SOLVABILITY OF MULTIPOINT DIFFERENTIAL OPERATORS OF FIRST ORDER

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1 Electronic Journal of Differential Equations, Vol , No. 36, pp ISSN: URL: or ftp ejde.math.txstate.edu SOLVABILITY OF MULTIPOINT DIFFERENTIAL OPERATORS OF FIRST ORDER ZAMEDDIN I. ISMAILOV, PEMBE IPEK In memory of acad. M. G. Gasymov Abstract. Based on methods of operator theory, we describe all boundedly solvable extensions of the minimal operator generated by a linear multipoint functional differential-operator expression of first order, in a direct sum of Hilbert space of vector-functions. We also study the structure of spectrum of these extensions. 1. Introduction The general theory of extension of densely defined linear operator in Hilbert spaces was started by von Neumann with his important work [10] in Later in 1949 and 1952 Vishik in [15, 16] studied the boundedly compact, regular and normal invertible extensions of any unbounded linear densely defined operator in Hilbert spaces Generalization of these results to the nonlinear and complete additive Hausdorff topological spaces in abstract terms have been done by Kokebaev, Otelbaev and Synybekov in [7, 12]. By Dezin [2] another approach to the description of regular extensions for some classes of linear differential operators in Hilbert spaces of vector-functions at finite interval has been offered. On other hand the role of the two point and multipoint theory of functional differential equations in our lives is indisputable. The general theory of delay differential equations is presented in many books for example [4, 14]. Applications of this theory can be found in economy, biology, control theory, electrodinamics, chemistry, ecology, epidemiology, tumor growth, neural networks and etc. see [3, 11]. In addition note that oscillation and boundness properties of solutions of different classes of dynamic equations have been investigated in the book by Agarwal, Bohner and Li [1]. Let us remember that an operator S : DS H H on Hilbert spaces is called boundedly invertible, if S is one-to-one, SDS = H and S 1 LH. The main goal of this work is to describe all boundedly solvable extensions of the minimal operator generated by linear multipoint functional differential-operator expression for first order in the direct sum of Hilbert spaces of vector-functions at 2000 Mathematics Subject Classification. 47A10. Key words and phrases. Differential operators; solvable extension; spectrum. c 2015 Texas State University - San Marcos. Submitted January 2, Published February 10,

2 2 Z. I. ISMAILOV, P. IPEK EJDE-2015/36 finite intervals and investigate the structure of spectrum of these extensions. In the second section all boundedly solvable extensions of the above mentioned minimal operator are described. Structure of spectrum of these extensions is studied in third section Finally, the obtained results are illustrated by applications. 2. Description of Boundedly Solvable Extensions Throughout this work H n is a separable Hilbert space, H n = L 2 H n, n, n =, b n R for each n 1 with property < inf < sup b n <, inf n > 0 and H = n=1h n. We consider the linear multipoint functional differential-operator expression of first order in H of the form where: lu = l n u n, u = u n, 1 l n u n = u nt + A n tuα n t, n 1; 2 operator-function A n : [, b n ] LH n, n 1 is continuous on the uniformly operator topology and sup sup t n A n t < ; 3 For any n 1, α n : [, b n ] [, b n ] is invertible and α n, αn 1 C[, b n ], also sup αn 1 t 1/2 < By standard way the minimal L n0 and maximal L n operators corresponding to the differential expression l n in H n can be defined for any n 1. It is clear that for every n 1 domains of minimal L n0 and maximal L n operators in H n are in the forms DL n0 = o W 1 2 H n, n, and DL n = W 1 2 H n, n, respectively. For any scalar function ϕ : [, b n ] [, b n ] and n 1 now define an operator P ϕ in H n in the form P ϕ u n t = u n ϕt, u n H n. If a function ϕ C 1 [, b n ] and ϕ t > 0 < 0 for t [, b n ], then for any u n H n, P ϕ u n 2 H n = = ϕbn ϕ ϕbn u n ϕt 2 H n dt ϕ u n ϕx 2 H n ϕ 1 xdx u n x 2 H n ϕ 1 x dx ϕ 1 u n x 2 H n dx = ϕ 1 u n 2 H n Consequently, for any strictly monotone function ϕ C 1 [, b n ], the operator P ϕ belongs to LH n and P ϕ ϕ 1. In actually, the expression l in H can be written in the form lu = u t + A α tut, 2.1

3 EJDE-2015/36 SOLVABILITY OF MULTIPOINT DIFFERENTIAL OPERATORS 3 where u = u n, A 1 t At = A 2 t A n t P α1 P α2 0, P α =... 0 P αn and A α t = AtP α, A α LH. The operators L 0 M 0 and LM be a minimal and maximal operators corresponding to 2.1 with m = d dt in H, respectively. Let us define L 0 = M 0 + A α t, L = M + A α t, L 0 : DL 0 H H, L : DL H H, DL 0 = u n H : u n W o 1 2 H n, n, n 1, L n0 u n 2 H n <, DL = u n H : u n W 1 2 H n, n, n 1, n=1 L n u n 2 H n <, The main goal in this section is to describe all boundedly solvable extensions of the minimal operator L 0 in H in terms in the boundary values. Before that, we prove the validity the following assertion. Lemma 2.1. The kernel and image sets of L 0 in H satisfy ker L 0 = 0 and ImL 0 H. Proof. Firstly, we prove that for any n 1 ker L n0 = 0. Consider the boundary values problems u nt + A n tp αn u n t = 0, n=1 u n = u n b n = 0, n 1 The general solution of these differential equations are t u n t = exp A n sp αn ds f n, n 1 Then from boundary value conditions we have f n = 0 for n 1. Hence ker L n0 = 0 for n 1. Then ker L 0 = 0. To show that ImL 0 H it is sufficient to show that ImL n0 H n for some n 1. So we consider the boundary value problem L n0u n = u nt + A n tp αn u n t = 0, The solutions of this equation are t u n t = exp A n sp αn ds g n, u n H n g n H n Consequently, H n ker L n0, n 1 This means that for any n 1 ImL n0 H n. Then ImL 0 H.

4 4 Z. I. ISMAILOV, P. IPEK EJDE-2015/36 Theorem 2.2. If L is any extension of L 0 in H, then L = n=1 L n, where L n is a extension of L n0, n 1, in H n. Proof. Indeed, in this case for any n 1 the linear manifold contains DL n0. That is, Then the operator M n := u n DL n : u n D L DL n0 M n DL n, n 1 L n u n = l n u n, u n M n, n 1 is an extensions of L n0. Consequently, for any n 1, From this, we obtain L n : D L n = M n H n H n L = n=1 L n. For the boundedly solvable extensions of L 0 and L n0, n 1, the following statement holds. Theorem 2.3. For the boundedly solvability of any extension L = L n=1 n of the minimal operator L 0, the necessary and sufficient conditions are the boundedly solvability of the coordinate operators L n of the minimal operator L n0, n 1 and sup L 1 n <. Proof. It is clear that the extension L is one-to-one operator in H if and only if the all coordinate extensions L n, n 1 of L are one-to-one operators in H n, n 1. On the other hand for the boundedness of L 1 = 1 n=1 L n of H the necessary and sufficient condition is sup L 1 n < see [5, 9]. Now let U n t, s, t, s n, n 1, be the family of evolution operators corresponding to the homogeneous differential equation U n t t, sf + A ntp αn U n t, sf = 0, with the boundary condition U n s, sf = f, f H n. t, s n The operator U n t, s, t, s n, n 1 is linear continuous boundedly invertible in H n with the property see [8] U 1 n t, s = U n s, t, s, t n. Lets us introduce the operator U n : H n H n as U n z n t =: U n t, 0z n t, t n. It is clear that if L n is any extension of the minimal operator L n0, that is, L n0 L n L n, then Un 1 L n0 U n = M n0, M n0 Un 1 L n U n = M n M n, Un 1 L n U n = M n

5 EJDE-2015/36 SOLVABILITY OF MULTIPOINT DIFFERENTIAL OPERATORS 5 In addition, for any n 1, and U n = exp b n exp t A n sp αn ds A n s P αn ds exp n P αn sup A n t t n t Un 1 = exp A n sp αn ds a n exp n P αn sup A n t t n < <. Theorem 2.4. Let n 1. Each boundedly solvable extension L n of the minimal operator L n0 in H n is generated by the differential-operator expression l n and the boundary condition + E n u n = U n, b n u n b n, 2.2 where LH n and E n : H n H n is identity operator. The operator is determined by the extension L n uniquely, i.e. L n = L Kn. On the contrary, the restriction of the maximal operator L n in H n to the linear manifold of vector-functions satisfy the condition 2.2 for some bounded operator LH n, is a boundedly solvable extension of the minimal operator L n0. On other hand, for n 1 L 1 n n 1/2 exp 2 n P αn sup A n t t n Proof. For any n 1, the description the all boundedly solvable extensions of L n of the minimal operator L n0 in H n, n 1 have been given in work [6]. From the relation Un 1 L Kn U n = M Kn we obtain L 1 = U n M 1 Un 1, where M Kn u n t = u nt with the boundary condition Hence for f n L 2 H n, n, M 1 f n 2 H n = 2 + E n u n = u n b n, n f n s ds + t f n s ds 2 dt + 2 f n s ds 2 dt f n s 2 ds n dt b n f n s 2 ds dt n t f n s ds 2 dt

6 6 Z. I. ISMAILOV, P. IPEK EJDE-2015/36 = 2 2 n n 2 f n 2 H n = 2 n f n 2 H n. Hence, M 1 2 n /2, n 1 From this and the properties of evolution operators, the validity of inequality is clear. This completes proof of theorem. Theorem 2.5. Let L Kn be a boundedly solvable extension of L n0 in H n and M Kn = U 1 n L Kn U n, for n 1. To have sup L 1 <, the necessary and sufficient condition is Theorem 2.6. Assumed that sup M 1 <. M Kn : W 1 2 H n, n L 2 H n, n L 2 H n, n, M Kn u n t = u nt, + E n u n 0 = u n 1 To have sup M 1 <, the necessary and sufficient condition is sup <. Proof. Indeed, from the proof of Theorem 2.4, M 1 2 n /2, n 1 From this, if sup <, then sup M 1 <. assumed that sup M 1 <. Then in the relation f n t dt = M 1 f n t t f n t dt, f n H n, n 1 choosing the functions f n t = f n, t n, f n H n, n 1, we have On the contrary, fn Hn n M 1 fn Hn + n fn Hn, n 1. Then 1 1 MK n n + 1. Consequently, from the above relation and the condition on n, n 1, we obtain 1 sup inf n sup M 1 K n + 1 <. Now using Theorems , we formulate an assertion on the description of all boundedly solvable extensions of to in H.

7 EJDE-2015/36 SOLVABILITY OF MULTIPOINT DIFFERENTIAL OPERATORS 7 Theorem 2.7. Each boundedly solvable extension L of the minimal operator L 0 in H is generated by differential-operator expression 2.1 and boundary conditions + E n u n = U n, b n u n b n, n 1, where LH n, K = n=1 L n=1 H n and En : H n H n is identity operator. The operator K is determined by the extension L uniquely, i.e. L = L K and vice versa. Remark 2.8. If in the 2.1, α n t = t α n t < t, t [, b n ] for any n 1, then this problem corresponds to the problem of theory of multipoint ordinary delay differential operators in Hilbert spaces of vector-functions. 3. Structure of spectrum of boundedly solvable extensions In this section the structure of spectrum of boundedly solvable extensions of minimal operator L 0 in H is investigated. First we consider the spectrum for the boundedly solvable extension L K, K = of the minimal operator L 0 in H; that is, L K u = λu + f, λ C, u = u n, f = f n H From it follows that n=1l Kn λe n u n = f n The last relation is equivalent to the equations That is, for any n 1, we have Therefore, L Kn λe n u n = f n, n 1, λ C, f n H n U n M Kn λe n Un 1 u n = f n σ p L Kn = σ p M n, σ c L Kn = σ c M n, σ r L Kn = σ r M n 3.1 Consequently, we consider the the spectrum parts of M Kn ; that is, Then from this we obtain M Kn u n = λu n + f n, λ C, f n H n, n 1 u n = λu n + f n, + E n u n = u n b n, n 1 Since the general solution of the above differential equation in H n has the form u n t, λ = expλt f 0 n + t expλt sf n s ds, t n, f 0 n H n, n 1, from the boundary condition it is obtained that E n + 1 expλ n f 0 n = expλb n sf n s ds, n 1 It is easy to show that λ n,m = 2mπi n ρm Kn, m Z, n 1. Then for λ n,m 2mπi/ n, m Z, n 1, we have 1 expλ n 1 E n f 0 n

8 8 Z. I. ISMAILOV, P. IPEK EJDE-2015/36 = 1 expλ n 1 Kn expλb n sf n s ds, f 0 n H n, f n H n, n 1. From this and relations 3.1 it follows the validity of following statement. Theorem 3.1. In order for λ to belong to σ p L Kn σ c L Kn, σ r L Kn, it is necessary and sufficient that 1 µ = expλ n 1 σ p σ c, σ r. Then a structure of spectrum of boundedly solvable extension L Kn can be formulated in the form Theorem 3.2. The point spectrum of the boundedly solvable extension L Kn has the form σ p L Kn = λ C : λ = 1 µ + 1 ln n µ + i argµ + 1 µ + 2mπi, µ σ p \0, 1, m Z Similarly propositions on the continuous σ c L Kn and residual σ r L Kn spectrums are true. Lastly, using the results on the spectrum parts of direct sum of operators in the direct sum of Hilbert spaces [13] in we can proved the following theorem. Theorem 3.3. For the parts of spectrum of the boundedly solvable extension L K = n=1l Kn, K = in Hilbert spaces H = n=1h n the following statements are true σ p L K = n=1σ p L Kn, c c σ c L K = n=1 σ p L Kn n=1 σ r L Kn λ n=1ρl Kn : sup R λ L Kn =, σ r L K = n=1 σ p L Kn c 4. Applications n=1 σ r L Kn In this section, we present an application of above results. n=1 σ c L Kn Example 4.1. For any n 1 let us H n = C,, = 0, b n = 1, A n t = c n, c n C, sup c n <, α n t = α n t, 0 < α n < 1 with property sup 1 α n <. Consider the pantograph type delay differential expression lu = u nt + c n u n α n t in H = n=1h n, where H n = L 2 0, 1. In this case by Theorem 2.7, all boundedly solvable extensions L k of the minimal operator L 0 generated by l in H are described by the differential expression l and the boundary conditions k n + 1u n 0 = k n U n 0, 1u n 1, where k n C, n 1, sup k n < and vice versa.

9 EJDE-2015/36 SOLVABILITY OF MULTIPOINT DIFFERENTIAL OPERATORS 9 On other hand in the case k n / 0, 1, for any n 1 the point, the continuous and residual spectrums of L kn in H n = L 2 0, 1 is of the form σ p L kn = λ C : λ = ln k n i arg k n mπi, m Z, k n k n σ c L kn = σ r L kn =. Hence by Theorem 3.3 spectrum parts of L k = n=1l kn has the form σ p L k = n=1 m Z ln k n i arg k n mπi, k n k n σ c L k = λ n=1ρl kn : sup R λ L kn =, σ r L k = Example 4.2. Let H n = C,,, a sequence of real numbers, sup <, b n, b n = + 1, n =, b n, H n = L 2 H n, n, l n u n = u n + u n α n t, α n t = t , t n, n 1, H = n=1h n, l = n=1l n. In this case for any n 1 the function α n is increase, invertible and αn 1 t = + t and α 1 n t 1 =. 2 t Hence sup αn In this case all boundedly solvable extensions of minimal operator L 0 in H are described by l and boundary conditions k n + 1u n = k n U n, b n u n b n, n 1, where sup k n <. On the other hand for k n / 0, 1 spectrum of each boundedly solvable extension L kn has the form σ p L kn = λ C : λ = ln k n i arg k n mπi, m Z, k n k n σ c L kn = σ r L kn =, n 1 Hence by Theorem 3.3, the spectrum parts of L k = n=1l kn have the form σ p L k = k n + 1 n=1 m Z ln + i arg k n mπi, k n k n σ c L k = λ n=1ρl kn : sup R λ L kn =, σ r L k = Remark 4.3. Similar to the problems in Example 4.1 and 4.2, we can investigate the case α n t = b n t, t b n, n 1. Acknowledgements. The authors are grateful to G. Ismailov Undergraduate Student of Marmara University, Istanbul for his helps in preparing the English version of this article and for the technical discussions. References [1] R. Agarwal, M. Bohner, W. T. Li; Nonoscillation and Oscillation Theory for Functional Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., [2] A. A. Dezin; General Problems in the Theory of Boundary Value Problems, Nauka Moscow, 1980 in Russian.

10 10 Z. I. ISMAILOV, P. IPEK EJDE-2015/36 [3] L. Edelstein-Keshet; Mathematical Models in Biology, McGraw-Hill, New York, [4] T. Erneux; Applied Delay Differential Equations, Springer-Verlag, [5] Z. I. Ismailov, E. Otkun Çevik, E. Unlüyol; Compact inverses of multipoint normal differential operators for first order, Electronic Journal of Differential Equations, 2011, 1-11, [6] Z. I. Ismailov, E. Otkun Çevik, B. O. Güler, P. Ipek; Structure of spectrum of solvable pantograph differential operators for the first order, AIP Conf. Proc., 1611, 89-94, [7] B. K. Kokebaev, M. Otelbaev, A. N. Shynybekov; On questions of extension and restriction of operator, English translation: Soviet Math. Dokl., 28, 1, , [8] S. G. Krein; Linear Differential Equations in Banach Space, emphtranslations of Mathematical Monographs, 29, American Mathematical Society, Providence, RI, [9] M. A. Naimark, S. V. Fomin; Continuous direct sums of Hilbert spaces and some of their applications, Uspehi Mat. Nauk, 10, 264, in Russian. [10] J. von Neumann; Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Math. Ann., 102, p , [11] J. R. Ockendon, A. B. Tayler; The dynamics of a current collection system for an electric locomotive, Proc. Roy. Soc. London Ser. A, 322, , [12] M. Otelbaev, A. N. Shynybekov; Well-posed problems of Bitsadze-Samarskii type, English translation: Soviet Math. Dokl., 26, 1, [13] E. Otkun Çevik, Z. I. Ismailov; Spectrum of the direct sum of operators, Electronic Journal of Differential Equations, 2012, 1-8, [14] H. Smith; An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer-Verlag, [15] M. I. Vishik; On linear boundary problems for differential equations, Doklady Akad. Nauk SSSR N.S. 65, , [16] M. I. Vishik; On general boundary problems for elliptic differential equations, Amer. Math. Soc. Transl. II, 24, , Zameddin I. Ismailov Karadeniz Technical University, Department of Mathematics, 61080, Trabzon, Turkey address: zameddin.ismailov@gmail.com Pembe Ipek Karadeniz Technical University, Institute of Natural Sciences, Trabzon, Turkey address: pembeipek@hotmail.com

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