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

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 ℝ^𝑛+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.

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 form𝑦^+(−𝛼+𝛽𝐵(𝑥))𝑦=0,(1.1) 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 equation𝑦^+(−𝑎+𝑏𝑞(𝑥))𝑦=0.(1.2) 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 equation𝑦^+(−𝑎+𝑏𝑝(𝑥))𝑦=0(1.3) 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 equation𝑦^+(−𝛼𝐴(𝑥)+𝛽𝐵(𝑥))𝑦=0(1.4) 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 explored

In 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 ℝ^𝑛+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 Section 2, 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. Let 𝕋 be a time scale (an arbitrary closed subset of real numbers). For 𝑡∈𝕋 one defines the forward jump operator 𝜎∶𝕋→𝕋 by 𝜎(𝑡)∶=inf{𝑠∈𝕋∶𝑠>𝑡},(2.1) while the backward jump operator 𝜌∶𝕋→𝕋 is defined by 𝜌(𝑡)∶=sup{𝑠∈𝕋∶𝑠<𝑡}.(2.2) 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 𝜇(𝑡)∶=𝜎(𝑡)−𝑡.(2.3)

Definition 2.2. One has 𝕋^𝜅=𝕋𝕋,−{𝑚},iiffssuupp𝕋𝕋=<∞∞,,(2.4) where 𝑚 is the left-scattered maximum.

Definition 2.3. Let 𝑓 be a function defined on 𝕋. One says that 𝑓 is delta differentiable at 𝑡∈𝕋^𝜅 provided there exists an 𝛼 such that for all 𝜖> 0 there is a neighborhood 𝒩 around 𝑡 with ||𝑓(𝜎(𝑡))−𝑓(𝑠)−𝛼(𝜎(𝑡)−𝑠)||≤𝜖||𝜎(𝑡)−𝑠||∀𝑠∈𝒩.(2.5)

Remark 2.4. For a function 𝑓∶𝕋→ℝ we will talk about the second derivative 𝑓^ΔΔ provided 𝑓^Δ is differentiable on 𝕋^𝜅2=(𝕋^𝜅)^𝜅 with derivative 𝑓^ΔΔ=(𝑓^Δ)^Δ∶𝕋^𝜅2→ℝ.

Definition 2.5. One defines the parameter space ℝ^𝑛+𝑚=𝜆₁,𝜆₂,…,𝜆_𝑛+𝑚,𝜆_𝑖∈ℝ,𝑖=1,2,…,𝑛+𝑚,𝑛,𝑚∈ℕ.(2.6)

3. Preliminary Results

Let 𝕋₁=[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=𝐾^𝜅2₁. 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):𝑦𝑦^1ΔΔ(𝑡)=𝑓₁𝑡,𝑦^𝜎1,𝑡∈𝕋₁,(3.1)_2ΔΔ(𝑡)=𝑓₂𝑡,𝑦^𝜎2,𝑡∈𝕋₂,(3.2) with the initial conditions𝑦𝑦₁(0)=𝑙,(3.3)_Δ1(0)=𝑚,where𝑙,𝑚>0,(3.4) followed by the matching interface conditions𝜌𝜌₁𝑦₁(𝑐)=𝜌₂𝑦₂(𝜎(𝑐)),(3.5)₃𝑦^Δ1(𝑐)=𝜌₄𝑦^Δ2(𝜎(𝑐)),𝜌_𝑖>0,𝑖=1,2,3,4.(3.6)

Definition 3.1. One calls a function 𝑦∶𝕋₁∪𝕋₂→ℝ to be a matching solution of the IVP-SIP-I if (i)the function 𝑦|_𝕋1=𝑦₁ and 𝑦₁ satisfies (3.1), (ii)the function 𝑦|_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 calls 𝑡∈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 denote 𝒞(𝕋_𝑖),𝑖=1,2, to be the set of continuous functions on time scales 𝕋₁ and 𝕋₂. Also, we denote 𝑓(𝑡)=𝑓¹𝑓₂(𝑡),𝑡∈𝕋₁,(𝑡),𝑡∈𝕋₂.(3.7)

Definition 3.4. For a given compact subinterval [𝑎,𝑐]∪[𝜎(𝑐),𝑏]⊂𝕋₁∪𝕋₂, one defines the space 𝐴([𝑎,𝑐]∪[𝜎(𝑐),𝑏])=𝜂₁,𝜂₂∶𝜂₁,𝜂₂∈𝒞[𝑎,𝑐]×𝒞[𝜎(𝑐),𝑏].(3.8)

For any (𝜂₁,𝜂₂)∈𝐴([𝑎,𝑐]∪[𝜎(𝑐),𝑏]), one defines the functional 𝜁𝜂₁,𝜂₂,𝑓₁,𝑓₂;𝑎,𝑏+=^𝑐𝑎𝜂^Δ1(𝑡)²+𝑓₁𝑡,𝑦^𝜎1𝜂₁(𝑡)²Δ𝑡_{𝑏𝜎(𝑐)}𝜂^Δ2(𝑡)²+𝑓₂𝑡,𝑦^𝜎2𝜂₂(𝑡)²Δ𝑡.(3.9)

Lemma 3.5. If IVP-SIP-I is disconjugate on every subset of 𝕋₁∪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 𝑦₁(𝑡)=^𝑡0^𝑠0𝑦₂𝑓₁𝑡,𝑦^𝜎1Δ𝑡Δ𝑠+𝑚𝑡+𝑙,𝑡∈𝕋₁,(𝑡)=^{𝑡𝜎(𝑐)}^{𝑠𝜎(𝑐)}𝑓₂𝑡,𝑦^𝜎2Δ𝑡Δ𝑠+^{𝑡𝜎(𝑐)}𝑐₂₁Δ𝑠+𝑐₂₂,𝑡∈𝕋₂,(3.10) where 𝑐₂₁=𝜌³𝜌₄^𝑐0𝑓₁𝑡,𝑦^𝜎1Δ𝑠,𝑐₂₂=𝜌¹𝜌₂^𝑐0^𝑠0𝑓₁𝑡,𝑦^𝜎1Δ𝑡Δ𝑠+𝑚𝑐+𝑙,(3.11) 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 on ℐ, then, for every closed subinterval of [𝑎,𝑐]∪[𝜎(𝑐),𝑏], the functional 𝜁((𝜂₁,𝜂₂),(𝑓₁,𝑓₂);𝑎,𝑏) is positive on 𝐴([𝑎,𝑐]∪[𝜎(𝑐),𝑏]).

Proof. Let us suppose that the IVP-SIP-I is disconjugate on ℐ. Then by Lemma 3.5 there is a positive solution (𝑢₁,𝑢₂) of the IVP-SIP-I.Assumption 3.7. We assume that 𝑓 is of the form 𝑓𝑓₁₂𝑡,𝑢^𝜎1=𝐴₁(𝑡)𝐵₁𝑢^𝜎1,𝑡∈[𝑎,𝑐]_𝕋,𝑡,𝑢^𝜎2=𝐴₂(𝑡)𝐵₂𝑢^𝜎2,𝑡∈[𝜎(𝑐),𝑏]_𝕋,(3.12) where 𝐴_𝑖,𝐵_𝑖 are functions on 𝕋_𝑖,𝒞(𝕋_𝑖),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 functional 𝜁((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.
If 𝐴₁,𝐴₂≤0, then we define𝜁𝜂₁,𝜂₂,𝑓₁,𝑓₂;𝑎,𝑏+=^𝑐𝑎𝜂^Δ1(𝑡)²−𝑓₁𝑡,𝑦^𝜎1𝜂₁(𝑡)²Δ𝑡_{𝑏𝜎(𝑐)}𝜂^Δ2(𝑡)²−𝑓₂𝑡,𝑦^𝜎2𝜂₂(𝑡)²Δ𝑡.(3.13) If 𝐴₁≤0,𝐴₂≥0, then we define 𝜁𝜂₁,𝜂₂,𝑓₁,𝑓₂;𝑎,𝑏+=^𝑐𝑎𝜂^Δ1(𝑡)²−𝑓₁𝑡,𝑦^𝜎1𝜂₁(𝑡)²Δ𝑡_{𝑏𝜎(𝑐)}𝜂^Δ2(𝑡)²+𝑓₂𝑡,𝑦^𝜎2𝜂₂(𝑡)²Δ𝑡.(3.14) If 𝐴₁≥0,𝐴₂≤0, then we define 𝜁𝜂₁,𝜂₂,𝑓₁,𝑓₂;𝑎,𝑏+=^𝑐𝑎𝜂^Δ1(𝑡)²+𝑓₁𝑡,𝑦^𝜎1𝜂₁(𝑡)²Δ𝑡_{𝑏𝜎(𝑐)}𝜂^Δ2(𝑡)²−𝑓₂𝑡,𝑦^𝜎2𝜂₂(𝑡)²Δ𝑡.(3.15) 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 that 𝐴₁,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 𝑦𝑦^1ΔΔ_2ΔΔ(𝑡)=𝜆₁₁𝑓₁𝑡,𝑦^𝜎1,𝑡∈𝕋₁,(𝑡)=−𝜆∞₁₂𝑓<₂𝜆𝑡,₁₁𝑦^𝜎2,𝑡∈𝕋₂,,𝜆₁₂<+∞,(3.16) 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. Let 𝑓(𝑡,𝑦^𝜎)=0, that is, 𝑓𝑓₁₁𝑡,𝑦^𝜎1=0,𝑡∈𝕋₁,𝑡,𝑦^𝜎1=0,𝑡∈𝕋₂.(3.17) Then we see that IVP-SIP-II reduces to the IVP-SIP-III 𝑦𝑦^1ΔΔ_2ΔΔ(𝑡)=0,𝑡∈𝕋₁,𝑦₁(𝑡)=0,𝑡∈𝕋₂,𝑦(0)=𝑙,_Δ1𝜌(0)=𝑚,₁𝜌𝑦₁(𝑐)=𝜌₂𝑦₂(𝜎(𝑐)),₃𝑦^Δ1(𝑐)=𝜌₄𝑦^Δ2(𝜎(𝑐)).(3.18) Simple calculations show that 𝑦𝑦₁(𝑡)=𝑚𝑡+𝑙,₂(𝑡)=𝜌³𝜌₄𝑚𝑡+𝜌¹𝜌₂(𝑐𝑚+𝑙)−𝜌³𝜌₄𝑚𝜎(𝑐).(3.19) We observe that 𝑦(𝑡)=0 whenever 𝑡=−𝑙𝑚or𝜎(𝑐)−𝜌¹𝜌₄𝜌₂𝜌₃𝑐+𝑚𝑙.(3.20) 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 every 𝜆₁₁,𝜆₁₂. From Lemma 3.6, we have𝜁𝜂₁,𝜂₂,𝑓₁,𝑓₂;𝑎,𝑏≥0.(3.21) Hence ^𝑐𝑎𝜂^Δ1(𝑡)²+𝜆₁₁𝑓₁𝑡,𝑦^𝜎1𝜂₁(𝑡)²Δ𝑡+^{𝑏𝜎(𝑐)}𝜂^Δ2(𝑡)²+𝜆₁₂𝑓₂𝑡,𝑦^𝜎2𝜂₂(𝑡)²Δ𝑡≥0.(3.22) So we have −𝜆₁₁^𝑐𝑎𝑓₁𝑡,𝑦^𝜎1𝜂₁(𝑡)²Δ𝑡−𝜆₁₂^{𝑏𝜎(𝑐)}≤𝑓₂𝑡,𝑦^𝜎2𝜂₂(𝑡)²Δ𝑡_𝑐𝑎𝜂_1Δ(𝑡)²+^{𝑏𝜎(𝑐)}𝜂_2Δ(𝑡)².(3.23) The previous equation is true for all −∞<𝜆₁₁,𝜆₁₂<+∞, which implies that ^𝑐𝑎𝑓₁𝑡,𝑦^𝜎1𝜂₁(𝑡)²Δ𝑡=0=^{𝑏𝜎(𝑐)}𝑓₂𝑡,𝑦^𝜎2𝜂₂(𝑡)²Δ𝑡.(3.24) 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 𝑦𝑦^1ΔΔ_2ΔΔ=𝜆₁₁𝑓₁₁𝑡,𝑦^𝜎1+𝜆₂₁𝑓₂₁𝑡,𝑦^𝜎1+⋯+𝜆_𝑛1𝑓_𝑛1𝑡,𝑦^𝜎1,=𝜆₁₂𝑓₁₂𝑡,𝑦^𝜎2+𝜆₂₂𝑓₂₂𝑡,𝑦^𝜎2+⋯+𝜆_𝑛2𝑓_𝑛2𝑡,𝑦^𝜎2,(3.25) along with (3.3)–(3.6). Then IVP-SIP-IV is disconjugate on ℐ for all  ((𝜆₁₁,𝜆₁₂),(𝜆₂₁,𝜆₂₂),…,(𝜆_𝑛1,𝜆_𝑛2))∈ℝ^2𝑛, then 𝑓_𝑖(𝑡,𝑦)=0,𝑖=1,2,…,𝑛, that is, 𝑓𝑓_𝑖1𝑖2𝑡,𝑦^𝜎1=0,for𝑡∈𝕋₁,𝑡,𝑦^𝜎2=0,for𝑡∈𝕋₂,(3.26) where (𝑓_𝑖1,𝑓_𝑖2) are nonlinear function tuples in 𝒞(𝕋₁×ℝ)×𝒞(𝕋₂×ℝ).

4. Disconjugacy Domain

We define IVP-SIP-V as𝑦𝑦^1ΔΔ_2ΔΔ(𝑡)=𝜆₁₁𝑓₁₁𝑡,𝑦^𝜎1+𝜆₂₁𝑓₂₁𝑡,𝑦^𝜎1+⋯+𝜆_𝑛1𝑓_𝑛1𝑡,𝑦^𝜎1,𝑡∈𝕋₁,(𝑡)=𝜆₁₂𝑓₁₂𝑡,𝑦^𝜎2+𝜆₂₂𝑓₂₂𝑡,𝑦^𝜎2+⋯+𝜆_𝑚2𝑓_𝑚2𝑡,𝑦^𝜎2,𝑡∈𝕋₂,(4.1) along with (3.3)–(3.6). Without any loss of generality, let us assume that 𝑛<𝑚, where 𝑛,𝑚∈ℕ and 𝑓_i1,𝑓_𝑗2,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚, belong to 𝒞(𝕋₁×ℝ), 𝒞(𝕋₂×ℝ), respectively.

Definition 4.1. One defines the disconjugacy domain 𝒟 of IVP-SIP-V as 𝒟=𝜆₁₁,𝜆₁₂,𝜆₂₁,𝜆₂₂,…,𝜆_𝑛1,𝜆_𝑛2,𝜆_(𝑛+1)2,𝜆_(𝑛+2)2,…,𝜆_𝑚2∈ℝ^n+𝑚(4.2) such that IVP-SIP-V is disconjugate on ℐ.

Theorem 4.2. The disconjugacy domain 𝒟 of IVP-SIP-V is the whole space ℝ^𝑛+𝑚 if and only if 𝑓𝑓_𝑖1=0,𝑖=1,2,…,𝑛,_𝑗1=0,𝑗=1,2,…,𝑚.(4.3)

Proof. For 𝜆𝜆₂₁=𝜆₃₁=⋯=𝜆_𝑛1=0,𝜆₁₁≠0,₂₂=𝜆₃₂=⋯=𝜆_𝑚2=0,𝜆₁₂≠0,(4.4) IVP-SIP-V reduces to IVP-SIP-II, and hence from Lemma 3.8 we have 𝑓₁₁=0=𝑓₁₂. If we choose 𝜆𝜆₁₁=𝜆₃₁=⋯=𝜆_𝑛1=0,𝜆₂₁≠0,₁₂=𝜆₃₂=⋯=𝜆_𝑚2=0,𝜆₂₂≠0,(4.5) we get 𝑓₂₁=0=𝑓₂₂. In similar lines we can show that 𝑓𝑓₃₁=𝑓₄₁=⋯=𝑓_𝑛1=0,₃₂=𝑓₄₂=⋯=𝑓_𝑛2=0.(4.6) Now let us choose 𝜆𝜆₁₁=𝜆₃₁=⋯=𝜆_(𝑛−1)1=0,𝜆_𝑛1≠0,₁₂=𝜆₃₂=⋯=𝜆_𝑛2=0=𝜆_(𝑛+2)2=⋯=𝜆_𝑚2,𝜆_(𝑛+1)2≠0.(4.7) We see that from Lemma 3.8 we have 𝑓_(𝑛+1)2=0. Similarly we can show that 𝑓_(𝑛+2)2=𝑓_(𝑛+3)2=⋯=𝑓_𝑚2=0.(4.8) Hence 𝑓𝑓_𝑖1=0,𝑖=1,2,…,𝑛,_𝑗1=0,𝑗=1,2,…,𝑚.(4.9) whenever the disconjugacy domain 𝒟 of IVP-SIP-V is the whole space ℝ^𝑛+𝑚.
On the other hand, if𝑓𝑓_𝑖1=0,𝑖=1,2,…,𝑛,_𝑗1=0,𝑗=1,2,…,𝑚,(4.10) by simple calculations, it can be seen that IVP-SIP-V is disconjugate.

Corollary 4.3. If at least one of the functions 𝑓_𝑖1 or 𝑓_𝑗2≠0, then 𝒟 is a proper subset of ℝ^𝑛+𝑚.

Proof. This is the contrapositive of Theorem 4.2.

Theorem 4.4. 𝒟 contains a proper subspace of the vector space ℝ^𝑛+𝑚 if the sets of functions {𝑓₁₁,𝑓₂₁,…,𝑓_𝑛1} and {𝑓₁₂,𝑓₂₂,…,𝑓_𝑚2} are linearly dependent.

Proof. Let {𝑓₁₁,𝑓₂₁,…,𝑓_𝑛1} and {𝑓₁₂,𝑓₂₂,…,𝑓_𝑚2} 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 𝑓𝑓₁₁₁₂𝑡,𝑦^𝜎1=𝑐₁₁𝑓₂₁𝑡,𝑦^𝜎1+𝑐₂₁𝑓₃₁𝑡,𝑦^𝜎1+⋯+𝑐_(𝑛−1)1𝑓_𝑛1𝑡,𝑦^𝜎2,for𝑡∈𝕋₁,𝑡,𝑦^𝜎2=𝑐₁₂𝑓₂₂𝑡,𝑦^𝜎2+𝑐₂₂𝑓₃₂𝑡,𝑦^𝜎2+⋯+𝑐_(𝑚−1)2𝑓_𝑚2𝑡,𝑦^𝜎2,for𝑡∈𝕋₂.(4.11) Therefore, the IVP-SIP-V becomes 𝑦^1ΔΔ(𝑡)=+𝜆𝜆₁₁₂₁𝑐₁₁𝑓₂₁+𝑐₂₁𝑓₃₁+⋯+𝑐_(𝑛−1)1𝑓_𝑛1𝑡,𝑦^𝜎1𝑦_2ΔΔ𝑓₂₁𝑡,𝑦^𝜎1+⋯+𝜆_𝑛1𝑓_𝑛1𝑡,𝑦^𝜎1,𝑡∈𝕋₁,(𝑡)=+𝜆𝜆₁₂₂₂𝑐₁₂𝑓₂₂+𝑐₂₂𝑓₃₂+⋯+𝑐_(𝑚−1)2𝑓_𝑚2𝑡,𝑦^𝜎2𝑓₂₂𝑡,𝑦^𝜎2+⋯+𝜆_𝑚2𝑓_𝑚2𝑡,𝑦^𝜎2,𝑡∈𝕋₂,(4.12) along with (3.3)–(3.6). We see that IVP-SIP-V now is to be 𝑦𝑦^1ΔΔ_2ΔΔ(𝑡)=𝜆₁₁𝑐₁₁+𝜆₂₁𝑓₂₁𝑡,𝑦^𝜎1+𝜆₁₁𝑐₂₁+𝜆₃₁𝑓₃₁𝑡,𝑦^𝜎1+⋯+𝜆₁₁𝑐_(𝑛−1)1+𝜆_𝑛1𝑓_𝑛1𝑡,𝑦^𝜎1,(𝑡)=𝜆₁₂𝑐₁₂+𝜆₂₂𝑓₂₂𝑡,𝑦^𝜎2+𝜆₁₂𝑐₂₂+𝜆₃₂𝑓₃₂𝑡,𝑦^𝜎2+⋯+𝜆₁₂𝑐_(𝑚−1)2+𝜆_𝑚2𝑓_𝑚2𝑡,𝑦^𝜎2,(4.13) along with (3.3)–(3.6). Hence 𝒟 contains the subspace 𝒮=𝜆₁₁,𝜆₁₂,𝜆₂₁,𝜆₂₂,…,𝜆_𝑛1,𝜆_𝑛2,𝜆_(𝑛+1)2,𝜆_(𝑛+2)2,…,𝜆_𝑚2∈ℝ^𝑛+𝑚,(4.14) where 𝜆𝜆₂₁₂₂=−𝜆₁₁𝑐₁₁,𝜆₃₁=−𝜆₁₁𝑐₂₁,…,𝜆_𝑛1=−𝜆₁₁𝑐_(𝑛−1)1,=−𝜆₁₂𝑐₁₂,𝜆₃₂=−𝜆₁₂𝑐₂₂,…,𝜆_𝑚2=−𝜆₁₂𝑐_(𝑚−1)2,(4.15) hence the proof.

Corollary 4.5. If the sets of functions {𝑓₁₁,𝑓₂₁,…,𝑓_𝑛1} and {𝑓₁₂,𝑓₂₂,…,𝑓_𝑚2} are linearly independent, then 𝒟 cannot contain a proper subspace of ℝ^𝑛+𝑚.

Proof. This is the contrapositive of Theorem 4.4.

Lemma 4.6. Let us consider the following two IVP-SIPs which are disconjugate on ℐ: 𝑦𝑦^1ΔΔ_2ΔΔ(𝑡)=𝑓₁₁𝑡,𝑦^𝜎1,𝑡∈𝕋₁,𝑦(𝑡)=𝑓₁₂𝑡,𝑦^𝜎2,𝑡∈𝕋₂,alongwith(3.3)-(3.6),_1ΔΔ𝑦_2ΔΔ(𝑡)=𝑓₂₁𝑡,𝑦^𝜎1,𝑡∈𝕋₁,(𝑡)=𝑓₂₂𝑡,𝑦^𝜎2,𝑡∈𝕋₂,alongwith(3.3)-(3.6).(4.16) Then the IVP-SIP-VI defined by 𝑦𝑦^1ΔΔ_2ΔΔ(𝑡)=(1−𝛾)𝑓₁₁𝑡,𝑦^𝜎1+𝛾𝑓₂₁𝑡,𝑦^𝜎1,𝑡∈𝕋₁,(𝑡)=(1−𝛾)𝑓₁₂𝑡,𝑦^𝜎2+𝛾𝑓₂₂𝑡,𝑦^𝜎2,𝑡∈𝕋₂,(4.17) along with (3.3)–(3.6) is also disconjugate on ℐ, for each 𝛾∈[0,1].

Proof. On account of Lemma 3.6 it is sufficient if we show that 𝜁𝜂₁,𝜂₂,(1−𝛾)𝑓₁₁+𝛾𝑓₂₁,(1−𝛾)𝑓₁₂+𝛾𝑓₂₂;𝑎,𝑏≥0(4.18) on 𝐴([𝑎,𝑐]∪[𝜎(𝑐),𝑏]) for every compact subinterval [𝑎,𝑐]∪[𝜎(𝑐),𝑏]⊂𝕋₁∪𝕋₂. We see that 𝜁𝜂=₁,𝜂₂,(1−𝛾)𝑓₁₁+𝛾𝑓₂₁,(1−𝛾)𝑓₁₂+𝛾𝑓₂₂;𝑎,𝑏_𝑐𝑎+𝜂^Δ1(𝑡)²+(1−𝛾)𝑓₁₁𝑡,𝑦^𝜎1+𝛾𝑓₂₁𝑡,𝑦^𝜎1𝜂₁(𝑡)²Δ𝑡_{𝑏𝜎(𝑐)}𝜂^Δ2(𝑡)²+(1−𝛾)𝑓₁₂𝑡,𝑦^𝜎1+𝛾𝑓₂₂𝑡,𝑦^𝜎1𝜂₂(𝑡)²Δ𝑡.(4.19) Now ^𝑐𝑎𝜂^Δ1=(𝑡)²+(1−𝛾)𝑓₁₁𝑡,𝑦^𝜎1+𝛾𝑓₂₁𝑡,𝑦^𝜎1𝜂₁(𝑡)²Δ𝑡_𝑐𝑎=(1−(1𝛾)−_𝑐𝑎𝛾)𝜂^Δ1(𝑡)²+𝛾𝜂^Δ1(𝑡)²+(1−𝛾)𝑓₁₁𝑡,𝑦^𝜎1+𝛾𝑓₂₁𝑡,𝑦^𝜎1𝜂₁(𝑡)²Δ𝑡𝜂^Δ1(𝑡)²+𝑓₁₁𝑡,𝑦^𝜎1𝜂₁(𝑡)²Δ𝑡+𝛾^𝑐𝑎𝜂^Δ1(𝑡)²+𝑓₂₁𝑡,𝑦^𝜎1𝜂₁(𝑡)²Δ𝑡.(4.20) Similarly it can be shown that ^{𝑏𝜎(𝑐)}𝜂^Δ2=(1−𝛾()𝑡)²_{𝑏𝜎(𝑐)}+(1−𝛾)𝑓₁₂𝑡,𝑦^𝜎1+𝛾𝑓₂₂𝑡,𝑦^𝜎1𝜂₂(𝑡)²Δ𝑡𝜂^Δ2(𝑡)²+𝑓₁₂𝑡,𝑦^𝜎2𝜂₂(𝑡)²Δ𝑡+𝛾^{𝑏𝜎(𝑐)}𝜂^Δ2(𝑡)²+𝑓₂₂𝑡,𝑦^𝜎2𝜂₂(𝑡)²Δ𝑡.(4.21) Hence we have 𝜁𝜂=₁,(𝜂1₂−,𝛾)(1𝜁−𝛾𝜂)₁𝑓₁₁+𝛾𝑓₂₁,(1−𝛾)𝑓₁₂+𝛾𝑓₂₂;𝑎,𝑏,𝜂₂,𝑓₁₁,𝑓₁₂;𝑎,𝑏+𝛾𝜁𝜂₁,𝜂₂,𝑓₂₁,𝑓₂₂;𝑎,𝑏≥0,(4.22) hence the proof.

Definition 4.7. Let (𝑓_𝑛1,𝑓_𝑛2),(𝑓₁,𝑓₂),𝑛∈ℕ be function tuples in 𝒞(𝕋₁×ℝ)×𝒞(𝕋₂×ℝ). One says that (𝑓_𝑛1,𝑓_𝑛2),(𝑓₁,𝑓₂) are close in uniform norm whenever sup_𝑡∈𝕋1||𝑓_𝑛1(𝑡)−𝑓₁(𝑡)||+sup_𝑡∈𝕋2||𝑓_𝑛2(𝑡)−𝑓₂(𝑡)||⟶0as𝑛⟶∞.(4.23)

Definition 4.8. Let (𝑦_𝑛1,𝑦_𝑛2),(𝑦₁,𝑦₂),𝑛∈ℕ be function tuples in 𝒞(𝕋₁)×𝒞(𝕋₂). One says that (𝑦_𝑛1,𝑦_𝑛2),(𝑦₁₁,𝑦₁₂) are close in uniform norm whenever sup_𝑡∈𝕋1||𝑦_𝑛1(𝑡)−𝑦₁(𝑡)||+sup_𝑡∈𝕋2||𝑦_𝑛2(𝑡)−𝑦₂(𝑡)||⟶0as𝑛⟶∞.(4.24)

Lemma 4.9. Let 𝑓_𝑛=(𝑓_𝑛1,𝑓_𝑛2),𝑓=(𝑓₁,𝑓₂),𝑛∈ℕ. Let the sequence of the IVP-SIPs defined by 𝑦𝑦^ΔΔ𝑛1_ΔΔ𝑛2(𝑡)=𝑓_𝑛1𝑡,𝑦^𝜎1,𝑡∈𝕋₁,(𝑡)=𝑓_𝑛2𝑡,𝑦^𝜎2,𝑡∈𝕋₂,(4.25) along with (3.3)–(3.6) be disconjugate on ℐ for each 𝑛=1,2,…. If 𝑓_𝑛→𝑓 uniformly, then the IVP-SIP 𝑦𝑦^1ΔΔ_2ΔΔ(𝑡)=𝑓₁𝑡,𝑦^𝜎1,𝑡∈𝕋₁,(𝑡)=𝑓₂𝑡,𝑦^𝜎2,𝑡∈𝕋₂,(4.26) 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 (𝑦_𝑛1,𝑦_𝑛2) be a solution such that 𝑦_𝑛1(𝑡₁)=0.Claim 1. The solutions of (4.25), (4.26) satisfying (3.3)–(3.6) are close in the uniform norm whenever (𝑓_𝑛1,𝑓_𝑛2) 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) are𝑦_𝑛1(𝑡)=^𝑡0^𝑘0𝑦𝑓_𝑛1𝑠,𝑦^𝜎1Δ𝑠Δ𝑘+𝑙𝑡+𝑚,_𝑛2(𝑡)=^{𝑡𝜎(𝑐)}^𝑘′_𝜎(𝑐)𝑓_𝑛2𝑠,𝑦^𝜎2Δ𝑠Δ𝑘^+^{𝑡𝜎(𝑐)}𝜌₃𝜌₄^𝑐0𝑓_𝑛1𝑠,𝑦^𝜎1Δ𝑠+𝑚Δ𝑘^+𝜌¹𝜌₂^𝑐0^𝑘0𝑦𝑓_𝑛1𝑠,𝑦^𝜎1Δ𝑠Δ𝑘+𝑙𝑐+𝑚,(4.27)₁(𝑡)=^𝑡0^𝑘0𝑦𝑓₁𝑠,𝑦^𝜎1Δ𝑠Δ𝑘+𝑙𝑡+𝑚,₂(𝑡)=^{𝑡𝜎(𝑐)}^𝑘′_𝜎(𝑐)𝑓₂𝑠,𝑦^𝜎2Δ𝑠Δ𝑘^+^{𝑡𝜎(𝑐)}𝜌₃𝜌₄^𝑐0𝑓₁𝑠,𝑦^𝜎1Δ𝑠+𝑚Δ𝑘^+𝜌¹𝜌₂^𝑐0^𝑘0𝑓₁𝑠,𝑦^𝜎1Δ𝑠Δ𝑘+𝑙𝑐+𝑚.(4.28) 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 (𝑓_𝑛1,𝑓_𝑛2) 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 sup_{𝑡∈[𝑡1,𝑐]}||𝑦_𝑛1(𝑡)−𝑦₁(𝑡)||+sup_{𝑡∈[𝜎(𝑐),𝑡2+𝛽]}||𝑦_𝑛2(𝑡)−𝑦₂(𝑡)||<𝜖(4.29) whenever sup_{𝑡∈[𝑡1,𝑐]}||𝑓_𝑛1(𝑡)−𝑓₁(𝑡)||+sup_{𝑡∈[𝜎(𝑐),𝑡2+𝛽]}||𝑓_𝑛2(𝑡)−𝑓₂(𝑡)||<𝛿.(4.30) Since 𝑦₂(𝑡) must change sign at 𝑡=𝑡₂, it follows that 𝑦_𝑛2(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 domain 𝒟 of IVP-SIP-V is a closed set of the parameter space ℝ^𝑛+𝑚.

Proof. Let (𝜆₁,𝜆₂,…,𝜆_𝑛+𝑚) be a limit point of the sequence (𝜆_𝑘1,𝜆_𝑘2,…,𝜆_{𝑘(𝑛+𝑚)})∈𝐷,𝑘=1,2,…. Then for every 𝜖>0 there exists sufficiently large 𝐾 such that for all 𝑘>𝐾 we have |𝜆_𝑘1−𝜆₁|<𝜖,|𝜆_𝑘2−𝜆₂|<𝜖,…,|𝜆_{𝑘(𝑛+𝑚)}−𝜆_𝑛+𝑚|<𝜖, and IVP-SIP-VII defined by 𝑦𝑦^1ΔΔ_2ΔΔ(𝑡)=𝜆_𝑘1𝑓₁₁+𝜆_𝑘3𝑓₂₁+⋯+𝜆_{𝑘(2𝑛−1)}𝑓_𝑛1,𝑡∈𝕋₁,(𝑡)=𝜆_𝑘2𝑓₁₂+𝜆_𝑘4𝑓₂₂+⋯+𝜆_𝑘(2𝑛)𝑓_𝑛2+𝜆_{𝑘(2𝑛+1)}𝑓_(𝑛+1)2+⋯+𝜆_{𝑘(𝑛+𝑚)}𝑓_𝑚2,𝑡∈𝕋₂,(4.31) along with (3.3)–(3.6) is disconjugate. Let 𝑦=(𝑦₁,𝑦₂) be a non trivial solution of IVP-SIP-V for 𝜆₁₁,𝜆₂₁,…,𝜆_𝑛1,𝜆₁₂,𝜆₂₂,…,𝜆_𝑚2=𝜆₁,𝜆₂,…,𝜆_𝑛+𝑚.(4.32) 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 (𝑦_𝑛1,𝑦_𝑛2) be a solution of IVP-SIP-VII such that 𝑦_𝑛1(𝑡₀)=0. From assumption we have that 𝑦_𝑛1(𝑡)≠0 for 𝑡≠𝑡₀. From Claim 1 we have (𝑦_𝑛1,𝑦_𝑛2) 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 domain 𝒟 of IVP-SIP-V is a convex set in the parameter space ℝ^𝑛+𝑚.

Proof. We need to show that for (𝜆₁₁,𝜆₁₂,…,𝜆_1(𝑛+𝑚)),(𝜆₂₁,𝜆₂₂,…,𝜆_2(𝑛+𝑚))∈𝐷 the convex combination (1−𝛾)𝜆₁₁,𝜆₁₂,…,𝜆_1(𝑛+𝑚)+𝛾𝜆₂₁,𝜆₂₂,…,𝜆_2(𝑛+𝑚)∈𝐷,(4.33) that is, (1−𝛾)𝜆₁₁+𝛾𝜆₂₁,(1−𝛾)𝜆₁₂+𝛾𝜆₂₂,…,(1−𝛾)𝜆_1(𝑛+𝑚)+𝛾𝜆_2(𝑛+𝑚)∈𝐷.(4.34) Since (𝜆₁₁,𝜆₁₂,…,𝜆_1(𝑛+𝑚)),(𝜆₂₁,𝜆₂₂,…,𝜆_2(𝑛+𝑚))∈𝐷, we have the IVP-SIPs 𝑦𝑦^1ΔΔ_2ΔΔ(𝑡)=𝜆₁₁𝑓₁₁+𝜆₁₃𝑓₂₁+⋯+𝜆_1(2𝑛−1)𝑓_𝑛1,𝑡∈𝕋₁,𝑦_1ΔΔ(𝑡)=𝜆₁₂𝑓₁₂+𝜆₁₄𝑓₂₂+⋯+𝜆_1(2𝑛)𝑓_𝑛2+𝜆_1(2𝑛+1)𝑓_(𝑛+1)2+⋯+𝜆_1(𝑛+𝑚)𝑓_𝑚2,𝑡∈𝕋₂,𝑦_2ΔΔ(𝑡)=𝜆₂₁𝑓₁₁+𝜆₂₃𝑓₂₁+⋯+𝜆_2(2𝑛−1)𝑓_𝑛1,𝑡∈𝕋₁,(𝑡)=𝜆₂₂𝑓₁₂+𝜆₂₄𝑓₂₂+⋯+𝜆_2(2𝑛)𝑓_𝑛2+𝜆_2(2𝑛+1)𝑓_(𝑛+1)2+⋯+𝜆_2(𝑛+𝑚)𝑓_𝑚2,𝑡∈𝕋₂,(4.35) along with (3.3)–(3.6) to be disconjugate. Now from Lemma 4.6 we have the IVP-SIP-VIII defined by 𝑦^1ΔΔ(𝑡)=+(1𝛾−𝜆𝛾)₂₁𝜆₁₁𝑓₁₁+𝜆₁₃𝑓₂₁+⋯+𝜆_1(2𝑛−1)𝑓_𝑛1𝑦_2ΔΔ𝑓₁₁+𝜆₂₃𝑓₂₁+⋯+𝜆_2(2𝑛−1)𝑓_𝑛1,𝑡∈𝕋₁,(𝑡)=+(1𝛾−𝜆𝛾)₂₂𝜆₁₂𝑓₁₂+𝜆₁₄𝑓₂₂+⋯+𝜆_1(2𝑛)𝑓_𝑛2+𝜆_1(2𝑛+1)𝑓_(𝑛+1)2+⋯+𝜆_1(𝑛+𝑚)𝑓_𝑚2𝑓₁₂+𝜆₂₄𝑓₂₂+⋯+𝜆_2(2𝑛)𝑓_𝑛2+𝜆_2(2𝑛+1)𝑓_(𝑛+1)2+⋯+𝜆_2(𝑛+𝑚)𝑓_𝑚2,𝑡∈𝕋₂,(4.36) along with (3.3)–(3.6) to be disconjugate; that, is we have the IVP-SIP 𝑦^1ΔΔ(𝑡)=(1−𝛾)𝜆₁₁𝑓₁₁+𝛾𝜆₂₁𝑓₁₁+(1−𝛾)𝜆₁₃𝑓₂₁+𝛾𝜆₂₃𝑓₂₁𝑦_2ΔΔ+⋯+(1−𝛾)𝜆_1(2𝑛−1)𝑓_𝑛1+𝛾𝜆_2(2𝑛−1)𝑓_𝑛1,𝑡∈𝕋₁,(𝑡)=(1−𝛾)𝜆₁₂𝑓₁₂+𝛾𝜆₂₂𝑓₁₂+(1−𝛾)𝜆₁₄𝑓₂₂+𝛾𝜆₂₄𝑓₂₂+⋯+(1−𝛾)𝜆_1(𝑛+𝑚)𝑓_𝑚2+𝛾𝜆_2(𝑛+𝑚)𝑓_𝑚2,𝑡∈𝕋₂,(4.37) along with (3.3)–(3.6) to be disconjugate. Hence (1−𝛾)𝜆₁₁,𝜆₁₂,…,𝜆_1(𝑛+𝑚)+𝛾𝜆₂₁,𝜆₂₂,…,𝜆_2(𝑛+𝑚)∈𝐷,(4.38) 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 domain 𝒩 of IVP-SIP-V as 𝒩=𝜆₁₁,𝜆₁₂,𝜆₂₁,𝜆₂₂,…,𝜆_𝑛1,𝜆_𝑛2,𝜆_(𝑛+1)2,𝜆_(𝑛+2)2,…,𝜆_𝑚2∈ℝ^𝑛+𝑚(5.1) such that IVP-SIP-V is non oscillatory on ℐ.

Lemma 5.3. If IVP-SIP-I is non oscillatory on ℐ, then there exists at least one positive or negative non trivial solution.

Proof. Let 𝑦(=(𝑦₁,𝑦₂)) be a solution of IVP-SIP-I. Let 𝑡₁,𝑡₂,…𝑡_𝑛 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>𝑡_𝑛. We now define 𝐾(𝑡)=0,𝑡≤𝑡^𝑛𝑦(𝑡),𝑡>𝑡_𝑛,.(5.2)
From Assumption 3.7 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 assume 𝐾(𝑡)≥0. Similar results can be developed when 𝐾(𝑡)≤0.

All the results discussed in Section 5 regarding 𝒟 can be easily extended to 𝒩. We just state two theorems without proof.

Theorem 5.4. The non oscillatory domain 𝒩 of IVP-SIP-V is a closed set of the parameter space ℝ^𝑛+𝑚.

Theorem 5.5. The non oscillatory domain 𝒩 of IVP-SIP-V is a convex set in the parameter space ℝ^𝑛+𝑚.

6. Applications

We define IVP-SIP-X by 𝑦𝑦^1ΔΔ_2ΔΔ(𝑡)=𝜆₁₁𝑓₁₁𝑡,𝑦^𝜎1+𝜒₁₁𝑓₂₁𝑡,𝑦^𝜎1+⋯+𝜒_(𝑛−1)1𝑓_𝑛1𝑡,𝑦^𝜎1,𝑡∈𝕋₁,(𝑡)=𝜆₁₂𝑓₁₂𝑡,𝑦^𝜎2+𝜒₁₂𝑓₂₂𝑡,𝑦^𝜎2+⋯+𝜒_(𝑚−1)2𝑓_𝑚2𝑡,𝑦^𝜎2,𝑡∈𝕋₂,(6.1) 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 (𝜆_𝑎1,𝜆_𝑏1)∈ℝ or (𝜆_𝑎2,𝜆_𝑏2)∈ℝ such that IVP-SIP-X is non oscillatory for either 𝜆₁₁∈𝜆_𝑎1,𝜆_𝑏1,𝜆₁₂=𝑐₁,(6.2)or 𝜆₁₁=𝑐₂,𝜆₁₂∈𝜆_𝑎2,𝜆_𝑏2,𝑐₁,𝑐₂∈ℝ.(6.3)For every other combinations of (𝜆₁₁,𝜆₁₂)∈ℝ², IVP-SIP-X is oscillatory.(iv)There exists either 𝜆₁ or 𝜆₂∈ℝ such that IVP-SIP-X is non oscillatory (resp., oscillatory) for either 𝜆₁₁∈𝜆₁,∞,𝜆₁₂=𝑐₁,(6.4)or 𝜆₁₁=𝑐₂,𝜆₁₂∈𝜆₂,∞,(6.5)and oscillatory (resp., non oscillatory) for either 𝜆₁₁∈−∞,𝜆₁,𝜆₁₂=𝑐₁,(6.6)or𝜆₁₁=𝑐₂,𝜆₁₂∈−∞,𝜆₂.(6.7)(v)IVP-SIP-X is non oscillatory for
either𝜆₁₁∈ℝ,𝜆₁₂=𝑐₁,(6.8)or𝜆₁₁=𝑐₂,𝜆₁₂∈ℝ.(6.9)(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 𝑐₃≤𝜆₁₁≤𝑐₄,𝑐_3≤𝜆₁₂≤𝑐_4(6.10)Semi-infinite plane 𝑐₃≤𝜆₁₁≤𝑐₄,−∞≤𝜆₁₂≤𝑐_5𝑐𝑐₃≤𝜆₁₁≤𝑐₄,𝑐_6≤𝜆₁₂≤+∞₃𝑐≤𝜆₁₁≤𝑐₄,−∞≤𝜆₁₂≤+∞₆≤𝜆₁₁≤+∞,𝑐_3≤𝜆₁₂≤𝑐_4𝑐₆≤𝜆₁₁≤+∞,−∞≤𝜆₁₂≤𝑐_5𝑐𝑐₆≤𝜆₁₁≤+∞,𝑐_6≤𝜆₁₂≤+∞₆−≤∞𝜆≤₁₁𝜆≤₁₁+∞,−∞≤𝜆₁₂≤+∞≤𝑐₅,𝑐_3≤𝜆₁₂≤𝑐_4−∞≤𝜆₁₁≤𝑐₅,−∞≤𝜆₁₂≤𝑐_5−−∞∞≤≤𝜆𝜆₁₁₁₁≤𝑐₅,𝑐_6≤𝜆₁₂≤+∞−∞≤𝜆₁₁≤𝑐₅,−∞≤𝜆₁₂≤+∞≤+∞,𝑐_3≤𝜆₁₂≤𝑐_4−∞≤𝜆₁₁≤+∞,−∞≤𝜆₁₂≤𝑐_5−∞≤𝜆₁₁≤+∞,𝑐_6≤𝜆₁₂≤+∞(6.11)infinite plane −∞≤𝜆₁₁≤+∞,−∞≤𝜆₁₂≤+∞,(6.12) where 𝑐₃,𝑐₄,𝑐₅,𝑐₆,𝑐_3,𝑐_4,𝑐_5,𝑐_6∈ℝ.

Proof. Clearly we see that IVP-SIP-X is a special case of IVP-SIP-V with 𝜆𝜆₂₁₂₂=𝜒₁₁,𝜆₃₁=𝜒₂₁,…,𝜆_𝑛1=𝜒_(𝑛−1)1,=𝜒₁₂,𝜆₃₂=𝜒₂₂,…,𝜆_𝑚2=𝜒_(𝑚−1)2.(6.13) Let ℒ={𝜆₁₁,𝜒₁₁,𝜒₂₁,…,𝜒_(𝑛−1)1,𝜆₁₂,𝜒₁₂,𝜒₂₂,…,𝜒_(𝑚−1)2} be a subset of the parameter space ℝ^𝑛+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 ℝ^𝑛+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 𝐿₁𝑢₁=𝑑²𝑢₁𝑑𝑥²𝐿₂+𝑘²¹𝑢²¹=0,0≤𝑥≤𝑙₁,𝑢₂=𝑑²𝑢₂𝑑𝑥²+𝑘²²𝑢²²=0,𝑙₁≤𝑥≤𝑙₂,(6.14) where 𝑘^2𝑖=𝑃/𝐸𝐼_𝑖,𝐸 is the modulus of elasticity, 𝑃 is the load applied, 𝐼_𝑖 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 𝑢₁(0)=𝑢^1(0)=0,𝑢₁𝑙₁=𝑢₂𝑙₁,𝑢^1𝑙₁=𝑢^2𝑙₁.(6.15) Here we see that 𝕋₁=[0,𝑙₁],𝕋₂=[𝑙₁,𝑙₂], 𝜌₁=1=𝜌₂, 𝜌₃=1=𝜌₄. We see that 𝜒₁₁𝜆₁₁=−𝑘²¹,𝜆₁₂=−𝑘²²,,𝜒₂₁,…,𝜒_(𝑛−1)1=0=𝜒₁₂,𝜒₂₂,…,𝜒_(𝑚−1)2.(6.16) Hence from Theorem 6.1 we see that if the point {−𝑘²¹,0,0,…,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. Let 𝑑₁,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 𝐿₁𝑢₁=𝑑𝑑𝑧𝑑1₁𝑑𝑢₁𝑑𝑧+𝑘𝑑¹²₁−𝜆𝑑₁𝑢¹𝐿=0,0≤𝑧≤𝑎,₂𝑢₂=𝑑𝑑𝑧𝑑1₂𝑑𝑢₂𝑑𝑧+𝑘𝑑²²₂−𝜆𝑑₂𝑢²=0,𝑎≤𝑧≤𝑏,(6.17) together with the mixed boundary conditions given by 𝑢₁𝑢(0)=𝑢_2(𝑏)=0,₁1(𝑎)=𝑢₂(𝑎),𝑑₁𝑢_1(𝑎)=1𝑑₂𝑢_2(𝑎),(6.18)𝑢₁ and 𝑢₂ denote the depth eigenfunctions corresponding to the eigenvalue 𝜆.
Here we see that 𝕋₁=[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 that𝜆𝜒₁₁₂₁=𝜆,𝜒₁₁=−𝑘₁₂,𝜆₁₂=𝜆,𝜒₁₂=−𝑘₂₂,,𝜒₃₁,…,𝜒_(𝑛−1)1=0=𝜒₂₂,𝜒₃₂,…,𝜒_(𝑚−1)2.(6.19) 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 lengths 𝑎 and 𝑏 and different uniform densities 𝑑₁ and 𝑑₂, respectively, having a tension T stretched between the points 𝑥=0 and 𝑥=𝑎+𝑏: 𝐿𝐿₁𝑢₁=𝑐₁₂−𝑢^1=𝜆𝑢₁,0≤𝑥≤𝑎,₂𝑢₂=𝑐₂₂−𝑢^2=𝜆𝑢₂,𝑎≤𝑥≤𝑎+𝑏,(6.20) with the mixed boundary conditions given by 𝑢𝑢₁(𝑎)=𝑢₂(𝑎),_1(𝑎)=𝑢^2(𝑎),(6.21) and 𝑢₁(0)=𝑢₂(𝑎+𝑏)=0, where 𝑐_𝑖2=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 𝜆₁₁=−𝜆𝑐₁₂,𝜆₁₂=−𝜆𝑐₂₂𝜒₁₁,,𝜒₂₁,…,𝜒_(𝑛−1)1=0=𝜒₁₂,𝜒₂₂,…,𝜒_(𝑚−1)2.(6.22) 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.

Acknowledgments

The authors dedicate this work to the Founder Chancellor of Sri Sathya Sai Institute of Higher Learning, Bhagwan Sri Sathya Sai Baba. This study is funded under the Research Project no. ERIP/ER/0803728/M/01/1158, by DRDO, Ministry of Defence, Government of INDIA.