Boichuk, Alexander

Medved, M.

Zhuravliov, Valerii

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Electronic Journal of Qualitative Theory of Differential Equations2015, No. 23, 1–9; http://www.math.u-szeged.hu/ejqtde/Fredholm boundary-value problems for linear delaysystems defined by pairwise permutable matricesAlexander A. BoichukB 1, Milan Medved’*2 andValerii P. Zhuravliov**31Institute of Mathematics of the National Academy of Sciences,3 Tereschenkivska Street, Kyiv, 01601, Ukraine2Department of Mathematical Analysis and Numerical Mathematics,Faculty of Mathematics, Physics and Informatics, Comenius University,Mlynská dolina, Bratislava, 842 48, Slovakia3Department of Higher Mathematics and Applied Mechanics Faculty of Engineering and Technology,Zhitomyr National Agroecological University, Zhitomyr, 7 Bulvar Starij, 10008, UkraineReceived 12 December 2014, appeared 8 May 2015Communicated by Michal FecˇkanAbstract. The paper deals with a Fredholm boundary value problem for a linear de-lay system with several delays defined by pairwise permutable constant matrices. Theinitial value condition is given on a finite interval and the boundary condition is givenby a linear vector functional. A sufficient condition for the existence of solutions ofthis type of boundary value problem is proved. Moreover, a family of linearly indepen-dent solutions in an explicit general analytic form is constructed under the assumptionthat the number of boundary conditions (determined by the dimension of linear vectorfunctional) do not coincide with the number of unknowns of the system of the delay dif-ferential equations. The proof of this result is based on a representation of solutions byusing the so-called multi-delayed matrix exponential and a method of a pseudo-inversematrix of the Moore–Penrose type.Keywords: boundary-value problem, multi-delayed system, Moore–Penrose pseudo-inverse matrix.2010 Mathematics Subject Classification: 34B05, 34B27, 34K10.1 IntroductionThe aim of the paper is to prove an existence result for the following boundary-value problem:z˙(t) = Az(t) + B1z(t− τ1) + · · ·+ Bnz(t− τn) + g(t), t ∈ [0, b],z(s) = ψ(s), if s ∈ [−τ, 0], (1.1)lz(·) = α ∈ Rm, (1.2)BCorresponding author. Email: boichuk.aa@gmail.com*Email: Milan.Medved@fmph.uniba.sk**Email: vfz2008@ukr.net2 A. A. Boichuk, M. Medved’ and V. P. Zhuravliovwhere τ1, . . . , τn > 0, (n > 0), τ := max{τ1, . . . , τn} and A, B1, . . . , Bn are N × N constantpermutable matrices such that ABi = Bi A, BiBj = BjBi for each i, j ∈ {1, . . . , n} and g(t)is an N-dimensional column-vector, with components in the space Lp[0, b] (1 < p < ∞)being functions integrable on [0, b]; ψ : R \ [0, b] → RN is a given N-dimensional column-vector function; α is an m-dimensional constant vector-column, l is an m-dimensional linearvector-functional, defined on the space Dp[0, b] of n-dimensional vector-functions absolutelycontinuous on [0, b]: l = col(l1, . . . , lm) : Dp[0, b] → Rm, li : Dp[0, b] → R. It is not verydifficult to prove that in this space such problems for functional-differential equations are ofFredholm’s type with nonzero index (see, e.g., [1, 4, 5]).First of all we consider initial value problems for a system of linear differential equationswith delays defined by pairwise permutable matrices:z˙(t) = Az(t) + B1z(t− τ1) + · · ·+ Bnz(t− τn) + g(t), t ∈ [0, b],z(s) = ψ(s), if s ∈ [−τ, 0]. (1.3)Using the notations(Shi z)(t) :={z(hi(t)) if hi(t) := t− τi ∈ [0, b],0 if hi(t) := t− τi 6∈ [0, b],(1.4)ψhi(t) :={0 if hi(t) ∈ [0, b],ψ(hi(t)) if hi(t) 6∈ [0, b],(1.5)it is possible to rewrite initial value problems for (1.3) as an operator equation(Lz)(t) := z˙(t)− Az(t)−n∑i=1Bi(Shi z)(t) = ϕ(t), (1.6)where (Shi z)(t) is an N-dimensional column-vector and ϕ(t) is an N-dimensional column-vector defined by the formulaϕ(t) := g(t) +n∑i=1Biψhi(t) ∈ Lp[0, b].The operator Shi : Dp → Lp admits the following representation:(Shi z)(t) =∫ b0χhi(t, s)z˙(s) ds + χhi(t, 0)z(0),where χhi(t, s) is the characteristic function of the setΩ = {(t, s) ∈ [0, b]× [0, b] : 0 ≤ s ≤ hi(t) ≤ b},defined byχhi(t, s) ={1, (t, s) ∈ Ω,0, (t, s) /∈ Ω.We will investigate the equation (1.6) assuming that the operator L maps a Banach spaceDp[0, b] of absolutely continuous functions z : [0, b]→ RN with the norm‖z(t)‖Dp = ‖z˙(t)‖Lp + ‖z(0)‖RNFredholm boundary-value problems for linear delay systems 3into the Banach space Lp[0, b] (1 < p < ∞) of functions ϕ : [0, b] → RN integrable on [0, b],equipped with the standard norms for these spaces. It is well-known [1] that, in the con-sidered spaces, problem (1.6) is equivalent to initial value problem (1.3). The transforma-tions (1.4), (1.5) allow to add the initial function ψ(s), s < 0 to an inhomogeneity and thus togenerate an additive and homogeneous operation not depending on ψ, and without a classicalassumption regarding the continuous connection of solution z(t) with the initial function ψ(t)at the point t = 0. A solution of differential system (1.6) is defined as a vector-function z(t) ∈ Dp[0, b]absolutely continuous on [0, b] with z˙(t) ∈ Lp[0, b], if it satisfies the system (1.6) almost everywhereon [0, b]. Such a treatment makes it possible to apply to the equation (1.6) with the linear andbounded operator L well developed methods of linear functional analysis. It is well-known(see, e.g., [1, 3, 4]) that an inhomogeneous operator equation (1.6) with delayed arguments issolvable for an arbitrary right-hand side ϕ(t) ∈ Lp[0, b] and has an N-dimensional family ofsolutions (dim ker L = N) in the formz(t) = X(t)c +∫ b0K(t, s)ϕ(s) ds, for all c ∈ RN (1.7)where the kernel K(t, s) of the integral is an (N×N)-dimensional Cauchy matrix K(t, s) being,for every fixed s, a solution of the matrix Cauchy problem:(LK( · , s))(t) := ∂K(t, s)∂t− AK(t, s)−n∑i=1Bi(Shi K(·, s))(t) = 0, K(s, s) = I.In the following we assume that the matrix K(t, s) is defined in the square [0, b]×[0, b] andK(t, s)≡ 0 if 0 ≤ t < s ≤ b. A fundamental (n× n)-dimensional matrix for the homogeneous(ϕ(t) ≡ 0) equation (1.6) has the form X(t) = K(t, 0) (see [1]).A disadvantage of this approach, when investigating the above-formulated problem, is thenecessity to find analytically a fundamental X(t) and the Cauchy K(t, s) matrices [5, 7]. Belowwe consider the case of a system with delays, when this problem can be directly solved. In thiscase the problem of how to construct the Cauchy matrix is successfully solved analytically dueto a delayed matrix exponential defined in [6] and generalized to the case of several delays in[8].2 Multi-delay matrix exponentialWe recall the definition of the multi-delay matrix exponential defined in [8].Definition 2.1. Let B1, . . . , Bn be pairwise permutable N × N matrices, i.e., BiBj = BjBi foreach i, j ∈ {1, . . . , n}. For each j = 2, . . . , n we define N × N multi-delayed matrix exponentialcorresponding to delays τj > 0 and matrices B1, . . . , Bj as followseB1,...,Bjtτ1,...,τj :=Θ, if t < −τj,Xj−1(t + τj), if τj ≤ t < 0,...Xj−1(t + τj) + Bj∫ t0 Xj−1(t− s1)Xj−1(s1) ds1+ · · ·+ Bkj∫ t(k−1)τj∫ s1(k−1)τj . . .∫ sk−1(k−1)τj Xj−1(t− s1)×∏k−1i=1 Xj−1(si − si+1)Xj−1(sk − (k− 1)τj) dsk . . . ds1, if (k− 1)τj≤ t < kτj,k = 1, 2, . . . ,(2.1)4 A. A. Boichuk, M. Medved’ and V. P. Zhuravliovwhere: Xj−1(t) = eB1,...,Bj−1(t−τj−1)τ1,...,τj−1 , Θ is the null N × N matrix, function eB1,...,Bntτ1,...,τn has the prop-erties well described in [8, Lemma 7].Using the multi-delayed matrix exponential (2.1) we can represent a solution z(t) of acorresponding linear system (1.6) with multiple delays and pairwise permutable matrices inthe form (1.7), whereK(t, s) := Y(t− s) if 0 ≤ s ≤ t ≤ b, K(t, s) ≡ 0 if 0 ≤ t < s ≤ b (2.2)andY(t) = eAt eB˜1,...,B˜n(t−τn)τ1,...,τn , B˜i = e−Aτi Bi, i = 1, . . . , n,X(t) := K(t, 0) = Y(t) =[eAt eB˜1,...,B˜n(t−τn)τ1,...,τn].3 Fredholm boundary-value problemUsing the results [3, 4], it is easy to derive results for a general boundary-value problem ifthe number m of boundary conditions does not coincide with the number N of unknowns ina differential system with a delay. We derive such results in an explicit analytical form. Weconsider the boundary-value problemz˙(t)− Az(t)−n∑i=1Bi(Shi z)(t) = ϕ(t), t ∈ [0, b] , (3.1)lz(·) = α ∈ Rm, (3.2)where α is an m-dimensional constant vector-column, l = col(l1, . . . , lm) : Dp[0, b] → Rm,(li : Dp[0, b] → R) is an m-dimensional linear vector-functional defined on the space Dp[0, b]of N-dimensional vector-functions absolutely continuous on [0, b]. As above, we state that, inthe spaces considered, this problem is equivalent to problem (1.1), (1.2), whereϕ(t) := g(t) +n∑i=1Biψhi(t) ∈ Lp[0, b].We will derive sufficient and necessary conditions, and a representation of the solutionsz ∈ Dp[0, b], z˙(t) ∈ Lp[0, b] of the boundary-value problem (3.1), (3.2).Substituting the general solution (1.7) of the equation (3.1) into the boundary condi-tion (3.2), in accordance with (2.2), we will have the algebraic systemQc = α− l∫ b0K( · , s)ϕ(s) ds (3.3)with the constant m× N dimension matrixQ = lX(·) = l[eA· eB˜1,...,B˜n(·−τn)τ1,...,τn].Preserving the above used notation [4], we have: rank Q = n1 ≤ min(m, N), PQ := IN −Q+Q is an N × N-dimensional matrix (orthogonal projection) projecting the space RN to thekernel (ker Q) of the matrix Q, PQ∗ := Im −QQ+ in an m×m-dimensional matrix (orthogonalprojection) projecting the space Rm to the kernel Q∗ of the transposed matrix Q∗ = QT. Usingthe propertyrank PQ∗ = m− rank Q∗ = d = m− n1Fredholm boundary-value problems for linear delay systems 5we will denote by PQ∗d a d× m-dimensional matrix constructed from d linearly independentrows of the matrix PQ∗ . Using the propertyrank PQ = N − rank Q = r = N − n1we will denote by PQr an N × r-dimensional matrix constructed from r linearly independentcolumns of the matrix PQ.Then (see [4, p. 79]) the conditionPQ∗d{α− l∫ b0K(·, s)ϕ(s) ds}= 0 (3.4)is necessary and sufficient for algebraic system (3.3) to be solvable and if such condition istrue, system (3.3) has a solutionc = PQr cr + Q+{α− l∫ b0K(·, s)ϕ(s) ds}for all cr ∈ Rr, (3.5)where Q+ is an N ×m-dimensional matrix pseudo-inverse with respect to the m× N-dimen-sional matrix Q.Substituting the constant c ∈ RN defined by (3.5) into (1.7), we get a formula for thegeneral solution of problem (3.1), (3.2):z(t, cr) = Xr(t)cr + (Gϕ)(t) + X(t)Q+α, (3.6)where Xr(t) = X(t)PQr ,(Gϕ)(t) :=∫ b0G(t, s)ϕ(s) dsis a generalized Green operator, andG(t, s) := K(t, s)− X(t)Q+lK( · , s)is a generalized Green matrix, corresponding to the boundary-value problem (3.1), (3.2).Therefore, the following theorem holds.Theorem 3.1. If rank Q = n1 ≤ min(m, N), then the homogeneous problem corresponding to prob-lem (3.1), (3.2) (with ϕ(t) = 0, α = 0) has exactly r (where r = N− n1) linearly independent solutionsin the space Dp[0, b]. The inhomogeneous problem (3.1), (3.2) is solvable in the space Dp[0, b] if andonly if ϕ(t) ∈ Lp[0, b] and α ∈ Rm satisfy d linearly independent conditions (3.4). Then it has anr-dimensional family of linearly independent solutions z(t, cr) : z(·, cr) ∈ Dp[0, b] , z˙(·, cr) ∈ Lp[0, b],represented in an explicit form (3.6).The case of rank Q = N implies the inequality m ≥ N, i.e., the boundary-value problem isoverdetermined, the number of boundary conditions is not less than the number of unknowns,Theorem 3.1 has the following corollary.Corollary 3.2. If rank Q = N, then the homogeneous problem has only the trivial solution. Inhomoge-neous problem (3.1), (3.2) is solvable if and only ifPQ∗d{α− l∫ b0K( · , s)ϕ(s) ds}= 0where d = m− N. Then the unique solution can be represented asz(t) = (Gϕ)(t) + X(t)Q+α.6 A. A. Boichuk, M. Medved’ and V. P. ZhuravliovThe case of rank Q = m is interesting as well. Then the inequality m ≤ N holds, i.e.,the boundary-value problem is underdetermined. In this case, Theorem 3.1 has the followingcorollary.Corollary 3.3. If rank Q = m, then the boundary-value problem has an r-dimensional (r = N −m)family of solutions. The inhomogeneous problem (3.1), (3.2) is solvable for arbitrary ϕ(t) ∈ Lp[0, b]and α ∈ Rm and has an r-parametric family of solutionsz(t, cr) = Xr(t)cr + (Gϕ)(t) + X(t)Q+α.Finally, combining both particular cases mentioned above, we get the following.Corollary 3.4. If rank Q = N = m, then the homogeneous problem has only the trivial solution.The inhomogeneous boundary-value problem (3.1), (3.2) is solvable for arbitrary ϕ(t) ∈ Lp[0, b] andα ∈ RN , and has a unique solutionz(t) = (Gϕ)(t) + X(t)Q−1α.Corollary 3.5. If A = 0 and i = 1 , then from Theorem 3.1 we obtain the result published in [2].4 ExampleConsider the boundary value problem with two delays [8, p. 3350]z˙(t) = b1z(t− 34)+ b2z(t− 1) + ϕ(t), t ∈ [0, 1], (4.1)`z(·) = α, (4.2)where ` = col(l1, l2) is a two-dimensional vector functional:l1z(·) := −b12 z(0) + 2z(1),l2z(·) :=(2+b14)z(0)− z(1),α = col(α1, α2) ∈ R2.The general solution of the equation (4.1) has the formz(t) = Y(t)c +∫ t0Y(t− s)ϕ(s) ds, (4.3)where Y(t) is the solution of the corresponding homogeneous (4.1) equation on the interval[0, 1] [8, p. 3351]Y(t) = eb1,b2(t−τ2)τ1,τ2 =0, t < 0,1, 0 ≤ t < 34 ,1+ b1(t− 34 ), 34 ≤ t ≤ 1.Substituting the general solution (4.3) into the boundary conditions (4.2), we obtain an alge-braic equationQc =[α1α2]−21∫0Y(1− s)ϕ(s) ds−1∫0Y(1− s)ϕ(s) ds . (4.4)Fredholm boundary-value problems for linear delay systems 7For boundary value problem (4.1), (4.2) the matrix Q has the formQ = `Y(·) =[− b12 Y(0) + 2Y(1)(2+ b14 ) Y(0)−Y(1)]=[21].ThenPQ = 0, PQ∗ =15−25−2545 , PQ∗d = [ 15 −25], Q+ =[2515].The equation (4.4), and hence the boundary value problem (4.1), (4.2) is solvable if and onlyif conditionPQ∗d{[α1α2]−[2∫ 10 Y(1− s)ϕ(s) ds− ∫ 10 Y(1− s)ϕ(s) ds]}= 0is satisfied, and after the transformation that is of the formα1 − 2α2 − 4∫ 10Y(1− s)ϕ(s) ds = 0, (4.5)whereY(1− s) =1, 0 ≤ 1− s ≤ 34⇔ 14≤ s ≤ 1,1+ b1(14− s),34≤ 1− s ≤ 1⇔ 0 ≤ s ≤ 14.Since PQ = 0, then under the condition (4.5), equation (4.4) has a unique solutionc = Q+{[α1α2]−[2∫ 10 Y(1− s)ϕ(s) ds− ∫ 10 Y(1− s)ϕ(s) ds]}. (4.6)Substituting c from (4.6) in the formula (4.3) we have a unique solution of the boundary valueproblem (4.1), (4.2)z(t) =∫ t0Y(t− s)ϕ(s) ds−Y(t)Q+[2∫ 10 Y(1− s)ϕ(s) ds− ∫ 10 Y(1− s)ϕ(s) ds]+Y(t)Q+[α1α2],which after conversion has the formz(t) =∫ t0Y(t− s)ϕ(s) ds− 35Y(t)∫ 10Y(1− s)ϕ(s) ds +Y(t)[2α15+α25]and the generalized Green matrix, corresponding to the boundary-value problem (4.1), (4.2),has the formG(t, s) =Y(t− s)− 35Y(t)Y(1− s), 0 ≤ s ≤ t,−35Y(t)Y(1− s), t < s ≤ 1.(4.7)For example, the condition (4.5) will be fulfilled for the inhomogeneities of the following form:ϕ(t) = t, α1 = 2, α2 = − b143 · 3.8 A. A. Boichuk, M. Medved’ and V. P. ZhuravliovOn the interval 0 ≤ t < 34 , we have in the Green matrix (4.7) Y(t) = 1, Y(t− s) = 1 and thesolution of the boundary value problem (4.1), (4.2) forϕ(t) = t, α1 = 2, α2 = − b143 · 3will have the formz1(t) =t22− 35[12+b143 · 6]+25α1 +15α2.On the interval 34 ≤ t ≤ 1 we have in the Green matrix (4.7)Y(t) = 1+ b1(t− 34),Y(t− s) = 1+ b1(t− 34− s)and the solution of the boundary value problem (4.1), (4.2) will have the formz2(t) =t22+ b1t(t− 34)22− b1 t33+ b13t28− b1 943 · 2− 35[1+ b1(t− 34)] [12+b143 · 6]+[1+ b1(t− 34)] [25α1 +15α2].AcknowledgementsThe work of the second author was supported by the Slovak Grant Agency VEGA-SAV-MŠ,No. 1/0071/14.References[1] N. V. Azbelev, V. P. Maksimov, Equations with retarded arguments, Differ. Equations18(1983), 1419–1441; Engl. transl. from Differ. Uravn. 18(1982), 2027–2050. MR683997[2] A. A. Boichuk, J. Diblík, D. Ya. Khusainov, M. 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Fredholm boundary-value problems for linear delay systems defined by pairwise permutable matrices

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