# Disconjugacy and Nonoscillation Domains for Nonlinear Singular Interface Problems on Semi-Infinite Time Scales

Authors: D. K. K. Vamsi; Pallav Kumar Baruah
Journal: ISRN Mathematical Analysis
 (2011)

---

## Abstract

We define and discuss the disconjugacy (𝒟) and nonoscillation (𝒩) domains for a pair of dynamic equations along with matching interface conditions on the semi-infinite time scale [0,c]𝕋∪[σ(c),∞]𝕋. We show that these domains are closed and convex subsets of the parameter space ℝn+m. The theory developed is used to discuss the oscillatory behavior of initial and boundary value problems associated with interface problems in the fields of applied elasticity, acoustic wave guides in ocean, and transverse vibrations in strings.


---

## Body

## 1. Introduction

The study of waves plays an important role in physical sciences. Waves of simple nature oscillate with a fixed frequency and wave length. The study of these simple sinusoidal waves forms the basis for the study of almost all forms of linear and nonlinear complex wave motions. The oscillation nature of waves can be modelled by differential equations specifically by ordinary Sturm-Liouville operators. Many of the eigenvalue (Sturm-Liouville) problems may be cast in the form(1.1)y′′+(-α+βB(x))y=0,
where α,β are real parameters and x varies over a subinterval of ℝ. Fixing α and allowing β to be the parameter we get weighted Sturm-Liouville equation. A survey on this is done in [1]. In [2], Moore exploited the relation between the nonoscillation and periodicity of solutions of the equation(1.2)y′′+(-a+bq(x))y=0.
Here a,b are real parameters and q(x) is real-valued, continuous, and periodic function. A special region R a subset of the ab-plane for which the equation has nonoscillatory solutions is introduced. R is shown to be a closed, convex and is also shown to be entirely contained in the half-plane a>0. In [3], Markus and Moore found the disconjugacy domain 𝒟 (a subset of the (a,b)-parameter plane) for which the equation(1.3)y′′+(-a+bp(x))y=0
has disconjugate solutions. D was shown to be closed and bounded. Influenced by this work, Mingarelli and Halvorsen [4] formalised this study. They defined the disconjugacy and non oscillation domains (denoted by 𝒟 and 𝒩) for the general equation(1.4)y′′+(-αA(x)+βB(x))y=0
on the closed half-line I=[0,∞),A,B:[0,∞)→ℝ. 𝒟 consists of all the values of (α,β)⊂ℝ2 such that solutions of the general equation are disconjugate. 𝒩 consists of all the values of (α,β)⊂ℝ2 such that solutions of the general equation are non oscillatory. They discussed the properties of 𝒟 and 𝒩 such as closedness and convexity. To our knowledge the concepts of disconjugacy and non oscillation domains for nonlinear equations seem to be less exploredIn the late 1980s, Hilger [5], then a graduate student at the Augsburg in Germany, developed a calculus called measure chains that unifies discrete and continuous analysis. For many purposes in analysis it is sufficient to consider a special case of a measure chain, a so-called time scale, which simply is a closed subset of the real numbers. A survey of this calculus can be found in the paper by Agarwal et al. [6] and also in the books by Bohner and Peterson [7, 8]. In the literature of time scales, we see that substantial amount of work has been done on oscillation behaviour of nonlinear dynamic equations on time scales. We refer the reader to [9–14] and references therein. We see that the concept of disconjugacy and non oscillation domains for linear and nonlinear dynamic equations on time scales has not yet been defined.In study of acoustic wave guides in ocean [15], transverse vibrations of strings [16], one-dimensional scattering in quantum theory [17], optical fiber transmission [18], and applied elasticity [19], we encounter problems wherein two different differential equations are defined on two adjacent intervals with a common point of interface, and the solutions satisfy matching conditions at the point of interface. We observe that the above problem for the regular case has been discussed in [20–25] and references therein. The problem of having singularity at the end boundary points is dealt within [20]. But the problem of having a singularity at the point of interface remained unexplored. The singularity at the point of interface in the domain of definition could be of the following three types satisfying certain matching conditions at the singular interface:
(1.5)To describe the singularities in the domain of definition we take help of the terminology used on time scale [8]. The new framework of the dynamic equations on time scale with facilities of the two jump operators with various definitions of continuity and derivatives make one's job simple to study these singular interface problems. This problem of having singularity at the point of interface for linear interface problems is discussed in [26–28] and for nonlinear case is discussed in [29, 30].From the above we see that the concepts of disconjugacy, non oscillatory domains have been less explored for nonlinear equations and has not yet been defined for linear and nonlinear dynamic equations on time scales. Also we note that the nonlinear singular interface problems and problems having singularity at the boundary are less explored.In this paper we extend the concepts of disconjugacy, non oscillatory domains to nonlinear dynamic equations on time scales and also discuss the oscillatory behaviour of nonlinear singular interface problems on semi-infinite time scales. In brief, we study the oscillation theory for an IVP associated with nonlinear singular interface problem on the semi infinite time scale[0,c]𝕋∪[σ(c),∞)𝕋. We define and discuss the disconjugacy (𝒟) and non oscillation (𝒩) domains for this IVP associated with nonlinear singular interface problems on semi infinite time scales. We show that these domains are closed and convex subsets of the parameter space ℝn+m. The theory developed is used to discuss the oscillatory nature of problems in the fields of applied elasticity, acoustic wave guides in ocean, and transverse vibrations in strings.In Section2, we give few mathematical preliminaries, which we use through the rest of the paper, and, in Section 3, we give few preliminary results. In Section 4, we discuss the disconjugacy domain for an IVP associated with nonlinear singular interface problems. Non oscillation domain for the IVP is discussed in Section 5. Finally, in Section 6, the oscillatory behaviour of initial and boundary value problems associated with interface problems in the fields of applied elasticity, acoustic wave guides in ocean, and transverse vibrations in strings is discussed.


## 2. Mathematical Preliminaries

Definition 2.1.
Let𝕋 be a time scale (an arbitrary closed subset of real numbers). For t∈𝕋 one defines the forward jump operator σ:𝕋→𝕋 by
(2.1)σ(t):=inf⁡{s∈T:s>t},
while the backward jump operator ρ:𝕋→𝕋 is defined by
(2.2)ρ(t):=sup⁡{s∈T:s<t}.
If σ(t)>t, we say that t is right scattered, while if ρ(t)<t we say that t is left scattered. Points that are right scattered and left scattered at the same time are called isolated. Also, if t<sup𝕋 and σ(t)=t, then t is called right dense, and if t>inf𝕋 and ρ(t)=t, then t is called left dense. Points that are right dense and left dense at the same time are called dense. Finally, the graininess function μ:𝕋→[0,∞) is defined by
(2.3)μ(t):=σ(t)-t.Definition 2.2.
One has(2.4)Tκ={T-{m},ifsup⁡T<∞,T,ifsup⁡T=∞,
where m is the left-scattered maximum.Definition 2.3.
Letf be a function defined on 𝕋. One says that f is delta differentiable at t∈𝕋κ provided there exists an α such that for all ϵ> 0 there is a neighborhood 𝒩 around t with
(2.5)|f(σ(t))-f(s)-α(σ(t)-s)|≤ϵ|σ(t)-s|∀s∈N.Remark 2.4.
For a functionf:𝕋→ℝ we will talk about the second derivative fΔΔ provided fΔ is differentiable on 𝕋κ2=(𝕋κ)κ with derivative fΔΔ=(fΔ)Δ:𝕋κ2→ℝ.Definition 2.5.
One defines the parameter space(2.6)Rn+m={(λ1,λ2,…,λn+m),λi∈R,i=1,2,…,n+m,n,m∈N}.


## 3. Preliminary Results

Let𝕋1=[0,c]𝕋(a time scale with end points 0 and c), K1=[σ(c),∞)𝕋 a time scale with one end being σ(c), 𝕋2=K1κ2. Also let (f1,f2) be nonlinear function tuple in 𝒞(𝕋1×ℝ)×𝒞(𝕋2×ℝ). And let (f1,f2) be positive. In this section we consider the following IVP associated with singular interface problem (IVP-SIP-I):(3.1)y1ΔΔ(t)=f1(t,y1σ),t∈T1,(3.2)y2ΔΔ(t)=f2(t,y2σ),t∈T2,
with the initial conditions(3.3)y1(0)=l,(3.4)y1Δ(0)=m,wherel,m>0,
followed by the matching interface conditions(3.5)ρ1y1(c)=ρ2y2(σ(c)),(3.6)ρ3y1Δ(c)=ρ4y2Δ(σ(c)),ρi>0,i=1,2,3,4.Definition 3.1.
One calls a functiony:𝕋1∪𝕋2→ℝ to be a matching solution of the IVP-SIP-I if (i)
the functiony|𝕋1=y1 and y1 satisfies (3.1),(ii)
the functiony|𝕋2=y2 and y2 satisfies (3.2),(iii)
y1 and y2 satisfy the initial and interface conditions (3.3)-(3.4) and (3.5)-(3.6), respectively.Definition 3.2.
One callst∈𝕋1∪𝕋2 to be a zero of the IVP-SIP-I if y(t)=0.Definition 3.3.
One calls an IVP associated with a singular interface problem to be disconjugate if every nontrivial solution of the IVP has at most one zero.We denote𝒞(𝕋i),i=1,2, to be the set of continuous functions on time scales 𝕋1 and 𝕋2. Also, we denote
(3.7)f(t)={f1(t),t∈T1,f2(t),t∈T2.Definition 3.4.
For a given compact subinterval[a,c]∪[σ(c),b]⊂𝕋1∪𝕋2, one defines the space
(3.8)Â([a,c]∪[σ(c),b])={(η1,η2):(η1,η2)∈C[a,c]×C[σ(c),b]}.For any(η1,η2)∈Â([a,c]∪[σ(c),b]), one defines the functional (3.9)ζ((η1,η2),(f1,f2);a,b)=∫ac[{η1Δ(t)}2+f1(t,y1σ){η1(t)}2]Δt+∫σ(c)b[{η2Δ(t)}2+f2(t,y2σ){η2(t)}2]Δt.Lemma 3.5.
If IVP-SIP-I is disconjugate on every subset of𝕋1∪𝕋2 (denoted by ℐ), then there exists a positive non trivial solution but no oscillatory solution for the IVP-SIP-I.Proof.
It is easy to see that IVP-SIP-I is equivalent to the integral equations(3.10)y1(t)=∫0t∫0sf1(t,y1σ)ΔtΔs+mt+l,t∈T1,y2(t)=∫σ(c)t∫σ(c)sf2(t,y2σ)ΔtΔs+∫σ(c)tc21Δs+c22,t∈T2,
where
(3.11)c21=ρ3ρ4∫0cf1(t,y1σ)Δs,c22=ρ1ρ2∫0c∫0sf1(t,y1σ)ΔtΔs+mc+l,
which give the required positive solution. Moreover, IVP-SIP-I is disconjugate implying that (y1,y2) is a non oscillatory solution.Lemma 3.6.
If the IVP-SIP-I is disconjugate onℐ, then, for every closed subinterval of [a,c]∪[σ(c),b], the functional ζ((η1,η2),(f1,f2);a,b) is positive on Â([a,c]∪[σ(c),b]).Proof.
Let us suppose that the IVP-SIP-I is disconjugate onℐ. Then by Lemma 3.5 there is a positive solution (u1,u2) of the IVP-SIP-I.Assumption 3.7.
We assume thatf is of the form
(3.12)f1(t,u1σ)=A1(t)B1(u1σ),t∈[a,c]T,f2(t,u2σ)=A2(t)B2(u2σ),t∈[σ(c),b]T,
where Ai,Bi are functions on 𝕋i,𝒞(𝕋i),i=1,2. We also assume that B1,B2 are positive basing on the fact that (u1,u2) is a positive solution of the IVP-SIP-I.
From the definition of the functionalζ((η1,η2),(f1,f2);a,b)we easily see that ζ((η1,η2),(f1,f2);a,b)≥0 whenever A1,A2≥0. For the cases when (A1,A2≤0),(A1≤0,A2≥0),(A1≥0,A2≤0) we define the functional ζ((η1,η2),(f1,f2);a,b) in the following manner.
IfA1,A2≤0, then we define(3.13)ζ((η1,η2),(f1,f2);a,b)=∫ac[{η1Δ(t)}2-f1(t,y1σ){η1(t)}2]Δt+∫σ(c)b[{η2Δ(t)}2-f2(t,y2σ){η2(t)}2]Δt.
If A1≤0,A2≥0, then we define
(3.14)ζ((η1,η2),(f1,f2);a,b)=∫ac[{η1Δ(t)}2-f1(t,y1σ){η1(t)}2]Δt+∫σ(c)b[{η2Δ(t)}2+f2(t,y2σ){η2(t)}2]Δt.
If A1≥0,A2≤0, then we define
(3.15)ζ((η1,η2),(f1,f2);a,b)=∫ac[{η1Δ(t)}2+f1(t,y1σ){η1(t)}2]Δt+∫σ(c)b[{η2Δ(t)}2-f2(t,y2σ){η2(t)}2]Δt.
From the above definitions of ζ((η1,η2),(f1,f2);a,b), we see that ζ((η1,η2),(f1,f2);a,b)≥0 irrespective of the sign of A1 and A2.Note 1.
Through the rest of the paper we assume thatA1,A2≥0. Similar results can be obtained for the cases when (A1,A2<0),(A1<0,A2>0),(A1>0,A2<0) by using the respective definitions for ζ((η1,η2),(f1,f2);a,b).Lemma 3.8.
Let IVP-SIP-II be defined by(3.16)y1ΔΔ(t)=λ11f1(t,y1σ),t∈T1,y2ΔΔ(t)=λ12f2(t,y2σ),t∈T2,-∞<λ11,λ12<+∞,
along with (3.3)–(3.6). Then IVP-SIP-II is disconjugate for every λ11,λ12 if and only if f(t,y)=0 on ℐ.Proof.
Letf(t,yσ)=0, that is,
(3.17)f1(t,y1σ)=0,t∈T1,f1(t,y1σ)=0,t∈T2.
Then we see that IVP-SIP-II reduces to the IVP-SIP-III
(3.18)y1ΔΔ(t)=0,t∈T1,y2ΔΔ(t)=0,t∈T2,y1(0)=l,y1Δ(0)=m,ρ1y1(c)=ρ2y2(σ(c)),ρ3y1Δ(c)=ρ4y2Δ(σ(c)).
Simple calculations show that
(3.19)y1(t)=mt+l,y2(t)=ρ3ρ4mt+ρ1ρ2(cm+l)-ρ3ρ4mσ(c).
We observe that y(t)=0 whenever
(3.20)t=-lmorσ(c)-ρ1ρ4ρ2ρ3(c+lm).
Clearly t≠-l/m and so t=σ(c)-(ρ1ρ4/ρ2ρ3)(c+(l/m)) is the only zero possible. Hence, IVP-SIP-III is disconjugate on ℐ.
Now let IVP-SIP-II be disconjugate for everyλ11,λ12. From Lemma 3.6, we have(3.21)ζ((η1,η2),(f1,f2);a,b)≥0.
Hence
(3.22)∫ac[{η1Δ(t)}2+λ11f1(t,y1σ){η1(t)}2]Δt+∫σ(c)b[{η2Δ(t)}2+λ12f2(t,y2σ){η2(t)}2]Δt≥0.
So we have
(3.23)-λ11∫acf1(t,y1σ){η1(t)}2Δt-λ12∫σ(c)bf2(t,y2σ){η2(t)}2Δt≤∫ac{η1Δ(t)}2+∫σ(c)b{η2Δ(t)}2.
The previous equation is true for all -∞<λ11,λ12<+∞, which implies that
(3.24)∫acf1(t,y1σ){η1(t)}2Δt=0=∫σ(c)bf2(t,y2σ){η2(t)}2Δt.
Since the above equation holds true for all η1 and η2, we must have f1=0=f2, hence the proof.Corollary 3.9.
Let IVP-SIP-IV be defined by(3.25)y1ΔΔ=λ11f11(t,y1σ)+λ21f21(t,y1σ)+⋯+λn1fn1(t,y1σ),y2ΔΔ=λ12f12(t,y2σ)+λ22f22(t,y2σ)+⋯+λn2fn2(t,y2σ),
along with (3.3)–(3.6). Then IVP-SIP-IV is disconjugate on ℐ for all  ((λ11,λ12),(λ21,λ22),…,(λn1,λn2))∈ℝ2n, then fi(t,y)=0,i=1,2,…,n, that is,
(3.26)fi1(t,y1σ)=0,fort∈T1,fi2(t,y2σ)=0,fort∈T2,
where (fi1,fi2) are nonlinear function tuples in 𝒞(𝕋1×ℝ)×𝒞(𝕋2×ℝ).


## 4. Disconjugacy Domain

We define IVP-SIP-V as(4.1)y1ΔΔ(t)=λ11f11(t,y1σ)+λ21f21(t,y1σ)+⋯+λn1fn1(t,y1σ),t∈T1,y2ΔΔ(t)=λ12f12(t,y2σ)+λ22f22(t,y2σ)+⋯+λm2fm2(t,y2σ),t∈T2,
along with (3.3)–(3.6). Without any loss of generality, let us assume that n<m, where n,m∈ℕ and fi1,fj2,i=1,2,…,n,j=1,2,…,m, belong to 𝒞(𝕋1×ℝ), 𝒞(𝕋2×ℝ), respectively.Definition 4.1.
One  defines the disconjugacy domain𝒟 of IVP-SIP-V as
(4.2)D={(λ11,λ12,λ21,λ22,…,λn1,λn2,λ(n+1)2,λ(n+2)2,…,λm2)∈Rn+m}
such that IVP-SIP-V is disconjugate on ℐ.Theorem 4.2.
The disconjugacy domain𝒟 of IVP-SIP-V is the whole space ℝn+m if and only if
(4.3)fi1=0,i=1,2,…,n,fj1=0,j=1,2,…,m.Proof.
For(4.4)λ21=λ31=⋯=λn1=0,λ11≠0,λ22=λ32=⋯=λm2=0,λ12≠0,
IVP-SIP-V reduces to IVP-SIP-II, and hence from Lemma 3.8 we have f11=0=f12. If we choose
(4.5)λ11=λ31=⋯=λn1=0,λ21≠0,λ12=λ32=⋯=λm2=0,λ22≠0,
we get f21=0=f22. In similar lines we can show that
(4.6)f31=f41=⋯=fn1=0,f32=f42=⋯=fn2=0.
Now let us choose
(4.7)λ11=λ31=⋯=λ(n-1)1=0,λn1≠0,λ12=λ32=⋯=λn2=0=λ(n+2)2=⋯=λm2,λ(n+1)2≠0.
We see that from Lemma 3.8 we have f(n+1)2=0. Similarly we can show that
(4.8)f(n+2)2=f(n+3)2=⋯=fm2=0.
Hence
(4.9)fi1=0,i=1,2,…,n,fj1=0,j=1,2,…,m.
whenever the disconjugacy domain 𝒟 of IVP-SIP-V is the whole space ℝn+m.
On the other hand, if(4.10)fi1=0,i=1,2,…,n,fj1=0,j=1,2,…,m,
by simple calculations, it can be seen that IVP-SIP-V is disconjugate.Corollary 4.3.
If at least one of the functionsfi1 or fj2≠0, then 𝒟 is a proper subset of ℝn+m.Proof.
This is the contrapositive of Theorem4.2.Theorem 4.4.
𝒟 contains a proper subspace of the vector space ℝn+m if the sets of functions {f11,f21,…,fn1} and {f12,f22,…,fm2} are linearly dependent.Proof.
Let{f11,f21,…,fn1} and {f12,f22,…,fm2} be sets of linearly dependent functions Then there exists constants {c11,c21,…,c(n-1)1}, {c12,c22,…,c(m-1)2}, not all zero, such that
(4.11)f11(t,y1σ)=c11f21(t,y1σ)+c21f31(t,y1σ)+⋯+c(n-1)1fn1(t,y2σ),fort∈T1,f12(t,y2σ)=c12f22(t,y2σ)+c22f32(t,y2σ)+⋯+c(m-1)2fm2(t,y2σ),fort∈T2.
Therefore, the IVP-SIP-V becomes
(4.12)y1ΔΔ(t)=λ11[c11f21+c21f31+⋯+c(n-1)1fn1](t,y1σ)+λ21f21(t,y1σ)+⋯+λn1fn1(t,y1σ),t∈T1,y2ΔΔ(t)=λ12[c12f22+c22f32+⋯+c(m-1)2fm2](t,y2σ)+λ22f22(t,y2σ)+⋯+λm2fm2(t,y2σ),t∈T2,
along with (3.3)–(3.6). We see that IVP-SIP-V now is to be
(4.13)y1ΔΔ(t)=(λ11c11+λ21)f21(t,y1σ)+(λ11c21+λ31)f31(t,y1σ)+⋯+(λ11c(n-1)1+λn1)fn1(t,y1σ),y2ΔΔ(t)=(λ12c12+λ22)f22(t,y2σ)+(λ12c22+λ32)f32(t,y2σ)+⋯+(λ12c(m-1)2+λm2)fm2(t,y2σ),
along with (3.3)–(3.6). Hence 𝒟 contains the subspace
(4.14)S={(λ11,λ12,λ21,λ22,…,λn1,λn2,λ(n+1)2,λ(n+2)2,…,λm2)∈Rn+m},
where
(4.15)λ21=-λ11c11,λ31=-λ11c21,…,λn1=-λ11c(n-1)1,λ22=-λ12c12,λ32=-λ12c22,…,λm2=-λ12c(m-1)2,
hence the proof.Corollary 4.5.
If the sets of functions{f11,f21,…,fn1} and {f12,f22,…,fm2} are linearly independent, then 𝒟 cannot contain a proper subspace of ℝn+m.Proof.
This is the contrapositive of Theorem4.4.Lemma 4.6.
Let us consider the following two IVP-SIPs which are disconjugate onℐ:
(4.16)y1ΔΔ(t)=f11(t,y1σ),t∈T1,y2ΔΔ(t)=f12(t,y2σ),t∈T2,along with(3.3)–(3.6),y1ΔΔ(t)=f21(t,y1σ),t∈T1,y2ΔΔ(t)=f22(t,y2σ),t∈T2,along with(3.3)–(3.6).
Then the IVP-SIP-VI defined by
(4.17)y1ΔΔ(t)=(1-γ)f11(t,y1σ)+γf21(t,y1σ),t∈T1,y2ΔΔ(t)=(1-γ)f12(t,y2σ)+γf22(t,y2σ),t∈T2,
along with (3.3)–(3.6) is also disconjugate on ℐ, for each γ∈[0,1].Proof.
On account of Lemma3.6 it is sufficient if we show that
(4.18)ζ((η1,η2),((1-γ)f11+γf21,(1-γ)f12+γf22);a,b)≥0
on A([a,c]∪[σ(c),b]) for every compact subinterval [a,c]∪[σ(c),b]⊂𝕋1∪𝕋2. We see that
(4.19)ζ((η1,η2),((1-γ)f11+γf21,(1-γ)f12+γf22);a,b)=∫ac[{η1Δ(t)}2+[(1-γ)f11(t,y1σ)+γf21(t,y1σ)]{η1(t)}2]Δt+∫σ(c)b[{η2Δ(t)}2+[(1-γ)f12(t,y1σ)+γf22(t,y1σ)]{η2(t)}2]Δt.
Now
(4.20)∫ac[{η1Δ(t)}2+[(1-γ)f11(t,y1σ)+γf21(t,y1σ)]{η1(t)}2]Δt=∫ac[(1-γ){η1Δ(t)}2+γ{η1Δ(t)}2+[(1-γ)f11(t,y1σ)+γf21(t,y1σ)]{η1(t)}2{η1Δ(t)}2]Δt=(1-γ)∫ac[{η1Δ(t)}2+f11(t,y1σ){η1(t)}2]Δt+γ∫ac[{η1Δ(t)}2+f21(t,y1σ){η1(t)}2]Δt.
Similarly it can be shown that
(4.21)∫σ(c)b[{η2Δ(t)}2+[(1-γ)f12(t,y1σ)+γf22(t,y1σ)]{η2(t)}2]Δt=(1-γ)∫σ(c)b[{η2Δ(t)}2+f12(t,y2σ){η2(t)}2]Δt+γ∫σ(c)b[{η2Δ(t)}2+f22(t,y2σ){η2(t)}2]Δt.
Hence we have
(4.22)ζ((η1,η2),((1-γ)f11+γf21,(1-γ)f12+γf22);a,b)=(1-γ)ζ((η1,η2),(f11,f12);a,b)+γζ((η1,η2),(f21,f22);a,b)≥0,
hence the proof.Definition 4.7.
Let(fn1,fn2),(f1,f2),n∈ℕ be function tuples in 𝒞(𝕋1×ℝ)×𝒞(𝕋2×ℝ). One says that (fn1,fn2),(f1,f2) are close in uniform norm whenever
(4.23)supt∈T1|fn1(t)-f1(t)|+supt∈T2|fn2(t)-f2(t)|⟶0asn⟶∞.Definition 4.8.
Let(yn1,yn2),(y1,y2),n∈ℕ be function tuples in 𝒞(𝕋1)×𝒞(𝕋2). One says that (yn1,yn2),(y11,y12) are close in uniform norm whenever
(4.24)supt∈T1|yn1(t)-y1(t)|+supt∈T2|yn2(t)-y2(t)|⟶0asn⟶∞.Lemma 4.9.
Letfn=(fn1,fn2),f=(f1,f2),n∈ℕ. Let the sequence of the IVP-SIPs defined by
(4.25)yn1ΔΔ(t)=fn1(t,y1σ),t∈T1,yn2ΔΔ(t)=fn2(t,y2σ),t∈T2,
along with (3.3)–(3.6) be disconjugate on ℐ for each n=1,2,…. If fn→f uniformly, then the IVP-SIP
(4.26)y1ΔΔ(t)=f1(t,y1σ),t∈T1,y2ΔΔ(t)=f2(t,y2σ),t∈T2,
is disconjugate on ℐ.Proof.
Now let us assume that (4.26) along with (3.3)–(3.6) is not disconjugate. Let us assume that it has a solution y(t) with two zeros t1,t2. Let us assume that t1∈𝕋1 and t2∈𝕋2. The other cases when t1,t2∈𝕋1, t1,t2∈𝕋2 can be worked in similar lines. Now let (yn1,yn2) be a solution such that yn1(t1)=0.Claim 1.
The solutions of (4.25), (4.26) satisfying (3.3)–(3.6) are close in the uniform norm whenever (fn1,fn2) and (f1,f2) are close in uniform norm on ℐ.
Simple calculations show that the solution of (4.25), (4.26) satisfying (3.3)–(3.6) are(4.27)yn1(t)=∫0t∫0kfn1(s,y1σ)ΔsΔk+lt+m,yn2(t)=∫σ(c)t∫σ(c)k′fn2(s,y2σ)ΔsΔk′+∫σ(c)tρ3ρ4[∫0cfn1(s,y1σ)Δs+m]Δk′+ρ1ρ2∫0c∫0kfn1(s,y1σ)ΔsΔk+lc+m,(4.28)y1(t)=∫0t∫0kf1(s,y1σ)ΔsΔk+lt+m,y2(t)=∫σ(c)t∫σ(c)k′f2(s,y2σ)ΔsΔk′+∫σ(c)tρ3ρ4[∫0cf1(s,y1σ)Δs+m]Δk′+ρ1ρ2∫0c∫0kf1(s,y1σ)ΔsΔk+lc+m.
From the above equations we clearly see that the solutions of (4.25), (4.26) along with (3.3)–(3.6) are close in the uniform norm whenever (fn1,fn2) and (f1,f2) are close in uniform norm on ℐ, so the claim. Hence for β>0 we can find ϵ(>0) and δ(>0) such that
(4.29)supt∈[t1,c]|yn1(t)-y1(t)|+supt∈[σ(c),t2+β]|yn2(t)-y2(t)|<ϵ
whenever
(4.30)supt∈[t1,c]|fn1(t)-f1(t)|+supt∈[σ(c),t2+β]|fn2(t)-f2(t)|<δ.
Since y2(t) must change sign at t=t2, it follows that yn2(t) must also change sign near t=t2 for sufficiently large n. Thus, for such n, (4.25) along with (3.3)–(3.6) is not disconjugate which leads us to a contradiction.Theorem 4.10.
The disconjugacy domain𝒟 of IVP-SIP-V is a closed set of the parameter space ℝn+m.Proof.
Let(λ1,λ2,…,λn+m) be a limit point of the sequence (λk1,λk2,…,λk(n+m))∈D,k=1,2,…. Then for every ϵ>0 there exists sufficiently large K such that for all k>K we have |λk1-λ1|<ϵ,|λk2-λ2|<ϵ,…,|λk(n+m)-λn+m|<ϵ, and IVP-SIP-VII defined by
(4.31)y1ΔΔ(t)=λk1f11+λk3f21+⋯+λk(2n-1)fn1,t∈T1,y2ΔΔ(t)=λk2f12+λk4f22+⋯+λk(2n)fn2+λk(2n+1)f(n+1)2+⋯+λk(n+m)fm2,t∈T2,
along with (3.3)–(3.6) is disconjugate. Let y=(y1,y2) be a non trivial solution of IVP-SIP-V for
(4.32)(λ11,λ21,…,λn1,λ12,λ22,…,λm2)=(λ1,λ2,…,λn+m).
Then we see that either y(t) never vanishes in which case y(t) is disconjugate or let y(t0)=0 for some t0. In the latter case we let (yn1,yn2) be a solution of IVP-SIP-VII such that yn1(t0)=0. From assumption we have that yn1(t)≠0 for t≠t0. From Claim 1 we have (yn1,yn2) uniformly approximating (y1,y2). Hence y(t) can change sign only at t=t0 and so y(t)≠0 for all t≠t0. Hence y(t) is disconjugate, so the result.Theorem 4.11.
The disconjugacy domain𝒟 of IVP-SIP-V is a convex set in the parameter space ℝn+m.Proof.
We need to show that for(λ11,λ12,…,λ1(n+m)),(λ21,λ22,…,λ2(n+m))∈D the convex combination
(4.33)(1-γ)(λ11,λ12,…,λ1(n+m))+γ(λ21,λ22,…,λ2(n+m))∈D,
that is,
(4.34)[(1-γ)λ11+γλ21,(1-γ)λ12+γλ22,…,(1-γ)λ1(n+m)+γλ2(n+m)]∈D.
Since (λ11,λ12,…,λ1(n+m)),(λ21,λ22,…,λ2(n+m))∈D, we have the IVP-SIPs
(4.35)y1ΔΔ(t)=λ11f11+λ13f21+⋯+λ1(2n-1)fn1,t∈T1,y2ΔΔ(t)=λ12f12+λ14f22+⋯+λ1(2n)fn2+λ1(2n+1)f(n+1)2+⋯+λ1(n+m)fm2,t∈T2,y1ΔΔ(t)=λ21f11+λ23f21+⋯+λ2(2n-1)fn1,t∈T1,y2ΔΔ(t)=λ22f12+λ24f22+⋯+λ2(2n)fn2+λ2(2n+1)f(n+1)2+⋯+λ2(n+m)fm2,t∈T2,
along with (3.3)–(3.6) to be disconjugate. Now from Lemma 4.6 we have the IVP-SIP-VIII defined by
(4.36)y1ΔΔ(t)=(1-γ)(λ11f11+λ13f21+⋯+λ1(2n-1)fn1)+γ(λ21f11+λ23f21+⋯+λ2(2n-1)fn1),t∈T1,y2ΔΔ(t)=(1-γ)(λ12f12+λ14f22+⋯+λ1(2n)fn2+λ1(2n+1)f(n+1)2+⋯+λ1(n+m)fm2)+γ(λ22f12+λ24f22+⋯+λ2(2n)fn2+λ2(2n+1)f(n+1)2+⋯+λ2(n+m)fm2),t∈T2,
along with (3.3)–(3.6) to be disconjugate; that, is we have the IVP-SIP
(4.37)y1ΔΔ(t)=(1-γ)λ11f11+γλ21f11+(1-γ)λ13f21+γλ23f21+⋯+(1-γ)λ1(2n-1)fn1+γλ2(2n-1)fn1,t∈T1,y2ΔΔ(t)=(1-γ)λ12f12+γλ22f12+(1-γ)λ14f22+γλ24f22+⋯+(1-γ)λ1(n+m)fm2+γλ2(n+m)fm2,t∈T2,
along with (3.3)–(3.6) to be disconjugate. Hence
(4.38)(1-γ)(λ11,λ12,…,λ1(n+m))+γ(λ21,λ22,…,λ2(n+m))∈D,
so the proof.


## 5. Nonoscillation Domain

Definition 5.1.
One calls an IVP associated with a singular interface problem to be non oscillatory if every non trivial solution of the IVP has at most finite number of zeros.Definition 5.2.
One defines the non oscillation domain𝒩 of IVP-SIP-V as
(5.1)N={(λ11,λ12,λ21,λ22,…,λn1,λn2,λ(n+1)2,λ(n+2)2,…,λm2)∈Rn+m}
such that IVP-SIP-V is non oscillatory on ℐ.Lemma 5.3.
If IVP-SIP-I is non oscillatory onℐ, then there exists at least one positive or negative non trivial solution.Proof.
Lety(=(y1,y2)) be a solution of IVP-SIP-I. Let t1,t2,…tn be the zeros of the IVP-SIP-I (since y(t) is non oscillatory). By the nature of y(t) we see that either y(t)<0 or y(t)>0 for t>tn. We now define
(5.2)K(t)={0,t≤tn,y(t),t>tn.
From Assumption3.7 we see that K(t) is a solution of IVP-SIP-I. We also see that either K(t)≤0 or K(t)≥0.Note 2.
We assumeK(t)≥0. Similar results can be developed when K(t)≤0.All the results discussed in Section5 regarding 𝒟 can be easily extended to 𝒩. We just state two theorems without proof.Theorem 5.4.
The non oscillatory domain𝒩 of IVP-SIP-V is a closed set of the parameter space ℝn+m.Theorem 5.5.
The non oscillatory domain𝒩 of IVP-SIP-V is a convex set in the parameter space ℝn+m.


## 6. Applications

We define IVP-SIP-X by(6.1)y1ΔΔ(t)=λ11f11(t,y1σ)+χ11f21(t,y1σ)+⋯+χ(n-1)1fn1(t,y1σ),t∈T1,y2ΔΔ(t)=λ12f12(t,y2σ)+χ12f22(t,y2σ)+⋯+χ(m-1)2fm2(t,y2σ),t∈T2,
where χ11,χ21,…,χ(n-1)1,χ12,χ22,…,χ(m-1)2∈ℝ, along with (3.3)–(3.6). We discuss the oscillatory behaviour of the IVP-SIP-X using the non oscillatory domain for IVP-SIP-V.Theorem 6.1.
One of the following cases can occur for IVP-SIP-X for every(λ11,λ12)∈ℝ2.(i)
IVP-SIP-X is oscillatory for every(λ11,λ12).(ii)
IVP-SIP-X is oscillatory for every(λ11,λ12) except at some unique (λ11,λ12) = (λ01,λ02).(iii)
There exists a finite interval(λa1,λb1)∈ℝ or (λa2,λb2)∈ℝ such that IVP-SIP-X is non oscillatory for either
(6.2)λ11∈(λa1,λb1),λ12=c1,or(6.3)λ11=c2,λ12∈(λa2,λb2),c1,c2∈R.For every other combinations of(λ11,λ12)∈ℝ2, IVP-SIP-X is oscillatory.(iv)
There exists eitherλ1 or λ2∈ℝ such that IVP-SIP-X is non oscillatory (resp., oscillatory) for either
(6.4)λ11∈(λ1,∞),λ12=c1,or(6.5)λ11=c2,λ12∈(λ2,∞),and oscillatory (resp., non oscillatory) for either(6.6)λ11∈(-∞,λ1),λ12=c1,or(6.7)λ11=c2,λ12∈(-∞,λ2).(v)
IVP-SIP-X is non oscillatory for
either(6.8)λ11∈R,λ12=c1,or(6.9)λ11=c2,λ12∈R.(vi)
The non oscillatory domain for IVP-SIP-X can be a finite, semi infinite, or infinite plane in one of the following ways:
finite plane(6.10)c3≤λ11≤c4,c3′≤λ12≤c4′Semi-infinite plane(6.11)c3≤λ11≤c4,-∞≤λ12≤c5′c3≤λ11≤c4,c6′≤λ12≤+∞c3≤λ11≤c4,-∞≤λ12≤+∞c6≤λ11≤+∞,c3′≤λ12≤c4′c6≤λ11≤+∞,-∞≤λ12≤c5′c6≤λ11≤+∞,c6′≤λ12≤+∞c6≤λ11≤+∞,-∞≤λ12≤+∞-∞≤λ11≤c5,c3′≤λ12≤c4′-∞≤λ11≤c5,-∞≤λ12≤c5′-∞≤λ11≤c5,c6′≤λ12≤+∞-∞≤λ11≤c5,-∞≤λ12≤+∞-∞≤λ11≤+∞,c3′≤λ12≤c4′-∞≤λ11≤+∞,-∞≤λ12≤c5′-∞≤λ11≤+∞,c6′≤λ12≤+∞infinite plane(6.12)-∞≤λ11≤+∞,-∞≤λ12≤+∞,
where c3,c4,c5,c6,c3′,c4′,c5′,c6′∈ℝ.Proof.
Clearly we see that IVP-SIP-X is a special case of IVP-SIP-V with(6.13)λ21=χ11,λ31=χ21,…,λn1=χ(n-1)1,λ22=χ12,λ32=χ22,…,λm2=χ(m-1)2.
Let ℒ={λ11,χ11,χ21,…,χ(n-1)1,λ12,χ12,χ22,…,χ(m-1)2} be a subset of the parameter space ℝn+m. The above claims are consequences of the intersection of ℒ with 𝒩, the non oscillation domain for IVP-SIP-V. We recall that 𝒩 is convex in ℝn+m. We see that ℒ intersects 𝒩 in one of the following ways: (i)
ℒ∩𝒩 = ϕ,(ii)
ℒ∩𝒩 = a single point,(iii)
ℒ∩𝒩 = a line segment,(iv)
ℒ∩𝒩 = a one-sided ray,(v)
ℒ∩𝒩 = a full ray,(vi)
ℒ∩𝒩 = a finite plane in ℝ2,(vii)
ℒ∩𝒩 = a semi infinite plane in ℝ2,(viii)
ℒ∩𝒩 = an infinite plane in ℝ2, hence the proof.Application I (see, Wang [19]) Applied Elasticity
In the branch of applied elasticity, we encounter the problem of buckling of columns of variable cross-sections given by(6.14)L1u1=d2u1dx2+k12u12=0,0≤x≤l1,L2u2=d2u2dx2+k22u22=0,l1≤x≤l2,
where ki2=P/EIi,E is the modulus of elasticity, P is the load applied, Ii are moments of inertia, i=1,2, and u1,u2 are the displacements of cross-sections for the thinner and the thicker portions of the column, respectively. The physical conditions are given by
(6.15)u1(0)=u1′(0)=0,u1(l1)=u2(l1),u1′(l1)=u2′(l1).
Here we see that 𝕋1=[0,l1],𝕋2=[l1,l2], ρ1=1=ρ2, ρ3=1=ρ4. We see that
(6.16)λ11=-k12,λ12=-k22,χ11,χ21,…,χ(n-1)1=0=χ12,χ22,…,χ(m-1)2.
Hence from Theorem 6.1 we see that if the point {-k12,0,0,…,0,-k22,0,0,…,0} intersects 𝒩 of IVP-SIP-V then we have the problem to be non oscillatory otherwise it will be oscillatory. The set of λ’s for which the problem is non oscillatory can be one of the sets discussed in Theorem 6.1.Application II (see, Allan Boyles [15]) Acoustic Wave Guides in Oceans
In the study of acoustic wave guides in ocean we encounter the following problem. The ocean is considered to be consisting of two homogeneous layers bounded by a pressure-release surface above and a rigid bottom below. Letd1,c1 and d2,c2 be the constant density and sound velocity in layers 1 and 2, respectively. Let k1 and k2 be the wave vectors which are given by k1=ω/c1,k2=ω/c2, where ω is the angular frequency. The governing problem is given by
(6.17)L1u1=ddz(1d1du1dz)+(k12d1-λd1)u1=0,0≤z≤a,L2u2=ddz(1d2du2dz)+(k22d2-λd2)u2=0,a≤z≤b,
together with the mixed boundary conditions given by
(6.18)u1(0)=u2′(b)=0,u1(a)=u2(a),1d1u1′(a)=1d2u2′(a),u1 and u2 denote the depth eigenfunctions corresponding to the eigenvalue λ.
Here we see that𝕋1=[0,a],𝕋2=[a,b], ρ1=1=ρ2, ρ3=1/d1,ρ4=1/d2. Though we have an extra boundary condition at b, similar theory can be developed for the above problem. We see that(6.19)λ11=λ,χ11=-k12,λ12=λ,χ12=-k22,χ21,χ31,…,χ(n-1)1=0=χ22,χ32,…,χ(m-1)2.
Hence we see that the set of λ’s for which the problem is non oscillatory can be one of the sets discussed in Theorem 6.1.Application III (see, Ghosh [16]) Transverse Vibrations in Strings
We encounter the following problem in the study of transverse vibrations of strings consisting of two portions of lengthsa and b and different uniform densities d1 and d2, respectively, having a tension T stretched between the points x=0 and x=a+b:
(6.20)L1u1=c12(-u1′′)=λu1,0≤x≤a,L2u2=c22(-u2′′)=λu2,a≤x≤a+b,
with the mixed boundary conditions given by
(6.21)u1(a)=u2(a),u1′(a)=u2′(a),
and u1(0)=u2(a+b)=0, where ci2=T|di,i=1,2. u1 and u2 are eigenfunctions corresponding to the eigenvalue λ. Here we see that 𝕋1=[0,a],𝕋2=[a,a+b], ρ1=1=ρ2, ρ3=1=ρ4. Though we have an extra boundary condition at a+b, similar theory can be developed for the above problem. We see that
(6.22)λ11=-λc12,λ12=-λc22,χ11,χ21,…,χ(n-1)1=0=χ12,χ22,…,χ(m-1)2.
Hence we see that the set of λ’s for which the problem is non oscillatory can be one of the sets discussed in Theorem 6.1.Remark 6.2.
The results presented here are generalization for the nonlinear problems of corresponding linear problems studied in [20–25] and references therein. A pair of nonlinear ordinary differential equations with matching interface conditions is a special case of the problem considered here, and our results hold true by considering ρ(c)=σ(c)=c and the delta derivative becomes the ordinary derivative.



---

*Source: 254695-2011-11-24.xml*