Electronic Journal: Southwest Journal o Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 83-464 Issue 2, December, 23, pp. 26 35. Submitted: December 24, 22. Published: December 3 23. NON-AUTONOMOUS INHOMOGENEOUS BOUNDARY CAUCHY PROBLEMS AND RETARDED EQUATIONS M. Filali and M. Moussi Abstract. In this paper we prove the existence and the uniqueness o classical solution o non-autonomous inhomogeneous boundary Cauchy problems, this solution is given by a variation o constants ormula. Then, we apply this result to show the existence o solution o a retarded equation. A.M.S. MOS Subject Classiication Codes. 34G; 47D6 Key Words and Phrases. Boundary Cauchy problem, Evolution amilies, Classical solution, Wellposedness, Variation o constants ormula, Retarded equation Introduction Consider the ollowing Cauchy problem with boundary conditions d ut = Atut, s t T, dt IBCP Ltut = φtut + t, s t T, us = u. This type o problems presents an abstract ormulation o several natural equations such as retarded dierential equations, retarded dierence equations, dynamical population equations and neutral dierential equations. In the autonomous case At = A, Lt = L, φt = φ the Cauchy problem IBCP was studied by Greiner [2,3]. He used a perturbation o domain o generator o semigroups, and showed the existence o classical solutions o IBCP via variation o constants ormula. In the homogeneous case =, Kellermann [6] and Nguyen Lan [8] have showed the existence o an evolution amily Ut, s t s as the classical solution o the problem IBCP. Department o Mathematics, Faculty o Sciences, Oujda, Morocco E-mail Address: ilali@sciences.univ-oujda.ac.ma moussi@sciences.univ-oujda.ac.ma c 23 Cameron University 26 Typeset by AMS-TEX
NON-AUTONOMOUS BOUNDARY CAUCHY PROBLEMS 27 The aim o this paper is to show well-posedness in the general case and apply this result to get a solution o a retarded equation. In Section 2 we prove the existence and uniqueness o the classical solution o IBCP. For that purpose, we transorm IBCP into an ordinary Cauchy problem and prove the equivalence o the two problems. Moreover, the solution o IBCP is explicitly given by a variation o constants ormula similar to the one given in [3] in the autonomous case. We note that the operator matrices method was also used in [4, 8, 9] or the investigation o inhomogeneous Cauchy problems without boundary conditions. Section 3 is devoted to an application to the retarded equation { vt = Ktvt + t, t s, RE v s = ϕ. We introduce now the ollowing basic deinitions which will be used in the sequel. A amily o linear unbounded operators At t T on a Banach space X is called a stable amily i there are constants M, ω R such that ]ω, [ ρat or all t T and R, At i M ω k or > ω i= or any inite sequence t... t k T. A amily o bounded linear operators Ut, s s t on X is said an evolution amily i Ut, t = I d and Ut, rur, s = Ut, s or all s r t, 2 the mapping { t, s R 2 + : t s} t, s Ut, s is strongly continuous. For evolution amilies and their applications to non-autonomous Cauchy problems we reer to [,5,]. 2 Well-posedness o Cauchy problem with boundary coditions Let D, X and Y be Banach spaces, D densely and continuously embedded in X, consider amilies o operators At LD, X, Lt LD, Y and φt LX, Y, t T. In this section we will use the operator matrices method in order to prove the existence o classical solution or the non-autonomous Cauchy problem with inhomogeneous boundary conditions d ut = Atut, s t T, dt IBCP Ltut = φtut + t, s t T, us = u, it means that we will transorm this Cauchy problem into an ordinary homogeneous one. In all this section we consider the ollowing hypotheses : H t Atx is continuously dierentiable or all x D; H 2 the amily A t t T, A t := At kerlt, is stable, with M, ω constants o stability; H 3 the operator Lt is surjective or every t [, T ] and t Ltx is continuously dierentiable or all x D;
28 SOUTHWEST JOURNAL OF PURE AND APPLIED MATHEMATICS H 4 t φtx is continuously dierentiable or all x X; H 5 there exist constants γ > and ω R such that Ltx Y γ ω x X or x ker At, > ω and t [, T ]. Deinition 2.. A unction u : [s, T ] X is called a classical solution o IBCP i it is continuously dierentiable, ut D, t [s, T ] and u satisies IBCP. I IBCP has a classical solution, we say that it is well-posed. We recall the ollowing results which will be used in the sequel. Lemma 2.. [6,7] For t [, T ] and ρa t we have the ollowing properties i D = DA t ker At. ii Lt ker At is an isomorphism rom ker At onto Y. iii t L,t := Lt ker At is strongly continuously dierentiable. As consequences o this lemma we have LtL,t = I dy, L,t Lt and I L,t Lt are the projections in D onto ker At and DA t respectively. In order to get the homogenization o IBCP, we introduce the Banach space E := X C [, T ], Y Y, where C [, T ], Y is the space o continuously dierentiable unctions rom [, T ] into Y equipped with the norm g := g + g, or g C [, T ], Y. Let A φ t be a matrix operator deined on E by A φ t :=, DA φ t := D C [, T ], Y {}, t [, T ], At Lt φt δ t here δ t : C[, T ], Y Y is such that δ t g = gt. To the amily A φ we associate the homogeneous Cauchy problem d NCP dt Ut = Aφ tut, s t T, u Us =. In the ollowing proposition we give an equivalence between solutions o IBCP and those o NCP. u Proposition 2.. Let D C [, T ], Y. ut u 2t i I the unction t Ut := is a classical solution o NCP with initial value. Then t u t is a classical solution o IBCP with initial u value u. ii Let u be a classical solution o IBCP with initial value u. Then, the unction t Ut = is a classical solution o NCP with initial value. ut u
NON-AUTONOMOUS BOUNDARY CAUCHY PROBLEMS 29 Proo.. i Since U is a classical solution, then, rom Deinition 2., u is continuously dierentiable and u t D, or t [s, T ]. Moreover we have Thereore U t = u t u 2 t = A φ tut = Atu t Ltu t φtu t δ tu 2t u t = Atu t and u 2t =.. 2. This implies that u 2 t = u 2 s =, t [s, T ], hence the equation 2. yields to The initial value condition is obvious. The assertion ii is obvious. Ltu t = φtu t + t, s t T. Now we return to the study o the Cauchy problem NCP. For that aim, we recall the ollowing result. Theorem 2.. [], Theorem.3 Let At t T be a stable amily o linear operators on a Banach space X such that i the domain D := DAt, D is a Banach space independent o t, ii the mapping t Atx is continuously dierentiable in X or every x D. Then there is an evolution amily Ut, s s t T on D. Moreover Ut, s has the ollowing properties : Ut, sds Dt or all s t T, where Dr is deined by Dr := { x D : Arx D }, 2 the mapping t Ut, sx is continuously dierentiable in X on [s, T ] and d Ut, sx = AtUt, sx or all x Ds and t [s, T ]. dt In order to apply Theorem 2., we need the ollowing lemma. Lemma 2.2. The amily o operators A φ t t T is stable. Proo.. For t [, T ], we write A φ t as A φ t = At +, φt δ t where At = At, with domain DAt = DA φ t. Lt
3 SOUTHWEST JOURNAL OF PURE AND APPLIED MATHEMATICS Since A φ t is a perturbation o At by a linear bounded operator on E, hence, in view o a perturbation result [], Thm. 5.2.3 it is suicient to show the stability o At t T. x Let > ω and y, we have R, A t L,t At x y = AtR, A tx AtL,t y LtR, A tx + LtL,t y Since R, A tx DA t = kerlt, L,t y ker At and LtL,t = I dy, we obtain At R, A t L,t = I de. 2.2 x On the other hand, or DAt, we have R, A t L,t x At = R, A t Atx + L,t Ltx. From Lemma 2., let x DA t and x 2 ker At such that x = x +x 2. Then R, A t Atx + L,t Ltx = R, A t Atx + x 2 + L,t Ltx + x 2 = R, A t Atx + L,t Ltx 2 = x + x 2 = x. As a consequence, we get R, A t L,t At = I DAt. From 2.2 and 2.3, we obtain that the resolvent o At is given by R, At = R, A t L,t. Hence, by a direct computation one can obtain, or a inite sequence t... t k T, R, A t i R, A t i L,t R, At i = i= i=2 i=. k
NON-AUTONOMOUS BOUNDARY CAUCHY PROBLEMS 3 From the hypothesis H 5, we conclude that L,t γ ω or all t [, T ] and > ω. Deine ω = maxω, ω. Thereore, by using H 2, we obtain or E x y i= R, At i x y = R, A t i x R, A t i L,t y + k i= i=2 M ω k x + M ω k γ ω y + ω k x M ω k, y where M := maxm, Mγ. Thus the lemma is proved. Now we are ready to state the main result. Theorem 2.2. Let be a continuously dierentiable unction on [, T ] onto Y. Then, or all initial value u D, such that Lsu = φsu + s, the Cauchy problem IBCP has a unique classical solution u. Moreover, u is given by the variation o constants ormula ut = Ut, si L,s Lsu + L,t t, ut t [ + Ut, r L,r r, ur L,r r, ur ] dr, 2.4 s where Ut, s t s is the evolution amily generated by A t and t, ut := φtut + t. Proo. First, or the existence o Ut, s we reer to [7]. Since A φ t t T is stable and rom assumptions H, H 3 and H 4, A φ t t T satisies all conditions o Theorem 2., then there exists an evolution amily U φ t, s such that, ut or all initial value DA φ s, the unction u 2t := U φ t, s is u a classical solution o NCP. Thereore, rom i o Proposition 2., u is a classical solution o IBCP. The uniqueness o u comes rom the uniqueness o the solution o NCP and Proposition 2.. Let u be a classical solution o IBCP, at irst, we assume that φt =, then u 2 t := L,t Ltut = L,t t, and u t := I L,t Ltut are dierentiable on t and we have u t = u t u 2 t = Atu t + u 2 t L,t t = A tu t + L,t t L,t t. u
32 SOUTHWEST JOURNAL OF PURE AND APPLIED MATHEMATICS I we deine t := L,t t L,t t, we obtain u t = Ut, su s + t Replacing u s by I L,s Lsu, we obtain consequently, u t = Ut, si L,s Lsu + ut = Ut, si L,s Lsu +L,t t+ t s s Ut, r r dr. t s Ut, r r dr, [ Ut, r L,r r L,r r ] dr. 2.5 Now in the case Φt, since, u is continuously dierentiable, similar arguments are used to obtain the ormula 2.5 or :=, u which is exactly 2.4. 3 Retarded equation On the Banach space CE := C [ r, ], E, where r > and E is a Banach space, we consider the retarded equation { vt = Ktvt + t, s t T, RE v s = ϕ C E. Where v t τ := vt + τ, or τ [ r, ], and : [, T ] E. Deinition 3.. A unction v : [s r, T ] E is said a solution o RE, i it is continuously dierentiable, Ktv t is well deined, t [, T ] and v satisies RE. In this section we are interested in the study o the retarded equation RE, we will apply the abstract result obtained in the previous section in order to get a solution o RE. As a irst step, we show that this problem can be written as a boundary Cauchy one. More precisely, we show in the ollowing theorem that solutions o RE are equivalent to those o the boundary Cauchy problem d ut = Atut, s t T, 3. IBCP dt Ltut = φtut + t, 3.2 us = ϕ. 3.3 Where At is deined by { Atu := u D := DAt = C [ r, ], E, Lt : D E : Ltϕ = ϕ and φt : C[ r, ], E E : φtϕ = Ktϕ. Note that the spaces D, X and Y in section 2, are given here by D := C [ r, ], E, X := C[ r, ], E and Y := E. We have the ollowing theorem
NON-AUTONOMOUS BOUNDARY CAUCHY PROBLEMS 33 Theorem. i I u is a classical solution o IBCP, then the unction v deined by { ut, s t T, vt := ϕt s, r + s t s, is a solution o RE. ii I v is a solution o RE, then t ut := v t is a classical solution o IBCP. Proo. i Let u be a classical solution o IBCP, then rom Deinition 2., v is continuously dierentiable. On the other hand, 3. and 3.3 implies that u veriies the translation property utτ = { ut + τ, s t + τ T ϕt + τ s, r + s t + τ s, which implies that v t = ut. Thereore, rom 3.2, we obtain vt = ut = Ltut + t = Ktut + t = Ktv t + t. Hence v satisies RE. ii Now, let v be a solution o RE. From Deinition 3., ut C [ r, ], E = DAt, or s t T. Moreover, The equation 3. is obvious. Ltut = ut = vt = Ktvt + t = φtut + t. This theorem allows us to concentrate our sel on the problem IBCP. So, it remains to show that the hypotheses H H 5 are satisied. The hypotheses H, H 3 are obvious and H 4 can be obtained rom the assumptions on the operator Kt. For H 2, let ϕ DA t = { ϕ C [ r, ], E ; ϕ = } and C[ r, ], E such that A tϕ =, or >. Then Since ϕ =, we get ϕτ = e τ ϕ + R, A tτ = τ e τ σ σ dσ, τ [ r, ]. τ e τ σ σ dσ.
34 SOUTHWEST JOURNAL OF PURE AND APPLIED MATHEMATICS By induction, we can show that R, A t i τ = k! i= or a inite sequence t... t k T. Hence Thereore R, A t i τ k! i= τ σ τ k e τ σ σ dσ, = e τ i k τ i i! i=k = eτ τ i k i! i=k τ σ τ k e τ σ dσ, or τ [ r, ]. k R, A t i, >. k i= This proves the stability o A t t [,T ]. Now, i ϕ ker At, then ϕτ = e τ ϕ, or τ [ r, ]. Hence since Ltϕ = ϕ = e τ ϕτ, e lim + = +, in C E, we can take c > such that e Ltϕ c ϕ, t [, T ]. So H 5 holds. As a conclusion, we get the ollowing corollary c, thereore Corollary 3.. Let be a continuously dierentiable unction rom [, T ] onto E, then or all ϕ CE such that, ϕ = Ksϕ + s, the retarded equation RE has a solution v, moreover, v satisies v t = T t sϕ e ϕ + e t, v t + t where T t t is the c -semigroup generated by A t. s Reerences T t re [r, v r r, v r ] dr,. K.J. Engel and R. Nagel, One-Parameter Semigroups or Linear Evolution Equations, Graduate Texts in Mathematics 94, Springer-Verlag, 2. 2. G. Greiner, Perturbing the boundary conditions o a generator, Houston J. Math. vol 3, 23-229 987.
NON-AUTONOMOUS BOUNDARY CAUCHY PROBLEMS 35 3. G. Greiner, Semilinear boundary conditions or evolution equations o hyperbolic type, Lecture Notes in Pure and Appl. Math. vol 6 Dekker, New York, 27-53 989. 4. R. Grimmer and E. Sinestrari, Maximum norm in one-dimensional hyperbolic problems, Di. Integ. Equat. vol 5, 42-432 992. 5. T. Kato,, Linear evolution equations o hyperbolic type, J. Fac. Sci. Tokyo. vol 7, 24-258 97. 6. H. Kellermann, Linear evolution equations with time dependent domain, Tübingerbericht Funktionalanalysis, 5-44 986-987. 7. N. T. Lan, On nonautonomous Functional Dierential Equations, J. Math. Anal. Appl. vol 239, 58-74 999. 8. N. T. Lan, G. Nickel, Time-Dependent operator matrices and inhomogeneous Cauchy Problems, Rendiconti Del circolo matematico di Palermo. XLVII, 5-24 998. 9. R. Nagel and E. Sinestrari, Inhomogeneous Volterra integrodierential equations or Hille- Yosida operators, Marcel Dekker, Lecture Notes Pure Appl. Math. vol 5, 5-7 994.. A. Pazy, Semigroups o linear operators and applications to partial dierential equations, New york Springer, 983.. N. Tanaka, Quasilinear evolution equations with non-densely deined operators, Di. Integral Eq. vol 9, 67-6 996.