ISRN Mathematical Analysis

Volume 2011, Article ID 254695, 23 pages

http://dx.doi.org/10.5402/2011/254695

## 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. Carpio and J.-F. Colombeau

Copyright © 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 () and nonoscillation () domains for a pair of dynamic equations along with matching interface conditions on the semi-infinite time scale . We show that these domains are closed and convex subsets of the parameter space . 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 form where are real parameters and varies over a subinterval of . Fixing and allowing 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 Here are real parameters and is real-valued, continuous, and periodic function. A special region a subset of the -plane for which the equation has nonoscillatory solutions is introduced. is shown to be a closed, convex and is also shown to be entirely contained in the half-plane . In [3], Markus and Moore found the disconjugacy domain (a subset of the -parameter plane) for which the equation has disconjugate solutions. 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 and ) for the general equation on the closed half-line . consists of all the values of such that solutions of the general equation are disconjugate. consists of all the values of such that solutions of the general equation are non oscillatory. They discussed the properties of and 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 . We define and discuss the disconjugacy () and non oscillation () 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 . 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 be a time scale (an arbitrary closed subset of real numbers). For one defines the forward jump operator by
while the backward jump operator is defined by
If , we say that is right scattered, while if we say that is left scattered. Points that are right scattered and left scattered at the same time are called isolated. Also, if and , then is called right dense, and if and , then is called left dense. Points that are right dense and left dense at the same time are called dense. Finally, the graininess function is defined by

*Definition 2.2. *One has
where is the left-scattered maximum.

*Definition 2.3. * Let be a function defined on . One says that is delta differentiable at provided there exists an such that for all 0 there is a neighborhood around with

*Remark 2.4. * For a function we will talk about the second derivative provided is differentiable on with derivative .

*Definition 2.5. *One defines the parameter space

#### 3. Preliminary Results

Let (a time scale with end points 0 and ), a time scale with one end being , . Also let be nonlinear function tuple in . And let be positive. In this section we consider the following IVP associated with singular interface problem (IVP-SIP-I): with the initial conditions followed by the matching interface conditions

*Definition 3.1. *One calls a function to be a matching solution of the IVP-SIP-I if (i)the function and satisfies (3.1), (ii)the function and satisfies (3.2), (iii) and satisfy the initial and interface conditions (3.3)-(3.4) and (3.5)-(3.6), respectively.

*Definition 3.2. *One calls to be a zero of the IVP-SIP-I if .

*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 , to be the set of continuous functions on time scales and . Also, we denote

*Definition 3.4. *For a given compact subinterval , one defines the space

For any , one defines the functional

Lemma 3.5. *If IVP-SIP-I is disconjugate on every subset of (denoted by ), 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
where
which give the required positive solution. Moreover, IVP-SIP-I is disconjugate implying that is a non oscillatory solution.

Lemma 3.6. *If the IVP-SIP-I is disconjugate on , then, for every closed subinterval of , the functional is positive on .*

*Proof. *Let us suppose that the IVP-SIP-I is disconjugate on . Then by Lemma 3.5 there is a positive solution of the IVP-SIP-I.*Assumption 3.7. *We assume that is of the form
where are functions on . We also assume that are positive basing on the fact that is a positive solution of the IVP-SIP-I.

From the definition of the functional we easily see that whenever . For the cases when we define the functional in the following manner.

If , then we define
If , then we define
If , then we define
From the above definitions of , we see that irrespective of the sign of and .

*Note 1. *Through the rest of the paper we assume that . Similar results can be obtained for the cases when by using the respective definitions for .

Lemma 3.8. *Let IVP-SIP-II be defined by
**
along with (3.3)–(3.6). Then IVP-SIP-II is disconjugate for every if and only if on .*

*Proof. *Let , that is,
Then we see that IVP-SIP-II reduces to the IVP-SIP-III
Simple calculations show that
We observe that whenever
Clearly and so is the only zero possible. Hence, IVP-SIP-III is disconjugate on .

Now let IVP-SIP-II be disconjugate for every . From Lemma 3.6, we have
Hence
So we have
The previous equation is true for all , which implies that
Since the above equation holds true for all and , we must have , hence the proof.

Corollary 3.9. *Let IVP-SIP-IV be defined by
**
along with (3.3)–(3.6). Then IVP-SIP-IV is disconjugate on for all , then , that is,
**
where are nonlinear function tuples in .*

#### 4. Disconjugacy Domain

We define IVP-SIP-V as along with (3.3)–(3.6). Without any loss of generality, let us assume that , where and , belong to , , respectively.

*Definition 4.1. *One defines the disconjugacy domain of IVP-SIP-V as
such that IVP-SIP-V is disconjugate on .

Theorem 4.2. *The disconjugacy domain of IVP-SIP-V is the whole space if and only if
*

*Proof. *For
IVP-SIP-V reduces to IVP-SIP-II, and hence from Lemma 3.8 we have . If we choose
we get . In similar lines we can show that
Now let us choose
We see that from Lemma 3.8 we have . Similarly we can show that
Hence
whenever the disconjugacy domain of IVP-SIP-V is the whole space .

On the other hand, if
by simple calculations, it can be seen that IVP-SIP-V is disconjugate.

Corollary 4.3. *If at least one of the functions or , then is a proper subset of .*

*Proof. *This is the contrapositive of Theorem 4.2.

Theorem 4.4. * contains a proper subspace of the vector space if the sets of functions and are linearly dependent.*

*Proof. *Let and be sets of linearly dependent functions Then there exists constants , , not all zero, such that
Therefore, the IVP-SIP-V becomes
along with (3.3)–(3.6). We see that IVP-SIP-V now is to be
along with (3.3)–(3.6). Hence contains the subspace
where
hence the proof.

Corollary 4.5. *If the sets of functions and are linearly independent, then cannot contain a proper subspace of .*

*Proof. *This is the contrapositive of Theorem 4.4.

Lemma 4.6. *Let us consider the following two IVP-SIPs which are disconjugate on :
**
Then the IVP-SIP-VI defined by
**
along with (3.3)–(3.6) is also disconjugate on , for each .*

*Proof. *On account of Lemma 3.6 it is sufficient if we show that
on for every compact subinterval . We see that
Now
Similarly it can be shown that
Hence we have
hence the proof.

*Definition 4.7. *Let be function tuples in . One says that are close in uniform norm whenever

*Definition 4.8. *Let be function tuples in . One says that are close in uniform norm whenever

Lemma 4.9. *Let . Let the sequence of the IVP-SIPs defined by
**
along with (3.3)–(3.6) be disconjugate on for each . If uniformly, then the IVP-SIP
**
is disconjugate on .*

*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 with two zeros . Let us assume that and . The other cases when , can be worked in similar lines. Now let be a solution such that .*Claim 1. *The solutions of (4.25), (4.26) satisfying (3.3)–(3.6) are close in the uniform norm whenever and are close in uniform norm on .

Simple calculations show that the solution of (4.25), (4.26) satisfying (3.3)–(3.6) are
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 and are close in uniform norm on , so the claim. Hence for we can find and such that
whenever
Since must change sign at , it follows that must also change sign near for sufficiently large . Thus, for such , (4.25) along with (3.3)–(3.6) is not disconjugate which leads us to a contradiction.

Theorem 4.10. *The disconjugacy domain of IVP-SIP-V is a closed set of the parameter space .*

*Proof. *Let be a limit point of the sequence . Then for every there exists sufficiently large such that for all we have , and IVP-SIP-VII defined by
along with (3.3)–(3.6) is disconjugate. Let be a non trivial solution of IVP-SIP-V for
Then we see that either never vanishes in which case is disconjugate or let for some . In the latter case we let be a solution of IVP-SIP-VII such that . From assumption we have that for . From Claim 1 we have uniformly approximating . Hence can change sign only at and so for all . Hence is disconjugate, so the result.

Theorem 4.11. *The disconjugacy domain of IVP-SIP-V is a convex set in the parameter space .*

*Proof. *We need to show that for the convex combination
that is,
Since , we have the IVP-SIPs
along with (3.3)–(3.6) to be disconjugate. Now from Lemma 4.6 we have the IVP-SIP-VIII defined by
along with (3.3)–(3.6) to be disconjugate; that, is we have the IVP-SIP
along with (3.3)–(3.6) to be disconjugate. Hence
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 of IVP-SIP-V as
such that IVP-SIP-V is non oscillatory on .

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

*Proof. *Let be a solution of IVP-SIP-I. Let be the zeros of the IVP-SIP-I (since is non oscillatory). By the nature of we see that either or for . We now define

From Assumption 3.7 we see that is a solution of IVP-SIP-I. We also see that either or .

*Note 2. *We assume . Similar results can be developed when .

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

Theorem 5.4. *The non oscillatory domain of IVP-SIP-V is a closed set of the parameter space .*

Theorem 5.5. *The non oscillatory domain of IVP-SIP-V is a convex set in the parameter space .*

#### 6. Applications

We define IVP-SIP-X by where , 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 .*(i)*IVP-SIP-X is oscillatory for every .*(ii)*IVP-SIP-X is oscillatory for every except at some unique = .*(iii)*There exists a finite interval or such that IVP-SIP-X is non oscillatory for either
**or
**For every other combinations of , IVP-SIP-X is oscillatory.*(iv)*There exists either or such that IVP-SIP-X is non oscillatory (resp., oscillatory) for either
**or
**and oscillatory (resp., non oscillatory) for either
**or*(v)*IVP-SIP-X is non oscillatory for**either**or*(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
Semi-infinite plane
infinite plane
where .*

*Proof. *Clearly we see that IVP-SIP-X is a special case of IVP-SIP-V with
Let be a subset of the parameter space . The above claims are consequences of the intersection of with , the non oscillation domain for IVP-SIP-V. We recall that is convex in . We see that intersects in one of the following ways: (i) = , (ii) = a single point, (iii) = a line segment, (iv) = a one-sided ray, (v) = a full ray, (vi) = a finite plane in , (vii) = a semi infinite plane in , (viii) = an infinite plane in , 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
where is the modulus of elasticity, is the load applied, are moments of inertia, , and are the displacements of cross-sections for the thinner and the thicker portions of the column, respectively. The physical conditions are given by
Here we see that , , . We see that
Hence from Theorem 6.1 we see that if the point intersects of IVP-SIP-V then we have the problem to be non oscillatory otherwise it will be oscillatory. The set of ’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 and be the constant density and sound velocity in layers 1 and 2, respectively. Let and be the wave vectors which are given by , where is the angular frequency. The governing problem is given by
together with the mixed boundary conditions given by
and denote the depth eigenfunctions corresponding to the eigenvalue .

Here we see that , , . Though we have an extra boundary condition at , similar theory can be developed for the above problem. We see that
Hence we see that the set of ’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 and and different uniform densities and , respectively, having a tension T stretched between the points and :
with the mixed boundary conditions given by
and , where . and are eigenfunctions corresponding to the eigenvalue . Here we see that , , . Though we have an extra boundary condition at , similar theory can be developed for the above problem. We see that
Hence we see that the set of ’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 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.

#### References

- A. B. Mingarelli, “A survey of the regular weighted Sturm-Liouville problem—the non-definite case,” in
*Proceedings of the Workshop on Applied Differential Equations*, Tsinghua University, Beiing, China, 1985. - R. A. Moore, “The least eigenvalue of Hill's equation,”
*Journal d'Analyse Mathématique*, vol. 5, pp. 183–196, 1956/57. View at Publisher · View at Google Scholar - L. Markus and R. A. Moore, “Oscillation and disconjugacy for linear differential equations with almost periodic coefficients,”
*Acta Mathematica*, vol. 96, pp. 99–123, 1956. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. B. Mingarelli and S. G. Halvorsen,
*Nonoscillation Domains of Differential Equations with Two Parameters*, vol. 1338, Springer, Berlin, Germany, 1988. View at Zentralblatt MATH - S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,”
*Results in Mathematics. Resultate der Mathematik*, vol. 18, no. 1-2, pp. 18–56, 1990. View at Google Scholar · View at Zentralblatt MATH - R. Agarwal, M. Bohner, D. O'Regan, and A. Peterson, “Dynamic equations on time scales: a survey,”
*Journal of Computational and Applied Mathematics*, vol. 141, no. 1-2, pp. 1–26, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Bohner and A. Peterson,
*Advances in Dynamic Equations on Time Scales*, Birkhäuser, Boston, Mass, USA, 2003. View at Zentralblatt MATH - M. Bohner and A. Peterson,
*Dynamic Equations on Time Scales*, Birkhäuser, Boston, Mass, USA, 2001. View at Zentralblatt MATH - M. Bohner and S. H. Saker, “Oscillation of second order nonlinear dynamic equations on time scales,”
*The Rocky Mountain Journal of Mathematics*, vol. 34, no. 4, pp. 1239–1254, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. Akin-Bohner and J. Hoffacker, “Oscillation properties of an Emden-Fowler type equation on discrete time scales,”
*Journal of Difference Equations and Applications*, vol. 9, no. 6, pp. 603–612, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Erbe, A. Peterson, and S. H. Saker, “Kamenev-type oscillation criteria for second-order linear delay dynamic equations,”
*Dynamic Systems and Applications*, vol. 15, no. 1, pp. 65–78, 2006. View at Google Scholar · View at Zentralblatt MATH - L. Erbe and A. Peterson, “An oscillation result for a nonlinear dynamic equation on a time scale,”
*The Canadian Applied Mathematics Quarterly*, vol. 11, no. 2, pp. 143–157, 2003. View at Google Scholar · View at Zentralblatt MATH - O. Došlý and D. Marek, “Half-linear dynamic equations with mixed derivatives,”
*Electronic Journal of Differential Equations*, no. 90, p. 118, 2005. View at Google Scholar · View at Zentralblatt MATH - S. H. Saker, “Oscillation of second-order forced nonlinear dynamic equations on time scales,”
*Electronic Journal of Qualitative Theory of Differential Equations*, no. 23, p. 17, 2005. View at Google Scholar · View at Zentralblatt MATH - C. Allan Boyles,
*Acoustic Waveguides, Applications to Oceanic Sciences*, John Wiley and Sons, 1978. - P. K. Ghosh,
*The Mathematics of Waves and Vibrations*, MacMillan, India, Delhi, 1975. - G. L. Lamb, Jr.,
*Elements of Soliton Theory*, John Wiley & Sons, New York, NY, USA, 1980. - K. Noda,
*Optical Fibre Transmission*, vol. 6 of*Studies in Telecommuniations*, North Holland, Amsterdam, The Netherlands, 1986. - Wang,
*Applied Elasticity*, MacGraw Hill, 1953. - P. K. Baruah and D. J. Das, “Study of a pair of singular Sturm-Liouville equations for an interface problem,”
*International Journal of Mathematical Sciences*, vol. 3, no. 2, pp. 323–340, 2004. View at Google Scholar - P. K. Baruah and M. Venkatesulu, “Deficiency indices of a differential operator satisfying certain matching interface conditions,”
*Electronic Journal of Differential Equations*, vol. 2005, no. 38, p. 19, 2005. View at Google Scholar · View at Zentralblatt MATH - P. K. Baruah and M. Venkatesulu, “Number of linearly independent square integrable solutions of a pair of ordinary differential equations satisfying certain matching interface conditions,”
*International Journal of Mathematics and Analysis*, vol. 3, no. 2-3, pp. 131–144, 2006. View at Google Scholar - M. Venkatesulu and P. K. Baruah, “A classical approach to eigenvalue problems associated with a pair of mixed regular Sturm-Liouville equations—I,”
*Journal of Applied Mathematics and Stochastic Analysis*, vol. 14, no. 2, pp. 205–214, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Venkatesulu and P. K. Baruah, “A classical approach to eigenvalue problems associated with a pair of mixed regular Sturm-Liouville equations—II,”
*Journal of Applied Mathematics and Stochastic Analysis*, vol. 15, no. 2, pp. 197–203, 2002. View at Google Scholar · View at Zentralblatt MATH - T. G. Bhaskar and M. Venkatesulu, “Computation of Green's matrices for boundary value problems associated with a pair of mixed linear regular ordinary differential operators,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 18, no. 4, pp. 789–797, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. K. Baruah and D. K. K. Vamsi, “Oscillation theory for a pair of second order dynamic equations with a singular interface,”
*Electronic Journal of Differential Equations*, no. 43, pp. 1–7, 2008. View at Google Scholar · View at Zentralblatt MATH - P. K. Baruah and D. K. K. Vamsi, “Green matrix for a pair of dynamic equations with singular interface,”
*International Journal of Modern Mathematics*, vol. 4, no. 2, pp. 135–152, 2009. View at Google Scholar · View at Zentralblatt MATH - P. K. Baruah and D. K. K. Vamsi, “IVPs for singular interface problems,”
*Advances in Dynamical Systems and Applications*, vol. 3, no. 2, pp. 209–227, 2008. View at Google Scholar - D. K. K. Vamsi and P. K. Baruah, “Bounds for solutions of nonlinear singular interface problems on time scales using a monotone iterative method,”
*Electronic Journal of Differential Equations*, vol. 2010, no. 109, pp. 1–9, 2010. View at Google Scholar · View at Zentralblatt MATH - D. K. K. Vamsi and P. K. Baruah, “Existential results for nonlinear singular interface problems
involving second order nonlinear dynamic equations using picards iterative technique,” Accepted for publication,
*The Journal of Nonlinear Science and Applications*.