Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 101959, 19 pages doi:10.1155/2010/101959 Research Article A Mixed Problem for Quasilinear Impulsive Hyperbolic Equations with Non Stationary Boundary and Transmission Conditions Akbar B Aliev1 and Ulviya M Mamedova2 Azerbaijan Technical University, AZ 1073, Baki, Azerbaijan Institute of Mathematics and Mechanics of NAS of Azerbaijan, AZ 1141, Baku, Azerbaijan Correspondence should be addressed to Akbar B Aliev, aliyevagil@yahoo.com Received 10 March 2010; Revised 13 June 2010; Accepted 26 October 2010 Academic Editor: Toka Diagana Copyright q 2010 A B Aliev and U M Mamedova This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited The initial-boundary value problem for a class of linear and nonlinear equations in Hilbert space is considers We prove the existence and uniqueness of solution of this problem The results of this investigation are applied to solvability of initial-boundary value problems for quasilinear impulsive hyperbolic equations with non-stationary transmission and boundary conditions Abstract Model Initial Boundary Value Problem with Non Stationary Boundary and Transmission Conditions for the Impulsive Linear Hyperbolic Equations In paper there is given an abstract scheme of investigation of mixed problems for hyperbolic equations with non stationary boundary conditions In this direction, some results were obtained in In this paper, we offer the analogues abstract model of investigation of mixed problem with non stationary boundary and transmission conditions for impulsive linear and semilinear hyperbolic equations 1.1 Statement of the Problem and Main Theorem j i i Let H i , H0 , Xν , Yμ ν 1, 2, , si ; i 1, 2, , m; μ 1, 2, , rj ; j 1, 2, , m be Hilbert Spaces Consider the following abstract initial-boundary value problem: Advances in Dierence Equations ui t ă m ă Bi ui t A i t ui t i Ckν t uk t giν t , fi t , hyperbolic equations , 1.1 non stationary boundary and transmission conditions , k m i Dkμ uk t 0, stationary boundary and transmission conditions , k u0 , i ui u1 , i ui ˙ initial conditions , 1.2 where t ∈ 0, T , ui d2 ui /dt2 , ui dui /dt, Ai t are the linear closed operators in H i ; Biν are ¨ ˙ j i i i the linear operators from H i to Xν ; Ckν t are the linear operators from H k to Xν ; Dkμ are the j linear operators from H k to Yμ ; ν 1, , si , i 1, , m, μ 1, , rj , j 1, , m, k We will investigate this problem under the following conditions 1, , m i i i Let H0 ⊂ H i , and let H0 be densely in H i and continuously imbedded into it, i 1, 2, , m In the Hilbert space H i , it was defined the system of the inner products ·, · generate uniform equivalent norms, that is, −1 c1 u u Hi t Hi ≤ u u, u Hi t Hi t ≤ c1 u Hi , t ∈ 0, T , i , Hi t , which c1 > 0, 1.3 1, 2, , m For each u ∈ H i , the function t → u i t : 0, T → R is continuously differentiable, H i 1, 2, , m i In the Hilbert space Xν , it was defined the system of the inner products ·, · Xν , which i generate uniform equivalent norms, that is, −1 c2 v v i Xν t v, v i Xν i Xν t , ≤ v i Xν t ≤ c2 v t ∈ 0, T , ν i For each v ∈ Xν , the function t → v i Xν t i, Xν c2 > 0, 1, 2, , si , i 1.4 1, 2, , m : 0, T → R is continuously differentiable ii For each t ∈ 0, T and i 1, 2, , m, Ai t is a linear closed operator in H i whose i i domain is H0 ; Ai t acts boundedly from H0 to H i ; Ai t is strongly continuously differentiable i i i iii The linear operators Biν , that act from H1/2 to Xν , bounded, where H1/2 i i H0 , H i 1/2 is interpolation space between H0 and H i of order 1/2 ν 1, , si , i 1, , m see i i iv For each t ∈ 0, T , the linear operators Ckν t , that act from H k to Xν , are i bounded; Ckν t is strongly continuously differentiable ν 1, , si , i 1, , m; k 1, , m Advances in Difference Equations j j k v The linear operators Dkμ , from H1/2 into Yμ , act boundedly μ 1, , m; k 1, , m 1, , rj , j Let us introduce the following designations: H ⊕ · · · ⊕ H m, H i u1 , , um , ui ∈ H , i u:u H0 m j Dkμ uk 0, μ 1, , m; 1, , rj , j 1, , m , k i u1 , , um , ui ∈ H1/2 , i u:u H1/2 m j Dkμ uk 0, μ 1, , rj , j 1, , m; 1.5 1, , m , k H1 w:w w1 , , wm , wi ui , Bi1 ui , , Bisi ui , i 1, , m, where u1 , , um ∈ H0 , Hi i i H i ⊕ X1 ⊕ · · · ⊕ Xsi , m H Hi , H1/2 H1 , H 1/2 i From condition v , it follows that the space H1/2 with the norm m u ui H1/2 i 1.6 i H1/2 is a subspace of H1/2 i u1 , , um , ui ∈ H1/2 , i u:u 1, , m m H1/2 × · · · × H1/2 1.7 vi Let the linear manifold H0 be dense in H1/2 , and let linear manifold H1 be dense in H vii Green’s Identity For arbitrary u, v ∈ H0 and t ∈ 0, T , the following identity is valid: m ⎡ ⎤ i k i Ckν Hi t i m m ν ⎣ Ai t ui , vi si ⎡ ⎣ ui , Ai t vi ⎦ t uk , Biν vi i Xν si m Biν ui , Hi t ν k t ⎤ ⎦ i Ckν t vk i Xν t 1.8 Advances in Difference Equations u1 , , um ∈ H0 , the following inequality is fulfilled: viii For all u m c1 ui i ≤ m si Hi ⎡ i Xν Biν ui ν ⎤ m ν ⎣ A i t ui , ui si i k ⎦ ≤ c2 i Ckν t uk , Biν ui Hi t i Xν ui i t 1.9 m i , H1/2 where c1 ∈ R, c2 > ix For each t ∈ 0, T , an operator pencil Lt λ : u u1 , , um −→ Lt λ u Lt λ u, Lt λ u, , Lt λ u, , Lt λ u, Lt λ u, , Lt m λ u , ms 10 11 1s m0 m1 which acts boundedly from H0 to H, has a regular point λ Lt λ u i0 λui m Lt λ u iν A i t ui , i Ckν t uk , λBiν ui ν i 1.10 λ0 ∈ R, where 1, 2, , m, 1, 2, , si , i 1, 2, , m 1.11 k i i x u0 ∈ H0 , u1 ∈ H1/2 , i i i m k j Dkμ u0 k 1, 2, , m, μ xi fi · ∈ Wp 0, T; H i , p ≥ 1, i i giν · ∈ Wp 0, T; Xν , 0, m k j Dkμ u1 k 1, 2, , rj , j 1, 2, , m 1.12 1, , m, p ≥ 1, ν 1, , si , i 1, , m 1.13 Definition 1.1 The function t → u1 t , , um t is called a solution of problem 1.1 - 1.2 if u1 t , , um t from 0, T to H0 is continuous, and the function the function t → u t t −→ u1 t , B11 u1 t , , B1s1 u1 t , , um t , Bm1 um t , , Bmsm um t 1.14 from 0, T to H is twice continuously differentiable and 1.1 - 1.2 are satisfied Theorem 1.2 Let conditions (i)–(xi) are satisfied, then the problem 1.1 - 1.2 has a unique solution Advances in Difference Equations Proof We define the operator A t in the Hilbert space H in the following way: D A t A tw m H1 , Ck1 t uk , , A t u1 , k m Cks1 t uk , , A t m um , k m m m Ck1 t uk , , k 1.15 t ∈ 0, T , w ∈ H1 m Cksm t uk , k Then the problem 1.1 - 1.2 is represented as the Cauchy problem w ă w where w t A tw w0 , Φt, 1.16 w1 , w ˙ u1 t , B11 u1 t , , B1s1 u1 t , , um t , Bm1 um t , , Bmsm um t , Φ t f1 t , g11 t , , g1s1 t , , fm t , gm1 t , , gmsm t , w0 u0 , B11 u0 , , B1s1 u0 , , u0 , Bm1 u0 , , Bmsm u0 , m m m 1 w1 u1 , B11 u1 , , B1s1 u1 , , u1 , Bm1 u1 , , Bmsm u1 m m m 1 1.17 It is obvious that if u1 t , , um t is the solution of problem 1.1 - 1.2 , then w t is the solution of the problem 1.16 On the contrary, if w t ∈ C2 0, T ; H ∩ C1 0, T ; H1 , H 1/2 ∩ C 0, T ; H1 1.18 is the solution of problem 1.16 , then w t u1 t , B11 u1 t , , B1s1 u1 t , , um t , Bm1 um t , , Bmsm um t and u1 t , , um t is the solution of problem 1.1 - 1.2 Let us define the system of inner product in Hilbert space H in the following way: w1 , w2 m Ht wi1 , wi2 i m si Hi t Biν u1 , Biν u2 i i i 1ν i Xν t , t ∈ 0, T , l l w1 , , wm , wil uli , Bi1 uli , , Bisi uli , i 1, 2, , m, ul1 , , ulm ∈ H0 , l where wl We denote space H with inner product 1.19 by H t 1.19 1, We will prove later the following auxiliary results Statement 1.3 There exists such c3 > 0, that −1 c3 w H ≤ w Ht ≤ c3 w H, t ∈ 0, T , 1.20 Advances in Difference Equations Ht and the function t → w w, w H t : 0, T → R is continuously differentiable, where w Ht Statement 1.4 A t is a symmetric operator in H t for each t ∈ 0, T Statement 1.5 A t has a regular point for each t ∈ 0, T in R A t is symmetric and R A t λI H t , for some λ ∈ R; therefore, for each t ∈ 0, T , A t is a selfadjoint operator in H t see 4, chapter x Taking into account viii and Statement 1.3, we get A t w, w m Ht ⎡ Ht ⎤ k Hi t i ≥ c1 w m ν ⎣ A i t ui , ui si i Ckν ⎦ t uk , Biν ui i Xν 1.21 t , that is, A t is a lower semibounded selfadjoint operator in H t A t λ0 I is selfadjoint and positive definite, where λ0 > c1 Thus, the operator A0 t Problem 1.16 can be rewritten as w t ă A0 t w t w t w w0 , w ˙ F t , 1.22 w1 It is known that if w0 ∈ H1 and w1 ∈ H1/2 , then the problem 1.22 has a unique solution w ∈ C2 0, T ; H ∩ C1 0, T ; H1/2 ∩ C 0, T ; H1 see 5, To complete the proof of the theorem, we need to show that w0 ∈ H1 and w1 ∈ H1/2 j i By conditions of the theorem u0 ∈ H0 , m Dkμ u0 i 1, 2, , m; μ 1, 2, , rj , k i k i i j 1, 2, , m and Biν are bounded operators from H1/2 to Xν , ν Therefore, w0 u0 , B11 u0 , , B1s1 u0 , , u0 , Bm1 u0 , , Bmsm u0 ∈ H1 m m m 1 i On the other hand, u1 ∈ H1/2 and i i 1, 2, , m , therefore, Biν u1 ∈ Xν ν i w1 J 1, 2, , si , i m k j Dkμ u1 k 1, 2, , si , i i 1, 2, , m, μ 1, 2, , m 1.23 1, 2, , rj , j 1, 2, , m Consequently, u1 , B11 u1 , , B1s1 u1 , , u1 , Bm1 u1 , , Bmsm u1 ∈ J, m m m 1 w:w m j w1 , , wm , wi Dkμ uk k 0, i i ui , Bi1 ui , , Bisi ui , ui ∈ H1/2 , 1, , m, μ 1, , rj , j 1, , m 1.24 Advances in Difference Equations From the definition of interpolation spaces see 3, chapter , 7, chapter , we get the following inclusion: m H1 ⊂ H1/2 ⊂ H1/2 i i i H1/2 ⊕ X1 ⊕ · · · ⊕ Xsi 1.25 i By virtue of definition, the powers of positive selfadjoint operator see 8, chapter , 1/2 7, chapter , we have that D A0 t H1/2 and c−1 w H1/2 1/2 ≤ A0 t w Assume that w ∈ D A0 1/2 A0 t w m H1/2 , c > 1.26 H1 , then A0 t w, w Ht ≤c w Ht Ht ⎡ m ν ⎣ A i t ui , ui si k Hi t i ⎤ ⎦ i Ckν t uk , Biν ui i Xν 1.27 t si m ui , ui λ0 Biν ui , Biν ui Hi t i i Xν t ν By virtue of conditions ii , viii , 1.26 , and 1.27 , we get w H1/2 ≤c m i H1/2 ui i 1.28 Let w1 ∈ J By virtue of condition vi , H0 is dense in H1/2 ; therefore, there exists a p p sequence u p u1 , , um , such that u p ∈ H0 and u p − u1 m H1/2 ⊕···⊕H1/2 −→ 0, at p −→ ∞ 1.29 −→ at p, q −→ ∞ 1.30 Hence it follows, that u p −u q m H1/2 ⊕···⊕H1/2 Then from 1.28 and 1.30 it follows that {w w where w p p p p −w p q H1/2 p } is fundamental in H1/2 , that is, −→ 0, at p, q −→ ∞, p p 1.31 p u1 , B11 u1 , , B1s1 u1 , , um , Bm1 um , , Bmsm um , p 1, 2, Advances in Difference Equations Thus, there exists w ∈ H1/2 such that w p −w H1/2 −→ 0, at p −→ ∞ 1.32 at p −→ ∞ 1.33 On the other hand, H1/2 ⊂ H1/2 , therefore, w p −w H1/2 −→ 0, Hence, up −u where u −→ 0, m H1/2 ⊕···⊕H1/2 at p −→ ∞, u1 , that is, u1 , , um From this, by virtue of 1.29 , u w 1.34 u1 , B11 u1 , , B1s1 u1 , , u1 , Bm1 u1 , , Bmsm u1 m m m 1 w1 1.35 Thus, w1 ∈ H1/2 The theorem is proved 1.2 Proof of Auxiliary Results Validity of Statement 1.3 follows from condition i , the Statement 1.4 from condition vii Proof of Statement Consider in Hilbert space H the equation λw A tw F, t ∈ 0, T , 1.36 where F f1 , f11 , , f1s1 , , fm , fm1 , , fmsm ∈ H, λ ∈ R Equation 1.36 is equivalent to the following system of differential-operator equations: Lt λ u i0 λui A i t ui m Lt λ u iν i Ckν t uk λBiν ui giν , fi , t ∈ 0, T , i t ∈ 0, T , ν 1, 2, , m, 1, 2, , si , i k m j Dkμ uk 0, μ 1, 2, , rj , j 1, 2, , m, 1.37 1, 2, , m k By virtue of ix , problem 1.37 has a solution u for each t ∈ 0, T , R λI A t u1 , , um ∈ H0 for some λ ∈ R Thus, Ht, where I is an identity operator in H t , that is, A has a regular point 1.38 Advances in Difference Equations Abstract Model of Initial Boundary Value Problem with Non Stationary Boundary and Transmission Conditions for the Impulsive Semilinear Hyperbolic Equations Consider the following initial boundary value problem: ui t ă fi t, u t , u t , A i t ui t m Biν ui t ă ă gi t, u t , u t , i Ckν t uk t k 2.1 m i Dkμ uk t 0, k u0 , i ui u1 , i ui ˙ ˙ ¨ u1 , , um , u u1 , , um , A i t , ˙ ˙ where t ∈ 0, T , ν 1, , si , μ 1, , ri , i 1, , m, u i i Biν , Ckν t and Dkμ satisfy all conditions of Theorem 1.2 Assume, that the nonlinear operators fi and giν satisfy the following conditions xi Suppose that the nonlinear operators m ˙ ˙ t, u, u −→ fi t, u, u : 0, T × m × i H1/2 −→ H i , Hi i i m m ˙ ˙ t, u, u −→ giν t, u, u : 0, T × × i H1/2 2.2 Hi i i −→ Xν i satisfy the local Lipschitz conditions in the following sense: for arbitrary t1 , t2 0, T , u1 , v1 , u2 , v2 ∈ H1/2 × H, fi t1 , u1 , v − fi t2 , u2 , v ≤ ci r m |t1 − t2 | Hi u1 − u2 i i i giν t1 , u1 , v − giν t2 , u2 , v2 ≤ ciν r m |t1 − t2 | 1, , si , i m r i 1l vi1 − vi2 i H1/2 Hi , 2.3 i Xν u1 − u2 i i i where ci · , ciν ∈ C R , R , ν ∈ i H1/2 vi1 − vi2 Hi , 1, , m, uli i H1/2 vil Hi 2.4 10 Advances in Difference Equations Theorem 2.1 Let conditions (i)–(x) and (xi ) be satisfied, then there exists T ∈ 0, T , such that the problem 2.1 has a unique solution 0, T , H0 ∩ C1 u1 , , um ∈ C u 0, T , H1/2 ∩ C2 0, T , H 2.5 Additionally, if m E t ui t i ui t ˙ i H1/2 ≤ϕ Hi m u0 i i where ϕ · ∈ C R , R , then T u1 i i H1/2 Hi t ∈ 0, T , , 2.6 T Otherwise, there exists T0 ∈ 0, T , such that ∞ lim E t t → T0 −0 2.7 In the Hilbert space H, the problem 2.1 is represented as the Cauchy problem w ă w where w A0 t w w0 , F t, w, w , ˙ w ˙ 2.8 w1 , u1 , B11 u1 , , B1s1 u1 , , um , Bm1 um , , Bmsm um , w0 u0 , B11 u0 , , B1s1 u0 , , u0 , Bm1 u0 , , Bmsm u0 , m m m 1 w1 u1 , B11 u1 , , B1s1 u1 , , u1 , Bm1 u1 , , Bmsm u1 , m m m 1 F t, w, w ˙ F1 t, w, w ˙ λ0 w F1 t, w, w , ˙ 2.9 ˙ ˙ ˙ f1 t, u, u , g11 t, u, u , , g1s1 t, u, u , , ˙ ˙ ˙ fm t, u, u , gm1 t, u, u , , gmsm t, u, u From xi’ , it follows that, for arbitrary t1 , t2 ∈ 0, T , w1 , w2 ∈ H1/2 , z1 , z2 ∈ H, F t1 , w1 , z1 − F t2 , w2 , z2 where c · ∈ C R , R , r l H wl ≤ c r |t1 − t2 | H1/2 zl H w1 − w2 H1/2 z1 − z2 H , 2.10 Advances in Difference Equations 11 Thus, the nonlinear operator F satisfies the condition of local solvability of the Cauchy problem for the quasilinear hyperbolic equations in Hilbert space see 6, Taking this into account, the problem 2.8 has a unique solution w ∈ C2 0, T ; H ∩ C1 0, T ; H1/2 ∩ C 0, T ; H1 2.11 Initial Boundary Value Problem with Non Stationary Boundary and Transmission Condition for the Impulsive Semilinear Hyperbolic Equations Let a1 < a2 < · · · < am We consider in the domain 0, T × m , the following mixed i problem ˙ fi t, x, ui t, x , ui t, x , ui t, x , ϕi u, u ˙ ui t, x − pi t ui t, x ¨ t, x ∈ 0, T × , ui t, ui t, , qi t ui t, 1 ˙ g0 t, ψ0 u, u − ui t, ui 0, x qm t um t, am u0 x , i ui 0, x ˙ i 1, 2, , m, , t > 0, ˙ gi t, i u, u i um t, am ă , 1, 2, , m − 1, t > 0, i u1 t, a1 − q0 t u1 t, a1 ă ui t, ă , 3.1 1, 2, , m − 1, t > 0, ˙ gm t, ψm u, u u1 x , i , x ∈ , bi , i , t > 0, 1, 2, , m, ui /t, ui ui /x, ui ă ∂2 ui /∂t2 , ui ∂2 ui /∂x2 , u u1 , , um , u where ui ˙ ˙ u1 , , um , pi , qj , fi , gj , ui , ui are some functions, ϕi and ψj are some functionals, which will ˙ be specified below, i 1, , m, j 0, 1, , m Recently, differential equations with impulses are great interest because of the needs of modern technology, where impulsive automatic control systems and impulsive computing systems are very important and intensively develop broadening the scope of their applications in technical problems, heterogeneous by their physical nature and functional purpose see 10, chapter Assume that the following conditions are held: 10 pi t ∈ C1 0, T , qj t ∈ C1 0, T ; pi t > 0, qj t > 0, t ∈ 0, T , i 0, 1, , m, 20 fi · ∈ C1 0, T × , 30 gj · ∈ C1 0, T , R , j × R4 , i 0, 1, , m, 1, 2, , m, 1, , m, j 12 Advances in Difference Equations 40 ϕi · are nonlinear functionals acting from m W2 ak , ak × L2 ak , ak 3.2 k to R and for arbitrary u1 , v , u2 , v2 inequality holds m k ∈ W2 ak , ak × L2 ak , ak the following ϕ i u1 , v − ϕ i u2 , v ≤ ci r m u1 − u2 k k W2 ak ,ak k 1 vk − vk L2 ak ,ak i where r m k u1 k W2 ak ,ak u2 k 1 W2 ak ,ak ci · ∈ C R , R , vk L2 ak ,ak 0, ∞ , i R 3.3 , 1, 2, , m, vk L2 ak ,ak , 1, 2, , m, 3.4 50 ψj · are nonlinear functionals acting from m W2 ak , ak × L2 ak , ak 3.5 k to R and for arbitrary u1 , v , u2 , v2 inequality holds ψj u1 , v1 − ψj u2 , v2 ≤ cj r m k ∈ m W2 ak , ak u1 − u2 k k k where cj · ∈ C R , R , j 60 u0 ∈ W2 , i × L2 ak , ak 1 vk − vk L2 ak ,ak the following , 3.6 0, 1, , m, and r—is defined as in 3.3 , , u1 ∈ W2 , i u0 aj j 1 W2 ak ,ak 1 u1 aj j u1 j aj ,i 1, 2, , m, where u0 j 1 , j aj , 1, 2, , m − By applying Theorem 2.1, we obtain the following result 3.7 Advances in Difference Equations 13 Theorem 3.1 Let conditions (10 )–(60) be held, then there exists a T ∈ 0, T , such that the problem u1 , , um , where 3.1 has a unique solution u ui ∈ C2 0, T ; L2 , ∩ C1 ui t, , ui t, 0, T ; W2 , ∈ C2 0, T , R , ∩C i 0, T ; W2 , j h1 , h2 Hi −1 pi t t −pi t ui , ui ∈ D Ai t B11 u1 u1 a1 , Bj1 i ui −1 qi t h1 h2 , i X2 t ,i 1, 2, , m, and qj t , j , 1, 2, , m, j qm t um am 2, , m, , 0, for all other i, k, i Ci2 t ui qi t ui j ,i −qj t uj 1, 2, , m, Cj2 t uj i Ck2 t 1, 2, , m − 1, 0, for all other i, k, i Di1 ui i Dk1 −ui 1 , Dii 0, k / i, k / i aj ,j 1,1 ui ui , i 1, 2, , m − 1, We also define the nonlinear operators as follows: Fi t, u, v fi t, x, ui x , ui x , vi x , ϕi u, v , i G11 t, u, v g0 t, ψ0 u, v , Gi2 t, u, v gi t, ψi u, v , i Gi1 t, u, v 0, i 2, 3, , m 3.9 1, 2, , m W2 , m −q0 t u1 a1 , Cm1 t um C11 t u1 i Ck1 t 0, B2i ui 1, 2, , si , i uv dx, From differentiability of the functions pi t , i follows that the condition i is satisfied Let us define the following operators: A i t ui , ν h1 , h2 h1 , h2 ∈ , 3.8 −1 q0 t h1 h2 , X1 t , 1, 2, , m i i Proof Let us denote H i L2 , , H0 W2 , , Xν ,Yμ 1, 2, , m, μ 1, 2, , rj , j 1, 2, , m, where si 2,rj i In space H i and Xν are defined the following inner products: u, v 1, 2, , m, 1, 2, , m, 0, 1, , m it 14 Advances in Difference Equations k k It is easy to verify that linear operators Ai t , Biν , Ciν t , and Diμ and the nonlinear operators Fi , Gi1 , and Gi2 , i 1, , m satisfy the conditions of Theorem 2.1, and the problem 3.1 is represented as an abstract initial boundary-value problem in the following way: ui t ă i Ci2 t ui t i Ci 2,2 i Di1 ui 0, i ˙ Gi t, u, u , t ui t 3.10 ˙ Gm t, u, u , m Cm2 t um t ă Bm2 um t i Di1 ui G0 t, u, u , C11 t u1 t ă B11 u1 t Bi2 ui t ă Fi t, u, u , A i t ui t 1, 2, , m − We will show that conditions of Theorem 2.1 are satisfied Conditions i – v follow j i i immediately from definitions of spaces H i , Xν , and Yμ and operators Ai t , Biν , Ckν t , j and Dkμ , and traces theorems see 3, chapter , where k 1, 2, , m; ν i 1, 2, , m; μ 1, 2, , rj ; j 1, 2, , m The linear manifolds H0 and H1 are defined in the following way: H0 u1 , , um , ui ∈ W2 , u, u uj aj uj aj ,j ,i 1, 2, , si ; 1, , m, 1, , m − , 3.11 H1 w, w wi w1 , , wm , w1 ui , ui u1 , u1 a2 , u1 a1 , 2, , m, u ∈ H0 ,i We also define the spaces H1/2 u, u u1 , , um , ui ∈ W2 , , i H1/2 u, u u1 , , um , ui ∈ W2 , ,i uj aj uj aj ,j 1, , m , 1, , m, 3.12 1, , m − Statement 3.2 H1 is dense in H L2 a1 , b1 ⊕ ⊕ ⊕ m i L2 , bi ⊕ 3.13 Advances in Difference Equations 15 Proof Assume that u1 , α1 , α0 , u2 , α2 , , um , αm ∈ H Consider the following functions: − x αi−1 − u0 x i From definitions of u0 x , i i u0 i Let u x − αi , − x ∈ , ,i 1, , m 3.14 1, , m, we can see that u0 i αi , 1, 2, , m − i 3.15 u1 , , um ∈ H Consider the function z u1 − u0 , , um − u0 m z1 , , zm 3.16 m m It is obvious that z ∈ i L2 , On the other hand, i D , , ai , , where D , i 1, , m is a space of infinitely differentiable finite functions Therefore, for an arbitrary ε > 0, there exist the functions hi ∈ D , , i 1, , m, such that m i L2 m zi − hi < ε 3.17 i By denoting hi u0 i hi from 3.17 , we get m i where hi ∈ C∞ , Thus, , hi αi−1 , i ui − hi L2 ,ai < ε, 3.18 1, , m u1 , u1 a2 , u1 a1 , u2 , u2 a2 , , um , um am − h1 , α1 , α0 , h2 , α1 , , hm , αm H < ε 3.19 The following statement is proved in the same way Statement 3.3 H0 is dense H1/2 Now, we prove that the condition vi holds 16 Advances in Difference Equations Let u m v1 , , vm ∈ H0 , then u1 , , um , v ⎡ ⎤ si ν ⎣ Ai t ui , vi − m ui ⎦ t uk , Biν vi i Xν t −u1 a1 , v1 a1 ui vi dx i k i Ckν Hi t i m m − ui , vi − ui 1 , vi 1 um am , vm am i m ui , vi i m − m−1 ui vi dx − u1 a1 , v1 a1 i − ui vi i m−1 ui vi um am 1 vm am i m vi 1 m−1 − u1 a1 v1 a1 m vm am i m i ui vi dx i ui vi − i um am m i ui vi − ui ui vi ui vi dx 3.20 Similary, we obtain the following identity: m ⎡ ⎣ ui , Ai t vi si Hi ⎤ m t i ν ⎦ i Ckν t uk Biν νi , k i Xν t m i ui vi dx 3.21 Thus, by virtue of 3.20 - 3.21 , the condition vi holds From 3.20 or 3.21 , putting vi ui , we also obtain the identity m i m u2 dx i ⎡ that is, condition viii is satisfied, c1 c2 k 1 ⎦, i Ckν t uk , Biν ui Hi t i ⎤ m ν ⎣ A i t ui , ui si i Xν t 3.22 Advances in Difference Equations 17 Now, we verify fulfillment of condition ix To that end, we consider the mixed problem λui − pi t ui hi x , i 1, 2, , m, λu1 a1 − q0 t u1 a1 λui qi t ui λum am − ui h10 , hi0 , qm t um am 3.23 1, 2, , m − 1, i 3.24 hm0 , where hi ∈ L2 , ,i 1, , m; hj0 ∈ R, j 0, 1, , m, λ ∈ R Let hi x be the extend of function hi x to R We consider the system of the differential equations λui − pi t uixx hi x , i 3.25 1, 2, , m Hence, we have λui − k2 pi t uixx where g hi x , i 3.26 1, 2, , m, F g is a Fourier transformation of the function g x From 3.26 , we obtain F −1 ui F −1 hi / λ k2 pi t satisfy 3.25 , and u hi / λ k2 pi t , then functions ui their constrictions on , satisfy the 3.23 It is clear that ui ∈ W2 , Considering linearity of the problem 3.23 , 3.24 , the solution can be represented in the form vi ui where vi ui , 3.27 ui − ui is a solution of the following problem: λvi x − pi t vi x 0, λv1 a1 − q0 t v1 a1 λvi qi t vi λvm am where h10 h10 − λu1 a1 hi0 1 − vi 3.28 h10 , hi0 , qm t vm am 1, 2, , m − 1, i 3.29 hm0 , q0 t u1 a1 , hi0 − λui hm0 − qi t ui − ui hm0 − λum am 1 , i qm t um am 1, 2, , m − 1, 3.30 18 Advances in Difference Equations A general solution of a system 3.28 is found in the following form: √ ci1 e− x−ai vi x ci2 e− bi −x λ/pi t Then, for determination of ci1 , ci2 , i system of the algebraic equations: √ c12 e− a2 −a1 λ c11 λ ci1 e− √ −ai λ − pi ci1 e− 1 −ai λ cm1 e− am ci t √ 1,1 −am , c11 − c12 e− a2 −a1 − ci2 qi t λ pi t ci1 e− ci − −ai 1,2 e √ λ/pm t 1,1 − ci λ/pi 1,2 e t −ai √ hi0 , − −ai −cm1 e− am cm2 1, 2, , m 3.31 √ λ/pi t λ/pi t h0 , ci2 3.32 √ i 1, 2, , m, from 3.29 , we get the following λ pi t − ci2 − ci λ/pi t λ/pi t − q0 t λ/pi t λ/pi t √ √ λ/pi 1 −am 1, 2, , m − 1, i t 0, λ/pm t cm2 √ i 1, , m − 1, hm0 Let R λ be a matrix of coefficients of system 3.32 From 3.32 , it is clear that R λ R0 λ R1 λ , where det R0 λ → ∞ and det R1 λ → as λ → ∞ Thus, for sufficiently large λ, R λ is invertible and det R λ → ∞ Therefore, the system 3.32 has a unique solution Thus, for sufficiently positive large λ, the problem 3.23 - 3.24 has a unique solution u u , , um ∈ H Thus, the condition ix is satisfied The fulfillment of other conditions follows from 10 – 60 Now, let us consider a class of nonlinear equations, for which the large solvability theorem takes place Let ˙ fi t, x, ui , ui , ui , ϕ u, u ˙ ˙ g0 t, ψ0 u, u ˙ gi t, ψi u, u where ρi ≥ 0, τj ≥ 0, i −|ui |ρi ui −|u1 a1 |τ0 u1 a1 −|ui |τi ui 1, 2, , m; j 70 f1i , g1j, ϕi and ψj , i ˙ f1i t, x, ui , ui , ui , ϕi u, u ˙ ˙ gi1 t, ψi u, u , i , 3.33 1, 2, , m, 0, 1, , m and 1, 2, , m, j 80 |fi t, x, ui , vi , ξi , η | ≤ c ˙ g01 t, ψ0 u, u , |ui | ρi 1, 2, , m satisfy the conditions 20 − − 50 /2 |vi | |ξi | |η| , Advances in Difference Equations 90 |g0i t, η | ≤ c 10 where yi |η| , |ϕi u, v | ≤ c , i 19 n i |ui | ρi 0, 1, , m, ρ /2 |vi |2 max mini |ui yi | τi 1,2, ,m /2 , ρi , Theorem 3.4 Let conditions 70 – 100 be held and initial data satisfy the condition 60 , then the u1 , , um , where problem 3.1 has a unique solution u ui ∈ C2 0, T ; L2 , ∩ C1 ui t, , ui t, 1 0, T ; W2 , ∈C 0, T ; R , i ∩C 0, T ; W2 , , 3.34 1, 2, , m References Y Yakubov, “Hyperbolic differential-operator equations on a whole axis,” Abstract and Applied Analysis, no 2, pp 99–113, 2004 P Lancaster, A Shkalikov, and Q Ye, “Strongly definitizable linear pencils in Hilbert space,” Integral Equations and Operator Theory, vol 17, no 3, pp 338–360, 1993 J.-L Lions and E Magenes, Problemes aux Limites non Homogenes et Applications, vol 1, Dunod, Paris, France, 1968 M Reed and B Simon, Methods of Modern Mathematical Physics II Fourier Analysis, Self-Adjointness, Academic Press, New York, NY, USA, 1975 T Kato, “Linear evolution equations of “hyperbolic” type II,” Journal of the Mathematical Society of Japan, vol 25, pp 648–666, 1973 T J R Hughes, T Kato, and J E Marsden, “Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity,” Archive for Rational Mechanics and Analysis, vol 63, no 3, pp 273–294, 1977 S Yakubov and Y Yakubov, Differential-Operator Equations Ordinary and Partial Differential Equations, vol 103 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2000 S Q Krein, Linear Differential Equation in Banach Spaces, Nauka, Moscow, Russia, 1967 A B Aliev, “Solvability “in the large” of the Cauchy problem for quasilinear equations of hyperbolic type,” Doklady Akademii Nauk SSSR, vol 240, no 2, pp 249–252, 1978 10 A M Samo˘lenko and N A Perestyuk, Impulsive Differential Equations, vol 14 of World Scientific Series ı on Nonlinear Science Series A: Monographs and Treatises, World Scientific, River Edge, NJ, USA, 1995  ... elastodynamics and general relativity,” Archive for Rational Mechanics and Analysis, vol 63, no 3, pp 273–294, 1977 S Yakubov and Y Yakubov, Differential-Operator Equations Ordinary and Partial... Advances in Difference Equations Abstract Model of Initial Boundary Value Problem with Non Stationary Boundary and Transmission Conditions for the Impulsive Semilinear Hyperbolic Equations Consider... Differential Equations, vol 103 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2000 S Q Krein, Linear Differential Equation in Banach