Existence of Positive Solutions in Generalized Boundary Value Problem for -Laplacian Dynamic Equations on Time Scales

Abstract

We analytically establish the conditions for the existence of at least two or three positive solutions in the generalized -point boundary value problem for the -Laplacian dynamic equations on time scales by means of the Avery-Henderson fixed point theorem and the five functionals fixed point theorem. Furthermore, we illustrate the possible application of our analytical results with a concrete and nontrivial dynamic equation on specific time scales.

1. Introduction

Since the seminal work by Stefan Hilger in 1988, there has been a rapid development in the research of dynamic equations on time scales. The gradually maturing theory of dynamic equations not only includes the majority of the existing analytical results on both differential equations and difference equations with uniform time-steps but also establishes a solid foundation for the research of hybrid equations on different kinds of time scales. More importantly, with this foundation and those ongoing investigations, concrete applications of dynamic equations on time scales in mathematical modeling of real processes and phenomena, such as population dynamics, economic evolutions, chemical kinetics, and neural signal processing, have been becoming fruitful [1–8].

Recently, among the topics in the research of dynamic equations on time scales, the investigation of the boundary value problems for some specific dynamic equations on time scales has become a focal one that attained a great deal of attention from many mathematicians. In fact, systematic framework has been established for the study of the positive solutions in the boundary value problems for the second-order equations on time scales [9–15]. In particular, some results have been analytically obtained on the existence of positive solutions in some specific boundary value problems for the -Laplacian dynamic equations on time scales [16–19].

More specifically, He and Li [19], investigated the existence of at least triple positive solutions to the following -Laplacian boundary value problem:

(11)

Here and throughout, is supposed to be a time scale, that is, is any nonempty closed subset of real numbers in with order and topological structure defined in a canonical way. The closed interval in is defined as . Accordingly, the open interval and the half-open interval could be defined, respectively. In addition, it is assumed that , , , , and for some positive constants and . Moreover, is supposed to be the -Laplacian operator, that is, and , in which and . With these configurations and with the aid of the five functionals fixed point theorem [20], they established the criteria for the existence of at least triple positive solutions of the above boundary value problem.

Later on, Yaslan [21], investigated the following boundary value problem:

(12)

in which , , and . Indeed, Yaslan analytically established the conditions for the existence of at least two or three positive solutions in the above boundary value problem by means of the Avery-Henderson fixed point theorem and the Leggett-Williams fixed point theorem [22]. As a matter of fact, these analytical results are even new for those special equations on time scales, such as the difference equations with uniform time-step and the ordinary differential equations. Following the work in [21], Sun and Wang [23], further extended the results to the following boundary value problem:

(13)

In this paper, inspired by the aforementioned results and methods in dealing with those boundary value problems on time scales, we intend to analytically discuss the possible existence of multiple positive solutions for the following one-dimensional -Laplacian dynamic equation:

(14)

with -point boundary value conditions:

(15)

In the following discussion, we impose the following three hypotheses.

(H1) for , , and , where

(H2) is left dense continuous (-continuous), and there exists a such that . is continuous.

(H3) Both and are continuously odd functions defined on . There exist two positive numbers and such that, for any ,

(16)

Note that the definition of the -continuous function will be described in Definition 2.3 of Section 2. Also note that, together with conditions (1.5) and the above hypotheses (H)–(H), the dynamic equation (1.4) with conditions (1.5) not only includes the above-mentioned specific boundary value problems in literature but also nontrivially extends the situation to a much wider class of boundary value problems on time scales. A question naturally appears: "can we still establish some criteria for the existence of at least double or triple positive solutions in the generalized boundary value problems (1.4) and (1.5)?" In this paper, we will give a positive answer to this question by virtue of the Avery-Henderson fixed point theorem and the five functionals fixed point theorem. Particularly, those obtained criteria will significantly extend the results in literature [19, 21, 23].

The rest of paper is organized as follows. In Section 2, we preliminarily import some definitions and properties of time scales and introduce some useful lemmas which will be utilized in the following discussion. In Section 3, we analytically present a criteria for the existence of at least two positive solutions in the boundary value problems (1.4) and (1.5) by virtue of the Avery-Henderson fixed point theorem. In Section 4, we provide some sufficient conditions for the existence of at least three positive solutions in light of the five functionals fixed point theorem. Finally, we further provides concrete and nontrivial example to illustrate the possible application of the obtained analytical results on dynamic equations on time scales in Section 5.

2. Preliminaries

2.1. Time Scales

For the sake of self-consistency, we import some necessary definitions and lemmas on time scales. More details can be found in [4] and reference therein. First of all, a time scale is any nonempty closed subset of real numbers with order and topological structure defined in a canonical way, as mentioned above. Then, we have the following definition of the categories of points on time scales.

Definition 2.1.

For and , define the forward jump operator and the backward jump operator , respectively, by

(21)

Then, the graininess operator is defined as In addition, 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 has a right scattered minimum , denote by ; otherwise, set . If has a left scattered maximum , denote by ; otherwise, set .

The following definitions describe the categories of functions on time scales and the basic computations of integral and derivative.

Definition 2.2.

Assume that is a function and that . is supposed to be the number (provided it exists) with the property that given any ; there is a neighborhood of satisfying

(22)

for all . Then is said to be the delta derivative of at . Similarly, assume that is a function and that . Denote by the number (provided it exists) with the property that given any , there is a neighborhood of such that

(23)

for all . Then is said to be the nabla derivative of at .

Definition 2.3.

A function is left dense continuous (-continuous) provided that it is continuous at all left dense points of , and its right side limits exists (being finite) at right dense points of . Denote by the set of all left dense continuous functions on .

Definition 2.4.

Let be a function, and . If there exists a function such that for all , then is a delta antiderivative of . In this case the integral is given by the formula

(24)

Analogously, let be a function, and . If there exists a function such that for all , then is a nabla antiderivative of . In this case, the integral is given by the formula

(25)

2.2. Main Lemmas

This subsection aims to establish several lemmas which are useful in the proof of the main results in this paper. In particular, these lemmas focus on the following linear boundary value problems:

(26)

(27)

Lemma 2.5.

If , then, for , the linear boundary value problems (2.6) and (2.7) have a unique solution satisfying

(28)

. Here,

(29)

Proof.

It follows from (2.8) that

(210)

Thus, we obtain that

(211)

and that

(212)

Then, satisfies (2.6), which verifies that is a solution of the problems (2.6) and (2.7). Furthermore, in order to show the uniqueness, we suppose that both and are the solutions of the problems (2.6) and (2.7). Then, we have

(213)

(214)

(215)

In fact, (2.13) further yields

(216)

Hence, from (2.14) and (2.16), the assumption , and the definition of the -Laplacian operator, it follows that

(217)

This equation, with (2.15), further implies

(218)

which consequently leads to the completion of the proof, that is, specified in (2.8) is the unique solution of the problems (2.6) and (2.7).

Lemma 2.6.

Suppose that . If , then the unique solution of the problems (2.6) and (2.7) satisfies

(219)

Proof.

Observe that, for any

(220)

Thus, by (2.8) specified in Lemma 2.5, we get

(221)

Thus, is nondecreasing in the interval . In addition, notice that

(222)

The last term in the above estimation is no less than zero owing to those assumptions. Therefore, from the monotonicity of , we get

(223)

which consequently completes the proof.

Lemma 2.7.

Suppose that . If , then the unique positive solution of the problems (2.6) and (2.7) satisfies

(224)

for with

Proof.

Since is nondecreasing in the interval

(225)

On the other hand,

(226)

Hence,

(227)

This completes the proof.

Now, denote by and by , where . Then, it is easy to verify that endowed with becomes a Banach space. Furthermore, define a cone, denoted by , through

(228)

Also, for a given positive real number , define a function set by

(229)

Naturally, we denote by and by . With these settings and notations, we are in a position to have the following properties.

Lemma 2.8.

If then (i) for any ; (ii) for any pair of with .

The proof of this lemma, which could be found in [19, 21], is directly from the specific construction of the set . Next, let us construct a map through

(230)

for any . Here, . Thus, we obtain the following properties on this map.

Lemma 2.9.

Assume that the hypotheses are all fulfilled. Then, and is completely continuous.

Proof.

At first, arbitrarily pick up . Then it directly follows from Lemma 2.6 that for all . Moreover, direct computation yields

(231)

for all , and

(232)

for all . Thus, the latter inequality implies that is decreasing on . This implies that for . Consequently, we complete the proof of the first part of the conclusion that for any .

Secondly, we are to validate the complete continuity of the map . To approach this aim, we have to verify that is bounded, where is obviously bounded. It follows from the proof of Lemma 2.7 that

(233)

where . This manifests the uniform boundedness of the set . In addition, for any given with we have the following estimation:

(234)

This validates the equicontinuity of the elements in the set . Therefore, according to the Arzelà-Ascoli theorem on time scales [2], we conclude that is relatively compact. Now, let with . Then for all and Also, is uniformly valid on . These, with the uniform continuity of on the compact set , leads to a conclusion that is uniformly valid on . Hence, it is easy to verify that as tends toward positive infinity. As a consequence, we complete the whole proof.

3. At Least Two Positive Solutions in Boundary Value Problems

This section aims to prove the existence of at least two positive solutions in the boundary value problems (1.4) and (1.5) in light of the well-known Avery-Henderson fixed point theorem. Firstly, we introduce the Avery-Henderson fixed point theorem as follows.

Theorem 3.1 ([24]).

Let be a cone in a real Banach space . For each , set . If and are increasing nonnegative continuous functional on , and let be a nonnegative continuous functional on with such that, for some and ,

(31)

for all . Suppose that there exist a completely continuous operator and three positive numbers such that

(32)

and (i) for all ; (ii) for all ; (iii) and for all . Then, the operator has at least two fixed points, denoted by and , belonging to and satisfying with and with .

Secondly, let and select satisfying . Furthermore, set, respectively,

(33)

Then, we arrive at the following results.

Theorem 3.2.

Suppose that the hypotheses all hold, and that there exist positive real numbers , , such that

(34)

In addition, suppose that satisfies the following conditions:

(B1) for and ;

(B2) for and ;

(B3) for and

Then, the boundary value problems (1.4) and (1.5) have at least two positive solutions and such that

(35)

Proof.

Construct the cone and the operator as specified in (2.28) and (2.30), respectively. In addition, define the increasing, nonnegative, and continuous functionals , , and on , respectively, by

(36)

Obviously, for each .

Moreover, Lemma 2.8 manifests that for each . Hence, we have

(37)

for each . Also, notice that for and . Furthermore, from Lemma 2.9, it follows that the operator is completely continuous.

Next, we are to verify the validity of all the conditions in Theorem 3.1 with respect to the operator .

Let . Then, . This implies for , which, combined with (3.7), yields

(38)

for . Noticing the assumption (), we have for . Also noticing the particular form in (2.30), Lemma 2.8, the property , and the proof of Lemma 2.7, we get

(39)

Therefore, condition (i) in Theorem 3.1 is satisfied.

In what follows, let us consider . In such a case, we obtain , which means that for . Similarly, it follows from (3.7) that, for all ,

(310)

Hence, we have for . This, combined with the assumption (), yields for all . Therefore, from the proof of Lemma 2.7, we have

(311)

which consequently leads to the validity of condition (ii) in Theorem 3.1.

Last, let us notice that the constant functions . Then, . Take . We thus obtain . This, with the assumption , manifests that and for all . Analogously, we can get

(312)

which shows the validity of condition (iii) in Theorem 3.1.

Now, in the light of Theorem 3.1, we consequently arrive to the conclusion that the boundary value problems (1.4) and (1.5) admit at least two positive solutions, denoted by and , satisfying with , and with , respectively.

4. At Least Three Positive Solutions in Boundary Value Problems

By means of the five functionals fixed point theorem which is attributed to Avery [20], this section is to analytically prove the existence of at least three positive solutions in the boundary value problems (1.4) and (1.5).

Take as nonnegative continuous convex functionals on . Both and are supposed to be nonnegative continuous concave functionals on . Then, for nonnegative real numbers , , , , and , construct five convex sets, respectively, through

(41)

Theorem 4.1 ([20]).

Let be a cone in a real Banach space . Suppose that and are nonnegative continuous concave functionals on , and that , , and are nonnegative continuous convex functionals on such that, for some positive numbers and ,

(42)

for all . In addition, suppose that is a completely continuous operator and that there exist nonnegative real numbers with such that

• (i)
• (ii)
• (iii)
• (iv)

Then the operator admits at least three fixed points ,, satisfying , , and with , respectively.

In the light of this theorem, we can have the following result on the existence of at least three solutions in the boundary value problems (1.4) and (1.5).

Theorem 4.2.

Suppose that the hypotheses are all fulfilled. Also suppose that there exist positive real numbers , , and such that

(43)

Furthermore, let satisfies the following conditions:

(C1) for and ;

(C2) for and ;

(C3) for and .

Then, the boundary value problems (1.4) and (1.5) possess at least three solutions , , and , defined on , satisfying, respectively,

(44)

Proof.

Set the cone as constructed in (2.28) and the operator as defined in (2.30). Take, respectively, the nonnegative continuous concave functionals on the as follows:

(45)

Then, we get for . Besides, from Lemma 2.8, it follows that

(46)

for . In what follows, we aim to show the validity of all the conditions in Theorem 4.1 with respect to the operator .

To this end, arbitrarily take a function . Thus, , which, combined with (4.6), gives for and . Hence, we have for , due to the assumption (). Furthermore, since , in the light of the proof of Lemma 2.7, we have

(47)

So, according to Lemma 2.9, we have the complete continuity of the operator .

Moreover, the set

(48)

is not empty, since the constant function is contained in the set . Similarly, the set

(49)

is nonempty because of . For a particular , the implementation of (4.6) gives

(410)

for . The utilization of the assumption () leads us to the inequality

(411)

Thus, it follows from (4.11) and Lemmas 2.7 and 2.8 that

(412)

Clearly, we verify the validity of condition () in Theorem 4.1.

Next, consider . In such a case, we obtain

(413)

for . Imposing the assumption () produces . Moreover, by the proof of Lemma 2.7, we obtain

(414)

Therefore, we further verify the validity of condition () in Theorem 4.1.

Finally, we are to validate conditions () and () aside from conditions () and (). For this purpose, on the one hand, let us consider with . Then, we have

(415)

On the other hand, consider with . In this case, we get

(416)

Accordingly, both conditions () and () in Theorem 4.1 are satisfied. Now, in light of Theorem 4.1, the boundary value problems (1.4) and (1.5) have at least three positive solutions circumscribed on satisfying , , and with .

5. An Illustrative Example

This section will provide a nontrivial example to clearly illustrate the feasibility of the above-established analytical results on the dynamic equations on time scales.

First of all, construct a nontrivial time scale set as . Set all the parameters in problems (1.4) and (1.5) as follows: , and , so that . For simplicity but without loss of generality, set . we can obtain

(51)

In particular, set the function in dynamic equation as

(52)

This setting allows us to properly take the other parameters as , , and . It is clear that these parameters satisfy

(53)

To this end, we can verify the validity of conditions in Theorem 4.2. As a matter of fact, direct calculations produce

(54)

as and ,

(55)

as and , and

(56)

and . Accordingly, conditions in Theorem 4.2 are satisfied for the above specified functions and parameters. Now, by virtue of Theorem 4.2, we can approach a conclusion that the dynamic equation on the specified time scales

(57)

with the boundary conditions

(58)

possesses at least three positive solutions defined on satisfying , , and with .