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On the existence of positive solutions for the one-dimensional p-laplacian boundary value problems on time scales.

1. INTRODUCTION

Recently, dynamic equations on time scales have generated a considerable amount of interest and attracted many researchers. They can not only unify differential and difference equations [11], but also have exhibited much more complicated dynamics [5]. Further they have led to several important applications e.g., in the study of insect population models, stock market, wound healing, and epidemic models [6, 13, 16].

In this paper, by using different method, we will discuss the existence of at least three positive solutions to the following p-Laplacian boundary value problem on time scales

(1.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[phi].sub.p] u) is p-Laplacian operator, i.e., [[phi].sub.p](u) = [[absolute value of u].sup.p-2]u, for p > 1, with [([[phi].sub.p]).sup.-1] = and 1/p + 1/q = 1. The usual notation and terminology for time scales as can be found in [5, 6], will be used here.

Agarwal and O'Regan [2] studied the existence of one or more solutions to nonlinear equations on time scales. They established by using either a nonlinear alternative of Leray-Schauder type or Krasnoselski's fixed point theorem in a cone. Dogan et al. [7] considered some existence criteria for positive solutions of a higher order semi-positone multi-point boundary value problem on a time scale. We also discussed applications to some special problems. He and Jiang [10] investigated the existence of at least three positive solutions of boundary value problems for p- Laplacian dynamic equations on time scales by applying a new triple fixed-point theorem. Hong [12] presented sufficient conditions for the existence of at least three positive solutions of three-point boundary value problems for p-Laplacian dynamic equations on a time scales. To show his main results, he applied a new fixed point theorem due to Avery and Peterson. Su et al. [19] considered the three-point boundary value problem for p-Laplacian dynamic equations on time scales. They proved that the boundary value problem has at least three positive pseudo-symmetric solutions under some assumptions by using a pseudo-symmetric technique and the five-functionals fixed point theorem.

In recent years, there have been many papers working on the existence of positive solutions for p-Laplacian boundary value problems for differential equations on time scales, see, for example [3, 10, 12, 15, 17, 18, 19, 20, 21]. However, to best of our knowledge, there are not much concerning the p-Laplacian boundary value problems on time scales when the nonlinear term f is involved with the first-order delta derivative [8, 9].

We assume the following conditions hold through the paper:

(H1) f [member of] [C.sub.ld]((0,T) X [R.sup.2], (0, [infinity])), 0, T [member of] T,

(H2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(H3) [B.sub.0] is continuous function defined on R and satisfies that there exist A [greater than or equal to] 1 and B > 0 such that Bv [less than or equal to] [B.sub.0](u) [less than or equal to] Av, for all v [member of] [0, + [infinity]).

2. PRELIMINARIES

Let E = [C.sup.1.sub.ld][0,T] with the norm

[parallel]u[parallel] = max{[[parallel]u[parallel].sub.0], [[parallel]u.sup.[DELTA][parallel].sub.0]}

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; clearly E is Banach space. Choose the cone P [subset] E defined

by

P = {u [member of] E : u(t) [greater than or equal to] 0, for t [member of] [[0, T].sub.T]; [u.sup.[DELTA][nabla]](t) [less than or equal to] 0, [u.sup.[DELTA]](t) [greater than or equal to] 0, for t [member of] [[0, T].sub.T]}.

We note that u(t) is a solution of (1.1), if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Define a completely continuous integral operator F : P [right arrow] E

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Clearly, [parallel]Fu[parallel] = max{(Fu)(0), [absolute value of [(Fu).sup.[DELTA]](T)]} = [T.sub.0] (Fu)(0), where [T.sub.0] = max {T, 1}.

Lemma 2.1. FP [subset] P

Proof. In fact

[(Fu).sup.DELTA]](t) = [[phi].sub.q] ([[integral].sup.T.sub.t] a(r)f(r,u(r),[u.sup.[DELTA]](r))[nabla]r)[greater than or equal to] 0.

Moreover, [[phi].sub.q] (x) is a monotone decreasing and continuously differential function and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

we have [(Fu).sup.[delta][nabla]](t) [less than or equal to] 0, therefore FP [subset] P.

Lemma 2.2. F : P [right arrow] P is completely continuous.

Proof. Firstly, we will show that F maps a bounded set into itself. Suppose c > 0 is a constant and u [member of] [[bar.P].sub.c] = {u [member of] P : [parallel]u[parallel] [less than or equal to] c}, and then [absolute value of u] [less than or equal to] c, [absolute value of v] [less than or equal to] c; notice that f (t,u,v) is continuous, therefore there exist a constant C > 0 such that f (t,u,v) [less than or equal to] [[phi].sub.p](C), and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

That is F [bar.P] is uniformly bounded. On the other hand,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

therefore F is equicontinuous on [[0,T].sub.T]; then by applying the Arzela-Ascoli theorem on time scales [1], we know that F[bar.P] is relatively compact. Using Lebesque's dominated convergence theorem on time scales [4], F is completely continuous on [[0,T].sub.T].

Let a,b,r > 0 be constants, [P.sub.r] = {u [member of] P : [parallel]u[parallel] < r}, P([alpha],a,b) = {u [member of] P : [alpha](u) [greater than or equal to] a, [parallel]u[parallel] < b}.

To prove our main results, we need the following Leggett-William fixed point Theorem [14].

Theorem 2.3 (Leggett-Williams). Let F : [[bar.P].sub.c] [right arrow] [[bar.P].sub.c] be a completely continuous map and [alpha] be a nonnegative continuous concave functional on P such that [alpha](u) [greater than or equal to] [parallel]u[parallel], [for all] [member of] [[bar.P].sub.c]. Assume there exist a, b, d with 0 < a < b < d [less than or equal to] such that

(A1) {u [member of] P([alpha], b, d) : [alpha](u) > b}[not equal to] = [empty set], and [alpha(Fu) > b for all u [member of] P([alpha], b, d)A;

(A2) [parallel]Fu[parallel] < a, for all [member of] [[bar.P].sub.a];

(A3) [alpha](Fu) > b for all u [member of] P([alpha],b,c) with [parallel]Fu[parallel] > d.

Then F has at least three fixed points [u.sub.1],[u.sub.2],[u.sub.3] satisfying

[parallel][u.sub.1] [parallel] < a, b < [alpha]([u.sub.2]), [parallel][u.sub.3][parallel] > a, [alpha]([u.sub.3]) < b.

3. MAIN RESULTS

We define the nonnegative continuous concave functional [alpha] : P [right arrow] [0, [infinity]) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Clearly, the following two conclusions hold:

(i) [alpha](u) = u(l) [less than or equal to] [parallel]u[parallel], [for all]u [member of] P;

(ii) [alpha](Fu) = u(l).

Theorem 3.1. Assume that there exist constants a,b,c,d such that 0 < a < b [less than or equal to] (1+B)m/[MBT.sub.0] d < d [less than or equal o] c and suppose that f satisfies the following conditions:

(B1) f(t,u,v) [less than or equal to] [[phi].sub.p] ([aT.sup.1/1-p]/[MT.sup.0]) for (t,u,v) [member of] [[0,T].sub.T] x [0,a] x [-a,0];

(B2) f(t,u,v) [less than or equal to] [[phi].sub.p] ([cT.sup.1/1-p]/[MT.sup.0]) for (t,u,v) [member of] [[0,T].sub.T] x [0,a] x [-c,0];

(B3) f(t,u,v) > [[phi].sub.p] (b/lk) for (t,u,v) [member of] [[0,T-Z].sub.T] x [b, d] x [-d,0];

(B4) min{f(t,u,v)} [[phi].sub.p] (M/m) [integral].sup.T-1.sub.0] a(t)[nabla]t [greater than or equal to] max{f(t, u, v)}[[integral].sup.T.sub.0] a(t)[nabla]t for (t, u, v) [member of] [[0, T].sub.T] x [0, c] x [-c, 0].

Then the boundary value problem (1.1) has at least three positive solutions [u.sub.1],[u.sub.2] and [u.sub.3] satisfying

[parallel][u.sub.1][parallel] < a, b < [alpha]([u.sub.2]), [parallel][u.sub.3][parallel] > a, [alpha]([u.sub.3]) < b.

Proof. First, we show that there exists a positive number c > d such that F[[bar.P].sub.c] [subset] [[bar.P].sub.c], F[[bar.P].sub.a] [subset] [[bar.P].sub.a]. From Lemma 2.1 F[[bar.P].sub.c] [subset] [[bar.P].sub.c], and then [for all]u [member of] [[bar.P].sub.c], from B2, we have 0 [less than or equal to] u [less than or equal to] c, -c [less than or equal to] v [less than or equal to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similarly, Fu [member of] [[bar.P].sub.a] for all u [member of] [[bar.P].sub.a].

Second, we show {u [member of] P([alpha], b, d) : [alpha](u) > b} [not equal to] [empty set], and [alpha](Fu) > b for all u [member of] P([alpha], b, d). In fact, set u = b+d/2, [parallel]u[parallel] = = b+d/2 [less than or equal to] d and [alpha](u) > b. Therefore {u [member of] P([alpha], b, d) : [alpha](u) > b} [not equal to] [empty set]. On the other hand, [for all]u [member of] P([alpha], b, d), we get b [less than or equal to] u [less than or equal to] d, -d [less than or equal to] v [less than or equal to] 0, and for t [member of] [[0, T - l].sub.T], from B3,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence [alpha](Fu) > b for u [member of] P([alpha], b, d).

Finally, we show [alpha](Fu) > b for all u [member of] P([alpha], b, d) and [parallel]Fu[parallel] > d. If u [member of]G ([alpha], b, d) and [parallel]Fu[parallel] > d, then 0 [less than or equal to] u [less than or equal to] c, -c [less than or equal to] v [less than or equal to] 0, and from B4,

[[phi].sub.p] (M/m) [[integral].sup.T-l.sub.0] a(r)f(r, u(r), [u.sup.[DELTA]] (r))[nabla]r [greater than or equal to] [[integral].sup.T.sub.0] a(r)f(r, u(r), [u.sup.[DELTA]] (r)) [nabla]r,

i.e.

[[integral].sup.T-l.sub.0] a(r)f(r, u(r), [u.sup.[DELTA]] (r))[nabla]r [greater than or equal to] [[integral].sup.T.sub.0] a(r)f(r, u(r), [u.sup.[DELTA]](r)) [nabla]r/[[phi].sub.p] (M/m)

Because Fu [member of] P,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and then [alpha](Fu) > b for all u [member of] P ([alpha], b, d) and [parallel]Fu[parallel] > d. Hence, an application of Theorem 2.3 completes the proof.

Theorem 3.2. Assume that there exist constants a, b, c, d such that 0 < a < b [less than or equal to] (l+B)m/[MBT.sub.0] d < d [less than or equal to] c and suppose that f satisfies (B1)-(B3) and

(C1) a(t) is decreasing for t [member of] [[0, T].sub.T],

(C2) [[phi].sub.p](M/m) [greater than or equal to] (T-l)[f.sub.M]/[lf.sub.m],

where

[f.sub.M] = max{f (t, u, v)} for (t, u, v) [member of] [[l, T].sub.T] x [0, c] x [-c, 0],

[f.sub.m] = min{f (t, u, v)} for (t, u, v) [member of] [[0, l].sub.T] x [0, c] x [-c, 0].

Then the boundary value problem (1.1) has at least three positive solutions [u.sub.1], [u.sub.2] and [u.sub.3] satisfying

[parallel][u.sub.1][parallel] < a, b < [alpha]([u.sub.2]), [parallel][u.sub.3][parallel] > a, [alpha]([u.sub.3]) < b.

Proof. We only show [alpha](Fu) > b for all u [member of] P ([alpha], b, c) and [parallel]Fu[parallel] > d. [for all]u [member of] P ([alpha], b, c), we have 0 [less than or equal to] u [less than or equal to] c, -c [less than or equal to] v [less than or equal to] 0, and from (C1), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From (C2), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the rest of the proof is similar to final step of Theorem 3.1 so we omit it.

4. EXAMPLE

Let T = R, T = 1, p = 3, then [T.sub.0] = 1. Consider the following boundary value problem

(4.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By the definition of a(t), we know m = 1/2000 and M = 1. It is obvious that A = B = 1/2. Choose l = 11/25, a direct calculation shows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If we take a = 1/2, b = 1, d = 1250, c = 1260, then

0 < 1/2 < 1 < (11/25 + 1/2) x 1/2000/1/2 x 1250 < 1250 < 1260,

we have

[[phi].sub.3](a/M) = [(1/2).sup.2] 1/4, [[phi].sub.3](c/M) = [1260.sup.2], [[phi].sub.3](b/lK) = 25 x [22.sup.2] x [25.sup.2]/[11.sup.2] x [47.sup.2] x 14,

then the nonlinear term f satisfies

(B1) f(t, u, v) [less than or equal to] 10x[(1/2).sup.5] + 1/90 < 1/4 = [[phi].sub.3](a/M), for 0 [less than or equal to] t [less than or equal to] 1, 0 [less than or equal to] u [less than or equal to] 1/2, -1/2 [less than or equal to]v [less than or equal to] 0:

(B2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(B3) f(t, u, v) [less than or equal to] 10 + 1/180 > 25 x [22.sup.2] x [25.sup.2]/[11.sup.2] x [47.sup.2] x 14 = 19963 = [[phi].sub.3](b/lK) for 0 [less than or equal to] t [less than or equal to] 144/2255, 1 [less than or equal to] u [less than or equal to] 1250, -1250 [less than or equal to] v [less than or equal to] 0;

(B4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus by Theorem 3.1, we find that boundary value problem (4.1) has at least three positive solutions.

Acknowledgments. The author would like to thank the anoymous referees and editor for their helpful comments and suggestions.

Received April 2, 2015

ABDULKADIR DOGAN

Department of Applied Mathematics, Faculty of Computer Sciences, Abdullah Gul University, Kayseri, 38039 Turkey

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Author:Dogan, Abdulkadir
Publication:Dynamic Systems and Applications
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Date:Sep 1, 2015
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