z k Γ(µ + kρ 1 ) (ρ > 0) k=0

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1 Distribution of Zeros of Mittag-Leffler Function and Relevant Spectrum Analysis T.S. Aleroev The Academy of National Economy under the Government of the Russian Federation, Moscow, 957, Russia H.T. Aleroeva MTUCI, Moscow Technical University of Communications and Informatics, Moscow, 24, Russia Qiang Sun, Yifa Tang and Ruili Zhang LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 9 sunqiang@lsec.cc.ac.cn, tyf@lsec.cc.ac.cn, rlzhang@lsec.cc.ac.cn Abstract: This paper is devoted to the distribution of eigenvalues of operators generated by differential expressions of fractional order with Sturm- Liouville regional conditions. In particular it offers a way of estimating the first eigenvalues. Keywords: Eigenvalue of Operator; Fractional Derivative; Mittag-Leffler Function; Spectrum Analysis.. Introduction. It is well known that the Mittag-Leffler function [] E ρ (z, µ) = k= z k Γ(µ + kρ ) (ρ > ) has appeared in the complex function theory. For example, E ρ (z, ) is a generalization of the function e z, and more generally, E ρ (z, µ) is used in studying asymptotic properties of the whole functions in integral transformations. Undoubtedly, this function plays a key role in fractional calculation theory. Great interest has been drawn by a problem on the distribution of zeros of function E ρ (z, µ). The present paper is mainly devoted to this problem. It is necessary to note that many authors have been devoted to studying zeros of Mittag- Leffler function (see [2] and references therein), and the basic works in this direction are due to A. Veeman and M.M. Dzhrbashchjan (see [3-4]).

2 Their techniques remain the basic tools for research of zeros of the function E ρ (z, µ). In present work, as well as in works [5-8], zeros of function E ρ (z, µ) are studied by the methods based on fractional calculation. Here again we result only those results, which can be compared with results of other authors. 2. Basic concepts. The spectral analysis of operators of the form A [α,β] γ u(x) = c α (x t) α u(t)dt + c β,γ x β ( t) γ u(t)dt was carried out in []. Here α, β, γ, c α, c β,γ are real numbers, and α, β, γ are positive. These operators arise in the study of boundary value problems for differential equations of fractional order (see [5] and references therein, where the corresponding Green functions are constructed). The given paper is also devoted to studying the boundary value problems for the differential equations of fractional order and integrated operators of the form A [α;β] γ. In order to describe the problem in detail we must mention some basic concepts from fractional calculus. Let f(x) L (, ). Then, the function d α f(x) dx α Γ(α) (x t) α f(t)dt L (, ) is called the fractional integral of order α > with starting point x =, and the function d α f(x) d( x) α Γ(α) x (t x) α f(t)dt L (, ) is called the fractional integral of order α > with ending point x = (refer to [5]). Here Γ(α) is Euler s Gamma-function. It is clear that when α =, we identify both the fractional integrals with the function f(x). As we know (see [5]), the function g(x) L (, ) is called the fractional derivative of the function f(x) L (, ) of order α > with starting point x =, if f(x) = d α d dx g(x). For α α dx, it s representing fractional α integral when α <, while representing fractional derivative when α >. d The fractional derivative α d( x) of the function f(x) L α (, ) of order α > with the ending point x = is defined in a similar way. 2

3 Let {γ k } n be any set of real numbers, satisfying the condition < γ j ( j n). Assume that k σ k = γ j ; µ k = σ k + = k j= j= γ j ( k n), and n ρ = γ j = σ n = µ n >. j= According to M.M. Dzhrbashchjan (see [5]), we consider the integral-differential operators D (σ) f(x) d ( γ) f(x), dx ( γ) D (σ) f(x) d ( γ) d γ f(x), dx ( γ) γ dx d γ D (σ2) f(x) d ( γ2) d γ f(x), dx ( γ2) dx γ γ dx D (σn) f(x) d ( γn) d γn dx ( γn) dx d γ f(x). γn γ dx If γ = γ = = γ n =, then we are able to obtain D (σ k) f(x) = f (k) (x) (k =,, 2,, n). The object of our research will focus on regional problems for an equation of the following kind: D (σn) u [ λ + q(x) ] u =, < σ n <, () u + Dx α u + q(x)u = λu, < α <. (2) Let s consider a problem: To find a nontrivial solution in L 2 (, ) for the following equation which satisfies the condition u λ dα u dx α = ( < α < ), (3) u() =, u() =. (4) This problem was object of research of many authors (see [9] and references therein). Therefore we also will begin our researches from this problem. 3

4 Theorem. Problem (-2) has no eigenvalues in a circle with radius Γ(4 α) whose center is in the beginning of coordinates. Proof. It is known that a problem () - (2) is equivalent to the integral equation [5] λ u(x) x( ξ) α u(ξ)dξ (x ξ) α u(ξ)dξ =. Γ(2 α) λ is an eigenvalue of the above integral equation, iff λ is a zero of function E 2 α ( λ; 2). Assume that Au = Au + Au, where and A u = Γ(2 α) A u = Γ(2 α) x( ξ) α u(ξ)dξ (x ξ) α u(ξ)dξ. It is obviously that A, A H (i.e. are nuclear). Therefore we are able to obtain sp(a + A ) = spa + spa. It is easy to verify that spa =. Assume that spa = λ. The following work will contribute to determining λ. The characteristic number of equation u λ Γ(2 α) x( ξ) α u(ξ)dξ =, is uniquely determined by the Fredholm determinant where K = Hence we are able to obtain that d(λ) = λk, ξ( ξ) α dξ = Γ(2 α) λ = Γ(4 α). Γ(4 α). 4

5 Thus we have A H spa = All values λ satisfying the inequality Γ(4 α). λ Γ(4 α) A are regular. This completes theorem. Consequence. Function E ( λ; 2) have no zeros inside circle 2 α with radius Γ(4 α) whose center is in beginning of co-ordinates. Remark. This result may be obtained by another way [9] in the case < α < and from the proof of theorem, it follows that this result is true for all α <. Displacement operators of kind A [α,β] γ have been entered in [] by means of parametrization. Certainly, similar results can be obtained for other operators of kind A [α,β] γ. For example, let s consider operator x Au = Γ(ρ (x t) ρ u(t)dt x ρ ( t) ρ u(t)dt. ) This operator has been studied for the first time in [5-8], in connection with the research of a point-to-point regional problem for equation d Γ( α) dx u() =, u() = (3 ) u (t) (x t) α dt; λu =, < α <. (4 ) Theorem 2. Problem (3-4 ) has no characteristic numbers inside circle Γ(2 α) with radius whose center is in beginning of co-ordinates. Γ(4 2α) Proof. Proof of theorem 2 is similar to the proof of theorem. And λ is an eigenvalue of a problem (3-4 ) iff λ is a zero of function E 2 α. Consequence 2. Function E have no zeros inside circle with radius 2 α Γ(2 α) whose center is in beginning of co-ordinates. Γ(4 2α) 5

6 As shown in [] similar results can be obtained for any Mittag-Leffler function by means of parametrization of powers of operators A [α,β] γ. Let s note that theorems of kind, 2 play an exclusive role in the theory of quite continuous vector fields (where it is necessary to know, whether is unit eigenvalue of the objective operator) [9]. Further more it is shown in [] that a kernel of the integral operator A [ρ,ρ] x ρ u = Γ(ρ (x t) ρ u(t)dt x ρ ( t) ρ u(t)dt, ) where < ρ < 2 is non-negative. For ρ <, function E ρ (λ; ρ ) has at least one material zero. It is also established that operator A [ρ,ρ] x ρ u = Γ(ρ (x t) ρ u(t)dt x ρ ( t) ρ u(t)dt ) is simple when ρ >, from which it follows that all zeros of function are complex. Now we will allocate an area condition in the complex plane where the Mittag-Leffler type functions have no zero. In fact, theorem 3 which we will formulate is only a consequence of sectorial property of operator A [ρ,ρ] ρ. The sectorial property is very useful for the proof of completness for the system of eigenfunctions. It has been told in paper [] that theory developed in [9] can be enclosed to operator A [α,β] γ u(x) = c α (x t) α u(t)dt + c β,γ x β ( t) γ u(t)dt For example: Let K be a cone of non-negative functions in L 2. Definition: A) Operator A is called u -positive, if there exists such element u K(u ) that, for any v K(v ), it is possible to specify numbers α(v), β(v) >, for which we have α(v)u Av β(v)u. B) Operator A is called u -limited, if Av β(v)u. It is shown in paper [6] that operator 6

7 A [ρ,ρ] ρ u = Γ(ρ ) is u -positive, where x ρ ( t) ρ u(t)dt u (t) = t/ρ ( t) Γ(ρ. ) For ρ = 2, Simple calculations show that (x t) ρ u(t)dt Au (t) Γ(ρ ) u (t), A 2 u (t) 2Γ(ρ ) Au (t). For ρ > 2, operator A has been studied by our student Erokhin [7]. Theorem 3. Let λ be the first eigenvalue of operator A (ρ > 2 ). Then we have the following estimation for λ : B(α + 2, α + 2) λ αb(α +, 2) Proof. Our operator is u -positive. To find such u, it is enough to calculate an integral [9] K(t, s)ds. Simple calculations show that u (t) may take function t α ( t). In order to find α(x) and β(x), we shall first find the representation of such kind (see [9]): u (t)b (s) K(t, s) u (t)b 2 (s). Using the property of kernel K(t, s) in [6-7], we obtain the above expression (*), where b (s) = s( s) α and b 2 (s) = α. Applying the property mentioned in [9], we obtain α(u ) = β(u ) = b (t)u (t)dt = B(α + 2, α + 2), b 2 (t)u (t)dt = αb(α +, 2) ( ) 7

8 v 5 6 v v This completes the proof of theorem 3. To specify more accurate estimations we shall calculate A n u (t) Hence we are able to obtain series of estimations of spectral radius according to theorem 5.5 in [9]. So the borders for the zeros of corresponding Mittag-Leffler function can be easily determined. α α α v v v α v 2 v 2 v 4 3 α+ α+ α 3 α+ 2 α+ α 3 α+ 2 α α+ α α v 7 8 α α+ α 3 α+ 2 α α+ α α+ 3 α α+ α 3 α+ 2 α α+ α 5 α+ 4 M M M Now we give the general formulation of c n+ for arbitrary n. Assume 8

9 that ρ 2.The kernel of operator A: x ρ ( t) ρ (x t) ρ, t x K(x, t) = x ρ ( t) ρ, x t A is µ -positive, where µ (t) = t ρ ( t). The following work will contribute to the calculation of A n µ (t) for the purpose of estimation of the spectrum radius of operator A. Assume ρ = α. µ (t) = t ρ t ρ, and A n µ (t) = A n (t α ) A n (t α+ ). Next we only need to consider A n (t α ). A(t α ) = B(α +, α + )x α B(α +, α + )x α+ x α. For simplicity, we denote B(m, α + ) by B(m). Thus we have A(t α ) = B(α + )x α B(α + )x α+ x α. A 2 (t α ) = B(α + )B(α + )x α B(α + )B(α + )x 2α+ B(α + )B(2α + 2)x α +B(α + )B(2α + 2)x 3α+2. There is a full binary tree configuration connected with polynomial A n (t α ). Further more, a full binary tree contains n + layers (depth) corresponding to the operator A n. We assign every vertex v(ij) (where i n + and j 2 i ) a value w (v(i, j)) according to the following two rules: ()For all i n+, if j 2 i and j is odd, we have w (v(i, j)) = α. (2)For all i n+, if j 2 i and j is even, we have w (v(i, j)) = w ( v ( i, j 2)) + α +. There exists one and only one path from the root of the full binary tree to one leaf of it, and we denote the path by P (v(, ), v(n +, j)), where j 2 n. Now we will give a method to obtain the only path for arbitrary leaf v(n +, j), where j 2 n. The path is denoted by v (, t()) v (2, t(2)) v (3, t(3)) v (n +, t(n + )), where v (, t()) = v(, ), v (n +, t(n + )) = v(n +, j): ( ()If j mod 2 =, v (n, t(n)) = v n, t(n+) 2 ) = v ( n, 2) j ; If j mod 2 =, v(n, t(n)) = v(n, j+ 2 ); (2)Assume t(k) = p. If p mod 2 =, v(k, t(k )) = v(k, t(k) 2 ) = v(k, p p+ 2 ); If p mod 2 =, v(k, t(k )) = v(k, 2 ); 9

10 Thus we obtain the path the full binary tree P(v(, ), v(n +, j)). Assume S(j) = {v(i, t(i)) t(i) mod 2 = and i n + }, Card(S(j)) represents the number of set S(j). So we can represent the polynomial A n (t α ) as follows A n (t α ) = B (w (v(i, t(i)) + ))t w(v(n+,j)). Hence we have j 2 n ( ) Card(S(j)) A n (t α )dt = j 2 n i n ( ) Card(S(j)) So for all n, we have c n+ = α Γ(α+) A n (t α t α+ )dt = α Γ(α+) w(v(,))=α j 2 n i n ( ) Card(S(j)) w (v(n +, j)) + B (w (v(i, t(i))) + ) i n w (v(n +, j)) + B (w (v(i, t(i))) + ) w(v(,))=α+ ( )Card(S(j)) i n B (w (v(i, t(i))) + ) j 2 w (v(n +, j)) + n Theorem 4. All zeros of function E ρ (z; ρ ) lies in angle arg z π(2ρ ). 2ρ Proof. It is known that number λ is an eigenvalue of a problem Su = [ d x u ] (t)dt q(x) u = λu Γ(α) dx (x t) α u() =, u() = iff λ is a zero of function E (z; 2 α). Let s show that operator S is 2 α sectorial and all values of form Re(Su, u) lies in angle argλ (SU, u) = Γ( γ ) = [ d dx [ x Γ( γ ) x u ] (t) dt q(x) ū(x)dx γ (x t) u (t) dt γ (x t) ] ū (x)dx Γ( γ ). ]. π(2ρ ). 2ρ q(x)u(x)ū(x)dx.

11 Theorem of Matsaev-Polant. Let A is a dissipative operator. Then, values of form (A v f; f), ( v ) lies in angle argλ vπ. It has been shown in [8] that we are able to allocate an area condition in the complex plane in a similar way where there is no zero for a wide class of functions E ρ (z; µ). Theorem 5. All zeros of function E ρ (z; µ ), ( < ρ < ) are simple. Proof. λ is an eigenvalue of operator A [ρ,ρ] ρ iff λ is a zero of function E ρ (z; µ ). Thus it is necessary to study the spectrum of operator A [ρ,ρ] ρ. Let λ n (ρ) be the n-th eigenvalue of operator A. U ρ is a limited area with straightened border du ρ such that λ n (ρ) U ρ and (σ(a ρ )λ n (ρ)) U ρ = Ø. Assume that P λn(ρ)(a ρ ) = R λ (A ρ )dλ. 2πi du ρ So P λn(ρ) is a Riss projector for operator A ρ corresponding to eigenvalue λ n (ρ). Let s show that P ρ λ n continuously depends on parameter ρ. It is known that operators of fractional integration (J α f)(x) = Γ(α) (x t) α f(t)dt form in L p (, )(p ) half-group, continuous in uniform topology for all α > and strongly continuous for all α. As A ρ is continuous in uniform (operational) topology(i.e. A ρ A ρ for ρ ρ ), for sufficiently close values ρ, ρ, p ρ λ n p ρ λ n <. Therefore, according to theorem of B.S. Nadyja [], p ρ λ n L 2 (, ) and p ρ λ n L 2 (, ) have equal dimension. So we have that for all dimp /2 λ n L 2 (, ) = dimp λn L 2 (, ) =. Hence, all eigenvalues of operator A [ρ,ρ] ρ are simple, i.e. all zeros of E ρ (z; µ ) are simple. To finish the proof of theorem 4, let z, z 2,..z n,..- be the zeros of function E ρ (z; µ ) numbered in ascending order for their modules. As E ρ (z; µ ) - is a function with zeros, it is possible to write out value z k, whose value equals to Γ(α+ρ ). It is known that spa p equals to Γ(α+ρ ). If all eigenvalues of operator A are simple, for they coincide with

12 zeros of function E ρ (z; µ ), so it follows that all its zeros are simple. This completes our theorem. Note that all properties resulted from operators A p can be similarly transferred to any Mittag-Leffler function. Estimation for first eigenvalue of operators A [α,β] ρ has been found in paper [5], so as the first zero of functions E ρ (z, µ). These estimations can be specified as following. Let s consider problem N for q(x) =. Let χ n (x) be an eigenfunction of this problem. According to [8], this system is complete in L 2 (, ). Using symbol χ n (x), let s designate eigenfunctions of operator adjoined to operator B. Thus the bilinear expansion of kernel K(x, t) has the following form χ n (x)χ n K(x, t) = (7) λ n n= Equality (7) is understood in the following sense m χ n χ n lim m K(x, t) = λ n L2 n= And the resolvent R λ can be formulated as follows R λ v = (v, χ n )χ n λ λ n (8) Furthermore, the eigenfunctions of operator A has the following representation ( ) χ n(x) = x ρ E λ n x ρ ;. ρ And the eigenfunctions of operator A v = Γ(ρ ) [ x (t x) ρ u(t)dt ( x) ρ t ρ u(t)dt] can be formulated: ( ) χ n (x) = ( x) ρ E ρ λ n ( x) ρ ; ρ Thus formulas (7) and (8) can be reformulated as K(x, t) = ( ) x ρ Eρ (λ n x /ρ ; ρ )( t) /ρ E ρ λ n ( t) /ρ ; ρ (7 ) λ n n= 2

13 ) ( ) v(t)t ρ E (λ n t /ρ ; ρ ( x) /ρ E ρ λ n ( x) /ρ ; ρ R(v) = (8 ) λ λ n n= Using (7 ) and (8 ), we are able to obtain many useful formulas. 3. Movement of oscillator, being under the influence of elastic force, characteristic for viscoelastic environments. Let s consider an equation studied in [5]. The spectral analysis of a problem of Storm-Liouville type for equation () has been carried out in paper [5]. This equation describes various physical processes [see the book of Uchaikin], in particular movement of oscillator, being under the influence of elastic force, characteristic for viscoelastic environments. It is known that the first regional problem for the equation of fluctuation of a string with an adjustable fractional derivative on time is reduced to equation () of Sturm-Liouville type. Consider the following problem u + D α xu + uλ =, ( ) u() =, u () =. (2 ) It is shown in paper [4] that problem ( -2 ) is equivalent to the following equation [ ] (x t) α u(x) = + λ(x t) u(t)dt + x. Γ(2 α) Assume that (x t) α + λ(x t), t < x <, K (x, t) = Γ(2 α), x < t. According to [], the sequence of kernels K n (x, t) recurrent parities is defined by using K n+ (x, t) = K n (x, t )K (t, t)dt. t By an induction on n, we are able to obtain K n (x, t, λ) = k= c k n λn k (x t)2n kα Γ(αn kα) 3

14 Hereafter the resolvent for our integrated equation has the following formulation: [ n ] c k R(x, t; λ) = K n (x, t, λ) = nλ n k (x t)2n kα. Γ(αn kα) n= n= k= Hence, the solution of our equation can be rewritten as x u(x) = x + R(x, t; λ)tdt = x + n= k= + Theorem 6. λ is an eigenvalue for problem A iff it is a zero of function ω(λ) = + n= k= c n k λn k Γ(2n+4 (k+)α) x2n+3 (k+)α c n k λn+ k Γ(2n+4 kα) x2n+3 kα. n= k= u + D α xu + λu =, u() = ; u() = c k n λn+ k Γ(2n + 4 kα) + Eigenfunctions of problem A is given by χ i (x) = + n= k= n= k= c k nλ n+ k i x 2n+3 (k+)α Γ(2n + 4 (k + )α) + c k n λn k Γ(2n + 4 (k + )α). n= k= (3 ) c k nλ n+ k i x 2n+3 kα Γ(2n + 4 kα) where, λ i - are roots of function ω(λ). Proof of this theorem follows from parity (3 ). Let λ, λ 2,, λ n, be zeros of ω(λ), enumerated in the order of non-decrease modulus. let s show that the first eigenfunction χ (x) have no nodes (i.e. is not converse to zero in an interval (,)). Theorem 7. Function χ (x) have no nodes. Proof. Assume that point x of function χ (x) is converse to zero. Then we have χ i (x ) = + = n= k= c k n λn+ k i x 2n+3 (k+)α Γ(2n+4 (k+)α) + n= k= c k n λn+ k i x 2n+3 kα Γ(2n+4 kα) 4

15 In fact, the system of functions is complete. Proof of this fact can be found in []. Our proof is based on Lidskii s theorem, which states that the system of eigenfunctions is complete in A but not orthogonal. Now we shall consider an adjoined problem. The adjoined problem A in [] can be written as follows: To find the solution of equation z + D α xz + λu = u() = ; u() = The eigenfunction for Problem A can be denoted by Z j (x)(see []). Similarly, it is possible to write out eigenfunctions } Z j (x). It has been shown in [] that systems {χ i (x) i= {z } and j (x) j= are biorthogonal on interval [, ]. References [] T.S. Aleroev, About one class of operators associated with differential equations of fractional order, Siberian Mathematical Journal 46 (25), No. 6, [2] A.M. Nachushev, The fractional calculating and its application, M.: Fiz- matlit (23), 272. [3] M.M. Dzhrbashyan, A boundary value problem for a Sturm-Liouville type differential operator of fractional order (in Russian), Izv. Akad. Nauk Armyan. SSR, Ser. Mat. 5 (97), No. 2, [4] T.S. Aleroev, The boundary problems for differential equations with fractional derivatives, Dissertation, doctor of Physical and Mathematical Sciences, Moscow State Univercity, 2. [5] T.S. Aleroev, H.T. Aleroeva, N.M Nie and Y.F. Tang, Boundary Value Problems for Differential Equations of Fractional Order, Memoirs on Differential Equations and Mathematics Physics 49 (2), [6] T.S. Aleroev and A.I. Aleroev, Vestnik Chech. GPI (29). [7] S.V. Erokhin, Boundary Value Problems for Differential Equations of Fractional Order. Approximation of Inverse Operators by Matrices, Memoirs on Differential Equations and Mathematics Physics 49 (2), 9-9. [8] T.S. Aleroev and H.T. Aleroeva, A Problem on the Zeros of the Mittag- Leffler Function and the Spectrum of a Fractional-Order Differential Operator, EJQTDE (29), No. 25, -8. 5

16 [9] M.A. Krasnosel kii and G.M. Vayinikko, Approximate Solutions of Operator Equations. Nauka, Moscow (969). [] I.C. Gohberg and M.G. Krein, The theory of Volterra operators in Gilbert space and its applications (967). [] A.M. Gachaev, The boundary problems for differential equations of fractional order, Dissertation, Nalchik, Russia, 25. 6