Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 5, Issue 2, Article 38, 2004 BOUNDARY VALUE PROBLEM FOR SECOND-ORDER DIFFERENTIAL OPERATORS WITH MIXED NONLOCAL BOUNDARY CONDITIONS M. DENCHE AND A. KOURTA L ABORATOIRE E QUATIONS D IFFERENTIELLES , D EPARTEMENT DE M ATHEMATIQUES , FACULTE DES S CIENCES , U NIVERSITE M ENTOURI , 25000 C ONSTANTINE , A LGERIA . denech@wissal.dz Received 15 October, 2003; accepted 16 April, 2004 Communicated by A.G. Ramm A BSTRACT. In this paper, we study a second order differential operator with mixed nonlocal boundary conditions combined weighting integral boundary condition with another two point boundary condition. Under certain conditions on the weighting functions and on the coefficients in the boundary conditions, called non regular boundary conditions, we prove that the resolvent decreases with respect to the spectral parameter in Lp (0, 1), but there is no maximal decreasing at infinity for p ≥ 1. Furthermore, the studied operator generates in Lp (0, 1) an analytic semi group with singularities for p ≥ 1. The obtained results are then used to show the correct solvability of a mixed problem for a parabolic partial differential equation with non regular boundary conditions. Key words and phrases: Green’s Function, Non Regular Boundary Conditions, Regular Boundary Conditions, Semi group with Singularities. 2000 Mathematics Subject Classification. 47E05, 35K20. 1. I NTRODUCTION In the space Lp (0, 1) we consider the boundary value problem 00 L (u) = u = f (x) , 0 0 B1 (u) = a0 u (0) + b0 u (0) + c0 u (1) + d0 u (1) = 0, (1.1) R R 0 1 1 B2 (u) = 0 R (t) u (t) dt + 0 S (t) u (t) dt = 0, where the functions R, S ∈ C([0, 1] , C). We associate to problem (1.1) in space Lp (0, 1) the operator: Lp (u) = u00 , ISSN (electronic): 1443-5756 c 2004 Victoria University. All rights reserved. The authors are grateful to the editor and the referees for their valuable comments and helpful suggestions which have much improved the presentation of the paper. 147-03 2 M. D ENCHE AND A. KOURTA with domain D(Lp ) = {u ∈ W 2,p (0, 1) : Bi (u) = 0, i = 1, 2} . Many papers and books give the full spectral theory of Birkhoff regular differential operators with two point linearly independent boundary conditions, in terms of coefficients of boundary conditions. The reader should refer to [7, 10, 20, 21, 22, 28, 31, 33] and references therein. Few works have been devoted to the study of a non regular situation. The case of separated non regular boundary conditions was studied by W. Eberhard, J.W. Hopkins, D. Jakson, M.V. Keldysh, A.P. Khromov, G. Seifert, M.H. Stone, L.E. Ward (see S. Yakubov and Y. Yakubov [33] for exact references). A situation of non regular non-separated boundary conditions was considered by H. E. Benzinger [2], M. Denche [4], W. Eberhard and G. Freiling [8], M.G. Gasumov and A.M. Magerramov [12, 13], A.P. Khromov [18], Yu. A. Mamedov [19], A.A. Shkalikov [24], Yu. T. Silchenko [26], C. Tretter [29], A.I. Vagabov [30], S. Yakubov [32] and Y. Yakubov [34]. A mathematical model with integral boundary conditions was derived by [9, 23] in the context of optical physics. The importance of this kind of problem has been also pointed out by Samarskii [27]. In this paper, we study a problem for second order ordinary differential equations with mixed nonlocal boundary conditions combined with weighted integral boundary conditions and another two point boundary condition. Following the technique in [11, 20, 21, 22], we should bound the resolvent in the space Lp (0, 1) by means of a suitable formula for Green’s function. Under certain conditions on the weighting functions and on the coefficients in the boundary conditions, called non regular boundary conditions, we prove that the resolvent decreases with respect to the spectral parameter in Lp (0, 1), but there is no maximal decreasing at infinity for p ≥ 1. In contrast to the regular case this decreasing is maximal for p = 1 as shown in [11]. Furthermore, the studied operator generates in Lp (0, 1) an analytic semi group with singularities for p ≥ 1. The obtained results are then used to show the correct solvability of a mixed problem for a parabolic partial differential equation with non regular non local boundary conditions. 2. G REEN ’ S F UNCTION Let λ ∈ C, u1 (x) = u1 (x, λ ) and u2 (x) = u2 (x, λ ) be a fundamental system of solutions to the equation L (u) − λu = 0. Following [20], the Green’s function of problem (1.1) is given by (2.1) G(x, s; λ) = N (x, s; λ) , ∆(λ) where ∆(λ) it the characteristic determinant of the considered problem defined by (2.2) ∆(λ) = B1 (u1 ) B1 (u2 ) B2 (u1 ) B2 (u2 ) and (2.3) N (x, s; λ) = u1 (x) u2 (x) g(x, s, λ) B1 (u1 ) B1 (u2 ) B1 (g)x B2 (u1 ) B2 (u2 ) B2 (g)x for x, s ∈ [0, 1].The function g(x, s, λ) is defined as follows (2.4) g(x, s; λ) = ± 1 u1 (x)u2 (s) − u1 (s)u2 (x) , 2 u01 (s)u2 (s) − u1 (s)u02 (s) where it takes the plus sign for x > s and the minus sign for x < s. J. Inequal. Pure and Appl. Math., 5(2) Art. 38, 2004 http://jipam.vu.edu.au/ BVP Given an arbitrary δ ∈ π ,π 2 FOR 2 ND -O RDER D IFFERENTIAL O PERATORS 3 , we consider the sector Σδ = {λ ∈ C, |arg λ| ≤ δ, λ 6= 0} . P For λ ∈ δ , define ρ as the square root of λ with positive real part. For λ 6= 0, we can consider a fundamental system of solutions of the equation u00 = λu = ρ2 u given by u1 (t) = e−ρt and u2 (t) = eρt . In the following we are going to deduce an adequate formulae for ∆(λ) and G(x, s; λ). First of all, for j = 1, 2, we have j j B1 (uj ) = a0 + (−1)j b0 ρ + c0 e(−1) ρ + (−1)j d0 ρe(−1) ρ , Z 1 Z 1 j (−1)j ρt j B2 (uj ) = R(t)e dt + (−1) ρ S(t)e(−1) ρt dt. 0 0 so we obtain from (2.2) −ρ Z 1 (2.5) ∆(λ) = a0 − b0 ρ + c0 e − d0 ρe (R(t) + ρS(t))e dt 0 Z 1 ρ ρ −ρt − (a0 + b0 ρ + c0 e + d0 ρe ) (R(t) − ρS(t))e dt , −ρ ρt 0 and g(x, s; λ) has the form g(x, s; λ) = 1 ρ(x−s) (e − eρ(s−x) ) if x > s, 4ρ 1 ρ(s−x) (e − eρ(x−s) ) if x < s. 4ρ Thus we have B1 (g) = a0 − b0 ρ − c0 e−ρ + d0 ρe−ρ eρs 4ρ e−ρs + (−a0 − b0 ρ + c0 eρ + d0 ρeρ ) , 4ρ Z s Z 1 eρs −ρt −ρt B2 (g) = (R(t) − ρS(t))e dt + (−R(t) + ρS(t))e dt 4ρ 0 s Z s Z 1 e−ρs ρt ρt + − (R(t) + ρS(t))e dt + (R(t) + ρS(t))e dt . 4ρ 0 s After a long calculation, formula (2.3) can be written as N (x, s; λ) = ϕ(x, s; λ) + ϕi (x, s; λ), (2.6) where 1 (2.7) ϕ(x, s; λ) = 2ρ Z 1 (a0 + ρb0 ) (R (t) + ρS (t)) eρt dt Z s ρ ρt − e (c0 + ρd0 ) (R (t) + ρS (t)) e dt e−ρ(x+s) 0 Z 1 + (a0 − ρb0 ) (R (t) − ρS (t)) e−ρt dt s Z s ρ −ρt ρ(x+s) − e (c0 − ρd0 ) (R (t) − ρS (t)) e dt e 0 0 J. Inequal. Pure and Appl. Math., 5(2) Art. 38, 2004 http://jipam.vu.edu.au/ 4 M. D ENCHE AND A. KOURTA and the function ϕi (x, s; λ) is given by ϕi (x, s; λ) = (2.8) ϕ1 (x, s; λ) if x > s, ϕ (x, s; λ) if x < s, 2 with Z s eρ(s−x) (2.9) ϕ1 (x, s; λ) = (a0 + ρb0 + c0 eρ + ρeρ d0 ) (R (t) − ρS (t)) e−ρt dt 2ρ 0 Z 1 ρt − (a0 − ρb0 ) (R (t) + ρS (t)) e dt s Z s eρ(x−s) + a0 − ρb0 + c0 e−ρ − ρe−ρ d0 (R (t) + ρS (t)) eρt dt 2ρ 0 Z 1 −ρt − (a0 + ρb0 ) (R (t) − ρS (t)) e dt , 0 and 1 (2.10) ϕ2 (x, s; λ) = 2ρ Z 1 − (a0 + ρb0 + c0 eρ + ρeρ d0 ) (R (t) − ρS (t)) e−ρt dt s Z 1 −ρ ρt + (c0 − ρd0 ) e (R (t) + ρS (t)) e dt eρ(s−x) 0 Z 1 − a0 − ρb0 + c0 e−ρ − ρe−ρ d0 (R (t) + ρS (t)) eρt dt s Z 1 ρ −ρt ρ(x−s) − (c0 + ρd0 ) e (R (t) − ρS (t)) e dt e . 0 3. B OUNDS ON THE R ESOLVENT Every λ ∈ C such that ∆(λ) 6= 0 belongs to ρ(Lp ), and the associated resolvent operator R(λ, Lp ) can be expressed as a Hilbert Schmidt operator Z 1 −1 (3.1) (λI − Lp ) f = R(λ, Lp )f = − G(·, s; λ)f (s)ds, f ∈ Lp (0, 1). 0 Then, for every f ∈ Lp (0, 1) we estimate (3.1) kR(λ; Lp )f kLp (0,1) ≤ Z sup 0≤s≤1 and so we need to bound p1 Z 1 p = sup |G(x, s; λ)| dx 0≤s≤1 0 1 |G(x, s; λ)|p dx p1 kf kLp (0,1) , 0 1 |∆(λ)| Z p |N (x, s; λ)| dx sup 0≤s≤1 1 p1 . 0 3.1. Estimation of N (x, s; λ). We will denote by k·k the supremum norm which is defined by kRk = sup |R (s)|. Since 0≤s≤1 J. Inequal. Pure and Appl. Math., 5(2) Art. 38, 2004 http://jipam.vu.edu.au/ BVP FOR N (x, s; λ) = 2 ND -O RDER D IFFERENTIAL O PERATORS 5 ϕ (x, s; λ) + ϕ1 (x, s; λ) if x > s ; ϕ (x, s; λ) + ϕ2 (x, s; λ) if x < s then kN (x, s; λ)kLp ≤ kϕ (x, s; λ)kLp + kϕi (x, s; λ)kLp , (3.2) from (2.8), we have kϕi (x, s; λ)kLp ≤ 21/p (3.3) (Z s p |ϕ2 (x, s; λ)| dx 0 p1 Z 1 p |ϕ1 (x, s; λ)| dx + p1 ) . s From (2.7) we have Z 1 e−(x+s) Re ρ |ϕ (x, s; λ)| ≤ (kRk + |ρ| kSk) (|a0 | + |ρ| |b0 |) et Re ρ dt 2 |ρ| s Z s Re ρ t Re ρ + (kRk + |ρ| kSk) (|c0 | + |ρ| |d0 |) e e dt 0 Z 1 e(x+s) Re ρ (kRk + |ρ| kSk) (|a0 | + |ρ| |b0 |) + e−t Re ρ dt 2 |ρ| s Z s − Re ρ −t Re ρ +e (kRk + |ρ| kSk) (|c0 | + |ρ| |d0 |) e dt 0 then |ϕ (x, s; λ)| ≤ e−(x+s) Re ρ (kRk + |ρ| kSk) (|a0 | + |ρ| |b0 |) eRe ρ − es Re ρ 2 |ρ| Re ρ + (kRk + |ρ| kSk) (|c0 | + |ρ| |d0 |) e(s+1) Re ρ − eRe ρ e(x+s) Re ρ + (kRk + |ρ| kSk) (|a0 | + |ρ| |b0 |) e−s Re ρ − e− Re ρ 2 |ρ| Re ρ + (kRk + |ρ| kSk) (|c0 | + |ρ| |d0 |) e− Re ρ − e−(s+1) Re ρ and Z 0 1 |ϕ (x, s; λ)|p dx R1 p 2p 0 e−px Re ρ dx h ≤ (kRk + |ρ| kSk)p (|a0 | + |ρ| |b0 |)p e(1−s) Re ρ − 1 p (|ρ| Re ρ) p i + (kRk + |ρ| kSk)p × (|c0 | + |ρ| |d0 |)p eRe ρ − e(1−s) Re ρ R1 p 2p 0 epx Re ρ dx + [(kRk + |ρ| kSk)p × (|a0 | + |ρ| |b0 |)p 1 − e(s−1) Re ρ p (|ρ| Re ρ) i p p (s−1) Re ρ −1 Re ρ p + (kRk + |ρ| kSk) (|c0 | + |ρ| |d0 |) e −e , J. Inequal. Pure and Appl. Math., 5(2) Art. 38, 2004 http://jipam.vu.edu.au/ 6 M. D ENCHE AND A. KOURTA after calculation we obtain Z p1 1 p |ϕ (x, s; λ)| dx 0 2 21+ p eRe ρ ≤ 1+ p1 |ρ| (Re ρ) p1/p −p Re ρ × 1−e (kRk + |ρ| kSk) (|a0 | + |ρ| |b0 |) e−s Re ρ − e− Re ρ p1 −p Re ρ + 1−e p1 × (|c0 | + |ρ| |d0 |) h (s−1) Re ρ 1−e 1 − e−p Re ρ i p1 + 1 − e−p Re ρ p1 + (kRk + |ρ| kSk) 1 − e−s Re ρ e(s−1) Re ρ − e− Re ρ ii as Re ρ > 0 so Z (3.4) p |ϕ (x, s; λ)| dx sup 0≤s≤1 p1 1 0 2 ≤ 21+ p eRe ρ (kRk + |ρ| kSk) [(|a0 | + |c0 |) + |ρ| (|b0 | + |d0 |)] 1 |ρ| (Re ρ)1+ p p1/p . From (2.9) we have e(s−x) Re ρ |ϕ1 (x, s; λ)| ≤ (|a0 | + |ρ| |b0 |) + eRe ρ (|c0 | + |ρ| |d0 |) 2 |ρ| Z s × (kRk + |ρ| kSk) e−t Re ρ dt 0 Z 1 t Re ρ + (|a0 | + |ρ| |b0 |) (kRk + |ρ| kSk) e dt 0 e(x−s) Re ρ + (kRk + |ρ| kSk) [(|a0 | + |ρ| |b0 |) 2 |ρ| Z s Re ρ +e (|c0 | + |ρ| |d0 |)] et Re ρ dt 0 Z 1 −t Re ρ + (|a0 | + |ρ| |b0 |) (kRk + |ρ| kSk) e dt 0 then |ϕ1 (x, s; λ)| ≤ e(s−x) Re ρ (|a0 | + |ρ| |b0 |) (kRk + |ρ| kSk) eRe ρ − e−s Re ρ 2 |ρ| Re ρ + (|c0 | + |ρ| |d0 |) (kRk + |ρ| kSk) eRe ρ − e(1−s) Re ρ e(x−s) Re ρ + (kRk + |ρ| kSk) (|a0 | + |ρ| |b0 |) es Re ρ − e− Re ρ 2 |ρ| Re ρ + (|c0 | + |ρ| |d0 |) (kRk + |ρ| kSk) e(s−1) Re ρ − e− Re ρ J. Inequal. Pure and Appl. Math., 5(2) Art. 38, 2004 http://jipam.vu.edu.au/ BVP and Z s FOR 2 ND -O RDER D IFFERENTIAL O PERATORS 7 1 |ϕ1 (x, s; λ)|p dx R1 p 2p s e−px Re ρ dx h ≤ (|a0 | + |ρ| |b0 |)p (kRk + |ρ| kSk)p e(1+s) Re ρ − 1 p p |ρ| (Re ρ) p i + (|c0 | + |ρ| |d0 |)p (kRk + |ρ| kSk)p e(1+s) Re ρ − eRe ρ R1 p 2p s epx Re ρ dx h + (|a0 | + |ρ| |b0 |)p (kRk + |ρ| kSk)p 1 − e−(1+s) Re ρ p p |ρ| (Re ρ) p + (|c0 | + |ρ| |d0 |)p (kRk + |ρ| kSk)p e− Re ρ − e−(1+s) Re ρ , this yields Z 1 |ϕ1 (x, s; λ)|p dx ≤ 2p+1 ep Re ρ p p p p+1 [(|a0 | + |ρ| |b0 |) (kRk + |ρ| kSk) |ρ| (Re ρ) p × 1 − e−(1+s) Re ρ 1 − ep(s−1) Re ρ p + (|c0 | + |ρ| |d0 |)p (kRk + |ρ| kSk)p 1 − e−s Re ρ 1 − ep(s−1) Re ρ s as Re ρ > 0 we obtain Z (3.5) p |ϕ1 (x, s; λ)| dx sup 0≤s≤1 p1 1 s 2 ≤ 21+ p eRe ρ (kRk + |ρ| kSk) [(|a0 | + |c0 |) + |ρ| (|b0 | + |d0 |)] 1 |ρ| (Re ρ)1+ p p1/p . From (2.10) we have Z 1 e(x−s) Re ρ |ϕ2 (x, s; λ)| ≤ (kRk + |ρ| kSk) (|c0 | + |ρ| |d0 |) e(1−t) Re ρ dt 2 |ρ| 0 Z − Re ρ + (kRk + |ρ| kSk) (|a0 | + |ρ| |b0 |) + e (|c0 | + |ρ| |d0 |) 1 t Re ρ e dt 0 Z 1 e(s−x) Re ρ + [(|c0 | + |ρ| |d0 |) (kRk + |ρ| kSk) e(t−1) Re ρ dt 2 |ρ| 0 Z 1 Re ρ −t Re ρ + (|a0 | + |ρ| |b0 |) + e (|c0 | + |ρ| |d0 |) (kRk + |ρ| kSk) e dt s then |ϕ2 (x, s; λ)| ≤ ex Re ρ (kRk + |ρ| kSk) (|c0 | + |ρ| |d0 |) e(1−s) Re ρ − e− Re ρ 2 |ρ| Re ρ + (|a0 | + |ρ| |b0 |) (kRk + |ρ| kSk) e(1−s) Re ρ − 1 e−x Re ρ + (kRk + |ρ| kSk) (|c0 | + |ρ| |d0 |) eRe ρ − e(s−1) Re ρ 2 |ρ| Re ρ + (|a0 | + |ρ| |b0 |) (kRk + |ρ| kSk) 1 − e(s−1) Re ρ . J. Inequal. Pure and Appl. Math., 5(2) Art. 38, 2004 http://jipam.vu.edu.au/ 8 M. D ENCHE AND A. KOURTA So Z s 2p+1 ep Re ρ h p p (s−2) Re ρ p (kRk + |ρ| kSk) (|c | + |ρ| |d |) 1 − e 0 0 p |ρ|p (Re ρ)p+1 × 1 − e−ps Re ρ + (|a0 | + |ρ| |b0 |)p p i × (kRk + |ρ| kSk)p 1 − e−ps Re ρ 1 − e(s−1) Re ρ |ϕ2 (x, s; λ)|p dx ≤ 0 as Re ρ > 0 we obtain Z (3.6) p |ϕ2 (x, s; λ)| dx sup 0≤s≤1 p1 s 0 2 ≤ 21+ p eRe ρ (kRk + |ρ| kSk) [(|a0 | + |c0 |) + |ρ| (|b0 | + |d0 |)] 1 |ρ| (Re ρ)1+ p p1/p . From (3.4), (3.5) and (3.6), we obtain 2 kR (λ, Lp )k ≤ 21+ p eRe ρ (kRk + |ρ| kSk) [(|a0 | + |c0 |) + |ρ| (|b0 | + |d0 |)] 1 |4 (λ)| |ρ| (Re ρ)1+ p p1/p for ρ ∈ Σ δ = ρ ∈ C : |arg ρ| ≤ 2δ , ρ 6= 0 , we have (Re ρ)−1 < 2 1+ p2 2 1+ p2 2 kR (λ, Lp )k ≤ 1 , |ρ| cos( 2δ ) 1+ p2 2 +1 , then + 1 eRe ρ (kRk + |ρ| kSk) [(|a0 | + |c0 |) + |ρ| (|b0 | + |d0 |)] . 1+ p1 1 |4 (λ)| |ρ| |ρ|1+ p cos 2δ p1/p Finally, we obtain for λ = ρ2 ∈ Σδ (3.7) kR (λ, Lp )kLp ≤ c eRe ρ 1 |4 (λ)| |ρ|1+ p kRk + kSk [(|a0 | + |c0 |) + |ρ| (|b0 | + |d0 |)] , |ρ| where 1+ p2 2 c= p1/p 2 +1 1+ p1 . cos 2δ 1+ p2 3.2. Estimation of the Characteristic Determinant. The next step is to determine the cases for which |∆(λ)| remains bounded below. It will then be necessary to bound |∆(λ)| appropriately. However, formula (2.5) is not useful for this purpose. It will be then necessary to make some additional regularity hypotheses on the functions R and S, and so we assume that the functions R and S are in C 2 ([0, 1] , C). Integrating twice by parts in (2.5), we obtain h 0 0 ρ (3.8) ∆(λ) = e ρ(d0 S(0) − b0 S(1)) + (d0 S (0) + b0 S (1) + a0 S (1) 1 0 −b0 R (1) − d0 R (0) + c0 S (0)) + (a0 R (1) − c0 R (0) + b0 R (1) ρ 1 0 0 0 0 0 −a0 S (1) − d0 R (0) + c0 S (0)) + 2 (−a0 R (1) − c0 R (0)) + Φ(ρ) , ρ J. Inequal. Pure and Appl. Math., 5(2) Art. 38, 2004 http://jipam.vu.edu.au/ BVP FOR 2 ND -O RDER D IFFERENTIAL O PERATORS 9 where Z 1 R00 (t) S 00 (t) ρt −ρ −2ρ (3.9) Φ(ρ) = (a0 − ρb0 )e + (c0 − ρd0 )e − e dt ρ2 ρ 0 Z 1 R00 (t) S 00 (t) −ρt −ρ + (a0 + ρb0 )e + (c0 + ρd0 ) − e dt ρ2 ρ 0 0 −ρ 1 +e −b0 R0 (0) + d0 R0 (1) − c0 S 0 (1) + a0 S (0) + c0 R(1) − a0 R(0) ρ + ρ(b0 S(0) − d0 S(1))] + 2e−2ρ [(a0 + ρb0 )(R(1) − ρS(1)) 1 0 0 − (c0 − ρd0 )(R(0) + ρS(0)) + 2 (a0 + ρb0 )(R (1) − ρS (1)) ρ 1 0 0 + 2 (c0 − ρd0 )(R (0) + ρS (0)) . ρ After a straightforward calculation, we obtain the following inequality valid for ρ ∈ Σ δ ,with 2 |ρ| sufficiently large, Re ρ kR00 k kS 00 k e −1 − Re ρ −2 Re ρ |Φ(ρ)| ≤ (|a0 | + |ρ| |b0 |) e + (|c0 | + |ρ| |d0 |) e + |ρ| Re ρ |ρ|2 00 kR k kS 00 k 1 − e− Re ρ − Re ρ + (|a0 | + |ρ| |b0 |) e + (|c0 | + |ρ| |d0 |) + |ρ| Re ρ |ρ|2 1 0 0 0 0 + 2e− Re ρ −b0 R (0) + d0 R (1) − c0 S (1) + a0 S (0) + c0 R(1) − a0 R(0) |ρ| 1 −2 Re ρ + |ρ| |b0 S(0) − d0 S(1)|] + e (|a0 | + |ρ| |b0 |) (kRk + |ρ| kSk) |ρ| ! 0 0 R + |ρ| S 1 + (|c0 | + |ρ| |d0 |) (kRk + |ρ| kSk) + (|a0 | + |ρ| |b0 |) |ρ| |ρ|2 !# 0 0 R + |ρ| S + (|c0 | + |ρ| |d0 |) . |ρ|2 Then 00 (|a0 | + |c0 |) 2 kR k 00 |Φ(ρ)| ≤ + (|b0 | + |d0 |) + kS k |ρ| |ρ| |ρ| cos 2δ 1 1 0 0 0 0 + 2 −b0 R (0) + d0 R (1) − c0 S (1) + a0 S (0) + c0 R(1) − a0 R(0) δ |ρ| cos 2 |ρ| 1 |a0 | kRk + |b0 S(0) − d0 S(1)|] + + |b0 | + kSk |ρ| |ρ| |ρ| (cos 2δ )2 ! 0 0 R S |c0 | kRk |a0 | + + |d0 | + kSk + + |b0 | + |ρ| |ρ| |ρ| |ρ| |ρ|2 !# 0 0 R S |c0 | (3.10) + + |d0 | + . |ρ| |ρ| |ρ|2 J. Inequal. Pure and Appl. Math., 5(2) Art. 38, 2004 http://jipam.vu.edu.au/ 10 M. D ENCHE AND A. KOURTA Where we have used that Re(ρ) ≥ |ρ| cos( 2δ ), (1 − e− Re ρ ) < 1, e− Re ρ ≤ 2 1 ande−2 Re ρ ≤ , e− Re ρ < 1. 2 δ 2 |ρ| (cos 2 ) 2 |ρ| (cos 2δ )2 2 There are several cases to analyze depending on the functions R and S. • Case 1. Suppose that S 6= 0, d0 6= 0, b0 6= 0, d0 S (0) − b0 S (1) = 0 and 0 0 k1 = d0 S (0) + b0 S (1) + a0 S (1) − b0 R (1) − d0 R (0) + c0 S (0) 6= 0. From (3.8), we have for |ρ| sufficiently large h 0 0 |4 (λ)| ≥ eRe ρ d0 S (0) + b0 S (1) + a0 S (1) − b0 R (1) − d0 R (0) + c0 S (0) − 1 0 0 0 0 a0 R (1) − c0 R (0) + b0 R (1) − a0 S (1) − d0 R (0) + c0 S (0) |ρ| 1 0 0 − 2 a0 R (1) + c0 R (0) − |Φ(ρ)| . |ρ| From (3.10) we deduce for ρ ∈ Σ δ , |ρ| ≥ r0 > 0. 2 |Φ(ρ)| ≤ c(r0 ) . |ρ| Then, we have Re ρ |4 (λ)| ≥ e h 0 0 d0 S (0) + b0 S (1) + a0 S (1) − b0 R (1) − d0 R (0) + c0 S (0) 1 0 0 0 0 a0 R (1) − c0 R (0) + b0 R (1) − a0 S (1) − d0 R (0) + c0 S (0) |ρ| 1 c(r0 ) 0 0 − 2 a0 R (1) + c0 R (0) − . |ρ| |ρ| we can now choose r0 > 0, such that − 1 0 0 0 0 a0 R (1) − c0 R (0) + b0 R (1) − a0 S (1) − d0 R (0) + c0 S (0) r0 1 c(r0 ) 0 0 + 2 a0 R (1) + c0 R (0) + r0 |ρ| 1 0 0 ≤ d0 S (0) + b0 S (1) + a0 S (1) − b0 R (1) − d0 R (0) + c0 S (0) , 2 then, for ρ ∈ Σ δ , |ρ| ≥ r0 > 0, we get 2 eRe ρ |k1 | . 2 From (3.7), we deduce the following bound, valid for every |arg ρ| ≤ 2δ , ρ 6= 0 |a0 | + |c0 | kRk 2c + (|b0 | + |d0 |) + kSk , kR (λ, Lp )k ≤ 1 |ρ| |ρ| |ρ| p |k1 | then c1 kR (λ, Lp )k ≤ 1 , |λ| 2p |4 (λ)| ≥ J. Inequal. Pure and Appl. Math., 5(2) Art. 38, 2004 http://jipam.vu.edu.au/ BVP FOR 2 ND -O RDER D IFFERENTIAL O PERATORS 11 as |λ| −→ +∞, where c1 = 2c kSk (|b0 | + |d0 |) . |k1 | • Case 2. If b0 = d0 = 0, S 6= 0 and a0 S (1) + c0 S (0) = 0, with 0 0 k2 = a0 R (1) − c0 R (0) − a0 S (1) + c0 S (0) 6= 0, we have the following bound, valid for λ ∈ Σδ and |λ| big enough, 2c (|a0 | + |c0 |) kRk kR (λ, Lp )k ≤ + kSk , 1 |ρ| |ρ| p |k2 | then kR (λ, Lp )k ≤ c1 1 , |λ| 2p as |λ| −→ +∞, where c1 = 2c (|a0 | + |c0 |) kSk . |k2 | • Case 3. If S = 0, b0 6= 0, d0 6= 0 and b0 R(1) + d0 R (0) = 0, with 0 0 k3 = a0 R (1) − c0 R (0) + b0 R (1) − d0 R (0) 6= 0. Similarly, we get kR (λ, Lp )k ≤ 2c kRk 1 |ρ|1+ p |k3 | (|a0 | + |c0 |) + (|b0 | + |d0 |) , |ρ| then, we have kR (λ, Lp )k ≤ c1 1 , |λ| 2p as |λ| −→ +∞, where c1 = 2c kRk (|b0 | + |d0 |) . |k3 | • Case 4. If b0 = d0 = 0, S = 0 and a0 R (1) − c0 R (0) = 0 with 0 0 k4 = a0 R (1) − c0 R (0) 6= 0. Again in this case, we have kR (λ, Lp )k ≤ 2c (|a0 | + |c0 |) kRk 1 , |ρ| p |k4 | then kR (λ, Lp )k ≤ c1 1 , |λ| 2p as |λ| −→ +∞, where c1 = J. Inequal. Pure and Appl. Math., 5(2) Art. 38, 2004 c kRk (|a0 | + |c0 |) . |k4 | http://jipam.vu.edu.au/ 12 M. D ENCHE AND A. KOURTA Definition 3.1. The boundary value conditions in (1.1) are called non regular if the functions R, S ∈ C 2 ([0, 1] , C) and if and only if one of the following conditions holds 1-d0 S (0) − b0 (1) = 0, b0 6= 0, d0 6= 0, S 6= 0 and 0 0 d0 S (0) + b0 S (1) + a0 S (1) − b0 R (1) − d0 R (0) + c0 S (0) 6= 0 2- b0 = d0 = 0, kSk = 6 0, a0 S (1) + c0 S (0) = 0 with 0 0 a0 R (1) − c0 R (0) − a0 S (1) + c0 S (0) 6= 0 3-S = 0, b0 6= 0, d0 6= 0, b0 R(1) +0 R (0) 6= 0 with 0 0 a0 R (1) − c0 R (0) + b0 R (1) − d0 R (0) 6= 0 4-b0 = d0 = 0, S = 0, a0 R (1) − c0 R (0) = 0 with 0 0 a0 R (1) − c0 R (0) 6= 0. This proves the following theorem Theorem 3.1. If the boundary value conditions in (1.1) are non regular, then Σδ ⊂ ρ(Lp ) for sufficiently large |λ| and there exists c > 0 such that c kR (λ, Lp )k ≤ 1 . |λ| 2p Remark 3.2. From Theorem 3.1 it follows that the operator Lp , for p 6= ∞, generates an analytic semigroup with singularities [25] of type A (2p − 1, 4p − 1). 3.3. Application. In the following, we apply the above obtained results to the study of a class of a mixed problem for a parabolic equation with an weighted integral boundary condition combined with another two point boundary condition of the form ∂u(t, x) ∂ 2 u(t, x) − a = f (t, x) ∂t ∂x2 L (u) = a u (0, t) + b u0 (0, t) + c u (1, t) + d u0 (1, t) = 0, 1 0 0 0 0 (3.11) R R 0 1 1 L2 (u) = 0 R(ξ) u(t, ξ) dξ + 0 S(ξ) u (t, ξ) dξ = 0, u(0, x) = u0 (x), where (t, ξ) ∈ [0, T ] × [0, 1]. Boundary value problems for parabolic equations with integral boundary conditions are studied by [1, 3, 5, 6, 14, 15, 16, 17, 35] using various methods. For instance, the potential method in [3] and [17], Fourier method in [1, 14, 15, 16] and the energy inequalities method has been used in [5, 6, 35]. In our case, we apply the method of operator differential equation. The study of the problem is then reduced to a cauchy problem for a parabolic abstract differential equation, where the operator coefficients has been previously studied. For this purpose, let E, E1 , and E2 be Banach spaces. Introduce two Banach spaces ( ) Cµ ((0, T ] , E) = f /f ∈ C ((0, T ] , E) , kf k = sup ktµ f (t)k < ∞ , µ ≥ 0, t∈(0,T ] ( Cµγ ((0, T ] , E) = f /f ∈ C ((0, T ] , E) , kf k = sup ktµ f (t)k t∈(0,T ] + sup −γ µ kf (t + h) − f (t)k h t < ∞ , µ ≥ 0, γ ∈ (0, 1] , 0<t<t+h≤T J. Inequal. Pure and Appl. Math., 5(2) Art. 38, 2004 http://jipam.vu.edu.au/ BVP FOR 2 ND -O RDER D IFFERENTIAL O PERATORS 13 and a linear space C 1 ((0, T ] , E1 , E2 ) = f /f ∈ C ((0, T ] , E1 ) ∩ C 1 ((0, T ] , E2 ) , E1 ⊂ E2 , where C ((0, T ] , E) and C 1 ((0, T ] , E) are spaces of continuous and continuously differentiable, respectively, vector-function from (0, T ] into E. We denote, for a linear operator A in a Banach space E, by n 1 o E(A) = u/u ∈ D(A), kukE(A) = kAuk2 + kuk2 2 , and n o 0 C 1 ((0, T ] , E(A), E) = f /f ∈ C ((0, T ] , E(A)) , f ∈ C ((0, T ] , E)) . Let us derive a theorem which was proved by various methods in [25] and [31, 33]. Consider, in a Banach space E, the Cauchy problem ( 0 u (t) = Au(t) + f (t), t ∈ [0, T ] , (3.12) u(0) = u0 , where A is, generally speaking, unbounded linear operator in E, u0 is a given element of E, f (t) is a given vector-function and u(t) is an unknown vector-function in E. Theorem 3.3. Let the following conditions be satisfied: (1) A is a closed linear operator in a Banach space E and for some β ∈ (0, 1] , α > 0 π kR (λ, A)k ≤ C |λ|−β , |arg λ| ≤ + α, |λ| → ∞; 2 γ (2) f ∈ Cµ ((0, T ] , E) for some γ ∈ (1 − β, 1] , µ ∈ [0, β) ; (3) u0 ∈ D(A). Then the Cauchy problem (3.12) has a unique solution u ∈ C ([0, T ] , E) ∩ C 1 ((0, T ] , E(A), E) and for the solution u the following estimates hold ku(t)k ≤ C kAu0 k + ku0 k + kf kCµ ((0,t],E) , t ∈ (0, T ] , 0 u (t) + kAu(t)k ≤ C tβ−1 (kAu0 k + ku0 k) + tβ−µ−1 kf kCµγ ((0,t],E) , t ∈ (0, T ] . As a result of this we get the following theorem Theorem 3.4. Let the following conditions be satisfied π (1) a 6= 0, |arg a| < , 2 (2) the functions R(t), S(t) ∈ C 2 ([0, 1] , C) and one of the following conditions is satisfied d0 S (0) − b0 S (1) = 0, b0 6= 0, d0 6= 0, and S 6= 0 0 0 d0 S (0) + b0 S (1) + a0 S (1) − b0 R (1) − d0 R (0) + c0 S (0) 6= 0, or b0 = 0, d0 = 0, S 6= 0 , a0 S (1) + c0 S (0) = 0 with 0 0 a0 R (1) − c0 R (0) − a0 S (1) + c0 S (0) 6= 0, or b0 6= 0, d0 6= 0, S = 0, b0 R(1) + d0 R (0) = 0 with 0 0 a0 R (1) − c0 R (0) + b0 R (1) − d0 R (0) 6= 0, J. Inequal. Pure and Appl. Math., 5(2) Art. 38, 2004 http://jipam.vu.edu.au/ 14 M. D ENCHE AND A. KOURTA or b0 = d0 = 0, S = 0, a0 R (1) − c0 R (0) = 0 with 0 0 a0 R (1) − c0 R (0) 6= 0. i h 1 1 γ q (3) f ∈ Cµ ((0, T ] , L (0, 1)) for some γ ∈ 1 − 2q , 1 and some µ ∈ 0, 2q , (4) u0 ∈ Wq2 (0, 1) , Li u = 0, i = 1, 2 . Then the problem (3.11) has a unique solution u ∈ C ((0, T ] , Lq (0, 1)) ∩ C 1 (0, T ] , Wq2 (0, 1), Lq (0, 1) and for this solution we have the estimates: (3.13) ku(t, ·)kLq (0,1) ≤ c ku0 kWq2 (0,1) + kf kCµ ((0,t],Lq (0,1)) , t ∈ (0, T ] , (3.14) ku00 (t, ·)kLq (0,1) + ku0 (t, ·)kLq (0,1) 1 1 ≤ c t 2q −1 ku0 kWq2 (0,1) + t 2q −1−µ kf kCµγ ((0,t],Lq (0,1)) , t ∈ (0, T ] . Proof. We consider in the space Lq (0, 1) 1 ≤ q < ∞, the operator A defined by A(u) = au00 (x), D(A) = u ∈ Wq2 (0, 1), Li (u) = 0, i = 1, 2 . Then problem (3.11) can be written as ( 0 u (t) = Au(t) + f (t), u(0) = u0 , where u(t) = u(t, ·), f (t) = f (t, ·), and u0 = u0 (·) are functions with values in the Banach 1 π space Lq (0, 1). From Theorem 3.1 we conclude that kR(λ, A)k ≤ c |λ|− 2q , for |arg λ| ≤ +α, 2 as |λ| → ∞. Then, from Theorem 3.3 the problem (3.11) has a unique solution u ∈ C ((0, T ] , Lq (0, 1)) ∩ C 1 (0, T ] , Wq2 (0, 1), Lq (0, 1) and we have the following estimates (3.15) ku(t, .)kLq (0,1) ≤ c kAu0 kLq (0,1) + ku0 kLq (0,1) + kf kCµ ((0,t],Lq (0,1)) , (3.16) 0 + kAu(t, ·)kLq (0,1) 1 1 ≤ c t 2q −1 kAu0 kLq (0,1) + ku0 kLq (0,1) + t 2q −1−µ kf kCµγ ((0,t],Lq (0,1)) , u (t) Lq (0,1) where t ∈ [0, T ], from (3.15) we get 00 ku(t, ·)kLq (0,1) ≤ c ku0 kLq (0,1) + ku0 kLq (0,1) + kf kCµ ((0,t],Lq (0,1)) ≤ c ku0 kWq2 (0,1) + kf kCµ ((0,t],Lq (0,1)) , t ∈ [0, T ] . and from (3.16) we get 0 + ku00 (t, ·)kLq (0,1) 1 1 ≤ c t 2q −1 kAu0 kLq (0,1) + ku0 kLq (0,1) + t 2q −1−µ kf kCµγ ((0,t],Lq (0,1)) 1 1 −1 −1−µ 2q 2q ku0 kWq2 (0,1) + t kf kCµγ ((0,t],Lq (0,1)) , t ∈ [0, T ] . ≤c t u (t) Lq (0,1) J. Inequal. Pure and Appl. Math., 5(2) Art. 38, 2004 http://jipam.vu.edu.au/ BVP FOR 2 ND -O RDER D IFFERENTIAL O PERATORS 15 which gives the desired result. R EFERENCES [1] S.A. BEILIN, Existence of solutions for one-dimensional wave equations with nonlocal conditions, Electronic Journal of Differential Equations, 2001 (2001), Art. 76, 1–8. [2] H.E. BENZINGER, Green’s function for ordinary differential operators, J. Differen. Equations, 7 (1970), 478–496. [3] J.R. CANNON, Y. LIN AND J. van der HOEK, A quasi-linear parabolic equation with nonlocal boundary conditions, Rend. Mat. Appl., 9(7) (1989), 239–264. [4] M. DENCHE, Defect-Coerciveness for non regular transmission problem, Advances in Mathematical Sciences and Applications, 9(1) (1999), 229–241. [5] M. DENCHE AND A.L. MARHOUNE, Mixed problem with nonlocal boundary conditions for a third-order partial differential equation of mixed type. IJMMS, 26(7) (2001), 417–426. [6] M. DENCHE AND A.L. MARHOUNE, Mixed problem with integral boundary condition for a high order mixed type partial differential equation, J. Appl. Math. Stochastic Anal., 16(1) (2003), 69–79. [7] N. DUNFORD AND J.T. SCHWARTZ, Linear Operators. Part III. Spectral operators, Interscience, New York, 1971. [8] W. EBERHARD AND G. FREILING, Ston reguläre Eigenwert problem, Math.Zeit., 160(2) (1978), 139–161. [9] J. FARJAS, J.I. ROSELL, R. HERRERO, R. PONS, F. PI AND G. ORRIOLS, Equivalent low-order model for a nonlinear diffusion equation, Phys. D., 95 (1996), 107–127. [10] J.M. GALLARDO, Generation of analitic semi -groups by second-order differential operators with nonseparated boundary conditions, Rocky Mountain J. Math., 30(2) (2000), 869–899. [11] J.M. GALLARDO, Second-order differential operators with integral boundary conditions and generation of analitic semigroups, Rocky Mountain J. Math., 30(4) (2000), 1265–1291. [12] M.G. GASUMOV AND A.M. MAGERRAMOV, On fold-completeness of a system of eigenfunctions and associated functions of a class of differential operators, DAN Azerb. SSR, 30(3) (1974), 9–12. [13] M.G. GASUMOV AND A.M. MAGERRAMOV, Investigation of a class of differential operator pencils of even order, DAN SSR, 265(2) (1982), 277–280. [14] N.I. IONKIN, Solution of boundary value problem in heat conduction theory with nonlocal boundary conditions, Differ. Uravn., 13 (1977), 294–304. [15] N.I. IONKIN, Stability of a problem in heat conduction theory with nonlocal boundary conditions, Differ. Uravn., 15(7) (1979), 1279–1283. [16] N.I. IONKIN AND E.I. MOISEEV, A problem for the heat conduction equation with two-point boundary condition, Differ. Uravn., 15(7) (1979), 1284–1295. [17] N.I. KAMYNIN, A boundary value problem in the theory of heat conduction with non classical boundary condition, Theoret. Vychisl. Mat. Fiz., 4(6) (1964), 1006–1024. [18] A.P. KHROMOV, Eigenfunction expansions of ordinary linear differential operators in a finite interval, Soviet Math. Dokl., 3 (1962), 1510–1514. [19] Yu. A. MAMEDOV, On functions expansion to the series of solutions residue of irregular boundary value problems, in “Investigations on Differential Equations”, AGU named after S.M. Kirov (1984), 94–98. J. Inequal. Pure and Appl. Math., 5(2) Art. 38, 2004 http://jipam.vu.edu.au/ 16 M. D ENCHE AND A. KOURTA [20] M.A. NAIMARK, Linear Differential Operators, Vols. 1-2, Ungar Publishing, (1967). [21] M.L. RASULOV, Methods of Contour Integration, North-Holland, Amsterdam, 1967. [22] M.L. RASULOV, Applications of the contour Integral Method, Nauka, Moscow, 1975. [23] J.I. ROSELL, J. FARJAS, R. HERRERO, F. PI AND G. ORRIOLS, Homoclinic phenomena in opto-thermal bistability with localized absorption, Phys. D, 85 (1995), 509–547. [24] A.A. SHKALIKOV, On completenes of eigenfunctions and associated function of an ordinary differential operator with separated irregular boundary conditions, Func. Anal. i evo Prilozh, 10(4) (1976), 69–80. [25] J.T. SILCHENKO, Differential equations with non-densely defined operator coefficients, generating semigroups with singularities, Nonlinear Anal., Ser. A: Theory Methods, 36(3) (1999), 345–352. [26] Yu. T. SILCHENKO, An ordinary differential operator with irregular boundary conditions, (Russian) Sibirsk. Mat. Zh., 40(1) (1999), 183–190, iv; translation in Siberian Math. J., 40(1) (1999), 158–164. [27] A.A. SAMARSKII, Some problems in differential equations theory, Differ. Uravn., 16(11) (1980), 1925–1935. [28] J.D. TAMARKIN, About Certain General Problems of Theory of Ordinary Linear Differential Equations and about Expansions of Derivative Functions into Series, Petrograd, 1917. [29] C. TRETTER, On λ-Nonlinear Boundary Eigenvalue Problems, Akademic Verlag, Berlin, 1993. [30] A.I. VAGABOV, On eigenvectors of irregular differential pencils with non-separated boundary conditions, DAN SSSR, 261(2) (1981), 268–271. [31] S. YAKUBOV, Completness of Root Functions of Regular Differential Operators, Longman Scientific Technical, New York, (1994). [32] S. YAKUBOV, On a new method for solving irregular problems, J. Math. Anal. Appl., 220(1) (1998), 224–249. [33] S. YAKUBOV AND Y. YAKUBOV, Differential-operator equations, Ordinary and Partial Differential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 103. Chapman & Hall/CRC, Boca Raton, FL, (2000). [34] Y. YAKUBOV, Irregular boundary value problems for ordinary differential equations, Analysis (Munich) 18(4) (1998), 359–402. [35] N.I. YURCHUK, Mixed problem with an integral condition for certain parabolic equations, Differ. Uravn., 22(12) (1986), 2117–2126. J. Inequal. Pure and Appl. Math., 5(2) Art. 38, 2004 http://jipam.vu.edu.au/