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1 Transactions of NAS of Azerbaijan, Issue Mathematics, 36 4, Series of Physical-Technical and Mathematical Sciences. Solvability of a boundary value roblem for second order differential-oerator equations with a sectral arameter in both the equation and boundary conditions Bahram A. Aliev Nargul K. Kurbanova Received:..6 / Revised: 9.7.6/ Acceted:.8.6 Abstract. In ilbert sace we study noncoercive solvability of a boundary value roblem for second order ellitic differential equations with a sectral arameter in the equation and boundary conditions in the case when in one of boundary conditions in addition to the sectral arameter there exists a linear bounded oerator in the rincial art of the given condition Keywords. differential-oerator equations, ellitic equations, sectral arameter, izomorfizm, noncoercive estimation. Mathematics Subject Classification : 47E5, 47A75, 34L5, 34G, 35J5 Introduction Boundary value roblems for second order differential oerator equations in the case when one and the same sectral arameter enters into the equation and boundary conditions were studied in different asects in the aers of V.I. Gorbachuk and M.A. Rybak [4], M. A. Rybak [6], M. A.Denche [], B. A. Aliev [,3,4], B. A. Aliev and Ya. Yakubov [5], A. Aibeche, A. Favini and Mezoued [], M. Bayramoglu and N.M.Aslanova [], B. A. Aliev and N. K. Kurbanova [6], B. A. Aliev, N. K. Kurbanova and Ya.Yakubov [], Ya.Yakubov [7] and others. In the aer [8] B.A.Aliev and Ya.Yakubov, in UMD Banach sace E the following boundary value roblem was studied for a second order ellitic differential-oerator equation with a sectral arameter in the case when one of the boundary conditions contains a linear bounded oerator: L λ, D u := λu x u x Au x A u x = f x, x,,. B.A. Aliev Institute of Mathematics and Mechanics, Baku, Azerbaijan, Azerbaijan State Pedagogical University Baku, Azerbaijan aliyevbakhram@yandex.ru N.K. Kurbanova nargul q@mail.ru Institute of Mathematics and Mechanics, Baku, Azerbaijan

2 B.A. Aliev, N.K. Kurbanova 3 L u := αu Bu N γ j u x j T u = f, L u := βu N γ j u x j T u = f,. where λ is a comlex arameter, A is an inversible R- sectorial oerator in E: B is a linear unbounded oerator in subjected to the oerator A / in the certain sense; A is a linear bounded oerator in L, ; E ; T k, k =, are linear oerators from L, ; E in E: α, β, γ kj are any fixed comlex numbers, α, β ; x kj, ; N k some integers; D := d dx. In the aer [8] for roblem.,. at rather large λ from some angle containing a ositive semi- axis, a theorem on an isomorhism between the solutions and the right hand side of roblem.,. in sace L, ; E,,, was roved, and it was established that for boundary value roblem.,. it holds coercive solvability with resect to u. Solvability of boundary value roblems in ilbert sace for fourth order differentialoerator equations without a sectral arameter with unbounded oerators in one of boundary conditions was studied in the aers of A. R. Aliev and E. S. Rzayev [9], E.S.Al- Aidarous, A.R.Aliev, E. S. Rzayev and. A. Zedan []. In the resent aer, in searable ilbert sace we study solvability of a boundary value roblem of equation. with A =, with the following boundary conditions L λu := αu λbu N γ j u x j = f L u := βu N γ j u x j = f..3 As is seen, unlike boundary conditions., in boundary conditions.3 these is a sectral arameter in the rincial art of the boundary condition. Existence of a sectral arameter in boundary conditions.3 qualitatively influences on solvability of boundary value roblem.,.3. The matter is that when we look for the solution of roblem.,.3 belonging to W, ; A, the element f can not be taken from interolational sace A, // that is dictated by the theorem on traces, it is taken from narrower interolational sace, more exactly, from A, /, though the element f is taken from A, //. As a sectral arameter exists in boundary conditions.3, we can not take the oerator B unbounded as it was taken from boundary conditions., we take it only bounded. As the element f can t be found in interolational sace, A, // by studying the solvability of boundary value roblem.,.3, it is imossible to take the vector-valued function fx from the sace L, ;, ;, as it was done in the aer [8]. We have to take it from narrower sace, more exactly from L, ; A /. As a result, in the resent aer, for roblem.,.3 a theorem on isomorisim dosen t hold. For the solution of roblem.,.3 belonging to W, ; A, for rather large λ from the angle arg µ ϕ < π noncorrosive estimation with resect to u was obtained, where ϕ [, π is any fixed number, i.e. for boundary value roblem.,.3 the noncoercive estimation with resect to u was obtained.

3 4 Solvability of a boundary value roblem for second order... Note that similar cases haen also by studying solvability of boundary value roblems for equation. with the following Birkhoff-Tamarkin irregular boundary conditions: L u := αu βu γu δu = f, L u := αu βu = f..4 In S.Yakubov and Ya.Yakubov s monograh [8] boundary value roblem.,.4 is studied in ilbert sace. It is shown that by studying solvability of boundary value roblems of the form.,.4 in the sace L, ; it is not succeeded to take the elements f k, k =, from the natural interolational sace, they are taken from narrower interolational sace. There with, the right hand side of equation. i.e. fx, is taken not from L, ; as in [8], but from the sace L, ; A /. In the monograh [8] it is roved that if in.4 f k A, k// and fx L, ; are taken, then for rather large from the angle arg µ ϕ for the solution of roblem.,.4 belonging to W, ; A,, a weaker estimation is obtained. Note that solvability for boundary value roblems of tye.,.4 in UMD Banach sace were studied in A. Favini and Ya. Yakubov s aer [3]. Note that even if in boundary conditions.3 B = I a unit oerator is taken, then for boundary value roblem.,.3 the obtained result is also new and is stated first in this aer. Let us introduce definitions and notion that are used in the resent aer. Let E and E be Banach saces. The set u, v of all vectors of the form u E, where v E with ordinary coordinate linear oerations and with the norm u, v E E := u E v E is a Banach sace and is called the direct sum of Banach saces E and E. Let E and E be two Banach saces. Denote by BE, E a Banach sace of all linear bounded oerators acting from E to E with ordinary oerator norm. In secial case we assume BE := BE, E. Definition. Linear closed oerator A in ilbert sace will be called strongly ositive if the domain of definition DA is dense in for some ϕ [, π for all oints from the angle arg µ ϕ including µ = these exist the oerators A µi and for these µ it holds the estimation A µi B C µ, where I is a unit oerator in, C = const >. For ϕ =, the oerator A is said to be ositive. The simlest examle of ositive oerators are selfadjoint ositive-definite oerators acting in a ilbert sace. Note that the strong ositivity of the oerator A yields the strong ositivity of the oerator A α, α,. Let A be a strongly ositive oerator in. As the inverse oerator A is bounded in, then } A n := {u : u D A n, u A n = An u, n N, is a ilbert sace whose norm is equivalent to the norm of the grah of the oerator A n. If the oerator A is strongly ositive in, it is known that the oerator A is a generator of the semigrou e ta analytic for t >, and this semigrou exonentially decreases, i.e., there exist two numbers C >, σ > such that e ta Ce σ t, t <. By [5,theorem.5.5], the oerator A / generates an analytic semigrou for t >, decreasing at infinity.

4 B.A. Aliev, N.K. Kurbanova 5 Definition. [7, theorem.4.5] Let A is a strongly ositive oerator in, Then international saces A n, θ of ilbert saces A n and, are defined by the equality A n, θ := {u : u, u A n, := θ = t nθ A n e ta u } dt <, θ,, >, n N. We denote, A n, := A n, A n, :=. Denote by L, ; < < a Banach sace for = a ilbert sace of vector-functions x u x : [, ] strongly measurable and summable in th with the norm / u := L,; u x dx < and by W n, ; A n, := { u : A n u, u n L, ; } denote a Banach sace of vector-functions with the norm u u W n,;a n, := An u n L,;. L,; It is known that [7, theorem.8.] if u W n, ; A n, then, u j A n, j n, j =,..., n. omogeneous Equations As first, consider the following boundary value roblem in searable ilbert sace L λ, D u := λu x u x Au x =, x,,. L λ u := αu λbu N γ j u x j = f, L u := βu N γ j u x j = f. Theorem. Let the following conditions be fulfilled:. A is a strongly ositive oerator in ;.The linear oerator B is bounded from into and from A into A; 3. α, β, γ kj are some comlex numbers, and α, β ; x kj,.. Then for, f A,, f A, and for sufficiently large λ from the angle arg λ ϕ < π, roblem.,. has a unique solution u W, ; A,, such that u DB, and the following noncoercive estimate holds for this solution: λ u L,; u L,; Au L,; C k= f k A, λ k f k k..3

5 6 Solvability of a boundary value roblem for second order... Proof. By [8, lemma 5.4./6] for arg λ ϕ < π, there exists the analytic for x > and strongly continuous for semigrou e xaλi/. By [8,lemma 5.3./], for the function ux be the solution of equation. belonging to W, ; A,,, it is necessary and sufficient that for arg λ ϕ < π, u x = e xaλi/ g e xaλi/ g,.4 where g, g A,. By [7,theorem.8.] see also [8, theorem.7.7/] u A,. From condition and interolation theorem [7, theorem.3.3/a] it follows that the oerator B is bounded from A, θ into A, θ for any θ,. So, u D B. Require the function.4 to satisfy the boundary conditions.. Then we get the following system for the elements g and g. N α A λi / e AλI/ γ j e x jaλi / λb g [α A λi / λbe AλI/ N γ j A λi / e x jaλi / g = f, N β A λi / γ j A λi / e x jaλi / g [β A λi / e AλI/ N γ j A λi / e x jaλi / g = f..5 We write system.5 in sace := A, A, in the from of the oerator equation g f A λ R λ =,.6 g f. where A λ and R λ are oerator matrices of dimension : λb α A λi / A λ = β A λi /, and D A λ := A,. A, K λ K R λ := λ, D R λ :=, K λ K λ

6 where B.A. Aliev, N.K. Kurbanova 7 N K λ := α A λi / e AλI/ γ j A λi / e x jaλi /, N K λ := λbe AλI/ γ j A λi / e x jaλi / N K λ := γ j A λi / e x jaλi /, N K λ := β A λi / e AλI/ γ j A λi / e x jaλi /. Show that the oerator A λ in sace for λ from the angle arg λ ϕ < π, has bounded inverse A λ., acting from into A, A, and it holds the estimation A λ B. C,.7,A, A, where C > is a constant indeendent of λ. As formally A λ has the form A β A λi / λ = α A λi / αβ λ AλI / B AλI /, then for that it sufficiencies to show that: a the oerator A λi, for arg λ ϕ < π, is bounded from A, into A, and it holds the estimate A λi / B C,.8 A,,A, where C > is a constant indeendent on λ; b the oerator A λi, for arg λ ϕ < π, is bounded from A, into A, and it holds the estimate A λi B A, C λ,.9 where C > is a constant indeendent on λ; c oerator λ A λi B A λi /, for arg λ ϕ < π, bounded from A, into A,, and it holds the estimate λ A λi B A λi C,. where C > is a constant indeendent of λ. B A,

7 8 Solvability of a boundary value roblem for second order... Item a was roved in the aer [8]. Prove b, by [8, lemma 5.4./6], for arg λ ϕ < π, the oerator A λi is bounded from into and it holds the estimate A λi C λ.. B Then it is obvious that A λi = A A λi A BA B = A λi C λ.. B From. and., by the interolation theorem [7, theorem.3.3/a], it follows that the oerator A λi, for arg λ ϕ < π, bounded from A, θ into A, θ, for any θ,, and it holds the estimate A λi A BA,θ C λi θ B A λi θ A C λi C λ..3 BA B We take in.3 θ =. Then we get.9, i.e. b is roved. Prove c, by. and condition, for arg λ ϕ < π we have λ A λi B A λi λ A λi B B A B B λi C λ λ C,.4 B [ ] λ A λi B A λi = A λ A λi B A λi A BA = λ A λi ABA A λi λ A λi ABA B B A λi C λ λ C..5 B By the above interolational theorem, from estimations.4 and.5 it follows that the oerator λ A λi B A λi, for arg λ ϕ < π, bounded from A, θ into A, θ for any θ, in articular, for θ = it holds estimation., i.e. c is roved. Estimations.8,.9 and. yield.7. Then from equation.6 we have I A λ g R λ = A λ g f..6 f From reresentations A λ and R λ it is seen that the roduct A λ R λ is an oerator -matrix whose members are linear combinations of the oerators e AλI/, λ A λi Be AλI /, e x kjaλi /, e x kjaλi /, λ A λi Be x j AλI /, λ A λi Be x j AλI /.

8 B.A. Aliev, N.K. Kurbanova 9 We can show that all oerators in the oerator-matrix A λ R λ, for arg λ ϕ < π, bounded from A, into A,. For examle, we show this for the oerator λ A λi Be AλI /. By [8, lemma 5.4./6] and condition for arg λ ϕ < π, we have λ A λi Be AλI / B λ A λi B B B e AλI/ B C λ λ / e ω λ / C λ / e ω λ /, C >, ω > ;.7 λ A λi Be AλI / BA [ = λ A A λi / Be AλI/] A B λ A λi / B ABA B e AλI/ B C λ e ω λ /, C >, ω >..8 By interolational theorem [7, theorem.3.3/a], from estimations.7,.8 it follows that for arg λ ϕ < π, the oerator λ A λi / Be AλI/ is bounded from A, θ into A, θ, for any θ,, including from A, into A, for θ = and it holds the estimate λ A λi / Be AλI/ B A, C λ e ω λ /, C >, ω >..9 In a similar way it is roved that remaining members of the oerator-matrix A λ R λ bounded from A, into A, and for the norms of these members, for arg λ ϕ < π, the estimations of tye.9 hold. So, for rather large λ from the angle arg λ ϕ < π, the oerator A λ R λ is bounded from.. A, A, into A, A, and it holds the estimate A λ R λ. B A, A, C λ e ω λ <, C >, ω >.. ence, by the Neumann identity, for rather large λ from the angle arg λ ϕ < π, k I A λ R λ = I k A λ R λ,. k=

9 3 Solvability of a boundary value roblem for second order... where the series in the right hand side of. converges in the norm of the sace of. bounded oerators in A, A,. By. and. from.6 for arg λ ϕ < π and λ, we have g = I A λ g R λ A λ f. f Consequently, for sufficiently large λ from the angle arg λ ϕ < π, the elements g and g can be reresented in the from: where g k = C k λ R k λ f C k λ R k λ f, k =,,. C λ =, C λ = β A λi, C λ = α A λi, C λ = αβ λ A λi B A λi, R kj λ, k, j =, are some bounded oerators from A, into A,, for arg λ ϕ and λ. Moreover, using the estimaties.7 and., one can show that, for arg λ ϕ < π and λ, R kj λ C λ e ω λ, C, ω >..3 B A, From the reresentation A λ and A λ R λ it also follows that, for sufficiently large λ from the angle arg λ ϕ < π, for the oerators R kj λ the estimate: Substituting. in.4, we get R kj λ B C λ e ω λ, C, ω >..4 ux = k= { e xaλi/ C k λ R k λ e xaλi/ C k λ R k λ}f k. Then, for sufficiently large λ from the angle arg λ ϕ < π, we have C λ u L,; u L,; Au L,; { [ / λ e xaλi/ C k λ f k dx k= / e xaλi/ R k λ f k dx / e xaλi/ C k λ f k dx

10 B.A. Aliev, N.K. Kurbanova 3 / ] e xaλi/ R k λ f k dx A A λi [ / A λi e xaλi/ C k λ f k dx / A λi e xaλi/ C k λ f k dx / ] } A λi e xaλi/ R k λ f k dx..5 By [8, theorem 5.4./] and estimations. and.4 for sufficiently large λ from the angle arg λ ϕ, for the first term of the right hand side of inequality.5 for k = we have: / λ e xaλi/ C λ f dx = λ e xaλi/ αβ λ A λi / / B A λi f C λ A λi B / dx / A λi e xaλi/ λ A λi / B A λi / f dx C λ A λi / B A λi / f λ C A, λ A λi / B A λi / f f A, λ f. By [8, theorem 5.4./] for sufficiently large λ from the angle arg λ ϕ, for the first term of the right hand side of inequality.5 for k = we have / λ e xaλi/ C λ f dx / C λ e xaλi/ A λi / f dx C λ A λi / B A λi / e xaλi/ f dx

11 3 Solvability of a boundary value roblem for second order... / C A λi / e xaλi/ f dx C A λi / / B A λi e xaλi/ f dx C λ f / A, λ f C f A, λ f..6 Estimate the integral with C λ f in the third term of the right hand side of inequality.5. Again, according to [8, theorem 5.4./] and for the same λ we have / λ e xaλi/ C λ f dx = λ / e xaλi/ α A λi / f dx C λ A λi / B A λi / e xaλi/ f dx C f A, λ f. The remaining summands of the right hand side are estimated in the same way. Theorem. is roved. 3 Nonhomogeneous Equations Now we consider a boundary value roblem for an nonhomogeneous equation with a arameter in, i.e. the roblem L λ, D u := λu x u x Au x = f x, x,, 3. L λ u := αu λbu N γ ju x j = f, L u := βu N γ ju x j = f. 3. Theorem 3. Let the conditions of theorem. be fulfilled: Then, for f L, ; A /, f A,, f A, and for sufficiently large λ from the angle arg λ ϕ < π, roblem 3., 3. has a unique solution from W, ; A, and for this solution it holds the following noncoercive estimation λ u L,; u Au L,; L,; [ C λ f L,;A / ] f k A, λ k f k k. 3.3 k=

12 B.A. Aliev, N.K. Kurbanova 33 Proof. The uniqueness follows from theorem.. The solution of roblem 3.,3., belonging to W, ; A, in the form of the sum u x = u x u x, where u x if [, ] is the contraction on [,] of the solution of the equation L λ, D ũ x = f x, x R = ;, 3.4 where f x := f x, if x [, ], and f x =, if x / [, ], and u x is the solution of the roblem L λ, D u =, x,, L λ u = f L λ u, 3.5 L u = f L u. As is shown in the roof [3, theorem ] if f L R; A /, then for arg λ ϕ < π, the solution of equation 3.4 belongs to the sace W R; A 3/, A / and for the solution is holds the estimation ũ W R;A 3/,A / C f 3.6 LR;A / uniformly with resect to λ. ence it follows that u W, ; A 3/, A / W, ; A,. Then from 3.4 and 3.6 it follows that λ u L,; u Au L,; L;; C f L,;A /. 3.7 By [7, theorem.8.] see also [8, theorem.7.7/], for any fixed x [, ], u x A 3/, A /, u x A 3/, A /. By [7, lemma.7.3/,.7.3/6 and.7.3/5], for k =,, we have A 3/, A /, = Consequently, A / k = A 3/ k, A 3/ =, A k k 4 = A, 4 k 4. u x A, 4, u x A, 4 4. By [7, theorem.3.3/b and theorem.5./f], we have A, 4 =, A =, A = A,.

13 34 Solvability of a boundary value roblem for second order... On the other hand, at the beginning of the roof of theorem. it was shown that the oerator B is bounded from A, θ into A, θ for any θ,. ence Similarly L λ u A, A,. 3.8 L u A,. 3.9 Furthermore A, 4 4 A, 4 = A, A,. 3. Then by theorem., for sufficiently large λ from the angle arg ϕ < π, for the solution of roblem 3.5 we have u λ u L,; Au L,; L,; C [ f L λ u A, λ f L λ u C ] f L u A, λ f L u f A, f A, λ f u N u x j λ f A, λ Bu A, A, N u x j u A, A,. 3. By [7, theorem.8.] see. also [8, theorem.7.7/] and inequality 3.7 we have x [, ] u x C u W A,,;A, C f L,;A /. 3. Estimate the exression λ Bu A,. Above we have already noted that the oerator B is bounded from A, θ into A, θ, for any θ,. Then by [7, theorem.8.] see. also [ 8, theorem.7.7/] and estimation 3. and 3.7 we have λ Bu A, λ B B A, u A, C λ u A,

14 B.A. Aliev, N.K. Kurbanova 35 C λ u W,;A, C λ f L,;A /. 3.3 Now estimate the norms u x, for any fixed x [, ]. Using [7, A, theorem.5. and.5.4/], we can show A, 4 = A 3, A. Then by [7, theorem.8.] see. also [8, theorem.7.7/] and 3.6,we have u x u x = A, A, 4 = u x A 3/,A / C u W,;A 3/,A / C f L,;A /. 3.4 Taking into account estimations in in 3. for sufficiently large λ from the angle arg λ ϕ < π, we have u λ u L,; Au L,; L,; k= C [ λ f L,;A / ] k f k A, λ f k k. 3.5 From 3.7 and 3.5 it follows 3.3. Theorem 3. is roved. References. Aibeche, A., Favini, A., Mezoued, Ch: Deficient coerciveness estimate for an abstract differential equation with a arameter deendent boundary conditions. Boll Unione Mat. Ital. Sez. B Artic. Ric. Mat. 8 3, Aliev, B.A.: Asymtotic behavior of eigenvalues of a boundary value roblem with sectral arameter in the boundary conditions for the second order ellitic differentialoerator equation. Trans. NAS Azerbaijan. Ser. Phys-Tech. and Math. Sci. 5 7, Aliev, B.A.: Asymtotic behavior of eigenvalues of a boundary value roblem for a second order ellitic, differential-oerator equation. Ukr. Matem. Zhurnal. 58 8, in Russian. 4. Aliev, B.A.: Solvability of a boundary value roblem for second order ellitic differential-oerator equation with a sectral arameter in the equation and in the boundary conditions. Ukr.Matem.Zhurnal 6, 3-4 in Russian. 5. Aliev, B.A., Yakubov, Ya.:Ellitic differential-oerator roblems with a sectral arameter in both the equation and boundary-oerator conditions. Adv. Differential. Equat.,, 8-6, Erratum in 9, Aliev, B.A., Kurbanova, N.K.: Asymtotic behavior of eigenvalues of a boundary value roblem for a second order ellitic differential-oerator equation. Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 4, Secial Issue,

15 36 Solvability of a boundary value roblem for second order Aliev, B.A., Kurbanova, N.K., Yakubov, Ya.: Solvability of the abstract Regge boundary value roblem and asymtotic behavior of eigenvalues of one abstract sectral roblem. Rivista di Matematica della Universita di Parma, acceted. 8. Aliev, B.A., Yakubov, Ya.: Second order ellitic differential oerator equations with unbounded oerator boundary conditions in UMD Banach saces. Integral Equations and Oerator Theory, 69, Aliev, A.R, Rzayev, E.S.: Solvability of boundary value roblem for ellitic oeratordifferential equations of fourth order with oerator boundary conditions. Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan. 4, Secial Issue, Al-Aidarous, E.S., Aliev, A.R, Rzayev, E.S., Zedan,.A.: Fourth order ellitic oerator-differential equations with unbounded oerator boundary conditions in the Sobolev-tye saces. Boundary Value Problems, 5:9, 9, Bairamoglu, M., Aslanova, N.M.: Distribution of eigenvalues and trace formula for the Sturm-Liouville oerator equation, Ukr. Matem. Zhurnal, 6 7, in Russian.. Denche, M.: Abstract differential equation with a sectral arameter in the boundary conditions. Result. Math., 35, Favini, A., Yakubov, Ya.: Irregular boundary value roblems for second order ellitic differential-oerator equations in UMD Banach saces. Math. Ann., 348, Gorbachuk, V.I., Rybak, M.A.: On boundary value roblems for Strum-Liouville equation with a sectral arameter in both the equation and in the boundary condition. Direct and universe roblems of scattering theory, Kiev, in Russian. 5. Krein, S.G.: Linear differential equations in Banach sace. M.Nauka, 967 in Russian. 6. Rybak, M.A.: On asymtotic distribution of eigen values of boundary value roblems for Sturm- Liouville oerator equation. Ukr. Matem. Zhurnal, 3, in Russian. 7. Tribel, Kh.: Interolation theory. Functional saces.differential oerators. M. Mir 98 in Russian. 8. Yakubov, S., Yakubov, Ya.: Differential-Oerator Equations. Ordinary and Partial Differential Equations. Chaman and all / CRC, lacecityboca Raton,. 9. Yakubov, Ya.: Ellitic differential-oerator roblems with the sectral arameter in both the equation and boundary conditions and the corresonding abstract arabolic initial boundary value roblems, New Prosects in Direct, Inverse and Control Problems for Evolution Equations, Sringer In. series,