigenvalue problem by using the Krasnosel'skii fixed point theorem. Later, they investigated the existence of at least two positive solutions of the BVP (4) by using the Avery-Henderson fixed point theorem.
In [20], Han and Liu studied the existence and uniqueness of nontrivial solution for the following third-order -Laplacian -point eigenvalue problems on time scales: where ( ) is -Laplacian operator, that is, ( ) = | | −2 , > 1, > 0 is a parameter, and 0 < 1 < ⋅ ⋅ ⋅ < −2 < ( ). They obtained several sufficient conditions of the existence and uniqueness of nontrivial solution of (5) when is in some interval. Their approach is based on the Leray-Schauder nonlinear alternative.
Recently, in [21], Hu considered the following singular third-order three-point boundary value problem on time scales: where ∈ ( , ), and/or = and may have singularity at = 0. Some theorems on the existence of positive solutions of (6) were obtained by utilizing the fixed theorem of cone expansion and compression type.
We note that Yaslan [23] constructed the Green's function for the BVP (1) and obtained the following result.
One of the most frequently used tools for proving the existence of positive solutions to the integral equations is Krasnosel'skii's theorem on cone expansion and compression and its norm-type version due to Guo and Lakshmikantham (see [25]), and they have been applied extensively to all kinds of problems, such as ordinary differential equations, difference equations, and the general dynamic equations on time scales. In order to apply them easily, more and more authors have been dedicated to extending this theorem. In [26], the authors dealt with modifications of the classical Krasnosel'skii fixed point theorem concerning cone compression and expansion of norm type. For further abstract result, the reader is referred to the recent paper [27].
In [28], Zima proved the following fixed point theorem of Leggett-Williams type.
Theorem 2 (see [28]). Let be a real Banach space, a normal cone in , and the normal constant of .
We would also like to mention the results of Zhang and Sun [29,30], in [29,30]. Zhang and Sun continued to extend this theorem by replacing the norm with some convex functional on cone (let be a real Banach space and a cone in ). : → R is said to be a convex functional on if ( + (1 − ) ) ≤ ( ) + (1 − ) ( ) for all , ∈ , and ∈ [0, 1]. They obtained the following main results.
Theorem 5 (see [31] and that one of the two following conditions holds: If, moreover, is nondecreasing in the set where ≥ 1 is the normal constant of the cone, then there exists ∈ , ̸ = , such that = . In this paper, we will improve and extend Theorem 1 in two different directions. On the one hand, we will weaken the restriction on in (8) by means of Theorem 2 (see Theorem 10). On the other hand, by constructing appropriate convex functional shown in [29,30], the properties of on bounded sets will be considered. Furthermore, the new existence result is obtained by Theorem 3 (see Theorem 11).

Preliminaries
Let ( , ) be Green's function for the following boundary value problem: A direct calculation gives where To state and prove the main results of this paper, we need the following lemmas.

Existence Theorem of Positive Solutions
In this section, we apply Theorems 2 and 3 to establish two existence criteria for the problem (1).
Remark 12. In Theorem 10, we substitute condition ( 2 ) for condition (8) in Theorem 1, and thus the more general and comprehensive functions are incorporated. In Theorem 11, we substitute condition ( 3 ) or ( 4 ) for conditions (7) and (8) in Theorem 1, where we only require that the "heights" of the nonlinear term are appropriate on some bounded sets. Moreover, the existence is independent of the growth of outside these bounded sets.

Solvability of the Problem (2)
In this section, we apply Theorem 5 to obtain sufficient conditions for the existence of solutions for the problem (2). We note that the conspicuous advantage of Theorem 5 is that the conditions over the set of the lower solution are deleted. We define the cone for some 0 < ≤ −1 / −1 ( will be fixed later).
Since the problem (2) is equivalent to the following integral equation: we still define the operator : → B as follows: By the proof of Section 3, we can know that : → is completely continuous. On the other hand, it is easy to 6 Abstract and Applied Analysis check that is normal with normal constant = 1 and with nonempty interior. Define The main result is as follows.

Remark 14.
In general, the previous results ensure the solvability of the problem (2) only for "enough small" , that is, for 0 < < 1 , where 1 is a constant explicitly established. However, Theorem 13 implies that the problem (2) is solvable for all > 0 even if ( ) = 0 for some ∈ (0, ]. We should point out that the spectrum structure of the corresponding linear problem of (2) is still unknown so far; therefore, we can not directly apply the results shown in [32] due to Webb and Infante and in [33] due to Webb and Lan to the problem (2).

Some Examples
In this section, we present three examples to explain our results.
Therefore, condition ( 4 ) in Theorem 11 is satisfied, and the BVP (58) has at least one positive solution.

Remark 15.
From the expressions of nonlinear terms which we defined in Examples 1 and 2, we can see that Theorem 1 cannot be directly applied to Examples 1 and 2.