Table of Contents
ISRN Mathematical Analysis
Volumeย 2011ย (2011), Article IDย 254695, 23 pages
http://dx.doi.org/10.5402/2011/254695
Research Article

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

Department of Mathematics and Computer Science, Sri Sathya Sai Institute of Higher Learning, Prasanthi Nilayam, Ananthapur, Andhra Pradesh, Puttaparthi 515134, India

Received 18 July 2011; Accepted 6 September 2011

Academic Editors: A.ย Carpio and J.-F.ย Colombeau

Copyright ยฉ 2011 D. K. K. Vamsi and Pallav Kumar Baruah. 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.

Abstract

We define and discuss the disconjugacy (๐’Ÿ) and nonoscillation (๐’ฉ) domains for a pair of dynamic equations along with matching interface conditions on the semi-infinite time scale [0,๐‘]๐•‹โˆช[๐œŽ(๐‘),โˆž]๐•‹. We show that these domains are closed and convex subsets of the parameter space โ„๐‘›+๐‘š. 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 form๐‘ฆ๎…ž๎…ž+(โˆ’๐›ผ+๐›ฝ๐ต(๐‘ฅ))๐‘ฆ=0,(1.1) where ๐›ผ,๐›ฝ are real parameters and ๐‘ฅ varies over a subinterval of โ„. Fixing ๐›ผ and allowing ๐›ฝ 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 equation๐‘ฆ๎…ž๎…ž+(โˆ’๐‘Ž+๐‘๐‘ž(๐‘ฅ))๐‘ฆ=0.(1.2) Here ๐‘Ž,๐‘ are real parameters and ๐‘ž(๐‘ฅ) is real-valued, continuous, and periodic function. A special region ๐‘… a subset of the ๐‘Ž๐‘-plane for which the equation has nonoscillatory solutions is introduced. ๐‘… is shown to be a closed, convex and is also shown to be entirely contained in the half-plane ๐‘Ž>0. In [3], Markus and Moore found the disconjugacy domain ๐’Ÿ (a subset of the (๐‘Ž,๐‘)-parameter plane) for which the equation๐‘ฆ๎…ž๎…ž+(โˆ’๐‘Ž+๐‘๐‘(๐‘ฅ))๐‘ฆ=0(1.3) has disconjugate solutions. ๐ท 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 ๐’Ÿ and ๐’ฉ) for the general equation๐‘ฆ๎…ž๎…ž+(โˆ’๐›ผ๐ด(๐‘ฅ)+๐›ฝ๐ต(๐‘ฅ))๐‘ฆ=0(1.4) on the closed half-line ๐ผ=[0,โˆž),๐ด,๐ตโˆถ[0,โˆž)โ†’โ„. ๐’Ÿ consists of all the values of (๐›ผ,๐›ฝ)โŠ‚โ„2 such that solutions of the general equation are disconjugate. ๐’ฉ consists of all the values of (๐›ผ,๐›ฝ)โŠ‚โ„2 such that solutions of the general equation are non oscillatory. They discussed the properties of ๐’Ÿ and ๐’ฉ 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 [9โ€“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 [20โ€“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: 254695.fig.001(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 [26โ€“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 [0,๐‘]๐•‹โˆช[๐œŽ(๐‘),โˆž)๐•‹. We define and discuss the disconjugacy (๐’Ÿ) and non oscillation (๐’ฉ) 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 โ„๐‘›+๐‘š. 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 ๐•‹ be a time scale (an arbitrary closed subset of real numbers). For ๐‘กโˆˆ๐•‹ one defines the forward jump operator ๐œŽโˆถ๐•‹โ†’๐•‹ by ๐œŽ(๐‘ก)โˆถ=inf{๐‘ โˆˆ๐•‹โˆถ๐‘ >๐‘ก},(2.1) while the backward jump operator ๐œŒโˆถ๐•‹โ†’๐•‹ is defined by ๐œŒ(๐‘ก)โˆถ=sup{๐‘ โˆˆ๐•‹โˆถ๐‘ <๐‘ก}.(2.2) If ๐œŽ(๐‘ก)>๐‘ก, we say that ๐‘ก is right scattered, while if ๐œŒ(๐‘ก)<๐‘ก we say that ๐‘ก is left scattered. Points that are right scattered and left scattered at the same time are called isolated. Also, if ๐‘ก<sup๐•‹ and ๐œŽ(๐‘ก)=๐‘ก, then ๐‘ก is called right dense, and if ๐‘ก>inf๐•‹ and ๐œŒ(๐‘ก)=๐‘ก, then ๐‘ก is called left dense. Points that are right dense and left dense at the same time are called dense. Finally, the graininess function ๐œ‡โˆถ๐•‹โ†’[0,โˆž) is defined by ๐œ‡(๐‘ก)โˆถ=๐œŽ(๐‘ก)โˆ’๐‘ก.(2.3)

Definition 2.2. One has ๐•‹๐œ…=๎‚ป๐•‹โˆ’{๐‘š},ifsup๐•‹<โˆž,๐•‹,ifsup๐•‹=โˆž,(2.4) where ๐‘š is the left-scattered maximum.

Definition 2.3. Let ๐‘“ be a function defined on ๐•‹. One says that ๐‘“ is delta differentiable at ๐‘กโˆˆ๐•‹๐œ… provided there exists an ๐›ผ such that for all ๐œ–> 0 there is a neighborhood ๐’ฉ around ๐‘ก with ||||||||๐‘“(๐œŽ(๐‘ก))โˆ’๐‘“(๐‘ )โˆ’๐›ผ(๐œŽ(๐‘ก)โˆ’๐‘ )โ‰ค๐œ–๐œŽ(๐‘ก)โˆ’๐‘ โˆ€๐‘ โˆˆ๐’ฉ.(2.5)

Remark 2.4. For a function ๐‘“โˆถ๐•‹โ†’โ„ we will talk about the second derivative ๐‘“ฮ”ฮ” provided ๐‘“ฮ” is differentiable on ๐•‹๐œ…2=(๐•‹๐œ…)๐œ… with derivative ๐‘“ฮ”ฮ”=(๐‘“ฮ”)ฮ”โˆถ๐•‹๐œ…2โ†’โ„.

Definition 2.5. One defines the parameter space โ„๐‘›+๐‘š=๐œ†๎€ฝ๎€ท1,๐œ†2,โ€ฆ,๐œ†๐‘›+๐‘š๎€ธ,๐œ†๐‘–๎€พโˆˆโ„,๐‘–=1,2,โ€ฆ,๐‘›+๐‘š,๐‘›,๐‘šโˆˆโ„•.(2.6)

3. Preliminary Results

Let ๐•‹1=[0,๐‘]๐•‹(a time scale with end points 0 and c), ๐พ1=[๐œŽ(๐‘),โˆž)๐•‹ a time scale with one end being ๐œŽ(๐‘), ๐•‹2=๐พ๐œ…21. Also let (๐‘“1,๐‘“2) be nonlinear function tuple in ๐’ž(๐•‹1ร—โ„)ร—๐’ž(๐•‹2ร—โ„). And let (๐‘“1,๐‘“2) be positive. In this section we consider the following IVP associated with singular interface problem (IVP-SIP-I):๐‘ฆ1ฮ”ฮ”(๐‘ก)=๐‘“1๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ,๐‘กโˆˆ๐•‹1,๐‘ฆ(3.1)2ฮ”ฮ”(๐‘ก)=๐‘“2๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ,๐‘กโˆˆ๐•‹2,(3.2) with the initial conditions๐‘ฆ1๐‘ฆ(0)=๐‘™,(3.3)ฮ”1(0)=๐‘š,where๐‘™,๐‘š>0,(3.4) followed by the matching interface conditions๐œŒ1๐‘ฆ1(๐‘)=๐œŒ2๐‘ฆ2๐œŒ(๐œŽ(๐‘)),(3.5)3๐‘ฆฮ”1(๐‘)=๐œŒ4๐‘ฆฮ”2(๐œŽ(๐‘)),๐œŒ๐‘–>0,๐‘–=1,2,3,4.(3.6)

Definition 3.1. One calls a function ๐‘ฆโˆถ๐•‹1โˆช๐•‹2โ†’โ„ to be a matching solution of the IVP-SIP-I if (i)the function ๐‘ฆ|๐•‹1=๐‘ฆ1 and ๐‘ฆ1 satisfies (3.1), (ii)the function ๐‘ฆ|๐•‹2=๐‘ฆ2 and ๐‘ฆ2 satisfies (3.2), (iii)๐‘ฆ1 and ๐‘ฆ2 satisfy the initial and interface conditions (3.3)-(3.4) and (3.5)-(3.6), respectively.

Definition 3.2. One calls ๐‘กโˆˆ๐•‹1โˆช๐•‹2 to be a zero of the IVP-SIP-I if ๐‘ฆ(๐‘ก)=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 ๐’ž(๐•‹๐‘–),๐‘–=1,2, to be the set of continuous functions on time scales ๐•‹1 and ๐•‹2. Also, we denote ๎‚ป๐‘“๐‘“(๐‘ก)=1(๐‘ก),๐‘กโˆˆ๐•‹1,๐‘“2(๐‘ก),๐‘กโˆˆ๐•‹2.(3.7)

Definition 3.4. For a given compact subinterval [๐‘Ž,๐‘]โˆช[๐œŽ(๐‘),๐‘]โŠ‚๐•‹1โˆช๐•‹2, one defines the space ๎[]โˆช[]๐œ‚๐ด(๐‘Ž,๐‘๐œŽ(๐‘),๐‘)=๎€ฝ๎€ท1,๐œ‚2๎€ธโˆถ๎€ท๐œ‚1,๐œ‚2๎€ธ[][]๎€พโˆˆ๐’ž๐‘Ž,๐‘ร—๐’ž๐œŽ(๐‘),๐‘.(3.8)

For any (๐œ‚1,๐œ‚2๎)โˆˆ๐ด([๐‘Ž,๐‘]โˆช[๐œŽ(๐‘),๐‘]), one defines the functional ๐œ๐œ‚๎€ท๎€ท1,๐œ‚2๎€ธ,๎€ท๐‘“1,๐‘“2๎€ธ๎€ธ=๎€œ;๐‘Ž,๐‘๐‘๐‘Ž๎‚ƒ๎€ฝ๐œ‚ฮ”1๎€พ(๐‘ก)2+๐‘“1๎€ท๐‘ก,๐‘ฆ๐œŽ1๐œ‚๎€ธ๎€ฝ1๎€พ(๐‘ก)2๎‚„+๎€œฮ”๐‘ก๐‘๐œŽ(๐‘)๎‚ƒ๎€ฝ๐œ‚ฮ”2๎€พ(๐‘ก)2+๐‘“2๎€ท๐‘ก,๐‘ฆ๐œŽ2๐œ‚๎€ธ๎€ฝ2๎€พ(๐‘ก)2๎‚„ฮ”๐‘ก.(3.9)

Lemma 3.5. If IVP-SIP-I is disconjugate on every subset of ๐•‹1โˆช๐•‹2 (denoted by โ„), 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 ๐‘ฆ1(๎€œ๐‘ก)=๐‘ก0๎€œ๐‘ 0๐‘“1๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธฮ”๐‘กฮ”๐‘ +๐‘š๐‘ก+๐‘™,๐‘กโˆˆ๐•‹1,๐‘ฆ2๎€œ(๐‘ก)=๐‘ก๐œŽ(๐‘)๎€œ๐‘ ๐œŽ(๐‘)๐‘“2๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ๎€œฮ”๐‘กฮ”๐‘ +๐‘ก๐œŽ(๐‘)๐‘21ฮ”๐‘ +๐‘22,๐‘กโˆˆ๐•‹2,(3.10) where ๐‘21=๐œŒ3๐œŒ4๎€œ๐‘0๐‘“1๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธฮ”๐‘ ,๐‘22=๐œŒ1๐œŒ2๎€œ๐‘0๎€œ๐‘ 0๐‘“1๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธฮ”๐‘กฮ”๐‘ +๐‘š๐‘+๐‘™,(3.11) which give the required positive solution. Moreover, IVP-SIP-I is disconjugate implying that (๐‘ฆ1,๐‘ฆ2) is a non oscillatory solution.

Lemma 3.6. If the IVP-SIP-I is disconjugate on โ„, then, for every closed subinterval of [๐‘Ž,๐‘]โˆช[๐œŽ(๐‘),๐‘], the functional ๐œ((๐œ‚1,๐œ‚2),(๐‘“1,๐‘“2);๐‘Ž,๐‘) is positive on ๎๐ด([๐‘Ž,๐‘]โˆช[๐œŽ(๐‘),๐‘]).

Proof. Let us suppose that the IVP-SIP-I is disconjugate on โ„. Then by Lemma 3.5 there is a positive solution (๐‘ข1,๐‘ข2) of the IVP-SIP-I.Assumption 3.7. We assume that ๐‘“ is of the form ๐‘“1๎€ท๐‘ก,๐‘ข๐œŽ1๎€ธ=๐ด1(๐‘ก)๐ต1๎€ท๐‘ข๐œŽ1๎€ธ[],๐‘กโˆˆ๐‘Ž,๐‘๐•‹,๐‘“2๎€ท๐‘ก,๐‘ข๐œŽ2๎€ธ=๐ด2(๐‘ก)๐ต2๎€ท๐‘ข๐œŽ2๎€ธ[],๐‘กโˆˆ๐œŽ(๐‘),๐‘๐•‹,(3.12) where ๐ด๐‘–,๐ต๐‘– are functions on ๐•‹๐‘–,๐’ž(๐•‹๐‘–),๐‘–=1,2. We also assume that ๐ต1,๐ต2 are positive basing on the fact that (๐‘ข1,๐‘ข2) is a positive solution of the IVP-SIP-I.
From the definition of the functional ๐œ((๐œ‚1,๐œ‚2),(๐‘“1,๐‘“2);๐‘Ž,๐‘)we easily see that ๐œ((๐œ‚1,๐œ‚2),(๐‘“1,๐‘“2);๐‘Ž,๐‘)โ‰ฅ0 whenever ๐ด1,๐ด2โ‰ฅ0. For the cases when (๐ด1,๐ด2โ‰ค0),(๐ด1โ‰ค0,๐ด2โ‰ฅ0),(๐ด1โ‰ฅ0,๐ด2โ‰ค0) we define the functional ๐œ((๐œ‚1,๐œ‚2),(๐‘“1,๐‘“2);๐‘Ž,๐‘) in the following manner.
If ๐ด1,๐ด2โ‰ค0, then we define๐œ๐œ‚๎€ท๎€ท1,๐œ‚2๎€ธ,๎€ท๐‘“1,๐‘“2๎€ธ๎€ธ=๎€œ;๐‘Ž,๐‘๐‘๐‘Ž๎‚ƒ๎€ฝ๐œ‚ฮ”1๎€พ(๐‘ก)2โˆ’๐‘“1๎€ท๐‘ก,๐‘ฆ๐œŽ1๐œ‚๎€ธ๎€ฝ1๎€พ(๐‘ก)2๎‚„+๎€œฮ”๐‘ก๐‘๐œŽ(๐‘)๎‚ƒ๎€ฝ๐œ‚ฮ”2๎€พ(๐‘ก)2โˆ’๐‘“2๎€ท๐‘ก,๐‘ฆ๐œŽ2๐œ‚๎€ธ๎€ฝ2๎€พ(๐‘ก)2๎‚„ฮ”๐‘ก.(3.13) If ๐ด1โ‰ค0,๐ด2โ‰ฅ0, then we define ๐œ๐œ‚๎€ท๎€ท1,๐œ‚2๎€ธ,๎€ท๐‘“1,๐‘“2๎€ธ๎€ธ=๎€œ;๐‘Ž,๐‘๐‘๐‘Ž๎‚ƒ๎€ฝ๐œ‚ฮ”1๎€พ(๐‘ก)2โˆ’๐‘“1๎€ท๐‘ก,๐‘ฆ๐œŽ1๐œ‚๎€ธ๎€ฝ1๎€พ(๐‘ก)2๎‚„+๎€œฮ”๐‘ก๐‘๐œŽ(๐‘)๎‚ƒ๎€ฝ๐œ‚ฮ”2๎€พ(๐‘ก)2+๐‘“2๎€ท๐‘ก,๐‘ฆ๐œŽ2๐œ‚๎€ธ๎€ฝ2๎€พ(๐‘ก)2๎‚„ฮ”๐‘ก.(3.14) If ๐ด1โ‰ฅ0,๐ด2โ‰ค0, then we define ๐œ๐œ‚๎€ท๎€ท1,๐œ‚2๎€ธ,๎€ท๐‘“1,๐‘“2๎€ธ๎€ธ=๎€œ;๐‘Ž,๐‘๐‘๐‘Ž๎‚ƒ๎€ฝ๐œ‚ฮ”1๎€พ(๐‘ก)2+๐‘“1๎€ท๐‘ก,๐‘ฆ๐œŽ1๐œ‚๎€ธ๎€ฝ1๎€พ(๐‘ก)2๎‚„+๎€œฮ”๐‘ก๐‘๐œŽ(๐‘)๎‚ƒ๎€ฝ๐œ‚ฮ”2๎€พ(๐‘ก)2โˆ’๐‘“2๎€ท๐‘ก,๐‘ฆ๐œŽ2๐œ‚๎€ธ๎€ฝ2๎€พ(๐‘ก)2๎‚„ฮ”๐‘ก.(3.15) From the above definitions of ๐œ((๐œ‚1,๐œ‚2),(๐‘“1,๐‘“2);๐‘Ž,๐‘), we see that ๐œ((๐œ‚1,๐œ‚2),(๐‘“1,๐‘“2);๐‘Ž,๐‘)โ‰ฅ0 irrespective of the sign of ๐ด1 and ๐ด2.

Note 1. Through the rest of the paper we assume that ๐ด1,๐ด2โ‰ฅ0. Similar results can be obtained for the cases when (๐ด1,๐ด2<0),(๐ด1<0,๐ด2>0),(๐ด1>0,๐ด2<0) by using the respective definitions for ๐œ((๐œ‚1,๐œ‚2),(๐‘“1,๐‘“2);a,b).

Lemma 3.8. Let IVP-SIP-II be defined by ๐‘ฆ1ฮ”ฮ”(๐‘ก)=๐œ†11๐‘“1๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ,๐‘กโˆˆ๐•‹1,๐‘ฆ2ฮ”ฮ”(๐‘ก)=๐œ†12๐‘“2๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ,๐‘กโˆˆ๐•‹2,โˆ’โˆž<๐œ†11,๐œ†12<+โˆž,(3.16) along with (3.3)โ€“(3.6). Then IVP-SIP-II is disconjugate for every ๐œ†11,๐œ†12 if and only if ๐‘“(๐‘ก,๐‘ฆ)=0 on โ„.

Proof. Let ๐‘“(๐‘ก,๐‘ฆ๐œŽ)=0, that is, ๐‘“1๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ=0,๐‘กโˆˆ๐•‹1,๐‘“1๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ=0,๐‘กโˆˆ๐•‹2.(3.17) Then we see that IVP-SIP-II reduces to the IVP-SIP-III ๐‘ฆ1ฮ”ฮ”(๐‘ก)=0,๐‘กโˆˆ๐•‹1,๐‘ฆ2ฮ”ฮ”(๐‘ก)=0,๐‘กโˆˆ๐•‹2,๐‘ฆ1๐‘ฆ(0)=๐‘™,ฮ”1๐œŒ(0)=๐‘š,1๐‘ฆ1(๐‘)=๐œŒ2๐‘ฆ2๐œŒ(๐œŽ(๐‘)),3๐‘ฆฮ”1(๐‘)=๐œŒ4๐‘ฆฮ”2(๐œŽ(๐‘)).(3.18) Simple calculations show that ๐‘ฆ1๐‘ฆ(๐‘ก)=๐‘š๐‘ก+๐‘™,2๐œŒ(๐‘ก)=3๐œŒ4๐œŒ๐‘š๐‘ก+1๐œŒ2๐œŒ(๐‘๐‘š+๐‘™)โˆ’3๐œŒ4๐‘š๐œŽ(๐‘).(3.19) We observe that ๐‘ฆ(๐‘ก)=0 whenever ๐‘™๐‘ก=โˆ’๐‘š๐œŒor๐œŽ(๐‘)โˆ’1๐œŒ4๐œŒ2๐œŒ3๎‚€๐‘™๐‘+๐‘š๎‚.(3.20) Clearly ๐‘กโ‰ โˆ’๐‘™/๐‘š and so ๐‘ก=๐œŽ(๐‘)โˆ’(๐œŒ1๐œŒ4/๐œŒ2๐œŒ3)(๐‘+(๐‘™/๐‘š)) is the only zero possible. Hence, IVP-SIP-III is disconjugate on โ„.
Now let IVP-SIP-II be disconjugate for every ๐œ†11,๐œ†12. From Lemma 3.6, we have๐œ๐œ‚๎€ท๎€ท1,๐œ‚2๎€ธ,๎€ท๐‘“1,๐‘“2๎€ธ๎€ธ;๐‘Ž,๐‘โ‰ฅ0.(3.21) Hence ๎€œ๐‘๐‘Ž๎‚ƒ๎€ฝ๐œ‚ฮ”1๎€พ(๐‘ก)2+๐œ†11๐‘“1๎€ท๐‘ก,๐‘ฆ๐œŽ1๐œ‚๎€ธ๎€ฝ1๎€พ(๐‘ก)2๎‚„๎€œฮ”๐‘ก+๐‘๐œŽ(๐‘)๎‚ƒ๎€ฝ๐œ‚ฮ”2๎€พ(๐‘ก)2+๐œ†12๐‘“2๎€ท๐‘ก,๐‘ฆ๐œŽ2๐œ‚๎€ธ๎€ฝ2๎€พ(๐‘ก)2๎‚„ฮ”๐‘กโ‰ฅ0.(3.22) So we have โˆ’๐œ†11๎€œ๐‘๐‘Ž๐‘“1๎€ท๐‘ก,๐‘ฆ๐œŽ1๐œ‚๎€ธ๎€ฝ1๎€พ(๐‘ก)2ฮ”๐‘กโˆ’๐œ†12๎€œ๐‘๐œŽ(๐‘)๐‘“2๎€ท๐‘ก,๐‘ฆ๐œŽ2๐œ‚๎€ธ๎€ฝ2๎€พ(๐‘ก)2โ‰ค๎€œฮ”๐‘ก๐‘๐‘Ž๎€ฝ๐œ‚1ฮ”๎€พ(๐‘ก)2+๎€œ๐‘๐œŽ(๐‘)๎€ฝ๐œ‚2ฮ”๎€พ(๐‘ก)2.(3.23) The previous equation is true for all โˆ’โˆž<๐œ†11,๐œ†12<+โˆž, which implies that ๎€œ๐‘๐‘Ž๐‘“1๎€ท๐‘ก,๐‘ฆ๐œŽ1๐œ‚๎€ธ๎€ฝ1๎€พ(๐‘ก)2๎€œฮ”๐‘ก=0=๐‘๐œŽ(๐‘)๐‘“2๎€ท๐‘ก,๐‘ฆ๐œŽ2๐œ‚๎€ธ๎€ฝ2๎€พ(๐‘ก)2ฮ”๐‘ก.(3.24) Since the above equation holds true for all ๐œ‚1 and ๐œ‚2, we must have ๐‘“1=0=๐‘“2, hence the proof.

Corollary 3.9. Let IVP-SIP-IV be defined by ๐‘ฆ1ฮ”ฮ”=๐œ†11๐‘“11๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ+๐œ†21๐‘“21๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ+โ‹ฏ+๐œ†๐‘›1๐‘“๐‘›1๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ,๐‘ฆ2ฮ”ฮ”=๐œ†12๐‘“12๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ+๐œ†22๐‘“22๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ+โ‹ฏ+๐œ†๐‘›2๐‘“๐‘›2๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ,(3.25) along with (3.3)โ€“(3.6). Then IVP-SIP-IV is disconjugate on โ„ for allโ€‰โ€‰((๐œ†11,๐œ†12),(๐œ†21,๐œ†22),โ€ฆ,(๐œ†๐‘›1,๐œ†๐‘›2))โˆˆโ„2๐‘›, then ๐‘“๐‘–(๐‘ก,๐‘ฆ)=0,๐‘–=1,2,โ€ฆ,๐‘›, that is, ๐‘“๐‘–1๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ=0,for๐‘กโˆˆ๐•‹1,๐‘“๐‘–2๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ=0,for๐‘กโˆˆ๐•‹2,(3.26) where (๐‘“๐‘–1,๐‘“๐‘–2) are nonlinear function tuples in ๐’ž(๐•‹1ร—โ„)ร—๐’ž(๐•‹2ร—โ„).

4. Disconjugacy Domain

We define IVP-SIP-V as๐‘ฆ1ฮ”ฮ”(๐‘ก)=๐œ†11๐‘“11๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ+๐œ†21๐‘“21๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ+โ‹ฏ+๐œ†๐‘›1๐‘“๐‘›1๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ,๐‘กโˆˆ๐•‹1,๐‘ฆ2ฮ”ฮ”(๐‘ก)=๐œ†12๐‘“12๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ+๐œ†22๐‘“22๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ+โ‹ฏ+๐œ†๐‘š2๐‘“๐‘š2๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ,๐‘กโˆˆ๐•‹2,(4.1) along with (3.3)โ€“(3.6). Without any loss of generality, let us assume that ๐‘›<๐‘š, where ๐‘›,๐‘šโˆˆโ„• and ๐‘“i1,๐‘“๐‘—2,๐‘–=1,2,โ€ฆ,๐‘›,๐‘—=1,2,โ€ฆ,๐‘š, belong to ๐’ž(๐•‹1ร—โ„), ๐’ž(๐•‹2ร—โ„), respectively.

Definition 4.1. One defines the disconjugacy domain ๐’Ÿ of IVP-SIP-V as ๐œ†๐’Ÿ=๎€ฝ๎€ท11,๐œ†12,๐œ†21,๐œ†22,โ€ฆ,๐œ†๐‘›1,๐œ†๐‘›2,๐œ†(๐‘›+1)2,๐œ†(๐‘›+2)2,โ€ฆ,๐œ†๐‘š2๎€ธโˆˆโ„n+๐‘š๎€พ(4.2) such that IVP-SIP-V is disconjugate on โ„.

Theorem 4.2. The disconjugacy domain ๐’Ÿ of IVP-SIP-V is the whole space โ„๐‘›+๐‘š if and only if ๐‘“๐‘–1๐‘“=0,๐‘–=1,2,โ€ฆ,๐‘›,๐‘—1=0,๐‘—=1,2,โ€ฆ,๐‘š.(4.3)

Proof. For ๐œ†21=๐œ†31=โ‹ฏ=๐œ†๐‘›1=0,๐œ†11๐œ†โ‰ 0,22=๐œ†32=โ‹ฏ=๐œ†๐‘š2=0,๐œ†12โ‰ 0,(4.4) IVP-SIP-V reduces to IVP-SIP-II, and hence from Lemma 3.8 we have ๐‘“11=0=๐‘“12. If we choose ๐œ†11=๐œ†31=โ‹ฏ=๐œ†๐‘›1=0,๐œ†21๐œ†โ‰ 0,12=๐œ†32=โ‹ฏ=๐œ†๐‘š2=0,๐œ†22โ‰ 0,(4.5) we get ๐‘“21=0=๐‘“22. In similar lines we can show that ๐‘“31=๐‘“41=โ‹ฏ=๐‘“๐‘›1๐‘“=0,32=๐‘“42=โ‹ฏ=๐‘“๐‘›2=0.(4.6) Now let us choose ๐œ†11=๐œ†31=โ‹ฏ=๐œ†(๐‘›โˆ’1)1=0,๐œ†๐‘›1๐œ†โ‰ 0,12=๐œ†32=โ‹ฏ=๐œ†๐‘›2=0=๐œ†(๐‘›+2)2=โ‹ฏ=๐œ†๐‘š2,๐œ†(๐‘›+1)2โ‰ 0.(4.7) We see that from Lemma 3.8 we have ๐‘“(๐‘›+1)2=0. Similarly we can show that ๐‘“(๐‘›+2)2=๐‘“(๐‘›+3)2=โ‹ฏ=๐‘“๐‘š2=0.(4.8) Hence ๐‘“๐‘–1๐‘“=0,๐‘–=1,2,โ€ฆ,๐‘›,๐‘—1=0,๐‘—=1,2,โ€ฆ,๐‘š.(4.9) whenever the disconjugacy domain ๐’Ÿ of IVP-SIP-V is the whole space โ„๐‘›+๐‘š.
On the other hand, if๐‘“๐‘–1๐‘“=0,๐‘–=1,2,โ€ฆ,๐‘›,๐‘—1=0,๐‘—=1,2,โ€ฆ,๐‘š,(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 ๐‘“๐‘–1 or ๐‘“๐‘—2โ‰ 0, then ๐’Ÿ is a proper subset of โ„๐‘›+๐‘š.

Proof. This is the contrapositive of Theorem 4.2.

Theorem 4.4. ๐’Ÿ contains a proper subspace of the vector space โ„๐‘›+๐‘š if the sets of functions {๐‘“11,๐‘“21,โ€ฆ,๐‘“๐‘›1} and {๐‘“12,๐‘“22,โ€ฆ,๐‘“๐‘š2} are linearly dependent.

Proof. Let {๐‘“11,๐‘“21,โ€ฆ,๐‘“๐‘›1} and {๐‘“12,๐‘“22,โ€ฆ,๐‘“๐‘š2} be sets of linearly dependent functions Then there exists constants {๐‘11,๐‘21,โ€ฆ,๐‘(๐‘›โˆ’1)1}, {๐‘12,๐‘22,โ€ฆ,๐‘(๐‘šโˆ’1)2}, not all zero, such that ๐‘“11๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ=๐‘11๐‘“21๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ+๐‘21๐‘“31๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ+โ‹ฏ+๐‘(๐‘›โˆ’1)1๐‘“๐‘›1๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ,for๐‘กโˆˆ๐•‹1,๐‘“12๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ=๐‘12๐‘“22๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ+๐‘22๐‘“32๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ+โ‹ฏ+๐‘(๐‘šโˆ’1)2๐‘“๐‘š2๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ,for๐‘กโˆˆ๐•‹2.(4.11) Therefore, the IVP-SIP-V becomes ๐‘ฆ1ฮ”ฮ”(๐‘ก)=๐œ†11๎€บ๐‘11๐‘“21+๐‘21๐‘“31+โ‹ฏ+๐‘(๐‘›โˆ’1)1๐‘“๐‘›1๎€ป๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ+๐œ†21๐‘“21๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ+โ‹ฏ+๐œ†๐‘›1๐‘“๐‘›1๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ,๐‘กโˆˆ๐•‹1,๐‘ฆ2ฮ”ฮ”(๐‘ก)=๐œ†12๎€บ๐‘12๐‘“22+๐‘22๐‘“32+โ‹ฏ+๐‘(๐‘šโˆ’1)2๐‘“๐‘š2๎€ป๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ+๐œ†22๐‘“22๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ+โ‹ฏ+๐œ†๐‘š2๐‘“๐‘š2๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ,๐‘กโˆˆ๐•‹2,(4.12) along with (3.3)โ€“(3.6). We see that IVP-SIP-V now is to be ๐‘ฆ1ฮ”ฮ”(๎€ท๐œ†๐‘ก)=11๐‘11+๐œ†21๎€ธ๐‘“21๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ+๎€ท๐œ†11๐‘21+๐œ†31๎€ธ๐‘“31๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ๎€ท๐œ†+โ‹ฏ+11๐‘(๐‘›โˆ’1)1+๐œ†๐‘›1๎€ธ๐‘“๐‘›1๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ,๐‘ฆ2ฮ”ฮ”๎€ท๐œ†(๐‘ก)=12๐‘12+๐œ†22๎€ธ๐‘“22๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ+๎€ท๐œ†12๐‘22+๐œ†32๎€ธ๐‘“32๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ๎€ท๐œ†+โ‹ฏ+12๐‘(๐‘šโˆ’1)2+๐œ†๐‘š2๎€ธ๐‘“๐‘š2๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ,(4.13) along with (3.3)โ€“(3.6). Hence ๐’Ÿ contains the subspace ๐œ†๐’ฎ=๎€ฝ๎€ท11,๐œ†12,๐œ†21,๐œ†22,โ€ฆ,๐œ†๐‘›1,๐œ†๐‘›2,๐œ†(๐‘›+1)2,๐œ†(๐‘›+2)2,โ€ฆ,๐œ†๐‘š2๎€ธโˆˆโ„๐‘›+๐‘š๎€พ,(4.14) where ๐œ†21=โˆ’๐œ†11๐‘11,๐œ†31=โˆ’๐œ†11๐‘21,โ€ฆ,๐œ†๐‘›1=โˆ’๐œ†11๐‘(๐‘›โˆ’1)1,๐œ†22=โˆ’๐œ†12๐‘12,๐œ†32=โˆ’๐œ†12๐‘22,โ€ฆ,๐œ†๐‘š2=โˆ’๐œ†12๐‘(๐‘šโˆ’1)2,(4.15) hence the proof.

Corollary 4.5. If the sets of functions {๐‘“11,๐‘“21,โ€ฆ,๐‘“๐‘›1} and {๐‘“12,๐‘“22,โ€ฆ,๐‘“๐‘š2} are linearly independent, then ๐’Ÿ cannot contain a proper subspace of โ„๐‘›+๐‘š.

Proof. This is the contrapositive of Theorem 4.4.

Lemma 4.6. Let us consider the following two IVP-SIPs which are disconjugate on โ„: ๐‘ฆ1ฮ”ฮ”(๐‘ก)=๐‘“11๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ,๐‘กโˆˆ๐•‹1,๐‘ฆ2ฮ”ฮ”(๐‘ก)=๐‘“12๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ,๐‘กโˆˆ๐•‹2๐‘ฆ,alongwith(3.3)-(3.6),1ฮ”ฮ”(๐‘ก)=๐‘“21๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ,๐‘กโˆˆ๐•‹1,๐‘ฆ2ฮ”ฮ”(๐‘ก)=๐‘“22๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ,๐‘กโˆˆ๐•‹2,alongwith(3.3)-(3.6).(4.16) Then the IVP-SIP-VI defined by ๐‘ฆ1ฮ”ฮ”(๐‘ก)=(1โˆ’๐›พ)๐‘“11๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ+๐›พ๐‘“21๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ,๐‘กโˆˆ๐•‹1,๐‘ฆ2ฮ”ฮ”(๐‘ก)=(1โˆ’๐›พ)๐‘“12๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ+๐›พ๐‘“22๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ,๐‘กโˆˆ๐•‹2,(4.17) along with (3.3)โ€“(3.6) is also disconjugate on โ„, for each ๐›พโˆˆ[0,1].

Proof. On account of Lemma 3.6 it is sufficient if we show that ๐œ๐œ‚๎€ท๎€ท1,๐œ‚2๎€ธ,๎€ท(1โˆ’๐›พ)๐‘“11+๐›พ๐‘“21,(1โˆ’๐›พ)๐‘“12+๐›พ๐‘“22๎€ธ๎€ธ;๐‘Ž,๐‘โ‰ฅ0(4.18) on ๐ด([๐‘Ž,๐‘]โˆช[๐œŽ(๐‘),๐‘]) for every compact subinterval [๐‘Ž,๐‘]โˆช[๐œŽ(๐‘),๐‘]โŠ‚๐•‹1โˆช๐•‹2. We see that ๐œ๐œ‚๎€ท๎€ท1,๐œ‚2๎€ธ,๎€ท(1โˆ’๐›พ)๐‘“11+๐›พ๐‘“21,(1โˆ’๐›พ)๐‘“12+๐›พ๐‘“22๎€ธ๎€ธ=๎€œ;๐‘Ž,๐‘๐‘๐‘Ž๎‚ƒ๎€ฝ๐œ‚ฮ”1๎€พ(๐‘ก)2+๎€บ(1โˆ’๐›พ)๐‘“11๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ+๐›พ๐‘“21๎€ท๐‘ก,๐‘ฆ๐œŽ1๐œ‚๎€ธ๎€ป๎€ฝ1๎€พ(๐‘ก)2๎‚„+๎€œฮ”๐‘ก๐‘๐œŽ(๐‘)๎‚ƒ๎€ฝ๐œ‚ฮ”2๎€พ(๐‘ก)2+๎€บ(1โˆ’๐›พ)๐‘“12๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ+๐›พ๐‘“22๎€ท๐‘ก,๐‘ฆ๐œŽ1๐œ‚๎€ธ๎€ป๎€ฝ2๎€พ(๐‘ก)2๎‚„ฮ”๐‘ก.(4.19) Now ๎€œ๐‘๐‘Ž๎‚ƒ๎€ฝ๐œ‚ฮ”1๎€พ(๐‘ก)2+๎€บ(1โˆ’๐›พ)๐‘“11๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ+๐›พ๐‘“21๎€ท๐‘ก,๐‘ฆ๐œŽ1๐œ‚๎€ธ๎€ป๎€ฝ1๎€พ(๐‘ก)2๎‚„=๎€œฮ”๐‘ก๐‘๐‘Ž๎‚ƒ๎€ฝ๐œ‚(1โˆ’๐›พ)ฮ”1๎€พ(๐‘ก)2๎€ฝ๐œ‚+๐›พฮ”1๎€พ(๐‘ก)2+๎€บ(1โˆ’๐›พ)๐‘“11๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ+๐›พ๐‘“21๎€ท๐‘ก,๐‘ฆ๐œŽ1๐œ‚๎€ธ๎€ป๎€ฝ1๎€พ(๐‘ก)2๎‚„=๎€œฮ”๐‘ก(1โˆ’๐›พ)๐‘๐‘Ž๎‚ƒ๎€ฝ๐œ‚ฮ”1๎€พ(๐‘ก)2+๐‘“11๎€ท๐‘ก,๐‘ฆ๐œŽ1๐œ‚๎€ธ๎€ฝ1๎€พ(๐‘ก)2๎‚„๎€œฮ”๐‘ก+๐›พ๐‘๐‘Ž๎‚ƒ๎€ฝ๐œ‚ฮ”1๎€พ(๐‘ก)2+๐‘“21๎€ท๐‘ก,๐‘ฆ๐œŽ1๐œ‚๎€ธ๎€ฝ1๎€พ(๐‘ก)2๎‚„ฮ”๐‘ก.(4.20) Similarly it can be shown that ๎€œ๐‘๐œŽ(๐‘)๎‚ƒ๎€ฝ๐œ‚ฮ”2๎€พ(๐‘ก)2+๎€บ(1โˆ’๐›พ)๐‘“12๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ+๐›พ๐‘“22๎€ท๐‘ก,๐‘ฆ๐œŽ1๐œ‚๎€ธ๎€ป๎€ฝ2๎€พ(๐‘ก)2๎‚„๎€œฮ”๐‘ก=(1โˆ’๐›พ)๐‘๐œŽ(๐‘)๎‚ƒ๎€ฝ๐œ‚ฮ”2๎€พ(๐‘ก)2+๐‘“12๎€ท๐‘ก,๐‘ฆ๐œŽ2๐œ‚๎€ธ๎€ฝ2๎€พ(๐‘ก)2๎‚„๎€œฮ”๐‘ก+๐›พ๐‘๐œŽ(๐‘)๎‚ƒ๎€ฝ๐œ‚ฮ”2๎€พ(๐‘ก)2+๐‘“22๎€ท๐‘ก,๐‘ฆ๐œŽ2๐œ‚๎€ธ๎€ฝ2๎€พ(๐‘ก)2๎‚„ฮ”๐‘ก.(4.21) Hence we have ๐œ๐œ‚๎€ท๎€ท1,๐œ‚2๎€ธ,๎€ท(1โˆ’๐›พ)๐‘“11+๐›พ๐‘“21,(1โˆ’๐›พ)๐‘“12+๐›พ๐‘“22๎€ธ๎€ธ๐œ‚;๐‘Ž,๐‘=(1โˆ’๐›พ)๐œ๎€ท๎€ท1,๐œ‚2๎€ธ,๎€ท๐‘“11,๐‘“12๎€ธ๎€ธ๐œ‚;๐‘Ž,๐‘+๐›พ๐œ๎€ท๎€ท1,๐œ‚2๎€ธ,๎€ท๐‘“21,๐‘“22๎€ธ๎€ธ;๐‘Ž,๐‘โ‰ฅ0,(4.22) hence the proof.

Definition 4.7. Let (๐‘“๐‘›1,๐‘“๐‘›2),(๐‘“1,๐‘“2),๐‘›โˆˆโ„• be function tuples in ๐’ž(๐•‹1ร—โ„)ร—๐’ž(๐•‹2ร—โ„). One says that (๐‘“๐‘›1,๐‘“๐‘›2),(๐‘“1,๐‘“2) are close in uniform norm whenever sup๐‘กโˆˆ๐•‹1||๐‘“๐‘›1(๐‘ก)โˆ’๐‘“1||(๐‘ก)+sup๐‘กโˆˆ๐•‹2||๐‘“๐‘›2(๐‘ก)โˆ’๐‘“2||(๐‘ก)โŸถ0as๐‘›โŸถโˆž.(4.23)

Definition 4.8. Let (๐‘ฆ๐‘›1,๐‘ฆ๐‘›2),(๐‘ฆ1,๐‘ฆ2),๐‘›โˆˆโ„• be function tuples in ๐’ž(๐•‹1)ร—๐’ž(๐•‹2). One says that (๐‘ฆ๐‘›1,๐‘ฆ๐‘›2),(๐‘ฆ11,๐‘ฆ12) are close in uniform norm whenever sup๐‘กโˆˆ๐•‹1||๐‘ฆ๐‘›1(๐‘ก)โˆ’๐‘ฆ1||(๐‘ก)+sup๐‘กโˆˆ๐•‹2||๐‘ฆ๐‘›2(๐‘ก)โˆ’๐‘ฆ2||(๐‘ก)โŸถ0as๐‘›โŸถโˆž.(4.24)

Lemma 4.9. Let ๐‘“๐‘›=(๐‘“๐‘›1,๐‘“๐‘›2),๐‘“=(๐‘“1,๐‘“2),๐‘›โˆˆโ„•. Let the sequence of the IVP-SIPs defined by ๐‘ฆฮ”ฮ”๐‘›1(๐‘ก)=๐‘“๐‘›1๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ,๐‘กโˆˆ๐•‹1,๐‘ฆฮ”ฮ”๐‘›2(๐‘ก)=๐‘“๐‘›2๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ,๐‘กโˆˆ๐•‹2,(4.25) along with (3.3)โ€“(3.6) be disconjugate on โ„ for each ๐‘›=1,2,โ€ฆ. If ๐‘“๐‘›โ†’๐‘“ uniformly, then the IVP-SIP ๐‘ฆ1ฮ”ฮ”(๐‘ก)=๐‘“1๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ,๐‘กโˆˆ๐•‹1,๐‘ฆ2ฮ”ฮ”(๐‘ก)=๐‘“2๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ,๐‘กโˆˆ๐•‹2,(4.26) is disconjugate on โ„.

Proof. Now let us assume that (4.26) along with (3.3)โ€“(3.6) is not disconjugate. Let us assume that it has a solution ๐‘ฆ(๐‘ก) with two zeros ๐‘ก1,๐‘ก2. Let us assume that ๐‘ก1โˆˆ๐•‹1 and ๐‘ก2โˆˆ๐•‹2. The other cases when ๐‘ก1,๐‘ก2โˆˆ๐•‹1, ๐‘ก1,๐‘ก2โˆˆ๐•‹2 can be worked in similar lines. Now let (๐‘ฆ๐‘›1,๐‘ฆ๐‘›2) be a solution such that ๐‘ฆ๐‘›1(๐‘ก1)=0.Claim 1. The solutions of (4.25), (4.26) satisfying (3.3)โ€“(3.6) are close in the uniform norm whenever (๐‘“๐‘›1,๐‘“๐‘›2) and (๐‘“1,๐‘“2) are close in uniform norm on โ„.
Simple calculations show that the solution of (4.25), (4.26) satisfying (3.3)โ€“(3.6) are๐‘ฆ๐‘›1๎€œ(๐‘ก)=๐‘ก0๎€œ๐‘˜0๐‘“๐‘›1๎€ท๐‘ ,๐‘ฆ๐œŽ1๎€ธ๐‘ฆฮ”๐‘ ฮ”๐‘˜+๐‘™๐‘ก+๐‘š,๐‘›2๎€œ(๐‘ก)=๐‘ก๐œŽ(๐‘)๎€œ๐‘˜โ€ฒ๐œŽ(๐‘)๐‘“๐‘›2๎€ท๐‘ ,๐‘ฆ๐œŽ2๎€ธฮ”๐‘ ฮ”๐‘˜๎…ž+๎€œ๐‘ก๐œŽ(๐‘)๐œŒ3๐œŒ4๎‚ธ๎€œ๐‘0๐‘“๐‘›1๎€ท๐‘ ,๐‘ฆ๐œŽ1๎€ธ๎‚นฮ”๐‘ +๐‘šฮ”๐‘˜๎…ž+๐œŒ1๐œŒ2๎€œ๐‘0๎€œ๐‘˜0๐‘“๐‘›1๎€ท๐‘ ,๐‘ฆ๐œŽ1๎€ธ๐‘ฆฮ”๐‘ ฮ”๐‘˜+๐‘™๐‘+๐‘š,(4.27)1๎€œ(๐‘ก)=๐‘ก0๎€œ๐‘˜0๐‘“1๎€ท๐‘ ,๐‘ฆ๐œŽ1๎€ธ๐‘ฆฮ”๐‘ ฮ”๐‘˜+๐‘™๐‘ก+๐‘š,2(๎€œ๐‘ก)=๐‘ก๐œŽ(๐‘)๎€œ๐‘˜โ€ฒ๐œŽ(๐‘)๐‘“2๎€ท๐‘ ,๐‘ฆ๐œŽ2๎€ธฮ”๐‘ ฮ”๐‘˜๎…ž+๎€œ๐‘ก๐œŽ(๐‘)๐œŒ3๐œŒ4๎‚ธ๎€œ๐‘0๐‘“1๎€ท๐‘ ,๐‘ฆ๐œŽ1๎€ธ๎‚นฮ”๐‘ +๐‘šฮ”๐‘˜๎…ž+๐œŒ1๐œŒ2๎€œ๐‘0๎€œ๐‘˜0๐‘“1๎€ท๐‘ ,๐‘ฆ๐œŽ1๎€ธฮ”๐‘ ฮ”๐‘˜+๐‘™๐‘+๐‘š.(4.28) From the above equations we clearly see that the solutions of (4.25), (4.26) along with (3.3)โ€“(3.6) are close in the uniform norm whenever (๐‘“๐‘›1,๐‘“๐‘›2) and (๐‘“1,๐‘“2) are close in uniform norm on โ„, so the claim. Hence for ๐›ฝ>0 we can find ๐œ–(>0) and ๐›ฟ(>0) such that sup๐‘กโˆˆ[๐‘ก1,๐‘]||๐‘ฆ๐‘›1(๐‘ก)โˆ’๐‘ฆ1||(๐‘ก)+sup๐‘กโˆˆ[๐œŽ(๐‘),๐‘ก2+๐›ฝ]||๐‘ฆ๐‘›2(๐‘ก)โˆ’๐‘ฆ2||(๐‘ก)<๐œ–(4.29) whenever sup๐‘กโˆˆ[๐‘ก1,๐‘]||๐‘“๐‘›1(๐‘ก)โˆ’๐‘“1||(๐‘ก)+sup๐‘กโˆˆ[๐œŽ(๐‘),๐‘ก2+๐›ฝ]||๐‘“๐‘›2(๐‘ก)โˆ’๐‘“2||(๐‘ก)<๐›ฟ.(4.30) Since ๐‘ฆ2(๐‘ก) must change sign at ๐‘ก=๐‘ก2, it follows that ๐‘ฆ๐‘›2(๐‘ก) must also change sign near ๐‘ก=๐‘ก2 for sufficiently large ๐‘›. Thus, for such ๐‘›, (4.25) along with (3.3)โ€“(3.6) is not disconjugate which leads us to a contradiction.

Theorem 4.10. The disconjugacy domain ๐’Ÿ of IVP-SIP-V is a closed set of the parameter space โ„๐‘›+๐‘š.

Proof. Let (๐œ†1,๐œ†2,โ€ฆ,๐œ†๐‘›+๐‘š) be a limit point of the sequence (๐œ†๐‘˜1,๐œ†๐‘˜2,โ€ฆ,๐œ†๐‘˜(๐‘›+๐‘š))โˆˆ๐ท,๐‘˜=1,2,โ€ฆ. Then for every ๐œ–>0 there exists sufficiently large ๐พ such that for all ๐‘˜>๐พ we have |๐œ†๐‘˜1โˆ’๐œ†1|<๐œ–,|๐œ†๐‘˜2โˆ’๐œ†2|<๐œ–,โ€ฆ,|๐œ†๐‘˜(๐‘›+๐‘š)โˆ’๐œ†๐‘›+๐‘š|<๐œ–, and IVP-SIP-VII defined by ๐‘ฆ1ฮ”ฮ”(๐‘ก)=๐œ†๐‘˜1๐‘“11+๐œ†๐‘˜3๐‘“21+โ‹ฏ+๐œ†๐‘˜(2๐‘›โˆ’1)๐‘“๐‘›1,๐‘กโˆˆ๐•‹1,๐‘ฆ2ฮ”ฮ”(๐‘ก)=๐œ†๐‘˜2๐‘“12+๐œ†๐‘˜4๐‘“22+โ‹ฏ+๐œ†๐‘˜(2๐‘›)๐‘“๐‘›2+๐œ†๐‘˜(2๐‘›+1)๐‘“(๐‘›+1)2+โ‹ฏ+๐œ†๐‘˜(๐‘›+๐‘š)๐‘“๐‘š2,๐‘กโˆˆ๐•‹2,(4.31) along with (3.3)โ€“(3.6) is disconjugate. Let ๐‘ฆ=(๐‘ฆ1,๐‘ฆ2) be a non trivial solution of IVP-SIP-V for ๎€ท๐œ†11,๐œ†21,โ€ฆ,๐œ†๐‘›1,๐œ†12,๐œ†22,โ€ฆ,๐œ†๐‘š2๎€ธ=๎€ท๐œ†1,๐œ†2,โ€ฆ,๐œ†๐‘›+๐‘š๎€ธ.(4.32) Then we see that either ๐‘ฆ(๐‘ก) never vanishes in which case ๐‘ฆ(๐‘ก) is disconjugate or let ๐‘ฆ(๐‘ก0)=0 for some ๐‘ก0. In the latter case we let (๐‘ฆ๐‘›1,๐‘ฆ๐‘›2) be a solution of IVP-SIP-VII such that ๐‘ฆ๐‘›1(๐‘ก0)=0. From assumption we have that ๐‘ฆ๐‘›1(๐‘ก)โ‰ 0 for ๐‘กโ‰ ๐‘ก0. From Claim 1 we have (๐‘ฆ๐‘›1,๐‘ฆ๐‘›2) uniformly approximating (๐‘ฆ1,๐‘ฆ2). Hence ๐‘ฆ(๐‘ก) can change sign only at ๐‘ก=๐‘ก0 and so ๐‘ฆ(๐‘ก)โ‰ 0 for all ๐‘กโ‰ ๐‘ก0. Hence ๐‘ฆ(๐‘ก) is disconjugate, so the result.

Theorem 4.11. The disconjugacy domain ๐’Ÿ of IVP-SIP-V is a convex set in the parameter space โ„๐‘›+๐‘š.

Proof. We need to show that for (๐œ†11,๐œ†12,โ€ฆ,๐œ†1(๐‘›+๐‘š)),(๐œ†21,๐œ†22,โ€ฆ,๐œ†2(๐‘›+๐‘š))โˆˆ๐ท the convex combination ๎€ท๐œ†(1โˆ’๐›พ)11,๐œ†12,โ€ฆ,๐œ†1(๐‘›+๐‘š)๎€ธ๎€ท๐œ†+๐›พ21,๐œ†22,โ€ฆ,๐œ†2(๐‘›+๐‘š)๎€ธโˆˆ๐ท,(4.33) that is, ๎€บ(1โˆ’๐›พ)๐œ†11+๐›พ๐œ†21,(1โˆ’๐›พ)๐œ†12+๐›พ๐œ†22,โ€ฆ,(1โˆ’๐›พ)๐œ†1(๐‘›+๐‘š)+๐›พ๐œ†2(๐‘›+๐‘š)๎€ปโˆˆ๐ท.(4.34) Since (๐œ†11,๐œ†12,โ€ฆ,๐œ†1(๐‘›+๐‘š)),(๐œ†21,๐œ†22,โ€ฆ,๐œ†2(๐‘›+๐‘š))โˆˆ๐ท, we have the IVP-SIPs ๐‘ฆ1ฮ”ฮ”(๐‘ก)=๐œ†11๐‘“11+๐œ†13๐‘“21+โ‹ฏ+๐œ†1(2๐‘›โˆ’1)๐‘“๐‘›1,๐‘กโˆˆ๐•‹1,๐‘ฆ2ฮ”ฮ”(๐‘ก)=๐œ†12๐‘“12+๐œ†14๐‘“22+โ‹ฏ+๐œ†1(2๐‘›)๐‘“๐‘›2+๐œ†1(2๐‘›+1)๐‘“(๐‘›+1)2+โ‹ฏ+๐œ†1(๐‘›+๐‘š)๐‘“๐‘š2,๐‘กโˆˆ๐•‹2,๐‘ฆ1ฮ”ฮ”(๐‘ก)=๐œ†21๐‘“11+๐œ†23๐‘“21+โ‹ฏ+๐œ†2(2๐‘›โˆ’1)๐‘“๐‘›1,๐‘กโˆˆ๐•‹1,๐‘ฆ2ฮ”ฮ”(๐‘ก)=๐œ†22๐‘“12+๐œ†24๐‘“22+โ‹ฏ+๐œ†2(2๐‘›)๐‘“๐‘›2+๐œ†2(2๐‘›+1)๐‘“(๐‘›+1)2+โ‹ฏ+๐œ†2(๐‘›+๐‘š)๐‘“๐‘š2,๐‘กโˆˆ๐•‹2,(4.35) along with (3.3)โ€“(3.6) to be disconjugate. Now from Lemma 4.6 we have the IVP-SIP-VIII defined by ๐‘ฆ1ฮ”ฮ”(๎€ท๐œ†๐‘ก)=(1โˆ’๐›พ)11๐‘“11+๐œ†13๐‘“21+โ‹ฏ+๐œ†1(2๐‘›โˆ’1)๐‘“๐‘›1๎€ธ๎€ท๐œ†+๐›พ21๐‘“11+๐œ†23๐‘“21+โ‹ฏ+๐œ†2(2๐‘›โˆ’1)๐‘“๐‘›1๎€ธ,๐‘กโˆˆ๐•‹1,๐‘ฆ2ฮ”ฮ”๎€ท๐œ†(๐‘ก)=(1โˆ’๐›พ)12๐‘“12+๐œ†14๐‘“22+โ‹ฏ+๐œ†1(2๐‘›)๐‘“๐‘›2+๐œ†1(2๐‘›+1)๐‘“(๐‘›+1)2+โ‹ฏ+๐œ†1(๐‘›+๐‘š)๐‘“๐‘š2๎€ธ๎€ท๐œ†+๐›พ22๐‘“12+๐œ†24๐‘“22+โ‹ฏ+๐œ†2(2๐‘›)๐‘“๐‘›2+๐œ†2(2๐‘›+1)๐‘“(๐‘›+1)2+โ‹ฏ+๐œ†2(๐‘›+๐‘š)๐‘“๐‘š2๎€ธ,๐‘กโˆˆ๐•‹2,(4.36) along with (3.3)โ€“(3.6) to be disconjugate; that, is we have the IVP-SIP ๐‘ฆ1ฮ”ฮ”(๐‘ก)=(1โˆ’๐›พ)๐œ†11๐‘“11+๐›พ๐œ†21๐‘“11+(1โˆ’๐›พ)๐œ†13๐‘“21+๐›พ๐œ†23๐‘“21+โ‹ฏ+(1โˆ’๐›พ)๐œ†1(2๐‘›โˆ’1)๐‘“๐‘›1+๐›พ๐œ†2(2๐‘›โˆ’1)๐‘“๐‘›1,๐‘กโˆˆ๐•‹1,๐‘ฆ2ฮ”ฮ”(๐‘ก)=(1โˆ’๐›พ)๐œ†12๐‘“12+๐›พ๐œ†22๐‘“12+(1โˆ’๐›พ)๐œ†14๐‘“22+๐›พ๐œ†24๐‘“22+โ‹ฏ+(1โˆ’๐›พ)๐œ†1(๐‘›+๐‘š)๐‘“๐‘š2+๐›พ๐œ†2(๐‘›+๐‘š)๐‘“๐‘š2,๐‘กโˆˆ๐•‹2,(4.37) along with (3.3)โ€“(3.6) to be disconjugate. Hence ๎€ท๐œ†(1โˆ’๐›พ)11,๐œ†12,โ€ฆ,๐œ†1(๐‘›+๐‘š)๎€ธ๎€ท๐œ†+๐›พ21,๐œ†22,โ€ฆ,๐œ†2(๐‘›+๐‘š)๎€ธโˆˆ๐ท,(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 ๐’ฉ of IVP-SIP-V as ๐œ†๐’ฉ=๎€ฝ๎€ท11,๐œ†12,๐œ†21,๐œ†22,โ€ฆ,๐œ†๐‘›1,๐œ†๐‘›2,๐œ†(๐‘›+1)2,๐œ†(๐‘›+2)2,โ€ฆ,๐œ†๐‘š2๎€ธโˆˆโ„๐‘›+๐‘š๎€พ(5.1) such that IVP-SIP-V is non oscillatory on โ„.

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

Proof. Let ๐‘ฆ(=(๐‘ฆ1,๐‘ฆ2)) be a solution of IVP-SIP-I. Let ๐‘ก1,๐‘ก2,โ€ฆ๐‘ก๐‘› be the zeros of the IVP-SIP-I (since ๐‘ฆ(๐‘ก) is non oscillatory). By the nature of ๐‘ฆ(๐‘ก) we see that either ๐‘ฆ(๐‘ก)<0 or ๐‘ฆ(๐‘ก)>0 for ๐‘ก>๐‘ก๐‘›. We now define ๎‚ป๐พ(๐‘ก)=0,๐‘กโ‰ค๐‘ก๐‘›,๐‘ฆ(๐‘ก),๐‘ก>๐‘ก๐‘›.(5.2)
From Assumption 3.7 we see that ๐พ(๐‘ก) is a solution of IVP-SIP-I. We also see that either ๐พ(๐‘ก)โ‰ค0 or ๐พ(๐‘ก)โ‰ฅ0.

Note 2. We assume ๐พ(๐‘ก)โ‰ฅ0. Similar results can be developed when ๐พ(๐‘ก)โ‰ค0.

All the results discussed in Section 5 regarding ๐’Ÿ can be easily extended to ๐’ฉ. We just state two theorems without proof.

Theorem 5.4. The non oscillatory domain ๐’ฉ of IVP-SIP-V is a closed set of the parameter space โ„๐‘›+๐‘š.

Theorem 5.5. The non oscillatory domain ๐’ฉ of IVP-SIP-V is a convex set in the parameter space โ„๐‘›+๐‘š.

6. Applications

We define IVP-SIP-X by ๐‘ฆ1ฮ”ฮ”(๐‘ก)=๐œ†11๐‘“11๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ+๐œ’11๐‘“21๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ+โ‹ฏ+๐œ’(๐‘›โˆ’1)1๐‘“๐‘›1๎€ท๐‘ก,๐‘ฆ๐œŽ1๎€ธ,๐‘กโˆˆ๐•‹1,๐‘ฆ2ฮ”ฮ”(๐‘ก)=๐œ†12๐‘“12๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ+๐œ’12๐‘“22๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ+โ‹ฏ+๐œ’(๐‘šโˆ’1)2๐‘“๐‘š2๎€ท๐‘ก,๐‘ฆ๐œŽ2๎€ธ,๐‘กโˆˆ๐•‹2,(6.1) where ๐œ’11,๐œ’21,โ€ฆ,๐œ’(๐‘›โˆ’1)1,๐œ’12,๐œ’22,โ€ฆ,๐œ’(๐‘šโˆ’1)2โˆˆโ„, along with (3.3)โ€“(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 (๐œ†11,๐œ†12)โˆˆโ„2.(i)IVP-SIP-X is oscillatory for every (๐œ†11,๐œ†12).(ii)IVP-SIP-X is oscillatory for every (๐œ†11,๐œ†12) except at some unique (๐œ†11,๐œ†12) = (๐œ†01,๐œ†02).(iii)There exists a finite interval (๐œ†๐‘Ž1,๐œ†๐‘1)โˆˆโ„ or (๐œ†๐‘Ž2,๐œ†๐‘2)โˆˆโ„ such that IVP-SIP-X is non oscillatory for either ๐œ†11โˆˆ๎€ท๐œ†๐‘Ž1,๐œ†๐‘1๎€ธ,๐œ†12=๐‘1,(6.2)or ๐œ†11=๐‘2,๐œ†12โˆˆ๎€ท๐œ†๐‘Ž2,๐œ†๐‘2๎€ธ,๐‘1,๐‘2โˆˆโ„.(6.3)For every other combinations of (๐œ†11,๐œ†12)โˆˆโ„2, IVP-SIP-X is oscillatory.(iv)There exists either ๐œ†1 or ๐œ†2โˆˆโ„ such that IVP-SIP-X is non oscillatory (resp., oscillatory) for either ๐œ†11โˆˆ๎€ท๐œ†1๎€ธ,โˆž,๐œ†12=๐‘1,(6.4)or ๐œ†11=๐‘2,๐œ†12โˆˆ๎€ท๐œ†2๎€ธ,โˆž,(6.5)and oscillatory (resp., non oscillatory) for either ๐œ†11โˆˆ๎€ทโˆ’โˆž,๐œ†1๎€ธ,๐œ†12=๐‘1,(6.6)or๐œ†11=๐‘2,๐œ†12โˆˆ๎€ทโˆ’โˆž,๐œ†2๎€ธ.(6.7)(v)IVP-SIP-X is non oscillatory for
either๐œ†11โˆˆโ„,๐œ†12=๐‘1,(6.8)or๐œ†11=๐‘2,๐œ†12โˆˆโ„.(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 ๐‘3โ‰ค๐œ†11โ‰ค๐‘4,๐‘3๎…žโ‰ค๐œ†12โ‰ค๐‘4๎…ž(6.10)Semi-infinite plane ๐‘3โ‰ค๐œ†11โ‰ค๐‘4,โˆ’โˆžโ‰ค๐œ†12โ‰ค๐‘5๎…ž๐‘3โ‰ค๐œ†11โ‰ค๐‘4,๐‘6๎…žโ‰ค๐œ†12๐‘โ‰ค+โˆž3โ‰ค๐œ†11โ‰ค๐‘4,โˆ’โˆžโ‰ค๐œ†12๐‘โ‰ค+โˆž6โ‰ค๐œ†11โ‰ค+โˆž,๐‘3๎…žโ‰ค๐œ†12โ‰ค๐‘4๎…ž๐‘6โ‰ค๐œ†11โ‰ค+โˆž,โˆ’โˆžโ‰ค๐œ†12โ‰ค๐‘5๎…ž๐‘6โ‰ค๐œ†11โ‰ค+โˆž,๐‘6๎…žโ‰ค๐œ†12๐‘โ‰ค+โˆž6โ‰ค๐œ†11โ‰ค+โˆž,โˆ’โˆžโ‰ค๐œ†12โ‰ค+โˆžโˆ’โˆžโ‰ค๐œ†11โ‰ค๐‘5,๐‘3๎…žโ‰ค๐œ†12โ‰ค๐‘4๎…žโˆ’โˆžโ‰ค๐œ†11โ‰ค๐‘5,โˆ’โˆžโ‰ค๐œ†12โ‰ค๐‘5๎…žโˆ’โˆžโ‰ค๐œ†11โ‰ค๐‘5,๐‘6๎…žโ‰ค๐œ†12โ‰ค+โˆžโˆ’โˆžโ‰ค๐œ†11โ‰ค๐‘5,โˆ’โˆžโ‰ค๐œ†12โ‰ค+โˆžโˆ’โˆžโ‰ค๐œ†11โ‰ค+โˆž,๐‘3๎…žโ‰ค๐œ†12โ‰ค๐‘4๎…žโˆ’โˆžโ‰ค๐œ†11โ‰ค+โˆž,โˆ’โˆžโ‰ค๐œ†12โ‰ค๐‘5๎…žโˆ’โˆžโ‰ค๐œ†11โ‰ค+โˆž,๐‘6๎…žโ‰ค๐œ†12โ‰ค+โˆž(6.11)infinite plane โˆ’โˆžโ‰ค๐œ†11โ‰ค+โˆž,โˆ’โˆžโ‰ค๐œ†12โ‰ค+โˆž,(6.12) where ๐‘3,๐‘4,๐‘5,๐‘6,๐‘3๎…ž,๐‘4๎…ž,๐‘5๎…ž,๐‘6๎…žโˆˆโ„.

Proof. Clearly we see that IVP-SIP-X is a special case of IVP-SIP-V with ๐œ†21=๐œ’11,๐œ†31=๐œ’21,โ€ฆ,๐œ†๐‘›1=๐œ’(๐‘›โˆ’1)1,๐œ†22=๐œ’12,๐œ†32=๐œ’22,โ€ฆ,๐œ†๐‘š2=๐œ’(๐‘šโˆ’1)2.(6.13) Let โ„’={๐œ†11,๐œ’11,๐œ’21,โ€ฆ,๐œ’(๐‘›โˆ’1)1,๐œ†12,๐œ’12,๐œ’22,โ€ฆ,๐œ’(๐‘šโˆ’1)2} be a subset of the parameter space โ„๐‘›+๐‘š. The above claims are consequences of the intersection of โ„’ with ๐’ฉ, the non oscillation domain for IVP-SIP-V. We recall that ๐’ฉ is convex in โ„๐‘›+๐‘š. We see that โ„’ intersects ๐’ฉ in one of the following ways: (i)โ„’โˆฉ๐’ฉ = ๐œ™, (ii)โ„’โˆฉ๐’ฉ = a single point, (iii)โ„’โˆฉ๐’ฉ = a line segment, (iv)โ„’โˆฉ๐’ฉ = a one-sided ray, (v)โ„’โˆฉ๐’ฉ = a full ray, (vi)โ„’โˆฉ๐’ฉ = a finite plane in โ„2, (vii)โ„’โˆฉ๐’ฉ = a semi infinite plane in โ„2, (viii)โ„’โˆฉ๐’ฉ = an infinite plane in โ„2, 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 ๐ฟ1๐‘ข1=๐‘‘2๐‘ข1๐‘‘๐‘ฅ2+๐‘˜21๐‘ข21=0,0โ‰ค๐‘ฅโ‰ค๐‘™1,๐ฟ2๐‘ข2=๐‘‘2๐‘ข2๐‘‘๐‘ฅ2+๐‘˜22๐‘ข22=0,๐‘™1โ‰ค๐‘ฅโ‰ค๐‘™2,(6.14) where ๐‘˜2๐‘–=๐‘ƒ/๐ธ๐ผ๐‘–,๐ธ is the modulus of elasticity, ๐‘ƒ is the load applied, ๐ผ๐‘– are moments of inertia, ๐‘–=1,2, and ๐‘ข1,๐‘ข2 are the displacements of cross-sections for the thinner and the thicker portions of the column, respectively. The physical conditions are given by ๐‘ข1(0)=๐‘ข๎…ž1(0)=0,๐‘ข1๎€ท๐‘™1๎€ธ=๐‘ข2๎€ท๐‘™1๎€ธ,๐‘ข๎…ž1๎€ท๐‘™1๎€ธ=๐‘ข๎…ž2๎€ท๐‘™1๎€ธ.(6.15) Here we see that ๐•‹1=[0,๐‘™1],๐•‹2=[๐‘™1,๐‘™2], ๐œŒ1=1=๐œŒ2, ๐œŒ3=1=๐œŒ4. We see that ๐œ†11=โˆ’๐‘˜21,๐œ†12=โˆ’๐‘˜22,๐œ’11,๐œ’21,โ€ฆ,๐œ’(๐‘›โˆ’1)1=0=๐œ’12,๐œ’22,โ€ฆ,๐œ’(๐‘šโˆ’1)2.(6.16) Hence from Theorem 6.1 we see that if the point {โˆ’๐‘˜21,0,0,โ€ฆ,0,โˆ’๐‘˜22,0,0,โ€ฆ,0} intersects ๐’ฉ of IVP-SIP-V then we have the problem to be non oscillatory otherwise it will be oscillatory. The set of ๐œ†โ€™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 ๐‘‘1,๐‘1 and ๐‘‘2,๐‘2 be the constant density and sound velocity in layers 1 and 2, respectively. Let ๐‘˜1 and ๐‘˜2 be the wave vectors which are given by ๐‘˜1=๐œ”/๐‘1,๐‘˜2=๐œ”/๐‘2, where ๐œ” is the angular frequency. The governing problem is given by ๐ฟ1๐‘ข1=๐‘‘๎‚ต1๐‘‘๐‘ง๐‘‘1๐‘‘๐‘ข1๎‚ถ+๎‚ต๐‘˜๐‘‘๐‘ง12๐‘‘1โˆ’๐œ†๐‘‘1๎‚ถ๐‘ข1๐ฟ=0,0โ‰ค๐‘งโ‰ค๐‘Ž,2๐‘ข2=๐‘‘๎‚ต1๐‘‘๐‘ง๐‘‘2๐‘‘๐‘ข2๎‚ถ+๎‚ต๐‘˜๐‘‘๐‘ง22๐‘‘2โˆ’๐œ†๐‘‘2๎‚ถ๐‘ข2=0,๐‘Žโ‰ค๐‘งโ‰ค๐‘,(6.17) together with the mixed boundary conditions given by ๐‘ข1(0)=๐‘ข2๎…ž๐‘ข(๐‘)=0,1(๐‘Ž)=๐‘ข21(๐‘Ž),๐‘‘1๐‘ข1๎…ž(1๐‘Ž)=๐‘‘2๐‘ข2๎…ž(๐‘Ž),(6.18)๐‘ข1 and ๐‘ข2 denote the depth eigenfunctions corresponding to the eigenvalue ๐œ†.
Here we see that ๐•‹1=[0,๐‘Ž],๐•‹2=[๐‘Ž,๐‘], ๐œŒ1=1=๐œŒ2, ๐œŒ3=1/๐‘‘1,๐œŒ4=1/๐‘‘2. Though we have an extra boundary condition at ๐‘, similar theory can be developed for the above problem. We see that๐œ†11=๐œ†,๐œ’11=โˆ’๐‘˜12,๐œ†12=๐œ†,๐œ’12=โˆ’๐‘˜22,๐œ’21,๐œ’31,โ€ฆ,๐œ’(๐‘›โˆ’1)1=0=๐œ’22,๐œ’32,โ€ฆ,๐œ’(๐‘šโˆ’1)2.(6.19) Hence we see that the set of ๐œ†โ€™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 ๐‘Ž and ๐‘ and different uniform densities ๐‘‘1 and ๐‘‘2, respectively, having a tension T stretched between the points ๐‘ฅ=0 and ๐‘ฅ=๐‘Ž+๐‘: ๐ฟ1๐‘ข1=๐‘12๎€ทโˆ’๐‘ข1๎…ž๎…ž๎€ธ=๐œ†๐‘ข1๐ฟ,0โ‰ค๐‘ฅโ‰ค๐‘Ž,2๐‘ข2=๐‘22๎€ทโˆ’๐‘ข2๎…ž๎…ž๎€ธ=๐œ†๐‘ข2,๐‘Žโ‰ค๐‘ฅโ‰ค๐‘Ž+๐‘,(6.20) with the mixed boundary conditions given by ๐‘ข1(๐‘Ž)=๐‘ข2๐‘ข(๐‘Ž),๎…ž1(๐‘Ž)=๐‘ข๎…ž2(๐‘Ž),(6.21) and ๐‘ข1(0)=๐‘ข2(๐‘Ž+๐‘)=0, where ๐‘๐‘–2=๐‘‡|๐‘‘๐‘–,๐‘–=1,2. ๐‘ข1 and ๐‘ข2 are eigenfunctions corresponding to the eigenvalue ๐œ†. Here we see that ๐•‹1=[0,๐‘Ž],๐•‹2=[๐‘Ž,๐‘Ž+๐‘], ๐œŒ1=1=๐œŒ2, ๐œŒ3=1=๐œŒ4. Though we have an extra boundary condition at ๐‘Ž+๐‘, similar theory can be developed for the above problem. We see that ๐œ†11๐œ†=โˆ’๐‘12,๐œ†12๐œ†=โˆ’๐‘22,๐œ’11,๐œ’21,โ€ฆ,๐œ’(๐‘›โˆ’1)1=0=๐œ’12,๐œ’22,โ€ฆ,๐œ’(๐‘šโˆ’1)2.(6.22) Hence we see that the set of ๐œ†โ€™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 [20โ€“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 ๐œŒ(๐‘)=๐œŽ(๐‘)=๐‘ 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.

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