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.


Introduction
The study of dynamic equations on time scales goes back to the 1989 Ph.D. thesis of Stefan Hilger [23,24], and is currently an area of mathematics receiving considerable recent attention [2,20,21,31,41]. 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 [6,13,14,25,32].
Boundary value p-Laplacian problems for differential equations and finite difference equations have been studied extensively (see, e.g., [4] and references therein). Although many existence results for dynamic equations on time scales are available [8,26], there are not many results concerning p-Laplacian problems on time scales [9,35,36,39]. In this paper we prove new existence results for three classes of p-Laplacian boundary value problems on time scales. In contrast with our previous works [35] and [36], 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 [27,45] 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 [5], and is now recognized as an important tool to prove existence of positive solutions for boundary value problems on time scales [12,29,33,42,43,44].
The paper is organized as follows. In Section 2 we present some necessary results from the theory of time scales ( §2.1) and the theory of cones in Banach spaces ( §2.2). We end §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 in three parts. The purpose of the first part ( §3.1) is to prove existence of positive solutions to the nonlocal p-Laplacian dynamic equation on time scales satisfying the boundary conditions is the p-Laplacian operator defined by φ p (s) = |s| p−2 s, p > 1, and (φ p ) −1 = φ q with q the Holder conjugate of p, i.e., 1 p + 1 q = 1. The concrete value of p is connected with the application at hands. For p = 2, for example, problem (1)-(2) describes the operation of a device flowed by an electric current, e.g., thermistors [37], 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 flexible in size and shape. Constant λ in (1) is a dimensionless parameter that can be identified with the square of the applied potential difference at the ends of a conductor; f (u) is the temperature dependent resistivity of the conductor; and β in (2) is a transfer coefficient supposed to verify 0 < β < 1. For a more detailed discussion about the physical justification of equations (1)-(2) the reader is referred to [35]. Theoretical analysis (existence, uniqueness, regularity, and asymptotic results) for thermistor problems with various types of boundary and initial conditions have received significant attention in the last few years for the particular case T = R [10,16,19,34,40]. The second part of our results ( §3.2) is concerned with the following quasilinear elliptic problem: where η ∈ (0, T ) T . Results on existence of radially infinity many 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 f satisfying appropriate conditions [7,15]; (ii) methods based on phase-plane analysis and the shooting method [17]; (iii) the technique of time maps [18]. For p = 2, h ≡ 0, and T = R, 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 T, p = 2, and h not identically zero. Finally, the third part of our contribution ( §3.3) is devoted to the existence of positive solutions to the p-Laplacian dynamic equation on a time scale T such that 0, T ∈ T κ κ , −r ∈ T with −r ≤ 0 < T , and where λ > 0. This problem is considered in [38] where the authors apply the Krasnoselskii fixed point theorem to obtain one positive solution to (4).
Here we use the same conditions as in [38], but applying Leggett-Williams' theorem we are able to obtain more: we prove existence of at least three positive solutions.

Preliminaries
Here we just recall the basic concepts and results needed in the sequel. For an introduction to time scales the reader is refereed to [1,2,3,6,13,14] and references therein; for a good introduction to the theory of cones in Banach spaces we refer the reader to the book [22].

Time Scales.
A time scale T is an arbitrary nonempty closed subset of the real numbers R. The operators σ and ρ from T to T are defined in [23,24] as and are called the forward jump operator and the backward jump operator, respectively. A point t ∈ T is left-dense, left-scattered, right-dense, rightscattered if ρ(t) = t, ρ(t) < t, σ(t) = t, σ(t) > t, respectively. If T has a right scattered minimum m, define T κ = T − {m}; otherwise set T κ = T. If T has a left scattered maximum M , define T κ = T − {M }; otherwise set T κ = T. Following [30], we also introduce the set T κ κ = T κ ∩ T κ . Let f : T → R and t ∈ T κ (assume t is not left-scattered if t = sup T), then the delta derivative of f at the point t is defined to be the number f ∆ (t) (provided it exists) with the property that for each ǫ > 0 there is a neighborhood U of t such that Similarly, for t ∈ T κ (assume t is not right-scattered if t = inf T), the nabla derivative of f at the point t is defined in [11] to be the number f ∇ (t) (provided it exists) with the property that for each ǫ > 0 there is a neighborhood U of t such that A function f is left-dense continuous (i.e., ld-continuous), if f is continuous at each left-dense point in T and its right-sided limit exists at each We define right-dense continuous (rd-continuous) functions in a similar way.

Cones in Banach Spaces.
In this article T is a time scale with 0 ∈ T κ and T ∈ T κ . We use R + and R + 0 to denote, respectively, the set of positive and nonnegative real numbers.
Definition 2.1. Let E be a real Banach space. A nonempty, closed, convex set P ⊂ E is called a cone if it satisfies the following two conditions: (i) u ∈ P , λ ≥ 0, implies λu ∈ P ; (ii) u ∈ P , −u ∈ P , implies u = 0.
Every cone P ⊂ E induces an ordering in E given by Definition 2.2. Let E be a real Banach space and P ⊂ E be a cone. A function α : P → R + 0 is called a nonnegative continuous concave functional if α is continuous and for all x, y ∈ P and 0 ≤ t ≤ 1.
Let a, b, r > 0 be constants, P r = {u ∈ P | u < r}, P (α, a, b) = {u ∈ P | a ≤ α(u), u < b}. The following fixed point theorem provides the existence of at least three positive solutions. The origin in E is denoted by ∅. The proof of the Leggett-Williams fixed point theorem can be found in Guo and Lakshmikantham [22] or Leggett and Williams [28]. Theorem 2.3 (Leggett-Williams' theorem). Let P be a cone in a real Banach space E. Let G : P c → P c be a completely continuous map and α a nonnegative continuous concave functional on P such that α(u) ≤ u ∀u ∈ P c . Suppose there exist a, b, d > 0 with 0 < a < b < d ≤ c such that Then G has at least three fixed points u 1 , u 2 and u 3 satisfying

Main Results
We prove existence of three positive solutions to different p-Laplacian problems on time scales: in §3.1 we study problem (1)-(2); in §3.2 problem (3); and finally (4) in §3.3.
3.1. Nonlocal Thermistor Problem. By a solution u : T → R of (1)-(2) we mean a delta differentiable function such that u ∆ and |u ∆ | p−2 u ∆ ∇ are both continuous on T κ κ and u satisfies (1)- (2). We consider the following hypothesis: (H1) f : R → R + is a continuous function.
is non increasing, which implies with the monotonicity of φ p that u ∆ is a non increasing function on (0, T ) T . Hence, u is concave. In order to apply Theorem 2.3, let us define the cone P ⊂ E by We also define the nonnegative continuous concave functional α : P → R + 0 by α(u) = min It is easy to see that (1)-(2) has a solution u = u(t) if and only if u is a fixed point of the operator G : P → E defined by where g and B are as in Lemma 3.1.
(i) holds clearly from above. (ii) Suppose that D ⊆ P is a bounded set and let u ∈ D. Then, In the same way, we have It follows that As a consequence, we get Then G(D) is bounded on the whole bounded set D. Moreover, if t 1 , t 2 ∈ [0, T ] T and u ∈ D, then we have for a positive constant c that We see that the right hand side of the above inequality goes uniformly to zero when |t 2 −t 1 | → 0. Then by a standard application of the Arzela-Ascoli theorem we have that G : P → P is completely continuous.
We can also easily obtain the following properties: We now state the main result of §3.1.
Proof. The proof passes by several lemmas. We have already seen in Lemma 3.3 that the operator G is completely continuous. We now show that Proof. Obviously, GP a 1 ⊂ P . Moreover, ∀u ∈ P a 1 , we have 0 ≤ u(t) ≤ a 1 .
On the other hand we have and ∀u ∈ GP a 1 we have 0 ≤ u(t) ≤ a 1 . Then, Using (H2) it follows that φ q (g(s)) ≤ aT φ q 1 T .
Then we get |Gu| ≤ a 1 and GP a 1 ⊂ P a 1 .
Proof. Let u = b+d 2 . Then, u ∈ P, u = b+d 2 ≤ d and α(u) ≥ b+d 2 > b. The first part of the lemma is proved.

3.2.
Quasilinear Elliptic Problem. We are interested in this section in the study of the following quasilinear elliptic problem: where η ∈ (0, T ) T . We assume the following hypotheses: Similarly as in §3.1, we prove existence of solutions by constructing an operator whose fixed points are solutions to (8). The main ingredient is, again, the Leggett-Williams fixed point theorem (Theorem 2.3). We can easily see that (8) is equivalent to the integral equation On the other hand, we have −(φ p (u ∆ )) ∇ = f (u(t))+h(t). Since f , h ≥ 0, we have (φ p (u ∆ )) ∇ ≤ 0 and (φ p (u ∆ (t 2 ))) ≤ (φ p (u ∆ (t 1 ))) for any t 1 , t 2 ∈ [0, T ] T with t 1 ≤ t 2 . It follows that u ∆ (t 2 ) ≤ u ∆ (t 1 ) for t 1 ≤ t 2 . Hence, u ∆ (t) is a decreasing function on [0, T ] T . Then, u is concave. In order to apply Theorem 2.3 we define the cone P = {u ∈ E | u is nonnegative, increasing on [0, T ] T , and concave on E}.
For ξ ∈ 0, T 2 we also define the nonnegative continuous concave functional α : and the operator F : P → E by It is easy to see that (8) has a solution u = u(t) if and only if u is a fixed point of the operator F . For convenience, we introduce the following notation: and, in addition to (A1) and (A2), that f satisfies (A3) max 0≤u≤a f (u) ≤ φ p (aA); (A4) max 0≤u≤c f (u) ≤ φ p (cA); (A5) min b≤u≤d f (u) ≥ φ p (bB). Then problem (8) has at least three positive solutions u 1 , u 2 , and u 3 , verifying u 1 < a , b < α(u 2 ) , u 3 > a , and α(u 3 ) < b.
Proof. As done for Theorem 3.5, the proof is divided in several steps. We first show that F : P → P is completely continuous. Indeed, F is obviously continuous. Let U δ = {u ∈ P | u ≤ δ}. It is easy to see that for u ∈ U δ there exists a constant c > 0 such that |F u(t)| ≤ c. On the other hand, let t 1 , t 2 ∈ (0, T ) T , u ∈ U δ . Then there exists a positive constant c such that which converges uniformly to zero when |t 2 − t 1 | tends to zero. Using the Arzela-Ascoli theorem we conclude that F : P → P is completely continuous.
We now show that For all u ∈ P a 1 we have 0 ≤ u ≤ a 1 and

Using the elementary inequality
x p + y p ≤ 2 p−1 (x + y) p , and the form of A, it follows that Then F P a 1 ⊂ P a 1 . In a similar way we prove that F P c 1 ⊂ P c 1 .
Our following step consists in show that The first point is obvious. Let us prove (9).

A p-Laplacian Functional Dynamic Equation on
Time Scales with Delay. Let T be a time scale with 0, T ∈ T κ κ , −r ∈ T with −r ≤ 0 < T . We are concerned in this section with the existence of positive solutions to the p-Laplacian dynamic equation (12) φ p (u ∆ (t)) ∇ + λa(t)f (u(t), u(ω(t))) = 0 , t ∈ (0, T ) T , where λ > 0. We define X = C ld ([0, T ] T , R), which is a Banach space with the maximum norm u = max [0,T ] T |u(t)|. We note that u is a solution to (12) if and only if Let K = {u ∈ X | u is nonnegative and concave on E}. Clearly K is a cone in the Banach space X. For each u ∈ X, we extend u to [−r, 0] T with u(t) = ψ(t) for t ∈ [−r, 0] T . We also define the nonnegative continuous concave functional α : P → R + 0 by For t ∈ [0, T ] T , define Q : K → X as (13) λa(r)f (u(r), u(ω(r))) ∇r λa(r)f (u(r), u(ω(r)))∇r ∆s .   (14) is a positive solution to (12) satisfying
From (13) and (15) it follows that (i) Q(K) ⊂ K; (ii) Q : K → K is completely continuous; (iii) u(t) ≥ δ T +γ u , t ∈ [0, T ] T . Depending on the signature of the delay ω, we set the following two subsets of [0, T ] T : In the remainder of this section, we suppose that Y 1 is nonempty and Y 1 a(r)∇r > 0. For convenience we also denote l := φ p T 0 a(r)∇r λ q−1 (T + γ) , m := φ p T 0 a(r)∇r δλ q−1 .
Lemma 3.14. The following relations hold: QP a ⊂ P a , QP c ⊂ P c .
Then QP a ⊂ P a . Similarly one can show that QP c ⊂ P c . Proof. Applying hypothesis (C8) we have f (u, ψ(s)) > φ p (mu) for each s ∈ [−r, 0] T .