Solution Matching for a Second Order Boundary Value Problem on Time Scales

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Volume 2012, Article ID 465867,7pages doi:10.1155/2012/465867

Research Article

Solution Matching for a Second Order Boundary

Value Problem on Time Scales

Aprillya Lanz

1

_{and Ana Tameru}

2

1_{Department of Mathematics and Computer Science, Virginia Military Institute,}

Lexington, VA 24450, USA

2_{Department of Mathematics and Computer Science, Alabama State University,}

915 S. Jackson Street, Montgomery, AL 36101, USA

Correspondence should be addressed to Ana Tameru,[email protected]

Received 7 June 2012; Revised 9 August 2012; Accepted 9 August 2012

Academic Editor: Palle E. Jorgensen

Copyrightq2012 A. Lanz and A. Tameru. 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.

LetTbe a time scale such that a < b;a, b ∈ T. We will show the existence and uniqueness of solutions for the second-order boundary value problem yΔΔt ft, yt, yΔt, t ∈

a, bT, ya A, yb B, by matching a solution of the first equation satisfying boundary

conditions ona, c_Twith a solution of the first equation satisfying boundary conditions onc, b_T, wherec∈a, bT.

1. Introduction

The result discussed in this paper was inspired by the solution matching technique that was

first introduced by Bailey et al.1. In their work, they dealt with the existence and uniqueness

of solutions for the second-order conjugate boundary value problems

y_t_f_{t, y, y}_, _1.1

ya y1, yb y2. 1.2

As shown in their work, the uniqueness and existence of the solutions of 1.1,1.2were

obtained by matching a solutiony1of1.1satisfying the boundary condition

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with a solutiony2of1.1satisfying the boundary condition

y_c_m, _yb_y

2, 1.4

wheremy₁c y₂c.

Since the initial work by Bailey et al., there have been many studies utilizing the

solution matching technique on boundary value problems, see for example, Rao et al. 2,

Henderson3,4, Henderson and Taunton5.

Existence and Uniqueness for solutions of boundary value problems have quite a history for ordinary diﬀerential equations as well for diﬀerence equations, we mentioned

papers of Barr and Sherman 6, Hartman7, Henderson8,9, Henderson and Yin 10,

Moorti and Garner11, Rao et al.12and many others.

While many of the work mentioned above considered boundary value problems for diﬀerential and diﬀerence equations, our study applies the solution matching technique to

obtain a solution to a similar boundary value problem1.1,1.2on a time scale. The theory

of time scales was first introduced by Hilger13in 1990 to unify results in diﬀerential and

difference equations. Since then, there has been much activity focused on dynamic equations on time scales, with a good deal of this activity devoted to boundary value problems. Efforts have been made in the context of time scales, in establishing that some results for boundary value problems for ordinary differential equations and their discrete analogues are special cases of more general results on time scales. For the context of dynamic equations on time

scales, we mention the results by Bohner and Peterson14,15, Chyan16, Henderson4,

and Henderson and Yin17.

In this work,Tis assumed to be a nonempty closed subset ofRwith infT−∞and

supT ∞. We shall also use the convention on notation that for each intervalIofR,

ITI∩T. 1.5

For readers’ convenience, we state a few definitions which are basic to the calculus on

the time scaleT. The forward jump operatorσ:T → Ris defined by

σt inf{s > t|s∈T} ∈T. 1.6

Ifσt > t,tis said to be right-scattered, whereas, if σt t,tis said to be right-dense. The

backward jump operatorρ:T → Ris defined by

ρt sup{s < t|s∈T} ∈T. 1.7

Ifρt< t,tis said to be left-scattered, and ifρt t, thentis said to be left-dense. Ifg :T → R

andt∈T, then the delta derivative of g at t,gΔt, is defined to be the numberprovided that it

exists, with the property that, given anyε >0, there is a neighborhoodUoft, such that

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for alls ∈ U. In this definition,t ∈ TK, where this set is derived from the time scaleTas

follows: if Thas a left-scattered maximum m, then TK T\ {m}. Otherwise, we define

TK_T.

We say that the functionyhas a generalized zero attifyt 0 or ifyt·yσt<0.

In the latter case, we would say the generalized zero is in the real intervalt, σt.

Theorem 1.1Mean Value Theorem. Ify:T → Ris continuous andythas a generalized zero

ataandb,a, b∈T, then there exists a pointr∈a, b_Tsuch thatyΔhas a generalized zero atr.

Let T be a time scale such that a, b ∈ T. In this paper, we are concerned with the

existence and uniqueness of solutions of boundary value problems on the intervala, b_Tfor

the second-order delta derivative equation

yΔΔ_f_{t, y, y}Δ_, _1.9

satisfying the boundary conditions,

ya y1, yσb y2, 1.10

wherea < bandy1, y2∈R. Throughout this paper, we will assume

A1ft, r1, r2is a real-valued continuous function defined onT×R2.

We obtain solutions by matching a solution of1.9 satisfying boundary conditions

ona, c_T to a solution of1.9satisfying boundary conditions onc, b_T. In particular, we

will give suﬃcient conditions such that ify1tis a solution of1.9satisfying the boundary

conditions ya y1, yΔc mand y2t is a solution of 1.9 satisfying the boundary

conditionsyσb y2, yΔc m, the solutions of1.9is

yt y1t, t∈a, cT,

y2t, t∈c, bT.

1.11

Moreover, we will assume the following conditions throughout this paper.

A2Solutions of initial value problems for1.9are unique and extend throughoutT.

A3c∈Tis right dense and is fixed.

And the uniqueness of solutions assumptions are stated in terms of generalized zeros as follows:

A4For any t1 < t2 in T, if u and v are solutions of 1.9 such that u − v has a

generalized zero at t1 and u−vΔ has a generalized zero at t2, then u ≡ v on

T.

2. Uniqueness of Solutions

In this section, we establish that under conditions A1 through A4, solutions of the

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Theorem 2.1. LetA, B∈Rbe given and assume conditionsA1throughA4are satisfied. Then,

given m ∈ R, each of boundary value problems of 1.9 satisfying any of the following boundary

conditions has at most one solution.

ya;m A, yΔ_c;_m _{m, a < c,} _where_{a, c}_∈_T, _2.1

yb;m B, yΔ_c;_m_{m, c < b,} _where _{c, b}_∈_T. _2.2

Proof. Assume for somem∈R, there exists distinct solutionsαandβof1.9,2.1, and set

wα−β. Then, we have

wa 0, wΔ_c_0. _2.3

Clearly, sincea < c andw has a generalized zero ataandwΔ has a generalized zero atc,

this contradicts conditionA4. Hence, the boundary value problems1.9,2.1have unique

solutions.

Next, we will look at a special boundary value problem of 1.9 satisfying the

boundary condition

yσb;m C, yΔ_b;_m_m. _2.4

We will show the uniqueness of solutions of the boundary value problems1.9,2.4and use

it to obtain the uniqueness of solutions of the boundary value problems1.9,2.2.

Assume that for somem ∈Rthere are two distinct solutions,αandβ, of1.9,2.4.

Letwα−β. Then, we have

wσb;m 0, wΔ_b;_m_0. _2.5

By the uniqueness of solutions of initial value problems of 1.9,wb/0. Without loss of

generality, we may assumewb>0. We consider the two cases ofb.

Ifbis right-dense,σb b, then

wΔ_b;_m_lim

t→b

wt;m−wb;m

t−b 0. 2.6

Ifbis right scattered,σb> b, then

wΔ_b;_m wσb;m−wb;m

σb−b 0. 2.7

Regardless of whether b is right dense or right scattered, we have wσb;m

wb;m 0, which is a contradiction to conditionA2. Hence,α≡β.

The uniqueness of solutions of boundary value problems of1.9,2.4 implies the

uniqueness of solutions of boundary value problems of 1.9,2.2 because the boundary

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Theorem 2.2. LetA, B ∈Rbe given and assume conditionsA1throughA4are satisfied. Then

the boundary value problems1.9,1.10has at most one solution.

Proof. Again, we argue by contradiction. Assume for some valuesA, B ∈ R, there are two

distinct solutions, α and β, of 1.9, 1.10. Let w α−β. Then, we have wa 0 and

wb 0. By the uniqueness of solutions of initial value problems of1.9,wΔa/0 and

wΔ_b_/_{0. We may assume, without loss of generality,}_wΔ_a_>_{0 and}_wΔ_b_>_0.

Then, there exists a pointc,a < c < b, such thatwhas a generalized zero atc. That is,

eitherwc 0 orwc·wσc<0. But, by conditionA3,wc 0.

Since wa 0 and wc 0, there exists a point r1 ∈ a, cT such that wΔ has

generalized zero at r1. Since we also obtain thatw has a generalized zero ata, it implies

thatα≡β, and this contradicts conditionA4.

Similarly, sincewc 0 andwb 0, there exists a pointr2 ∈ c, bTsuch thatwΔ

has generalized zero atr2. Note thatc < r2 and we obtain thatwhas a generalized zero at c

andwΔhas a generalized zero atr2. This, again, implies thatα ≡β, and, hence, contradicts

conditionA4.

3. Existence of Solutions

In this section, we establish monotonicity of the derivative as a function ofm, of solutions

of 1.9satisfying each of the boundary conditions 2.1,2.2. We use these monotonicity

properties then to obtain solutions of1.9,1.10.

Theorem 3.1. Suppose that conditions A1throughA4 are satisfied and that for eachm ∈ R

there exists solutions of 1.9,2.1and1.9,2.2. Then,αΔc;mandβΔc;mare both strictly

increasing function ofmwhose range isR.

Proof. The strictness of the conclusion arises fromTheorem 2.1. Letm1> m2and let

wx αx;m1−αx;m2. 3.1

Then, byTheorem 2.1,

wa αa;m1−αa;m2 0,

wc αc;m1−αc;m2 0,

wΔ_c_αΔ_c;_m

1−αΔc;m2/0.

3.2

Suppose to the contrary thatwΔc<0. Then there exists a pointr1∈a, c_Tsuch that

wΔ_{has a generalized zero at}_r

1. This contradicts conditionA4. Thus,wΔc > 0 and as a

consequence,wΔc;mis a strictly increasing function ofm.

We now show that{αΔc;m|m∈R}R. Letk∈Rand consider the solutionux;k

of1.9,2.1, withuas defined above. Consider also the solutionαx;uΔc;kof1.9,2.1.

Hence, byTheorem 2.1,αx;uΔc;k≡uΔc;kand the range ofαΔc;mas a function ofm

is the set of real numbers.

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In a similar way, we also have a monotonicity result on the functions ut;m and

vt;m.

Theorem 3.2. Assume the hypotheses ofTheorem 3.1. Then,ut;mand vt;mare, respectively,

strictly increasing and decreasing functions ofmwith ranges all ofR.

We now provide our existence result.

Theorem 3.3. Assume the hypotheses of Theorem 3.1. Then, the boundary value problems 1.9,

1.10has a unique solution.

Proof. The existence is immediate from Theorem 3.1 or Theorem 3.2. Making use of

Theorem 3.1, there exists a uniquem0∈Rsuch thatuΔc;m0 vΔc;m0 m0. Then,

yt ut;m0, a≤t≤c,

vt;m0, c≤t≤b

3.3

is a solution of1.9,1.10, and byTheorem 2.2,ytis the unique solution.

References

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