Research Article | Open Access
Existence of Three Positive Solutions to Some p-Laplacian Boundary Value Problems
We obtain, by using the Leggett-Williams fixed point theorem, sufficient conditions that ensure the existence of at least three positive solutions to some p-Laplacian boundary value problems on time scales.
The study of dynamic equations on time scales goes back to the 1989 Ph.D. thesis of Hilger [1, 2] and is currently an area of mathematics receiving considerable attention [3–7]. Although the basic aim of the theory of time scales is to unify the study of differential and difference equations in one and the same subject, it also extends these classical domains to hybrid and in-between cases. A great deal of work has been done since the eighties of the XX century in unifying the theories of differential and difference equations by establishing more general results in the time scale setting [8–12].
Boundary value -Laplacian problems for differential equations and finite difference equations have been studied extensively (see, e.g.,  and references therein). Although many existence results for dynamic equations on time scales are available [14, 15], there are not many results concerning -Laplacian problems on time scales [16–19]. In this paper we prove new existence results for three classes of -Laplacian boundary value problems on time scales. In contrast with our previous works [17, 18], which make use of the Krasnoselskii fixed point theorem and the fixed point index theory, respectively, here we use the Leggett-Williams fixed point theorem [20, 21] obtaining multiplicity of positive solutions. The application of the Leggett-Williams fixed point theorem for proving multiplicity of solutions for boundary value problems on time scales was first introduced by Agarwal and O’Regan  and is now recognized as an important tool to prove existence of positive solutions for boundary value problems on time scales [23–28].
The paper is organized as follows. In Section 2 we present some necessary results from the theory of time scales (Section 2.1) and the theory of cones in Banach spaces (Section 2.2). We end Section 2.2 with the Leggett-Williams fixed point theorem for a cone-preserving operator, which is our main tool in proving existence of positive solutions to the boundary value problems on time scales we consider in Section 3. The contribution of the paper is Section 3, which is divided into three parts. The purpose of the first part (Section 3.1) is to prove existence of positive solutions to the nonlocal -Laplacian dynamic equation on time scales satisfying the boundary conditions where , is the -Laplacian operator defined by , , and with the Holder conjugate of , that is, . The concrete value of is connected with the application at hands. For , problem (1)-(2) describes the operation of a device flowed by an electric current, for example, thermistors , which are devices made from materials whose electrical conductivity is highly dependent on the temperature. Thermistors have the advantage of being temperature measurement devices of low cost, high resolution, and flexibility in size and shape. Constant in (1) is a dimensionless parameter that can be identified with the square of the applied potential difference at the ends of a conductor, is the temperature-dependent resistivity of the conductor, and in (2) is a transfer coefficient supposed to verify . For a more detailed discussion about the physical justification of (1)-(2) the reader is referred to . Theoretical analysis (existence, uniqueness, regularity, and asymptotic results) for thermistor problems with various types of boundary and initial conditions has received significant attention in the last few years for the particular case [30–34]. The second part of our results (Section 3.2) is concerned with the following quasilinear elliptic problem: where . Results on existence of infinitely many radial solutions to (3) are proved in the literature using (i) variational methods, where solutions are obtained as critical points of some energy functional on a Sobolev space, with satisfying appropriate conditions [35, 36]; (ii) methods based on phase-plane analysis and the shooting method ; (iii) the technique of time maps . For , , and , problem (3) becomes a well-known boundary value problem of differential equations. Our results generalize earlier works to the case of a generic time scale , , and not identically zero. Finally, the third part of our contribution (Section 3.3) is devoted to the existence of positive solutions to the -Laplacian dynamic equation on a time scale such that , with , where . This problem is considered in  where the author applies the Krasnoselskii fixed point theorem to obtain one positive solution to (4). Here we use the same conditions as in , but applying Leggett-Williams’ theorem we are able to obtain more: we prove existence of at least three positive solutions and we are able to localize them.
Here we just recall the basic concepts and results needed in the sequel. For an introduction to time scales the reader is referred to [3, 8–10, 40, 41] and references therein; for a good introduction to the theory of cones in Banach spaces we refer the reader to the book .
2.1. Time Scales
A time scale is an arbitrary nonempty closed subset of the real numbers . The operators and from to are defined in [1, 2] as and are called the forward jump operator and the backward jump operator, respectively. A point is left-dense, left-scattered, right-dense, and right-scattered if , and , respectively. If has a right-scattered minimum , define ; otherwise set . If has a left-scattered maximum , define ; otherwise set . Following , we also introduce the set .
Let and (assume is not left-scattered if ), then the delta derivative of at the point is defined to be the number (provided it exists) with the property that for each there is a neighborhood of such that Similarly, for (assume is not right-scattered if ), the nabla derivative of at the point is defined in  to be the number (provided it exists) with the property that for each there is a neighborhood of such that If , then . If , then is the forward difference operator while is the backward difference operator.
A function is left-dense continuous (i.e., -continuous), if is continuous at each left-dense point in and its right-sided limit exists at each right-dense point in . If is -continuous, then there exists such that for any . We then introduce the nabla integral by We define right-dense continuous (-continuous) functions in a similar way. If is -continuous, then there exists such that for any , and we define the delta integral by
2.2. Cones in Banach Spaces
In this paper is a time scale with and . We use and to denote, respectively, the set of positive and nonnegative real numbers. By we denote the set . Similarly, . Let . It follows that is a Banach space with the norm .
Definition 1. Let be a real Banach space. A nonempty, closed, and convex set is called a cone if it satisfies the following two conditions:(i), , implies ; (ii), , implies .
Every cone induces an ordering in given by
Definition 2. Let be a real Banach space and a cone. A function is called a nonnegative continuous concave functional if is continuous and for all and .
Let , and be positive constants, , . The following fixed point theorem provides the existence of at least three positive solutions. The origin in is denoted by . The proof of the Leggett-Williams fixed point theorem can be found in Guo and Lakshmikantham  or Leggett and Williams .
Theorem 3 (Leggett-Williams’ Theorem). Let be a cone in a real Banach space . Let be a completely continuous map and a nonnegative continuous concave functional on such that for all . Suppose there exist positive constants , and with such that(i) and for all ;(ii) for all ;(iii) for all with . Then has at least three fixed points , , and satisfying
3. Main Results
3.1. Nonlocal Thermistor Problem
Proof. We have . Then, . It follows that . Since , we also have If , then Consequently, for .
On the other hand, we have . Since , then . This means that , . Moreover, is nonincreasing, which implies with the monotonicity of that is a nonincreasing function on . Hence, is concave. In order to apply Theorem 3, let us define the cone by We also define the nonnegative continuous concave functional by It is easy to see that problem (1)-(2) has a solution if and only if is a fixed point of the operator defined by where and are as in Lemma 4.
Lemma 6. Let be defined by (19). Then,(i); (ii) is completely continuous.
Proof. (i)holds clearly from above. (ii)Suppose that is a bounded set and let . Then,
In the same way, we have It follows that As a consequence, we get Then is bounded on the whole bounded set . Moreover, if and , then we have for a positive constant We see that the right-hand side of the above inequality goes uniformly to zero when . Then by a standard application of the Arzela-Ascoli theorem we have that is completely continuous.
We can also easily obtain the following properties.
Lemma 7. (i) for all ;
We now state the main result of Section 3.1.
Theorem 8. Suppose that is verified and there exist positive constants , , , and such that (1/T, with . One further imposes to satisfy the following hypotheses:(H2) uniformly for all ;(H3) uniformly for all ;(H4) uniformly for all , where Then the boundary value problem (1)-(2) has at least three positive solutions , , and , verifying
Proof. The proof passes by several lemmas. We have already seen in Lemma 6 that the operator is completely continuous. We now show the following Lemma.
Lemma 9. The following relations hold:
Proof. Obviously, . Moreover, for all , we have . On the other hand we have and for all we have . Then, We have Using (H2) it follows that Then we get Similarly, using (H3) we get .
Lemma 10. The set is nonempty, and
Proof. Let . Then, , and . The first part of the lemma is proved. For we have . If , then Since , we have by using (H4) Using the fact that is nondecreasing we get Using the expression of
Lemma 11. For all with one has
Proof. If and , then . Using hypothesis (H3) and the fact that , it follows that Gathering Lemmas 4 to 11 and applying Theorem 3, there exist at least three positive solutions , , and to (1)-(2) verifying
Example 12. Let , where denotes the set of all nonnegative integers. Consider the -Laplacian dynamic equation satisfying the boundary conditions where , , , , , , and Choose , , , and . It is easy to see that , and Then, hypotheses (H1)–(H4) are satisfied. Therefore, by Theorem 8, problem (42)-(43) has at least three positive solutions.
3.2. Quasilinear Elliptic Problem
We are interested in this section in the study of the following quasilinear elliptic problem: where . We assume the following hypotheses: (A1) function is continuous;(A2) function is left dense continuous, that is, Similarly as in Section 3.1, we prove existence of solutions by constructing an operator whose fixed points are solutions to (46). The main ingredient is, again, the Leggett-Williams fixed point theorem (Theorem 3). We can easily see that (46) is equivalent to the integral equation On the other hand, we have . Since , , we have and for any with . It follows that for . Hence, is a decreasing function on . Then, is concave. In order to apply Theorem 3 we define the cone For we also define the nonnegative continuous concave functional by and the operator by It is easy to see that (46) has a solution if and only if is a fixed point of the operator . For convenience, we introduce the following notation:
Theorem 13. Suppose that hypotheses (A1) and (A2) are satisfied; there exist positive constants , , , and with and, in addition to (A1) and (A2), that satisfies (A3);(A4);(A5). Then problem (46) has at least three positive solutions , , and , verifying
Proof. As done for Theorem 8, the proof is divided in several steps. We first show that is completely continuous. Indeed, is obviously continuous. Let
It is easy to see that for there exists a constant such that . On the other hand, let , . Then there exists a positive constant such that
which converges uniformly to zero when tends to zero. Using the Arzela-Ascoli theorem we conclude that is completely continuous.
We now show that For all we have and Using the elementary inequality and the form of , it follows that Then . In a similar way we prove that .
Our following step is to show that The first point is obvious. Let us prove the second part of (61). For we have , if . Then, using we have Finally we prove that for all and : Using again the elementary inequality we get that By Theorem 3 there exist at least three positive solutions , , and to (46) satisfying , , , and .
Example 14. Let , where denotes the set of all nonnegative integers. Consider the -Laplacian dynamic equation satisfying the boundary conditions where , , , , , and Choose , , , and . It is easy to see that , , , , and Therefore, by Theorem 13, problem (65)-(66) has at least three positive solutions.
3.3. A p-Laplacian Functional Dynamic Equation on Time Scales with Delay
Let be a time scale with , with . We are concerned in this section with the existence of positive solutions to the -Laplacian dynamic equation where . We define , which is a Banach space with the maximum norm . We note that is a solution to (69) if and only if Let Clearly is a cone in the Banach space . For each , we extend to with for . We also define the nonnegative continuous concave functional by For , define as
Depending on the signature of the delay , we set the following two subsets of :
In the remainder of this section, we suppose that is nonempty and . For convenience we also denote
Theorem 16. Suppose that there exist positive constants , , , and such that . Assume that the following hypotheses (C1)–(C8) hold: (C1) is continuous;(C2) function is left-dense continuous;(C3) is continuous;(C4) is continuous, for all ;(C5) is continuous and there are such that
(C6), uniformly in ;(C7) ;(C8), uniformly in . Then, for each the boundary value problem (69) has at least three positive solutions , , and verifying
Proof. The proof passes by three lemmas.
Lemma 17. The following relations hold:
Proof. Using condition (C6) for such that , we have Applying condition (C7) we get for such that , . Put . Then for and from (75) we have Then . Similarly one can show that .
Lemma 18. The set is nonempty, and
Proof. Applying hypothesis we have