Some Properties of Eigenvalues and Generalized Eigenvectors of One Boundary Value Problem

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Filomat 3:3 018, 911 90 https://doi.org/10.98/fil1803911o Published by Facu

1 Filomat 3:3 018, Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Some Properties of Eigenvalues and Generalized Eigenvectors of One Boundary Value Problem Hayati Olǧar a, Oktay Sh. Mukhtarov a,b, Kadriye Aydemir c a Department of Mathematics, Faculty of Arts and Science, Gaziosmanpaşa University, 6050 Tokat, Turkey b Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan c Department of Mathematics, Faculty of Art and Science, Amasya University, Amasya, Turkey Abstract. We investigate a discontinuous boundary value problem which consists of a Sturm-Liouville equation with piecewise continuous potential together with eigenparameter dependent boundary conditions and supplementary transmission conditions. We establish some spectral properties of the considered problem. In particular, it is shown that the problem under consideration has precisely denumerable many eigenvalues λ 1, λ,..., which are real and tends to +. Moreover, it is proven that the generalized eigenvectors form a Riesz basis of the adequate Hilbert space. 1. Introduction Sturm-Liouville problems with eigenparameter appearing in the boundary conditions have been studied by many authors see, [4 6, 10, 0, 7] and corresponding references cited therein. The main goal of those papers is the analysis of the spectrum and justification of the eigenfunction expansion. The considerations of [10, 6] are based on the operator-theoretic formulation of such type Sturm-Liouville problems. In [6, 7], the residue calculus in a manner similar to [4] is employed. In recent years, there has been increasing interest of Sturm-Liouville type problems together with eigenparameter dependent boundary conditions and supplementary transmission conditions see, [ 4, 11 14, 18 ]. Such properties as isomorphism, coerciveness with respect to the spectral parameter, completeness and Abel basis property of a system of root functions, asymptotics of eigenvalues of some boundary value problems with transmission conditions and its applications to the corresponding initial boundary value problems for parabolic equations have been investigated in [11, 0 ]. The various physics applications of this kind of problems arise in heat and mass transfer problems [16, 17, 3], in vibrating string problems when the string loaded additionally with point masses [3], in diffraction problems [5], etc. Basis properties and eigenfunction expansions for Sturm-Liouville problems involving eigenparameter in the boundary conditions have been considered in [1, 5, 9, 10, 14, 7 9]. Aliyev and Kerimov [1] studied basisness of root functions of Sturm-Liouville problems with a boundary condition depending quadratically on the spectral parameter. In [5], it is shown that the problem of eigenoscillations of various mechanical 010 Mathematics Subject Classification. Primary 34B4; Secondary 34B09 Keywords. Boundary value problem, eigenvalues, generalized eigenvectors Received: 8 December 016; Revised: 10 May 017; Accepted: 10 May 017 Communicated by Allaberen Ashyralyev addresses: hayatiolgar@gmail.com Hayati Olǧar, omukhtarov@yahoo.com Oktay Sh. Mukhtarov, kadriyeaydemr@gmail.com Kadriye Aydemir

2 H. Olǧar et al. / Filomat 3:3 018, systems is formulated as the Sturm-Liouville problem with the eigenvalue parameter appearing in the boundary conditions. It is proven that the eigenfunctions form a Riesz basis of suitable Hilbert space. In [9], such properties as completeness, minimality and basis property for the eigenfunctions are investigated. In [9], for a non-linear eigenvalue problem similar to a linear Sturm-Liouville problem the properties of the eigenvalues and corresponding eigenfunctions are analysed and the system of eigenfunctions is shown to be a Riesz basis in L. In this paper we shall investigate the Sturm-Liouville equation on two disjoint intervals [ 1, 0 and 0, 1] given by f x + qx f x = λ f x x [ 1, 0 0, 1], 1 together with boundary conditions at the end-points x = 1, and x = 1 given by f 1 = 0, ln f 1 = aλ bλ + c, and two supplementary transmission conditions at the point of discontinuity x = 0 given by f 0 + f 0 = 0, f 0 + f 0 = δ f 0. 4 Here a, b, c, δ R, δ > 0, θ = ac > 0, λ C, the function qx is positively definite, measurable and bounded on [a, c c, b]. The purpose of this paper is to reduce the Sturm-Liouville problem to the operator polynomial equation in an appropriate Hilbert space and to prove that the corresponding eigenfunctions form a Riesz basis of this space. At first, we shall define some new Hilbert spaces and give some inequalities which is needed for further investigation. It is well-known that the standard Sobolev spaces play a fundamental role in studying various spectral properties of differential operators. Recall that the Sobolev space W k Ω, k = 0, 1,,... is the Hilbert space consisting of all functions f L Ω that have generalized derivatives f n L Ω for n = 1,,..., k with the inner product f, W k Ω := f x x + f x x + f f k x k xdx, Ω where Ω R is any bounded interval. Let Ω 1 := [ 1, 0, Ω := 0, 1], Ω = Ω 1 Ω and let f be any function defined on Ω = Ω 1 Ω. Then by f i x we shall denote the restriction of f x on the interval Ω i. Below, by H 0 Ω 1 H 0 Ω we denote the Hilbert space L Ω L Ω 1 L Ω with the inner product 3 f, 0 := f 1 x 1 x dx + Ω 1 Ω f x x dx. Since the embedding operators J : W 1 Ω i CΩ i i = 1, are compact, we can show that the linear space H 1 Ω 1 H 1 Ω = { f H 0 Ω 1 H 0 Ω f i W 1 Ω i i = 1,, f 1 0 = f 0 + } forms a Hilbert space with respect to the inner-product f, 1 := f1 x 1 x + f 1 x 1 x dx + Ω 1 Ω f x x + f x x dx.

3 H. Olǧar et al. / Filomat 3:3 018, In the same linear space H 1 Ω 1 H 1 Ω, we shall introduce another inner-product as < f, > := f, q 0 + f, 0, 5 with corresponding norm f := f,. 6 The function qx is positively definite and bounded, hence there exist positive constants m, M, independent of f, such that m f 1 < f < M f 1. Consequently, the inner product space { H 1 q Ω 1 H 1 q Ω,.,. } is also Hilbert space. It is clear that the functions in H 1 Ω 1 H 1 Ω are continuous on each [ 1, 0 and 0, 1], but their generalized derivatives can only be assumed to be elements of L Ω. From the well-known embedding theorems for Sobolev spaces see [16] we can derive easily following inequalities f 1 γ 1 f 0 + γ 1 f 0, 7 f 0 + γ f 0 + γ f 0, 8 f 1 0 γ 3 f 0 + γ 3 f 0, 9 f ξ Cξ f for any ξ Ω, 10 where γ j j = 1,, 3 are any positive real numbers that are small enough and the constant Cξ is independent of the function f, i.e. the constant Cξ is dependent only of ξ. Moreover, the inequality f 0 C 1 f 11 follows directly from the definition of the norms 5-6. Let us introduce to the consideration the Hilbert space H := H 1 Ω 1 H 1 Ω C, with the inner product for F := F, G H f z := f, + z w, G := w H which is the main space for this study.. The Generalized Eigenvectors of the Problem The concept of generalized eigenfunction for our problem 1 4 is the main object of this study. Note that this concept is based on the weak solutions of problem 1 4, which we shall define by the following procedure.

4 H. Olǧar et al. / Filomat 3:3 018, Multiplying both side of equation 1 by the complex conjugate of an arbitrary function µ Hq 1 Ω 1 Hq 1 Ω integrate by parts over the intervals [ 1, 0 and 0, 1] and applying the boundary and transmission conditions 4 produce the following. f 1 x µ 1 x + q 1 x f 1 x µ 1 x dx + f x µ x + q x f x µ x dx Ω 1 Ω a b f 1µ 1 + δ f 0 µ0 + b µ 1 = λ f 1 x µ 1 x dx + λ f x µ x dx. 1 Ω Ω 1 By defining a new unknown parameter by := a f 1 b f 1 and using the boundary condition we have 1 b f 1 ab = λ θ. 13 Thus, the following theorem is proven. Theorem.1. Let q i CΩ i i = 1, and suppose that there exists a finite one-hand limits q0 ± := lim x 0 ± qx. If the function f = f 0 x is the classical eigenfunction of problem 1-4, then two-component vector function f0 x satisfies equation 1 and equation 13 for any µ H q 1 Ω 1 Hq 1 Ω. By applying the standard procedure [16], we have the following definition of a generalized solution of boundary value transmission problem BVTP 1 4. f x Definition.. The two-component element F = of the Hilbert space H is said to be a generalized eigenvector of BVTP 1 4, if this element satisfies equation 1 and equation 13 for any µ H 1 Ω 1 H 1 Ω. It is easy to see that the next theorem is true. f x Theorem.3. Let F = H be generalized eigenvector of BVTP 1 4. If f i C Ω i, q i CΩ i with the finite one-hand limits f 0 ±, f 0 ±, f 0 ±, then the first component f x becomes a classical eigenfunction of BVTP 1 4. f x Proof. Let F = be generalized eigenvector of BVTP 1 4. Then for arbitrary µ H q 1 Ω 1 Hq 1 Ω the function f i C Ω i satisfy equalities Integrating by parts over the interval Ω i we have from 1 that the equality f 1 x + q 1x f 1 x µ 1 xdx + f x + q x f x µ xdx Ω 1 Ω = λ f 1 xµ 1 xdx + λ f xµ xdx Ω 1 is hold for arbitrary µ H 1 q Ω 1 H 1 q Ω. The arbitrariness of the function µ shows that the function f satisfies equation 1. From the equalities 1-13 it follows immediately that the solution of 1-13 satisfies conditions -4. The proof is complete. Remark.4. If qx is only measurable and bounded on Ω i.e. are not necessarily continuous functions on Ω, then it is not possible to define the classical eigenfunction. Nevertheless, even in this situation all terms of integral equation 1 and equation 13 are defined in the space H 1 q Ω 1 H 1 q Ω. Ω

5 H. Olǧar et al. / Filomat 3:3 018, Main Results Now we are ready to establish some important results. Particularly, based on the definitions and results of the previous section we shall reduce BVTP 1 4 into an operator-pencil equation with self-adjoint compact operators and then it will be proved that the generalized eigenvectors of BVTP 1 4 form a Riesz basis of the Hilbert space H. Let us introduce to consideration the following bilinear forms. l 0 f, µ := a b f 1µ 1 + δ f 0 µ0, for all f, µ H 1 Ω 1 H 1 Ω, 14 l 1 f, µ := f, µ H 0 = f 1 x µ 1 xdx + f x µ xdx, for all f, µ H 1 Ω 1 H 1 Ω, 15 Ω 1 Ω l, µ := µ 1, for all C, µ H 1 Ω 1 H 1 Ω. 16 b Theorem 3.1. There are bounded linear operators T 0, T 1 : H 1 Ω 1 H 1 Ω H 1 Ω 1 H 1 Ω and T : C H 1 Ω 1 H 1 Ω satisfying the following representations. l j f, µ = < T j f, µ > j = 0, 1 and l, µ = < T, µ >. 17 Proof. We prove firstly that the bilinear forms l j j = 0, 1 are continuous in µ H 1 Ω 1 H 1 Ω for any given f H 1 Ω 1 H 1 Ω. The proof is based on the following inequalities and l 0 f, µ C f µ, l 1 f, µ C 3 f µ. 18 Note that here, and below, the symbols C n for n = 1,,..., are used to denote different constants which do not depend on the functions under consideration and whose exact values are not important for the proof. By using the well-known Schwarz inequality and 9, we get l 0 f, µ C 4 f 0 µ 0 C 4 C 1 f 0 µ C 4 C 1 f µ. The proof of inequality 18 is completely similar. Further, from 16, it follows immediately that l, µ C 5 µ 1. By using interpolation inequalities 7-11, we have the following inequality l, µ C 6 µ 1 C 7 µ. To complete the proof it is enough to apply familiar Riesz Representation Theorem see, for example [15]. Theorem 3.. We have the following assertions: i. The operators T 0, T 1 : H 1 Ω 1 H 1 Ω H 1 Ω 1 H 1 Ω are self-adjoint and compact, ii. The operator T 1 is positive,

6 iii. The operator T : C H 1 Ω 1 H 1 Ω is compact. H. Olǧar et al. / Filomat 3:3 018, Proof. i. Firstly, let us show that the operator T 0 is self-adjoint. Let f, µ H 1 Ω 1 H 1 Ω be arbitrary functions. By 14 and 17, we have that f, T 0 µ = T 0 µ, f = a b f 1µ 1 + δ f 0 µ0 = l 0 µ, f = l 0 f, µ = T 0 f, µ. So the operator T 0 is self-adjoint in the Hilbert space H 1 Ω 1 H 1 Ω. The proof of the self-adjointness of the operator T 1 is completely similar. Now, we shall prove that the operators T 0 and T 1 are compact operators in the Hilbert space H 1 Ω 1 H 1 Ω. For this it is sufficient to show that any weakly convergent sequence { f k }k = 1,,... in H 1 Ω 1 H 1 Ω is transformed by T j j = 0, 1 into a strongly convergent sequence {T j f k }j = 0, 1. The boundedness of T j j = 0, 1 implies the weak convergence of {T j f k }j = 0, 1 to {T j f }j = 0, 1 in H 1 Ω 1 H 1 Ω, where f x is the weak limit of { f k }. Moreover, the compactness of the embedding operator from H 1 Ω 1 H 1 Ω into L Ω 1 L Ω [16] implies the strong convergence of the sequences { f k } and {T j f k } j = 0, 1 in L Ω 1 L Ω to f and T j f j = 0, 1, respectively. Further, the compactness of the embedding operator from H 1 Ω 1 H 1 Ω into CΩ 1 CΩ [16] implies the strong convergence of the sequences { f k d} and {T j f k d} j = 0, 1 in C to f d and T j f d j = 0, 1, respectively with d 1 = 1 or d = 0 or d 3 = +1. If we use definition 17 of the operators T j j = 0, 1, inequalities 7-11 and representations 14-17, then we have T 0 f k T 0 f m = T 0 f k f m, T 0 f k f m 3 = l 0 f k f m, T 0 f k f m c 8 f k f m d α T0 f k f m d α α=1 C 9 { fk +1 f m +1 T0 f k f m +1 + fk 0 f m 0 T0 f k f m 0 }, T 1 f k T 1 f m = T 1 f k f m, T 1 f k f m = l 1 f k f m, T 1 f k f m C 10 r i f k x f m x T 1 f k f m dx Ω i i=1 C 11 f k f m 0 T 1 f k f m. Consequently, T j f k f m 0 as k, m. Hence, the sequence {T j f k } j = 0, 1 converges strongly in the Hilbert space H 1 Ω 1 H 1 Ω, so the compactness of the operators T j j = 0, 1 are proven. ii. Since r 1 x and r x are positive definite functions, the positivity of the operator T 1 is obvious. iii. It is easy to verify that the adjoint operator of T is defined on whole H 1 Ω 1 H 1 Ω with action law T f = 1 b f 1. From this representation it follows that, the operator T from H 1 Ω 1 H 1 Ω to C is bounded, i.e there exists C 1 > 0 such that, for all f H 1 Ω 1 H 1 Ω T f C 1 f. Consequently, the operator T is bounded linear operator with finite dimensional range and therefore is compact. The proof is complete.

7 H. Olǧar et al. / Filomat 3:3 018, Let us define two new operators R and S in the Hilbert space H by the equalities R f x = f + T0 f x + T T f x c bθ and S f x = T1 f x θ respectively, in terms of which we shall define a linear operator-pencil Aλ by the equality Aλ = R + λ S. By using 1, 13, 14, 15 and 16 we have the following result. Lemma 3.3. The generalized eigenfunctions of the BVTP 1-4 satisfy the operator-polynomial equation Aλ f x in the Hilbert space H. = 0 19 The following result is very important for further consideration. Lemma 3.4. There exists c > 0, such that for all real λ 0 > c the operator polynomial Aλ 0 is positive definite. Proof. Equations and 19 give the formula f x f x Aλ 0, = f a b f 1 + δ f 0 c bθ + λ 0 θ + b Re f 1 H + λ 0 r 1 x Ω f1 x dx + r x f x dx. 1 Ω Applying the obvious inequality b Re f 1 1 ε + 1 b ε f 1 with an arbitrary parameter ε > 0 yields f x f x Aλ 0, f a + 1 f 1 + ε c b ε b b b θ + λ 0 θ H + δ f 0 + λ 0 r 1 x Ω f1 x dx + r x f x dx. 0 1 Ω Let use define the following functionals P f := f, f 0 = f 1 x dx + f x dx, Ω 1 Ω Q f := q f, f 0 = q 1 x f1 x dx + q x f x dx, Ω 1 Ω R f := r f, f 0 = r 1 x f1 x dx + r x f x dx. Ω 1 Ω

8 Then we have H. Olǧar et al. / Filomat 3:3 018, f = P f + Q f for all f H 1 Ω 1 H 1 Ω. 1 Since the functions qx and rx are positive and bounded, we have f 0 C 13P f, f 0 C 14Q f, r f, f 0 = R f C 15 Q f. By using inequalities 7-11, 0, and equality 1 we can show easily that f x f x Aλ 0, Ϝ 1 P f + Ϝ λ 0 Q f + Ϝ 3 λ 0, H where Ϝ 1 := 1 a + 1 γ1 C 13 + δγ C 13, b ε b Ϝ λ 0 := 1 a + 1 C 14 + δ C 14 + λ 0 C 15, b ε b γ 1 γ Ϝ 3 λ 0 := ε c b bθ + λ 0 θ. Since θ > 0, it is possible to choose the arbitrary positive parameters γ 1, γ and ε so small and the positive parameter λ 0 so large that the inequalities Ϝ 1 > 0, Ϝ λ 0 > 0 and Ϝ 3 λ 0 > 0 hold. Thus, we have that f x f x Aλ 0, min Ϝ 1, Ϝ λ 0, Ϝ 3 λ 0 Φ H H f x f x f x for all := Φ H and hence the quadratic form Aλ 0, is positive definite H for sufficiently large positive values of λ 0. Thus the operator pencil Aλ 0 is positive definite for sufficiently large λ 0 > 0. The proof is complete. By virtue of the Lemma 3.4 there exists c > 0, such that for all real λ 0 > c the operator polynomial Aλ 0 is positive definite. Moreover, we know that the operator Aλ 0 is also self-adjoint. Therefore there exists positive square root Aλ 0 which is invertible. Consequently, we can introduce to consideration a new operator χλ 0 defined by χλ 0 := Aλ 0 1 S Aλ0 1 in the Hilbert space H. Hence we have the following result. Theorem 3.5. The operator χλ 0 is positive, self-adjoint and compact in the Hilbert space H for sufficiently large positive λ 0. Theorem 3.6. Let λ 0 be any real positive number, such that Aλ 0 is positive defined operator in the Hilbert space H. Then the operator χλ 0 := Aλ 0 1 S Aλ0 1 has precisely denumerable many positive eigenvalues {η n } which tends to +. The corresponding eigenvectors form an orthogonal basis of H. Proof. The proof of this Theorem follows from the well-known Fredholm theorems for compact self-adjoint operators in a Hilbert spaces see, for example, [15]. Now, by using the well-known fact, that every bounded invertible operator transforms any orthonormal basis of a Hilbert space H into Riesz basis of H see, for example, [8] we get the following main result. Theorem 3.7. Let qx be positive defined, bounded and measurable function on Ω = Ω 1 Ω and suppose that ab > 0 and δ > 0. Then, BVTP 1-4 has precisely denumerable many eigenvalues λ 1, λ,..., which are real and tends to +. Moreover, the corresponding system of generalized eigenvectors forms a Riesz basis of H.

9 H. Olǧar et al. / Filomat 3:3 018, Conclusion The following important conclusions can be really drawn: i. There exist a real λ 0 such that the operator χλ 0 is positive, self-adjoint and compact in the Hilbert space H, ii. BVTP 1-4 has only point spectrum, iii. The eigenvalues form a real sequences with the only point of accumulation at +, iv. The generalized eigenvectors of BVTP 1-4 forms a Riesz basis of H. References [1] Y.N. Aliyev, N.B. Kerimov, The basis property of Sturm-Liouville problems with boundary conditions depending quadratically on the eigenparameter, Arabian J. Sci. Engin [] B.P. Allahverdiev, A dissipative singular Sturm-Liouville problem with a spectral parameter in the boundary condition, J. Math. Anal. Appl [3] B.P. Allahverdiev, E. Bairamov, E. Ugurlu, Eigenparameter dependent Sturm-Liouville problems in boudary conditions with transmission conditions, J. Math. Anal. Appl [4] K. Aydemir, Boundary value problems with eigenvalue depending boundary and transmission conditions, Boundary Value Problems :131. [5] B.P. Belinskiy, J.P. Dauer, Eigenoscillations of mechanical systems with boundary conditions containing the frequency, Quart. Appl. Math [6] C.T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edin. 77A [7] C.T. Fulton, Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburg A 1. [8] I.C. Gohberg, M.G. Krein, Introduction to The Theory of Linear Non-Selfadjoint Operators, Translation of Mathematical Monographs, vol. 18, Amer. Math. Soc., Providence, Rhode Island, [9] V.Y. Gulmamedov, Kh.R. Mamedov, On basis property for a boundary-value problem with a spectral parameter in the boundary condition, Çankaya Üniv. Fen-Edebiyat Fak. J. Arts Sci. Sayı: [10] D.B. Hinton, An expansion theorem for an eigenvalue problem with eigenvalue parameter contained in the boundary condition, Quart. J. Math. Oxford [11] M. Kadakal, O.Sh. Mukhtarov, SturmLiouville problems with discontinuities at two points, Comput. Math. Appl [1] M. Kandemir, Irregular boundary value problems for elliptic differential-operator equations with discontinuous coefficients and transmission conditions, Kuwait J. Sci. Eng. 39 1A [13] M. Kandemir, O. Mukhtarov, Asymptotic behavior of eigenvalues of the functional-many point boundary-value problem with discontinuous coefficients, Bull. Pure Appl. Sci. 19 E [14] N.B. Kerimov, Z.S. Aliev, On the basis property of the system of eigenfunctions of a spectral problem with spectral parameter in a boundary condition, Differ. Uravn [15] E. Kreyszig, Introductory Functional Analysis with Application, New-York, [16] O.A. Ladyzhenskaia, The Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York, [17] A.V. Likov, Yu.A. Mikhailov, The Theory Of Heat And Mass Transfer, Qosenergaizdat, 1963 in Russian. [18] K.R. Mamedov, F. A. Cetinkaya, Boundary value problem for a Sturm-Liouville operator with piecewise continuous coefficient, Hacettepe J. Math. Stat [19] Kh.R. Mamedov, On a basic problem for a second order differential equation with a discontinuous coefficient and a spectral parameter in the boundary conditions, Proc. Seventh Internat. Conf. Geometry, Integrability and Quantization, Institute of Biophysics and Biomedical Engineering Bulgarian Academy of Sciences 006 pp [0] O.Sh. Mukhtarov, H. Olǧar, K. Aydemir, Resolvent operator and spectrum of new type boundary value problems, Filomat [1] O.Sh. Mukhtarov, M. Kandemir, Asymptotic behaviour of eigenvalues for the discontinuous boundary-value problem with functional-transmissin conditions, Acta Math. Scientia B [] O.Sh. Mukhtarov, S. Yakubov, Problems for ordinary differential equations with transmission conditions, Appl. Anal [3] A.N. Tikhonov, A.A. Samarskii, Equations Of Mathematical Physics, Oxford and New York, Pergamon, [4] E.C. Titchmarsh, Eigenfunctions Expansion Associated with Second Order Differential Equations I, second ed., Oxford Univ. Press, London, 196. [5] N.N. Voitovich, B.Z. Katsenelbaum, A.N. Sivov, Generalized Method Of Eigen-Vibration In The Theory Of Diffraction, Nakua, Moscow, 1997 in Russian. [6] J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Z

10 H. Olǧar et al. / Filomat 3:3 018, [7] A. Wang, J. Sun, A. Zettl, Two interval Sturm-Liouville operators in modified Hilbert spaces, J. Math. Anal. Appl [8] A. Wang, J. Sun, X. Hao, S. Yao, Completeness of eigenfunctions of Sturm-Liouville problems with transmission conditions, Methods Appl. Anal [9] P.E. Zhidkov, Riesz basis property of the system of eigenfunctions for a nonlinear problem of Sturm-Liouville type, Sbornik Math

NONLOCAL STURM-LIOUVILLE PROBLEMS WITH INTEGRAL TERMS IN THE BOUNDARY CONDITIONS

Electronic Journal of Differential Equations, Vol. 2017 (2017, No. 11, pp. 1 12. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLOCAL STURM-LIOUVILLE PROBLEMS WITH INTEGRAL