Existence and Iteration of Positive Solutions to Third-Order BVP for a Class of 𝑝 -Laplacian Dynamic Equations on Time Scales

We investigate the existence and iteration of positive solutions for the following third-order 𝑝 -Laplacian dynamic equations on time scales: (𝜙 𝑝 (𝑢 ΔΔ (𝑡))) ∇ + 𝑞(𝑡)𝑓(𝑡,𝑢(𝑡),𝑢 ΔΔ (𝑡)) = 0, 𝑡 ∈ [𝑎, 𝑏],𝛼𝑢(𝜌(𝑎)) − 𝛽𝑢 Δ (𝜌(𝑎)) = 0, 𝛾𝑢(𝑏) + 𝛿𝑢 Δ (𝑏) = 0, 𝑢 ΔΔ (𝜌(𝑎)) = 0, where 𝜙 𝑝 (𝑠) is 𝑝 -Laplacian operator; that is, 𝜙 𝑝 (𝑠) = |𝑠| 𝑝−2 𝑠, 𝑝 > 1, 𝜙 −1𝑝 = 𝜙 𝑞 , and 1/𝑝 + 1/𝑞 = 1. By applying the monotone iterative technique and without the assumption of the existence of lower and upper solutions, we not only obtain the existence of positive solutions for the problem, but also establish iterative schemes for approximating the solutions.


Introduction
A time scale T is a nonempty closed subset of R. We make the blanket assumption that , are points in T. By an interval [ , ], we always mean the intersection of the real interval [ , ] with the given time scale; that is, [ , ] ∩ T.
In recent years, dynamic equations on time scales have found a considerable interest and attracted by many researchers; see, for example, [1][2][3][4][5][6][7][8][9][10][11]. The reason seems to be twofold. Theoretically, dynamic equations on time scales not only can unify differential and difference equations [12], but also have displayed much more complicated dynamics [13,14]. Moreover, the study of time scales has led to several important applications in the study of insect population models, neural networks, stock market, heat transfer, economic, wound healing, and epidemic models.
Recently, there is much attention paid to the existence of positive solutions for second-order nonlinear boundary value problems on time scales; see [2,6,15,16] and the references therein. On the one hand, higher-order nonlinear boundary value problems have been studied extensively; for details, see [4,10,17,18] and references therein. On the other hand, the boundary value problems with -Laplacian operator have also been discussed extensively in the literature; for example, see [3, 7-9, 11, 19-21]. However, to the best of our knowledge, there are not many results concerning the third-order -Laplacian dynamic equations on time scales.
In [22], Yang and Yan studied the following third-order Sturm-Liouville boundary value with -Laplacian: ( ( ( ))) + ( , ( )) = 0, ∈ (0, 1) , By using the fixed point index method, they established the existence of at least one or two positive solutions for the third-order Sturm-Liouville boundary value problem with -Laplacian.

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International Journal of Differential Equations In [8], Han and Kang were concerned with the existence of multiple positive solutions of third-order -Laplacian dynamic equation on time scales: where ( ) = | | −2 , > 1. By using the fixed point index method, they obtained the existence of multiple positive solutions for singular nonlinear boundary value problem.
Remark 1. Inequalities (5) and (6) are equivalent to the following inequalities, respectively: This paper is organized as follows. In Section 2, we state some basic time scale definitions and several preliminary results. Section 3 is devoted to the iteration and existence of positive solutions of for the third-order Sturm-Liouville boundary value problem with -Laplacian on time scales (3)- (4). Finally, in Section 4, we give an example to demonstrate our main result.

Definition 2.
A time scale T is a nonempty closed subset of real numbers R. For < sup T and > inf T, define the forward jump operator and backward jump operator , respectively, by for all , ∈ T. If ( ) > , is said to be right scattered, and if ( ) < , is said to be left scattered; if ( ) = , is said to be right dense, and if ( ) = , is said to be left dense. If T has a right scattered minimum , define Throughout this paper, we make the blanket assumption that ≤ are points in T and define the interval in T: Other types of intervals are defined similarly.
Definition 3. For : T → R and ∈ T , the delta derivative of at the point is defined to be the number Δ ( ) (provided it exists) with the property that, for each > 0, there is a neighborhood of such that for all ∈ . For : T → R and ∈ T , the nabla derivative of at is denoted by ∇ ( ) (provided it exists) with the property that, for each > 0, there is a neighborhood of such that for all ∈ .

Definition 4.
A function is left-dense-continuous (i.e., ldcontinuous), if is continuous at each left-dense point in T and its right-sided limit exists at each right-dense point in T. It is well known that if is ld-continuous, then there is a function ( ) such that ∇ ( ) = ( ). In this case, it is defined that International Journal of Differential Equations 3 Definition 5. If Δ ( ) = ( ), then we define the delta integral by If ∇ ( ) = ( ), then we define the nabla integral by , then we say that is concave down on [ , ].
By a function ( ) being a positive solution of (3)-(4), we mean that ( ) is positive on ( , ) and satisfies (3)-(4). Let ( , ) be Green's function of ΔΔ ( ) = 0 with respect to boundary condition (4); then where , , , ≥ 0 : In fact, on the one hand, from Green's function, if is kept fixed then ( , ) is a decreasing function of for > ( ) and increasing for < ( ). This leads to the conclusion that ( , ) has a global maximum at = ( ), so ( , ) ≤ ( ( ), ), for ∈ [ ( ), ]. On the other hand, we can easily know that ( , ) is nonnegative from the following two equalities: Let the Banach space = 2 [ ( ), ] be equipped with the norm  Definition 6. Let be a real Banach space. Let be a nonempty, convex closed set in . We say that is a cone it it satisfies the following properties: ∈ for ∈ , ≥ 0; (ii) , − ∈ implies = ( denotes the null element of ).
We define operator : and then all the fixed points of operator are the solutions for BVP (3)-(4).

Remark 7.
From the definition of the operator , we can easily get the following properties: (26)

The Main Results
We now give our results for the existence of positive solutions for BVP (3) and (4).
In fact, International Journal of Differential Equations 5 and from (23) and (24), we have Thus, we obtained that ‖ ‖ ≤ . So, we have shown that ⊂ .
It follows from ⊂ , that ∈ , = 1, 2, . . .. Since is completely continuous, we can assert that { } is a sequentially compact set. Since International Journal of Differential Equations we obtain By (22) and (H5), we have Hence by induction we have Thus, there exists * ∈ such that → * . Applying the continuity of and +1 ( ) = ( ), we get * ( ) = * ( ) which implies that * is a nonnegative concave solution of the BVP (3) and (4).
Since V 1 = V 0 = 0 ∈ , we have So, By an induction argument similar to the above we obtain Hence, there exists V * ∈ such that V → V * . Applying the continuity of and V +1 ( ) = V ( ), we obtain V * ( ) = V * ( ) which implies that V * is a nonnegative solution of the BVP (3) and (4). From the condition (H4), we know that the zero function is not the solution of the BVP (3) This means that * is a positive concave solution of the BVP (3) and (4). Thus the BVP (3) and (4) has two positive and concave solutions * , V * such that 0 ≤ ‖ * ‖ ≤ , 0 ≤ ‖V * ‖ ≤ , and, from the above proof, we know that the iterative sequences hold. The proof is completed.

Example
Now, we present an example to illustrate the main result: Theorem 9.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.