Oscillation Properties for Non-Classical Sturm-Liouville Problems with Additional Transmission Conditions

. This work is aimed at studying some comparison and oscillation properties of boundary value problems (BVP’s) of a new type, which diﬀer from classical problems in that they are deﬁned on two disjoint intervals and include additional transfer conditions that describe the interaction between the left and right intervals. This type of problems we call boundary value-transmission problems (BVTP’s). The main diﬃculty arises when studying the distribution of zeros of eigenfunctions, since it is unclear how to apply the classical methods of Sturm’s theory to problems of this type. We established new criteria for comparison and oscillation properties and new approaches used to obtain these criteria. The obtained results extend and generalizes the Sturm’s classical theorems on comparison and oscillation.

1 Introduction and statement of the problem is growing interest in deriving a new sufficient conditions for the oscillation or nonoscillation of the solutions of new type non-classical differential equations that have appeared in various fields of natural science in recent years. In particular, non-classical Sturm-Liouville type problems involving additional transmission conditions appear frequently in modeling of different problems encountered in physics such that electromagnetic processes in ferromagnetic media with different dielectric properties, in vibrating folded membranes, in elastic multi-structures, in problems of "hydraulic fracturing" in heat and mass transfer, in vibrations of mechanical systems etc., which, in general, are associated with nonsmooth or discontinuous properties of the medium (see for example, [11,19,29,30]).
The present article devoted to the investigation of comparison and oscillation properties of new type Sturm-Liouville problems involving additional transmission conditions. Our motivation for this work stems from a number of important problems of mathematical physics, in solving of which it is necessary to determine the oscillation properties of solutions. For example, in the study of free oscillations of the Earth, the equation determining the radial dependence of its toroidal oscillations is described by transfer problems for equations of the Sturm-Liouville form(see, [3]). Note that, our problem differs from the classical Sturm-Liouville problems(SLP's) in that it includes additional transmission conditions that describe the relationship between the left and right limits of the solutions and its derivatives at the initial point of interaction. The major difficulty arises in studying the existence and behavior of zeros of solutions and it is required to modify the classical methods such that they can be also applicable to the problems involving additional transmission conditions.
It is known that in studying the various properties of many physical phenomena, it is necessary to obtain sufficient conditions for oscillatory and nonoscillatory solutions, to estimate the minimum distance between consecutive zeros and the maximum number of zeros lying in a given interval, and also to find a connection between such properties of differential equations that arise during modeling such physical phenomena. Therefore the Sturmian comparison and oscillation theory is attracting increasing attention. Sturm's comparison theorem which compares the rapidly of the oscillation of the solutions of the two differential equations, is formulated usually as follows: Consider the differential equations and if x 1 , x 2 are two consecutive zeros of a solution u = u(x) of (1.1) then every solution υ = υ(x) of (1.2) has at least one zero in (x 1 , x 2 ). Note that the Sturm-Liouville equation is a typical of a large class of equations which arise in problems of many branches of natural science. By the above Sturm's comparison property an increment in |λ| produced a more rapid oscillation of the solution. This comparison theorem has some physical interpretations. For example, let a particle is attracted to u = 0 by force f 1 (t) = −h 1 (t)u and suppose that the degree freedom of this particle is equal to one, then the mathematical model of its motion has the Sturm-Liouville form where t stands for time. If another particle attracted to ν = 0 by force those motion is also modeled by the similar equation ν + f 2 (t)ν = 0), then as known from Quantum Theory the second particle will oscillated more rapidly. That is, if the first particle passed through u = 0 at successive times t 1 and t 2 then between these moments the second particle (that is a particle with a large force) will pass at least once. Similar investigation for two strings oscillating around an equilibrium line, which is a simple harmonic motion, gives Sturmian comparison and oscillation theorems for hyperbolic equations. This theory can also be utilized to study some qualitative properties of solutions that we cannot find in exact form. Therefore, the Sturmian theory is significant from the theoretical as well as practical point of view. Moreover when studying some problems of mathematical physics, it is required to find oscillatory and non-oscillatory solutions not only for ordinary differential equations (ODE's) but also for partial differential equations (PDE's). The first important results on the comparison and oscillatory properties of solutions of ordinary linear differential equations of second-orders were established by Sturm in 1836 [29]. For classical comparison and oscillation theory for ODE's we refer to Swanson's [30] and for PDE's we refer to Yoshida's [31]. In [19] has been established some important consequences of the Sturm's comparison theorem. There is also a huge literature on applications of this theory (see, for example, [10,29] and references cited therein). For the further developments and various type generalization we refer to [3,9,11,16,31,32] and references cited therein. In this work we will extend and generalize some well-known Sturm's comparison and oscillation theorems for problems of a new type, which consist of Sturm-Liouville equation Here we assume that, the real-valued function q(x) is continuous on each of the intervals [a, c) and (c, b] with a finite limits q(c ± 0) := lim →0 q(c ± | |) ; λ is a complex eigenparameter, δ, δ , γ, γ , α, α , β, β are real numbers, |α| + |β| = 0, |α | + |β | = 0. In recent years, the authors of this study and some other mathematicians have studied many other spectral properties of similar BTVP's (see, for example, [1,2,4,5,6,7,9,13,17,22,23,27,28]). BVTP's also find many important application in the study of various phenomena in physics and technology, such as vibration of string involving different types of loads (see, for example, [12,14,18,21,25,26]), heat transfer through a solid-liquid interface(see, for example, [8,15,20]), water vapor diffusion through a porous membrane (see, for example, [24]). have infinitely many real eigenvalues λ n = ( (2n+1)π 2 ) 2 , n = 0, 1, 2, ..., the BVTP (1.8)-(1.9) has only trivial solution y = 0 for all λ ∈ R, that is this problem has no real eigenvalue.
where, as usual W (y 1 , y 2 ; x) = y 1 (x)y 2 (x)−y 1 (x)y 2 (x) denotes the Wronskian's of the solutions y 1 , y 2 . Then between any two consecutive zeros of the solution y 1 (x) there is at least one zero of the solution y 2 (x).
Proof. Let x 1 and x 2 be two successive zeros of the solution y 1 (x). Since the cases x 1 , x 2 ∈ (a, c) and x 1 , x 2 ∈ (c, b) are totally similar to the classical case one (see, [30]), we will consider only the case Integrating the identity (2.2) and then using transmission conditions (2.1), yields we have that y 1 (x 1 ) ≥ 0. Similarly, y 1 (x 2 ) ≤ 0. Consequently, the right side of the identity (2.3) is positive, but the left side is negative. Thus we arrive at a contradiction which completes the proof.
From this comparison criteria there follows the following corollaries, which are important for further investigation. Then between any two successive zeros of the solution y(x, λ 1 ) has not fewer than one zero of the solution y(x, λ 2 ).
then ϕ(x, λ 2 ) has at least m zeros in the same domain (a, c) ∪ (c, b] and the i-th zero of ϕ(x, λ 2 ) is less than i-th zero of ϕ(x, λ 1 ).
Proof. Let x 1 , x 2 , ..., x m be consecutive zeros of the solution ϕ(x, λ 1 ) lying in (a, c) ∪ (c, b]. By virtue of the preceding theorem there is at least one zero of ϕ(x, λ 2 ) in each of intervals (x i , x i+1 ), i = 1, 2, ..., n−1. Therefore it is suffices to show that the solution ϕ(x, λ 2 ) has at least one zero inside (a, x 1 ). Case 1. Let x 1 < c. Assume the contrary i.e. assume that ϕ(x, λ 2 ) does not vanish inside [a, x 1 ]. Then without loss of generality we may assume that both ϕ(x, λ 1 ) and ϕ(x, λ 2 ) are positive inside [a, x 1 ]. Integrating the identity (2.2) from x = a to x = x 1 , yields Obviously, the right side of this identity is positive. Since ϕ(x 1 , λ 1 ) = 0 and ϕ(x, λ 1 ) > 0 inside [a, x 1 ], it follows that Therefore the left side of the identity (2.6) is not positive (in fact, the left side of (2.6) is negative, since ϕ x (x 1 , λ 1 ) can not vanish at x = x 1 , otherwise then it would follow from the well-know existence and uniqueness theorem from the theory of ordinary differential equations that ϕ(x, λ 1 ) ≡ 0 which is impossible). Thus we have a contradiction, which proves the theorem when the first zero x = x 1 of the function ϕ(x, λ 1 ) lying in the left interval (a, c) i.e. for the case x 1 < c. Case 2. Let x 1 > c. Assume the contrary, i.e. assume that ϕ(x, λ 2 ) = 0 in (a, c) ∪ (c, x 1 ). Integrating both sides of the identity (2.2) from a to x 1 and taking in view that the integral is improper at x = c yields Using the transmission conditions (2.5) the relation (2.7) is reduced to the following form Then both ϕ(x, λ 1 ) and ϕ(x, λ 2 ) does not change its sign in both intervals (a, c) and (c, x 1 ). For definiteness we put ϕ(x, λ 1 ) < 0 and ϕ(x, λ 2 ) < 0 in both (a, c) and (c, x 1 ). These conditions ensure that the right side of (2.8) is positive. On the other hand, since ϕ(x, λ 1 ) = 0 and ϕ(x, λ 1 ) < 0 for x ∈ (c, x 1 ) the function ϕ(x, λ 1 ) is decreasing at the point x = x 1 , i.e. ϕ x (x 1 , λ 1 ) ≤ 0. Therefore the left side of the identity (2.8) is negative. Thus we have a contradiction, which proves that the solution ϕ(x, λ 2 ) has at least one zero in (a, x 1 ). The proof is complete.

Oscillations properties of the solutions
For further consideration we need the next theorem which assert that the zeros of the solution ϕ(x, λ) are continuous functions with respect to the variable λ.

Remark 2.
The obtained results about zeros of the solution ϕ(x, λ) can be interpreted as follows. As λ varies, the solution ϕ(x, λ) can lose zero or acquire zero only if the zero gets inside or outside the intervals [a, c) and (c, b] through one of the endpoints x = a and x = b or through of the point of interaction x = c. In particular, as λ increases, the zeros of the solution ϕ(x, λ) move to left, but cannot cross the endpoint x = a, since the number of zeros does not decrease and new zeros enter through the right end point. Now consider the following auxiliary initial-transmission problem This problem has unique solution y = ϕ(x, λ), which is the entire function with respect to the complex variable λ for each x ∈ [a, c) ∪ (c, b](see, [6]).