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

Authors: Aprillya Lanz; Ana Tameru
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

Let𝕋 be 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 yΔΔ(t)=f(t,y(t),yΔ(t)),t∈[a,b]𝕋,y(a)=A,y(b)=B, by matching a solution of the first equation satisfying boundary conditions on [a,c]𝕋 with a solution of the first equation satisfying boundary conditions on [c,b]𝕋, where c∈(a,b)𝕋.


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## Body

## 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
(1.1)y′′(t)=f(t,y,y′),(1.2)y(a)=y1,y(b)=y2.
As shown in their work, the uniqueness and existence of the solutions of (1.1), (1.2) were obtained by matching a solution y1 of (1.1) satisfying the boundary condition
(1.3)y(a)=y1,y′(c)=m
with a solution y2 of (1.1) satisfying the boundary condition
(1.4)y′(c)=m,y(b)=y2,
where m=y1′(c)=y2′(c).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], Henderson [3, 4], Henderson and Taunton [5].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], Hartman [7], Henderson [8, 9], Henderson and Yin [10], Moorti and Garner [11], Rao et al. [12] and 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 problem (1.1), (1.2) on a time scale. The theory of time scales was first introduced by Hilger [13] in 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 Peterson [14, 15], Chyan [16], Henderson [4], and Henderson and Yin [17].In this work,𝕋 is assumed to be a nonempty closed subset of ℝ with inf𝕋=-∞ and sup𝕋=+∞. We shall also use the convention on notation that for each interval I of ℝ,
(1.5)IT=I∩T.For readers' convenience, we state a few definitions which are basic to the calculus on the time scale𝕋. The forward jump operatorσ:𝕋→ℝ is defined by
(1.6)σ(t)=inf⁡{s>t∣s∈T}∈T.
If σ(t)>t, t is said to be right-scattered, whereas, if σ(t)=t, t is said to be right-dense. The backward jump operatorρ:𝕋→ℝ is defined by
(1.7)ρ(t)=sup⁡{s<t∣s∈T}∈T.
If ρ(t)<t, t is said to be left-scattered, and if ρ(t)=t, then t is said to be left-dense. If g:𝕋→ℝ and t∈𝕋, then the delta derivative of g at t, gΔ(t), is defined to be the number (provided that it exists), with the property that, given any ε>0, there is a neighborhood U of t, such that
(1.8)|[g(σ(t))-g(s)]-gΔ(t)[σ(t)-s]|≤ε|σ(t)-s|,
for all s∈U. In this definition, t∈𝕋𝒦, where this set is derived from the time scale 𝕋 as follows: if 𝕋 has a left-scattered maximum m, then 𝕋𝒦=𝕋∖{m}. Otherwise, we define 𝕋𝒦=𝕋.We say that the functiony has a generalized zero at t if y(t)=0 or if y(t)·y(σ(t))<0. In the latter case, we would say the generalized zero is in the real interval (t,σ(t)).Theorem 1.1 (Mean Value Theorem).
Ify:𝕋→ℝ is continuous and y(t) has a generalized zero at a and b, a,b∈𝕋, then there exists a point r∈[a,b]𝕋 such that yΔ has a generalized zero at r.Let𝕋 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 interval [a,b]𝕋 for the second-order delta derivative equation
(1.9)yΔΔ=f(t,y,yΔ),
satisfying the boundary conditions,
(1.10)y(a)=y1,y(σ(b))=y2,
where a<b and y1,y2∈ℝ. Throughout this paper, we will assume  (A1)
f(t,r1,r2) is a real-valued continuous function defined on 𝕋×ℝ2.We obtain solutions by matching a solution of (1.9) satisfying boundary conditions on [a,c]𝕋 to a solution of (1.9) satisfying boundary conditions on [c,b]𝕋. In particular, we will give sufficient conditions such that if y1(t) is a solution of (1.9) satisfying the boundary conditions y(a)=y1,yΔ(c)=m and y2(t) is a solution of (1.9) satisfying the boundary conditions y(σ(b))=y2,yΔ(c)=m, the solutions of (1.9) is
(1.11)y(t)={y1(t),t∈[a,c]T,y2(t),t∈[c,b]T.Moreover, we will assume the following conditions throughout this paper.(A2)
Solutions of initial value problems for (1.9) are unique and extend throughout 𝕋.(A3)
c∈𝕋 is right dense and is fixed.And the uniqueness of solutions assumptions are stated in terms of generalized zeros as follows:(A4)
For anyt1<t2 in 𝕋, if u and v are solutions of (1.9) such that (u-v) has a generalized zero at t1 and (u-v)Δ 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 conjugate boundary value problems of this paper are unique, when they exist.Theorem 2.1.
LetA,B∈ℝ be given and assume conditions (A1) through (A4) are 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. 							(2.1)y(a;m)=A,yΔ(c;m)=m,a<c,wherea,c∈T,(2.2)y(b;m)=B,yΔ(c;m)=m,c<b,wherec,b∈T.Proof.
Assume for somem∈ℝ, there exists distinct solutions α and β of (1.9), (2.1), and set w=α-β. Then, we have
(2.3)w(a)=0,wΔ(c)=0.
Clearly, since a<c and w has a generalized zero at a and wΔ has a generalized zero at c, this contradicts condition (A4). Hence, the boundary value problems (1.9), (2.1) have unique solutions.
Next, we will look at a special boundary value problem of (1.9) satisfying the boundary condition
(2.4)y(σ(b);m)=C,yΔ(b;m)=m.
We will show the uniqueness of solutions of the boundary value problems (1.9), (2.4) and use it to obtain the uniqueness of solutions of the boundary value problems (1.9), (2.2).
Assume that for somem∈ℝ there are two distinct solutions, α and β, of (1.9), (2.4). Let w=α-β. Then, we have
(2.5)w(σ(b);m)=0,wΔ(b;m)=0.
By the uniqueness of solutions of initial value problems of (1.9), w(b)≠0. Without loss of generality, we may assume w(b)>0. We consider the two cases of b.
Ifb is right-dense, σ(b)=b, then
(2.6)wΔ(b;m)=lim⁡t→bw(t;m)-w(b;m)t-b=0.
Ifb is right scattered, σ(b)>b, then
(2.7)wΔ(b;m)=w(σ(b);m)-w(b;m)σ(b)-b=0.
Regardless of whetherb is right dense or right scattered, we have w(σ(b);m)=w(b;m)=0, which is a contradiction to condition (A2). Hence, α≡β.
The uniqueness of solutions of boundary value problems of (1.9), (2.4) implies the uniqueness of solutions of boundary value problems of (1.9), (2.2) because the boundary conditions are defined at c<b. This completes the proof.Theorem 2.2.
LetA,B∈ℝ be given and assume conditions (A1) through (A4) are satisfied. Then the boundary value problems (1.9), (1.10) has 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 w(a)=0 and w(b)=0. By the uniqueness of solutions of initial value problems of (1.9), wΔ(a)≠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 that w has a generalized zero at c. That is, either w(c)=0 or w(c)·w(σ(c))<0. But, by condition (A3), w(c)=0.
Sincew(a)=0 and w(c)=0, there exists a point r1∈(a,c)𝕋 such that wΔ has generalized zero at r1. Since we also obtain that w has a generalized zero at a, it implies that α≡β, and this contradicts condition (A4).
Similarly, sincew(c)=0 and w(b)=0, there exists a point r2∈(c,b)𝕋 such that wΔ has generalized zero at r2. Note that c<r2 and we obtain that w has a generalized zero at c and wΔ has a generalized zero at r2. This, again, implies that α≡β, and, hence, contradicts condition (A4).


## 3. Existence of Solutions

In this section, we establish monotonicity of the derivative as a function ofm, of solutions of (1.9) satisfying each of the boundary conditions (2.1), (2.2). We use these monotonicity properties then to obtain solutions of (1.9), (1.10).Theorem 3.1.
Suppose that conditions(A1) through (A4) are satisfied and that for each m∈ℝ there exists solutions of (1.9), (2.1) and (1.9), (2.2). Then, αΔ(c;m) and βΔ(c;m) are both strictly increasing function of m whose range is ℝ.Proof.
The strictness of the conclusion arises from Theorem2.1. Let m1>m2 and let
(3.1)w(x)=α(x;m1)-α(x;m2).
Then, by Theorem2.1,
(3.2)w(a)=α(a;m1)-α(a;m2)=0,w(c)=α(c;m1)-α(c;m2)=0,wΔ(c)=αΔ(c;m1)-αΔ(c;m2)≠0.
Suppose to the contrary thatwΔ(c)<0. Then there exists a point r1∈(a,c)𝕋 such that wΔ has a generalized zero at r1. This contradicts condition (A4). Thus, wΔ(c)>0 and as a consequence, wΔ(c;m) is a strictly increasing function of m.
We now show that{αΔ(c;m)∣m∈ℝ}=ℝ. Let k∈ℝ and consider the solution u(x;k) of (1.9), (2.1), with u as defined above. Consider also the solution α(x;uΔ(c;k)) of (1.9), (2.1). Hence, by Theorem 2.1, α(x;uΔ(c;k))≡uΔ(c;k) and the range of αΔ(c;m) as a function of m is the set of real numbers.
The argument forβΔ is quite similar. This completes the proof.In a similar way, we also have a monotonicity result on the functionsu(t;m) and v(t;m).Theorem 3.2.
Assume the hypotheses of Theorem3.1. Then, u(t;m) and v(t;m) are, respectively, strictly increasing and decreasing functions of m with ranges all of ℝ.We now provide our existence result.Theorem 3.3.
Assume the hypotheses of Theorem3.1. Then, the boundary value problems (1.9), (1.10) has a unique solution.Proof.
The existence is immediate from Theorem3.1 or Theorem 3.2. Making use of Theorem 3.1, there exists a unique m0∈ℝ such that uΔ(c;m0)=vΔ(c;m0)=m0. Then,
(3.3)y(t)={u(t;m0),a≤t≤c,v(t;m0),c≤t≤b
is a solution of (1.9), (1.10), and by Theorem 2.2, y(t) is the unique solution.



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*Source: 465867-2012-09-05.xml*