Dynamic Boundary Value Problems under Barrier Strips Condition

doi:10.1155/2011/378686

Research Article

On the Existence of Solutions for

Dynamic Boundary Value Problems under Barrier Strips Condition

Hua Luo

and Yulian An

1School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics, Dalian 116025, China

2Department of Mathematics, Shanghai Institute of Technology, Shanghai 200235, China

Correspondence should be addressed to Hua Luo,[email protected] Received 24 November 2010; Accepted 20 January 2011

Academic Editor: Jin Liang

Copyrightq2011 H. Luo and Y. An. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

By defining a new terminology, scatter degree, as the supremum of graininess functional value, this paper studies the existence of solutions for a nonlinear two-point dynamic boundary value problem on time scales. We do not need any growth restrictions on nonlinear term of dynamic equation besides a barrier strips condition. The main tool in this paper is the induction principle on time scales.

1. Introduction

Calculus on time scales, which unify continuous and discrete analysis, is now still an active area of research. We refer the reader to 1–5 and the references therein for introduction on this theory. In recent years, there has been much attention focused on the existence and multiplicity of solutions or positive solutions for dynamic boundary value problems on time scales. See6–17for some of them. Under various growth restrictions on nonlinear term of dynamic equation, many authors have obtained many excellent results for the above problem by using Topological degree theory, fixed-point theorems on cone, bifurcation theory, and so on.

In 2004, Ma and Luo18firstly obtained the existence of solutions for the dynamic boundary value problems on time scales

x^ΔΔt f

t, xt, x^Δt

, t∈0,1Ì,

x0 0, x^Δσ1 0 1.1

under a barrier strips condition. A barrier stripP is defined as follows. There are pairstwo or fourof suitable constants such that nonlinear termft, u, pdoes not change its sign on sets of the form0,1Ì×−L, L×P, whereLis a nonnegative constant, andP is a closed interval bounded by some pairs of constants, mentioned above.

The idea in18was from Kelevedjiev19, in which discussions were for boundary value problems of ordinary differential equation. This paper studies the existence of solutions for the nonlinear two-point dynamic boundary value problem on time scales

x^ΔΔt f

t, x^σt, x^Δt , t∈

a, ρ²b

Ì

,

x^Δa 0, xb 0, 1.2

where is a bounded time scale with a inf , b sup , anda < ρ²b. We obtain the existence of at least one solution to problem1.2without any growth restrictions onf but an existence assumption of barrier strips. Our proof is based upon the well-known Leray- Schauder principle and the induction principle on time scales.

The time scale-related notations adopted in this paper can be found, if not explained specifically, in almost all literature related to time scales. Here, in order to make this paper read easily, we recall some necessary definitions here.

A time scale is a nonempty closed subset of; assume that has the topology that it inherits from the standard topology on. Define the forward and backward jump operators

σ, ρ: → by

σt inf{τ > t|τ∈ }, ρt sup{τ < t|τ∈ }. 1.3

In this definition we put inf∅sup , sup∅inf . Setσ²t σσt, ρ²t ρρt. The sets ^kand kwhich are derived from the time scale are as follows:

k:

t∈ :tis not maximal or ρt t ,

k:{t∈ :tis not minimal orσt t}. 1.4

Denote intervalIon byIÌI∩ .

Definition 1.1. Iff : → is a function andt∈ ^k, then the delta derivative offat the point tis defined to be the numberf^Δt provided it existswith the property that, for eachε >0, there is a neighborhoodUoftsuch that

fσt−fs−f^Δtσt−s ε|σt−s| 1.5

for alls∈U. The functionfis calledΔ-differentiable on ^kiff^Δtexists for allt∈ ^k. Definition 1.2. IfF^Δfholds on ^k, then we define the CauchyΔ-integral by

t

sfτΔτFt−Fs, s, t∈ ^k. 1.6

Lemma 1.3see2, Theorem 1.16SUF. IffisΔ-differentiable att∈ ^k, then

fσt ft σt−tf^Δt. 1.7

Lemma 1.4see18, Lemma 3.2. Suppose thatf : a, bÌ → isΔ-differentiable ona, b^kÌ, then

ifis nondecreasing ona, bÌif and only iff^Δt≥0, t∈a, b^kÌ, iifis nonincreasing ona, bÌif and only iff^Δt≤0, t∈a, b^kÌ.

Lemma 1.5see4, Theorem 1.4. Let be a time scale withτ ∈ . Then the induction principle holds.

Assume that, for a family of statements At, t ∈ τ, ∞Ì, the following conditions are satisfied.

1Aτholds true.

2For eacht∈τ, ∞Ìwithσt> t, one hasAt⇒Aσt.

3For eacht∈τ, ∞Ìwithσt t, there is a neighborhoodUoftsuch thatAt⇒As for alls∈U, s > t.

4For eacht∈τ, ∞^Ìwithρt t, one hasAsfor alls∈τ, t^Ì⇒At.

ThenAtis true for allt∈τ, ∞Ì.

Remark 1.6. Fort∈−∞, τÌ, we replaceσtwithρtandρtwithσt, substitute<for>, then the dual version of the above induction principle is also true.

ByC²a, b, we mean the Banach space of second-order continuousΔ-differentiable functionsx:a, bÌ→ equipped with the norm

xmax

|x|₀,x^Δ

0,x^ΔΔ

0

, 1.8

where |x|₀ max_t∈a,bÌ|xt|, |x^Δ|₀ max_t∈a,ρb

Ì|x^Δt|, |x^ΔΔ|₀ max_t∈a,ρ²_b

Ì|x^ΔΔt|.

According to the well-known Leray-Schauder degree theory, we can get the following theorem.

Lemma 1.7. Suppose thatfis continuous, and there is a constantC >0, independent ofλ∈0,1, such thatx< Cfor each solutionxtto the boundary value problem

x^ΔΔt λf

t, x^σt, x^Δt , t∈

a, ρ²b

Ì

,

x^Δa 0, xb 0. 1.9

Then the boundary value problem1.2has at least one solution inC²a, b.

Proof. The proof is the same as18, Theorem 4.1.

2. Existence Theorem

To state our main result, we introduce the definition of scatter degree.

Definition 2.1. For a time scale , define the right direction scatter degreeRSDand the left direction scatter degreeLSDon by

r sup

σt−t:t∈ ^k , l sup

t−ρt:t∈ k ,

2.1

respectively. Ifr l , then we callr orl the scatter degree on .

Remark 2.2. 1If , then r l 0. If h : {hk : k ∈ , h > 0}, then r l h. If q^Æ : {q^k : k ∈ } and q > 1, thenr l ∞.2If is bounded, then bothr andl are finite numbers.

Theorem 2.3. Letf :a, ρbÌײ → be continuous. Suppose that there are constantsLi, i 1,2,3,4, withL2> L1≥0,L3< L4 ≤0 satisfying

H1L2> L1 Mr , L3< L4−Mr ,

H2ft, u, p≤0 fort, u, p∈a, ρbÌ×−L2b−a,−L3b−a×L1, L₂,ft, u, p≥0 fort, u, p∈a, ρbÌ×−L2b−a,−L3b−a×L3, L₄,

where

Msupf

t, u, p: t, u, p

∈ a, ρb

Ì×−L2b−a,−L3b−a×L3, L₂

. 2.2

Then problem1.2has at least one solution inC²a, b.

Remark 2.4. Theorem 2.3extends19, Theorem 3.2even in the special case . Moreover, our method to proveTheorem 2.3is different from that of19.

Remark 2.5. We can find some elementary functions which satisfy the conditions in Theorem 2.3. Consider the dynamic boundary value problem

x^ΔΔt −

x^Δt3 h

t, x^σt, x^Δt , t∈

a, ρ²b

Ì

,

x^Δa 0, xb 0, 2.3

whereht, u, p:a, ρbÌײ → is bounded everywhere and continuous.

Suppose thatft, u, p −p³ ht, u, p, then fort∈a, ρbÌ

f t, u, p

−→ −∞, ifp−→ ∞, f

t, u, p

−→ ∞, ifp−→ −∞. 2.4

It implies that there exist constantsLi, i1,2,3,4, satisfyingH1andH2inTheorem 2.3.

Thus, problem2.3has at least one solution inC²a, b.

Proof ofTheorem 2.3. DefineΦ: → as follows:

Φu

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−L2b−a, u≤ −L2b−a,

u, −L2b−a< u <−L3b−a,

−L3b−a, u≥ −L3b−a.

2.5

For allλ∈0,1, suppose thatxtis an arbitrary solution of problem x^ΔΔt λf

t,Φx^σt, x^Δt , t∈

a, ρ²b

Ì

,

x^Δa 0, xb 0. 2.6

We firstly prove that there existsC >0, independent ofλandx, such thatx< C.

We show at first that

L₃< x^Δt< L₂, t∈ a, ρb

Ì

. 2.7

LetAt:L₃ < x^Δt< L₂, t ∈a, ρbÌ. We employ the induction principle on time scalesLemma 1.5to show thatAtholds step by step.

1From the boundary conditionx^Δa 0 and the assumption ofL₃ < 0 < L₂,Aa holds.

2For eacht∈a, ρbÌwithσt> t, suppose thatAtholds, that is,L₃ < x^Δt <

L₂. Note that−L2b−a ≤ Φx^σt ≤ −L3b−a; we divide this discussion into three cases to prove thatAσtholds.

Case 1. IfL₄< x^Δt< L₁, then fromLemma 1.3,Definition 2.1, andH1there is x^Δσt x^Δt x^ΔΔtσt−t

< L1 Mr

< L₂.

2.8

Similarly,x^Δσt> L₄−Mr > L₃.

Case 2. IfL1≤x^Δt< L2, then similar to Case1we have x^Δσt x^Δt x^ΔΔtσt−t

> L₄−Mr

> L₃.

2.9

Suppose to the contrary thatx^Δσt≥L2, then

λf

t,Φx^σt, x^Δt

x^ΔΔt x^Δσt−x^Δt

σt−t >0, 2.10

which contradictsH2. Sox^Δσt< L2.

Case 3. IfL₃< x^Δt≤L₄, similar to Case2, thenL₃< x^Δσt< L₂holds.

Therefore,Aσtis true.

3For eacht∈a, ρbÌ, withσt t, andAtholds, then there is a neighborhood Uoftsuch thatAsholds for alls∈U, s > tby virtue of the continuity ofx^Δ. 4For each t ∈ a, ρbÌ, withρt t, and As is true for alls ∈ a, tÌ, since

x^Δt lims→t,s<tx^Δsimplies that

L₃≤x^Δt≤L₂, 2.11

we only show thatx^Δt/L₂andx^Δt/L₃. Suppose to the contrary thatx^Δt L₂. From

x^Δs< L2, s∈a, tÌ, 2.12

ρt t, and the continuity ofx^Δ, there is a neighborhoodVoftsuch that

L1< x^Δs< L2, s∈a, t^Ì∩V. 2.13

SoL₁ < x^Δs≤L₂, s∈a, tÌ∩V. Combining with−L2b−a≤Φx^σs≤ −L3b−a, s∈ a, tÌ∩V, we have fromH2,x^ΔΔs λfs,Φx^σs, x^Δs≤0, s∈a, tÌ∩V. So from Lemma 1.4

x^Δs≥x^Δt L₂, s∈a, tÌ∩V. 2.14

This contradiction shows thatx^Δt/L₂. In the same way, we claim thatx^Δt/L₃. Hence,At:L₃< x^Δt< L₂, t∈a, ρbÌ, holds. So

x^Δ

0< C₁:max{−L3, L₂}. 2.15

FromDefinition 1.2andLemma 1.3, we have fort∈a, ρbÌ

xt x ρb

− ^ρb

t

x^ΔsΔs

xb−x^Δ

ρb

b−ρb

− ^ρb

t x^ΔsΔs.

2.16

There are, fromxb 0 and2.7, xt<−L3

b−ρb

−L₃

ρb−t

≤ −L3b−a, xt>−L2

b−ρb

−L₂

ρb−t

≥ −L2b−a 2.17

fort∈a, ρbÌ. In addition,

−L2b−a< xb 0<−L3b−a. 2.18

Thus,

−L2b−a< xt<−L3b−a, t∈a, b^Ì, 2.19

that is,

|x|₀< C1b−a. 2.20

Moreover, by the continuity off, the equation in2.6,2.7and the definition ofΦ x^ΔΔ

0< M, 2.21

whereMis defined in2.2. Now letC max{C1, C₁b−a, M}. Then, from2.15,2.20, and2.21,

x< C. 2.22

Note that from2.19we have

−L2b−a< x^σt<−L3b−a, t∈ a, ρb

Ì

, 2.23

that is,Φx^σt x^σt, t∈a, ρbÌ. Soxis also an arbitrary solution of problem x^ΔΔt λf

t, x^σt, x^Δt , t∈

a, ρ²b

Ì

,

x^Δa 0, xb 0. 2.24

According to2.22andLemma 1.7, the dynamic boundary value problem1.2has at least one solution inC²a, b.

3. An Additional Result

Parallel to the definition of delta derivative, the notion of nabla derivative was introduced, and the main relations between the two operations were studied in7. Applying to the dual

version of the induction principle on time scalesRemark 1.6, we can obtain the following result.

Theorem 3.1. Letg:σa, bÌײ → be continuous. Suppose that there are constantsIi, i 1,2,3,4, withI2> I1≥0,I3< I4≤0 satisfying

S1I₂ > I₁ Nl , I3< I₄−Nl ,

S2gt, u, p≥0 fort, u, p∈σa, bÌ×I3b−a, I2b−a×I1, I₂,gt, u, p≤0 for t, u, p∈σa, b^Ì×I3b−a, I2b−a×I3, I4,

where

Nsupg

t, u, p: t, u, p

∈σa, b^Ì×I3b−a, I2b−a×I3, I2

. 3.1

Then dynamic boundary value problem x^∇∇t g

t, x^ρt, x^∇t , t∈

σ²a, b

Ì

,

xa 0, x^∇b 0 3.2

has at least one solution.

Remark 3.2. According toTheorem 3.1, the dynamic boundary value problem related to the nabla derivative

x^∇∇t

x^∇t3 k

t, x^ρt, x^∇t , t∈

σ²a, b

Ì

,

xa 0, x^∇b 0 3.3

has at least one solution. Herekt, u, p: σa, bÌײ → is bounded everywhere and continuous.

Acknowledgments

H. Luo was supported by China Postdoctoral Fund no. 20100481239, the NSFC Young Item no. 70901016, HSSF of Ministry of Education of Chinano. 09YJA790028, Program for Innovative Research Team of Liaoning Educational Committee no. 2008T054, and Innovation Method Fund of China no. 2009IM010400-1-39. Y. An was supported by 11YZ225 and YJ2009-16A06/1020K096019.

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