The periodic problem for the second order integro-differential equations with distributed deviation

2020 MATHEMATICA BOHEMICA 17 pp Online i s THE PERIODIC PROBLEM FOR THE SECOND ORDER INTEGRO-DIFFERENTIAL EQUATIONS WITH DISTRIBUTED DEVIATION Sulkhan Mukhigulash ili,Ve onika No o ná, B no Recei ed Ap il 19, 2019. Published online June 12, 2020. Communica ed by Leonid Be ezansky Abs ac . We s udy he ques ion o he unique sol abili y o he pe iodic ype p oblem o he second o de linea in eg o-di e en ial equa ion wi h dis ibu ed a gumen de ia ion u′′( ) = p0( )u( ) + Zω 0 p( , s)u(τ( , s)) ds+q( ), and on he basis o he ob ained esul s by he a p io i boundedness p inciple we p o e he new esul s on he sol abili y o pe iodic ype p oblem o he second o de nonlinea unc ional di e en ial equa ions, which a e close o he linea in eg o-di e en ial equa ions. The p o ed esul s a e op imal in some sense. Keywo ds: linea in eg o-di e en ial equa ion; pe iodic p oblem; dis ibu ed de ia ion; sol abili y MSC 2020: 34K06, 34K13, 34B15 1. In oduc ion On he in e al I= [0, ω], conside he second o de linea in eg o-di e en ial equa ion (1.1) u′′( ) = p0( )u( ) + Zω 0 p( , s)u(τ( , s)) ds+q( ), Fo S. Mukhigulash ili, he esea ch was suppo ed by he g an RVO: 67985840. Fo V. No o ná, he esea ch was suppo ed by he Czech Science Founda ion, p ojec No. GA16-03796S. c The au ho (s) 2020. This is an open access a icle unde he CC BY-NC-ND licence cbnd DOI: 10.21136/MB.2020.0061-19 1 and nonlinea unc ional di e en ial equa ion (1.2) u′′( ) = F(u)( ) + q( ) wi h he pe iodic ype wo poin bounda y condi ions (1.3) u(i−1)(ω)−u(i−1)(0) = ci, i = 1,2, whe e c1, c2∈R,p0, , q ∈L∞(I, R),p∈L∞(I2,R),τ:I2→Iis a measu able unc- ion, and F:C′(I, R)→L∞(I, R)is a con inuous ope a o . (The spaces C′(I, R) and L∞(I, R)a e de ined below.) We will say ha a unc ion u:I→Ris a solu ion o p oblem (1.2), (1.3) i i is absolu ely con inuous oge he wi h i s i s de i a i e, sa is ies equa ion (1.2) almos e e ywhe e on Iand sa is ies condi ions (1.3). I is well-known ha he e a e many subjec s in physics and echnology using ma hema ical me hods ha depend on he in eg o-di e en ial equa ions. Fo hese and o pu ely heo e ical easons ample in e es ing li e a u e is de o ed o he pe- iodic p oblem o he in eg o-di e en ial equa ions (see, e.g., [4], [6], [3], [10] and he e e ences he ein). Ou wo k is mo i a ed by some o iginal esul s o he unc ional di e en ial equa ions wi h a gumen de ia ion (see [1], [2], [9]), and he esul s o Nie o (see [10]), E be and Guo (see [4]), and Kuo-Shou Chiu (see [3]). Nie o in [10] s udied linea equa ion (1.1) on he in e al I= [0,2π]when p0≡M, p( , s) = Nk( , s)and τ( , s)≡s, i.e. he equa ion o he o m (1.4) u′′( ) = Mu( ) + N[Ku]( ) + q( ) unde condi ions (1.3) wi h c1=c2= 0,whe e [Ku]( ) = R2π 0k( , s)u(s) ds, k ∈ L2(I×I),M > 0and N∈R. In his pape , di e en su icien e icien condi ions o he unique sol abili y o linea p oblem (1.4), (1.3) a e es ablished, and one o hem, he condi ion kτk2<1, is op imal, whe e τ( , s) = R2π 0G( , )k( , s) d , and G is he G een’s unc ion o he pe iodic p oblem o he equa ion ′′( ) = M ( ).On he basis o hese esul s, he pe iodic p oblem o he nonlinea equa ion u′′( ) = ( , u( ),[Ku]( )) is s udied e en in he case when he ke nel kchanges i s sign. In [6] he au ho s de elop he mono one i e a i e me hod based on compa ison e- sul s, which is applicable o p oblem (1.4), (1.3) only i Kis Vol e a ope a o wi h nonnega i e ke nel. A mo e gene al case is conside ed in pape [4], he e he op- e a o Kis o he o m [Kx] = N[Tx] + N1[Sx], whe e Tis he in eg al ope a o o Vol e a ype and Sis he in eg al ope a o o F edholm ype wi h nonnega i e ke nels. Chiu in [3] in es iga es he exis ence o pe iodic solu ions o he sys ems o in eg o-di e en ial equa ions wi h piecewise al e na ely e a ded and ad anced 2Online i s a gumen o gene alized ype. In he men ioned pape he au ho p o es in e es ing esul s o he sol abili y and unique sol abili y, bu hese esul s do no ake in o accoun he e ec o a gumen de ia ion. In his pape we es ablish he heo ems which in some sense comple e and gene - alize he esul s o he wo ks ci ed abo e as well as some o he known esul s. We i s desc ibe some classes o unique sol abili y o linea p oblem (1.1), (1.3), and on he basis o hese esul s, by he a p io i boundedness p inciple, we p o e he exis ence heo ems o nonlinea p oblem (1.2), (1.3). The condi ions we ob ain ake in o accoun he e ec o a gumen de ia ion, and in some sense a e op imal (see Rema ks 2.1, 2.3). In e es ing esul s ollow om ou main p oposi ion o such special cases o equa- ions (1.1) and (1.2) as a e linea in eg o-di e en ial equa ions wi h dis ibu ed delay (see Co olla y 2.2), linea di e en ial equa ions wi h a gumen de ia ion (see Co ol- la y 2.3), o he nonlinea equa ion (1.5) u′′( ) =  , u( ),Zω 0 V(u)( , s)u(τ( , s)) ds+q( ), whe e :I×R2→Ris om he Ca a héodo y class, and V:C′(I, R)→L∞(I2,R) is a con inuous bounded ope a o . Also ou esul s allow o ob ain condi ions o unique sol abili y o a la ge class o he wo poin BVP o highe o de unc ional di e en ial equa ions. He e as an example o such p oblems we conside n h o de linea unc ional di e en ial equa ion wi h a gumen de ia ion (1.6) u(n)( ) = p1( )u(τ( )) + q( ) unde he wo poin bounda y condi ions (1.7) u(i−1)(ω)−u(i−1)(0) = ci, u(j−1)(0) = cj, i = 1,2, j = 3,...,n, i n⩾3,whe e ck∈R,k=1, n,p1∈L∞(I, R),and τ:I→Iis a measu able unc ion. Th oughou he pape we use he ollowing no a ions: R= ]−∞,∞[,R+= [0,∞[; C(I;R)is he Banach space o con inuous unc ions u:I→Rwi h he no m kukC= max{|u( )|: ∈I};C′(I;R)is he Banach space o he unc ions u:I→Rwhich a e con inuous oge he wi h hei i s de i a i es wi h he no m kukC′= max{|u( )|+ |u′( )|: ∈I};L(I;R)is he Banach space o he Lebesgue in eg able unc ions p:I→Rwi h he no m kpkL=Rω 0|p(s)|ds;L∞(I, R)is he space o he essen ially bounded measu able unc ions p:I→Rwi h he no m kpk∞= esssup{|p( )|: Online i s 3 ∈I};L∞(I2,R)is he se o such unc ions p:I2→R, ha o any ixed ∈I, p( , ·)∈L(I, R)and Rω 0|p(·, s)|ds∈L∞(I, R).Also o a bi a y p0, p1∈L∞(I, R), p∈L∞(I2,R),and measu able τ:I2→Iwe will use he no a ions l0(p0, p)( ) = |p0( )|+Zω 0 |p( , s)|ds, l1(p, τ) = 2π ωZω 0Zω 0 |p(ξ, s)||τ(ξ, s)−ξ|dsdξ1/2 . De ini ion 1.1. Le σ∈ {−1,1}and τ:I2→Ibe measu able unc ion. We will say ha he pai o unc ions (h0, h), whe e h0∈L∞(I, R+)and h∈L∞(I2,R+) belong o he se Pσ τi o a bi a y measu able unc ions p0:I→Rand p:I2→R such ha 0⩽σp0( )⩽h0( ),0⩽σp( , s)⩽h( , s) o , s ∈I,(1.8) p0( ) + Zω 0 p( , s) ds6≡ 0,(1.9) he homogeneous p oblem ′′( ) = p0( ) ( ) + Zω 0 p( , s) (τ( , s)) ds,(1.10) (i−1)(ω)− (i−1)(0) = 0, i = 1,2,(1.11) has no non i ial solu ion. 2. S a emen o he main esul s 2.1. Linea p oblem. P oposi ion 2.1. Le σ∈ {−1,1}, (2.1) h0∈L∞(I, R+), h ∈L∞(I2,R+), h0( ) + Zω 0 h( , s) ds6≡ 0, and o almos all ∈I he inequali y (2.2) 1−σ 2l0(h0, h)( ) + l1(h, τ)l1/2 0(h0, h)( )<4π2 ω2 holds. Then (2.3) (h0, h)∈Pσ τ. 4Online i s Theo em 2.1. Le σ∈ {−1,1}, σp0∈L∞(I, R+),σp ∈L∞(I2,R+)and condi- ion (1.9) be ul illed. Mo eo e , le o almos all ∈I he inequali y (2.4) 1−σ 2l0(p0, p)( ) + l1(p, τ)l1/2 0(p0, p)( )<4π2 ω2 hold. Then p oblem (1.1),(1.3) is uniquely sol able. R e m a k 2.1. Condi ion (2.4) is op imal in he sense ha o he equa ion (2.5) u′′( ) = p0( )u( ) o ∈[0,2π] when p0( )⩽0,condi ion (2.4) ans o ms in o he condi ion |p0( )|<1,which is op imal, because i p0≡ −1, hen sin is a nonze o solu ion o p oblem (2.5), (1.3) wi h c1=c2= 0. F om he las heo em i also ollows he well known ac ha i p0( )⩾0, hen p oblem (2.5), (1.3) wi h c1=c2= 0,has only he ze o solu ion. When in equa ion (1.1) he coe icien s p0and pa e nonnega i e, hen 1−σ= 0, and om Theo em 2.1 i ollows: Co olla y 2.1. Le (2.6) p0∈L∞(I, R+), p ∈L∞(I2,R+), p0( ) + Zω 0 p( , s) ds6≡ 0, and o almos all ∈Ile he inequali y Zω 0Zω 0 p(ξ, s)|τ(ξ, s)−ξ|dsdξp0( ) + Zω 0 p( , s) ds<4π2 ω2 hold. Then p oblem (1.1),(1.3) is uniquely sol able. Le now p0≡0,τ( , s)≡ −ν( , s),and (2.7) 0⩽ν( , s)⩽ o , s ∈I. Then equa ion (1.1) ans o ms in o he in eg o-di e en ial equa ion wi h dis ibu ed delay (2.8) u′′( ) = Zω 0 p( , s)u( −ν( , s)) ds+q( ), and om Co olla y 2.1 i ollows: Co olla y 2.2. Le p∈L∞(I2,R+),Rω 0p( , s) ds6≡ 0and o almos all ∈Ile he inequali y Zω 0Zω 0 p(ξ, s)ν(ξ, s) dsdξZω 0 p( , s) ds < 4π2 ω2 hold. Then p oblem (2.8),(1.3) is uniquely sol able. Online i s 5 I p0≡0and τ( , s) = τ( ) o , s ∈I, hen equa ion (1.1) ans o ms in o equa ion (1.6) wi h n= 2,p1( ) = Rω 0p( , s) ds, and hen om Co olla y 2.1 i ollows: Co olla y 2.3. Le p1∈L∞(I, R+)be such ha o almos all ∈I he inequali y p1( )Zω 0 p1(s)|τ(s)−s|ds < 4π2 ω2 holds. Then p oblem (1.6),(1.3) when n= 2, is uniquely sol able. Co olla y 2.4. Le n⩾3and he unc ion p1∈L∞(I, R+)be such ha o almos all ∈I he inequali y Zω 0Z 0 p1(s)|τ(s)− |dsd Zω 0 p1(s) ds⩽4π2((n−3)!)2 ω2(n−2) holds. Then p oblem (1.6),(1.7) is uniquely sol able. R e m a k 2.2. I in Co olla ies 2.1–2.3 we assume ha σp0=h0and σp =h, we ge he su icien e icien condi ions which gua an ee inclusion (2.3). 2.2. Nonlinea p oblem. Now we conside he heo ems on he sol abili y o nonlinea p oblem (1.2), (1.3). Fi s we will in oduce he e he de ini ions. De ini ion 2.1. We will say ha he ope a o Fbelongs o Ca a héodo y’s local class and w i e F∈K(C′, L∞)i F:C′(I, R)→L∞(I, R)is con inuous ope a o , and o an a bi a y > 0 sup{|F(x)( )|:kxkC′⩽ , x ∈C′(I, R)} ∈ L∞(I, R+). De ini ion 2.2. Le σ∈ {−1,1},inclusion (2.3) hold and he ope a o s V0: C′(I, R)→L∞(I, R), V :C′(I, R)→L∞(I2,R)be con inuous. Then we will say ha (V0, V )∈E(h0, h, Pσ τ)i o all x∈C′(I, R) he condi ions (2.9) 0⩽σV0(x)( )⩽h0( ),0⩽σV (x)( , s)⩽h( , s) o , s ∈I hold, and (2.10) in {kL(x, 1)kL:x∈C′(I, R)}>0, 6Online i s whe e (2.11) L(x, y)( ) = V0(x)( )y( ) + Zω 0 V(x)( , s)y(τ( , s)) ds. Also h oughou he pape we assume ha (2.12) sgn x=(1i x⩾0, −1i x < 0. Then he nex heo em is ue: Theo em 2.2. Le σ∈ {−1,1}and (2.13) (V0+e V0, V )∈E(h0, h, Pσ τ), whe e σV0(x)( )⩾0,σe V0(x)( )⩾0on I o all x∈C′(I, R). Mo eo e , le he cons an 0>0, he ope a o F∈K(C′, L∞)and he unc ion g0∈L(I, R+)be such ha he condi ions (2.14) g0( )⩽σ(F(x)( )−L(x, x)( )) sgn x( ) ⩽|e V0(x)( )x( )|+η( , kxkC′) o ∈I, kxkC′⩾ 0, and (2.15) |c2|⩽Zω 0 g0(s) ds−Zω 0 q(s) ds hold, whe e he unc ion η:I×R+→R+is summable in he i s a gumen , non- dec easing in he second one, and admi s o he condi ion (2.16) lim →∞ 1 Zω 0 η(s, ) ds= 0. Then p oblem (1.2),(1.3) has a leas one solu ion. R e m a k 2.3. Inequali y (2.15) canno be eplaced by he inequali y (2.17) |c2|⩽Zω 0 g0(s) ds−Zω 0 q(s) ds+ε, no ma e how small ε > 0would be. Indeed, i F≡0,q( )≡εω−1,g0≡0,c2= 0, hen ins ead o (2.15), inequali y (2.17) holds and all o he condi ions o Theo em 2.2 a e ul illed wi h L(x, y)≡0,η≡0,e V0≡h0≡1,σ= 1.Ne e heless, in ha case, p oblem (1.2), (1.3) is no sol able. Online i s 7 R e m a k 2.4. Le σ∈ {−1,1},(h0, h)∈Pσ τ, (2.18) V0(x)( ) = p0( ),e V0(x)( ) = ep0( ), whe e σp0, σep0∈L∞(I, R+), and V:C′(I, R)→L∞(I2,R)be he con inuous ope - a o . Then due o De ini ion 2.2 i is ob ious ha inclusion (2.13) holds i (2.19) 0<|p0( )|, σ(p0( ) + ep0( )) ⩽h0( ) o ∈I, 0⩽σV (y)( , s)⩽h( , s) o , s ∈I, y ∈C′(I, R). Co olla y 2.5. Le σ∈ {−1,1},inclusion (2.3) hold, he unc ions g0, σp0, σep0∈ L∞(I, R+),and he con inuous ope a o V:C′(I, R)→L∞(I2,R)be such ha inequali ies (2.15),(2.19) a e ul illed. Mo eo e , le (2.20) g0( )⩽σ( ( , x1, x2)−p0( )x1−x2) sgn x1 ⩽|ep0( )x1|+η( , |x1|) o ∈I, x1, x2∈R, whe e η:I×R+→R+be summable in he i s a gumen , nondec easing in he second one and admi s o condi ion (2.16). Then p oblem (1.5),(1.3) has a leas one solu ion. E x a m p l e 2.1. The in eg o-di e en ial equa ion wi h dis ibu i e delay (2.21) u′′( ) = αu( ) + β 1 + kukC′Z1 0 |u′(s)|u − 1 + sds+q( ) o ∈[0,1], whe e α, β ∈R+and α6= 0,unde condi ions (1.3) wi h ω= 1,c2= 0,has a leas one solu ion i R1 0q(s) ds= 0 and β(α+β)<8π2/ln 2 ≈113,91. Indeed, in iew o Co olla y 2.2 he las inequali y gua an ees he alidi y o inclusion (2.3), and hen all he assump ions o Co olla y 2.5 wi h σ= 1,p0≡h0≡α,V(y)( ) = β|y′( )|/(1 + kykC′),h≡β,g0≡ep0≡q( , )≡0a e ul illed. The sol abili y o p oblem (2.21), (1.3) does no ollow om he p e iously known esul s. 3. Auxilia y P oposi ions Now conside he modi ica ion o he well known Wi inge ’s inequali y (see The- o em 258 in [5]). P oposi ion 3.1. Le ′′ ∈L∞(I, R)and condi ions (1.11) hold. Then (3.1) Zω 0 ′2(s) ds⩽ω 2π2Zω 0 ( ′′(s))2ds. 8Online i s Lemma 3.1. Le all he condi ions o P oposi ion 2.1and condi ions (1.8),(1.9) hold. Then p oblem (1.10),(1.11) has only he i ial solu ion. P o o . On he con a y, assume ha p oblem (1.10), (1.11) has nonze o solu- ion . I ≡c(ob iously c6= 0), hen ′′ ≡0and in iew o (1.10) we ge he con a- dic ion wi h (1.9), i.e. 6≡ cons . The e o e due o (1.11) he inequali y ′6≡ cons holds and hen he e exis ∗, ∗∈Isuch ha ∗< ∗and ′( ∗)− ′( ∗)6= 0. The e o e om (1.10) by (1.8), Schwa z and Cauchy-Schwa z inequali ies i ollows ha 0<| ′( ∗)− ′( ∗)|⩽Zω 0p0(ξ) (ξ) + Zω 0 p(ξ, s) (τ(ξ, s)) dsdξ ⩽Zω 0 |p0(ξ)|dξZω 0 |p0(ξ)| 2(ξ) dξ1/2 +Zω 0Zω 0 |p(ξ, s)|dsdξZω 0Zω 0 |p(ξ, s)| 2(τ(ξ, s)) dsdξ1/2 ⩽δ1/2Zω 0 |p0(ξ)|dξ+Zω 0Zω 0 |p(ξ, s)|dsdξ1/2 , whe e δ=Rω 0δ0(ξ) dξ,δ0(ξ) = σRω 0p(ξ, s) 2(τ(ξ, s)) ds+p0(ξ) 2(ξ), and hen (3.2) δ > 0. Analogously om (1.10) by (1.8), Schwa z and Cauchy-Schwa z inequali ies we ge (3.3) Zω 0 ( ′′(ξ))2dξ⩽Zω 0|p0(ξ)|1/2(|p0(ξ)| 2(ξ))1/2 +Zω 0 |p(ξ, s)|ds1/2Zω 0 |p(ξ, s)| 2(τ(ξ, s)) ds1/22 dξ ⩽Zω 0 l0(h0, h)(ξ)δ0(ξ) dξ. Now no e ha in iew o (1.10), o δ he ep esen a ion is ue: (3.4) δ=σZω 0 (ξ) ′′(ξ) dξ+Zω 0Zω 0 |p(ξ, s)| (τ(ξ, s))Zτ(ξ,s) ξ ′(η) dηdsdξ. Due o (3.1) and (3.3), by in eg a ion by pa s and bounda y condi ions (1.11) we ind ha (3.5) σZω 0 (ξ) ′′(ξ) dξ=−σZω 0 ′2(ξ) dξ⩽1−σ 2Zω 0 ′2(ξ) dξ ⩽1−σ 2ω 2π2Zω 0 l0(h0, h)(ξ)δ0(ξ) dξ. Online i s 9 condi ion (2.13) and nonnega i i y o he unc ion η, we ge ha uis a solu ion o he equa ion (4.5) u′′( ) = L(u, u)( ) + (λ+ (1 −λ)ν( ))e V0(u)( )u( ) + η1( , kukC′), whe e η1( , kukC′) = σ(1 −λ)(1 + η( , kukC′))ν( ) sgn u( ) + (1 −λ)q( ), ν( ) = σ(F(u)( )−L(u, u)( )) sgn u( ) |e V0(u)( )u( )|+η( , kukC′) + 1 , and due o condi ion (2.14) he es ima ions (4.6) 0⩽ν( )<1,|η1( , kukC′)|⩽1 + η( , kukC′) + |q( )| a e alid on I. Now no e ha acco ding o condi ions (2.13), (4.6) and he nonneg- a i i y o he ope a o s σe V0(u)( )and σV0(u)( ), he es ima ion 0⩽σ(V0(u)( ) + (λ+ (1 −λ)ν( ))e V0(u)( )) ⩽σ(V0(u)( ) + e V0(u)( )) ⩽h0( ) is sa is ied on I. Consequen ly, due o inclusion (2.13), o a bi a y λ∈(0,1) he inclusion (V1, V )∈E(h0, h, Pσ τ),whe e V1(x)( ) = V0(x)( )+(λ+(1−λ)ν( ))e V0(x)( ), is alid oo. Then om (4.5) by Lemma 3.2 due o inequali y (4.6) we ge he es ima ion kukC′⩽0µ(u) + |c1|+|c2|+Zω 0 (|q(s)|+η(s, kukC′) + 1) ds, which in iew o (4.3) con adic s wi h inequali y (4.4), i.e. ou assump ion is in alid and es ima ion (3.27) holds. On he o he hand, om Lemma 3.4 due o inclusion (2.13) i ollows ha he pai o he ope a o e Land condi ions (1.11) belongs o he Opial class O2 0,and he e o e all he assump ions o Lemma 3.5 a e ul illed, om which he sol abili y o p oblem (1.1), (1.3) ollows.  P o o o Co olla y 2.5. Assume ha he ope a o s V0,e V0a e de ined by (2.18). Then due o Rema k 2.4 in iew o (2.19), inclusion (2.13) holds. Also, om (2.20) he alidi y o condi ions (2.14) ollows, wi h F(x)( ) =  , x( ),Zω 0 V(x)( , s)x(τ( , s)) ds, L(x, y)( ) = p0( )y( ) + Zω 0 V(x)( , s)y(τ( , s)) ds. The e o e all he assump ions o Theo em 2.2 a e ul illed om which alidi y o ou co olla y immedia ely ollows.  16 Online i s Re e ences [1] E. B a yi: On sol abili y o pe iodic bounda y alue p oblems o second o de linea unc ional di e en ial equa ions. Elec on. J. Qual. Theo y Di e . Equ. 2016 (2016), Pape No. 5, 18 pages. zbl MR doi [2] E. I. B a yi: On he bes cons an s in he sol abili y condi ions o he pe iodic bounda y alue p oblem o highe -o de unc ional di e en ial equa ions. Di e . Equ. 48 (2012), 779–786; T ansla ion om Di e . U a n. 48 (2012), 773–780. zbl MR doi [3] K.-S. Chiu: Pe iodic solu ions o nonlinea in eg o-di e en ial sys ems wi h piecewise cons an a gumen . Sci. Wo ld J. 2014 (2014), A icle ID 514854, 14 pages. doi [4] L. H. E be, D. Guo: Pe iodic bounda y alue p oblems o second o de in eg odi e en- ial equa ions o mixed ype. Appl. Anal. 46 (1992), 249–258. zbl MR doi [5] G. H. Ha dy, J. E. Li lewood, G. Pólya: Inequali ies. Camb idge Ma hema ical Lib a y. 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Sci. 18 (1995), 757–764. zbl MR doi Au ho s’ add esses:Sulkhan Mukhigulash ili, Ins i u e o Ma hema ics o he Czech Academy o Sciences, Žižko a 22, 616 62 B no, Czech Republic, e-mail: [email p o ec ed]; Ve onika No o ná, Facul y o Business and Managemen , B no Uni e si y o Technology, Kolejní 2906/4, 612 00 B no, Czech Republic, e-mail: [email p o ec ed]. Online i s 17