Existence and uniqueness of almost everywhere solutions of nonlocal problems to functional partial differential systems in diagonal form are investigated. The proof is based on the characteristics and fixed point methods

Marx, JJM

Peeters, PHM

Roest, M

van der A, DL

van der Hel, O

van der Schouw, YT

van Gils, CH

van Noord, PAH

Turo, J.

Diposit Digital de Documents de la UAB

Publicacions Matema`tiques, Vol 41 (1997), 507–517.NONLOCAL PROBLEMSFOR QUASILINEAR FUNCTIONALPARTIAL DIFFERENTIAL EQUATIONSOF FIRST ORDERJan TuroAbstractExistence and uniqueness of almost everywhere solutions of nonlo-cal problems to functional partial differential systems in diagonalform are investigated. The proof is based on the characteristicsand fixed point methods.1. IntroductionFor any metric spaces X and Y we denote by C(X,Y ) the set ofall continuous functions from X to Y . Let a0 > 0 be a given con-stant and Ia0 = [0, a0] × Rm. Write D = [−τ, 0] × [−b, b], where τ ∈R+ = [0,+∞) and b = (b1, . . . , bm) ∈ Rm+ . For z : [−τ, a0] × Rm → Rnand (x, y) = (x, y1, . . . , ym) ∈ Ia0 , we define z(x,y) : D → Rnby z(x,y)(s, t) = z(x + s, y + t), (s, t) ∈ D. Thus, we see that z(x,y)is a restriction of z to the rectangle [x − τ, x] × [y − b, y + b]. PutΩ = [0, a0]× Rm × C(D,Rm) and I = [−τ, 0]× Rm.We assume that  = [ij ] : Ω → Rnm, i = 1, . . . , n, j = 1, . . . ,m,f = (f1, . . . , fn) : Ω → Rn, hk = [hkij ] : I → Rnn, i, j = 1, . . . , n,k = 1, . . . , r, ϕ = (ϕ1, . . . , ϕn) : I → Rn are given functions.We consider quasilinear hyperbolic systems of functional partial dif-ferential equations(1) Dxzi(x, y) +m∑j=1ij(x, y, z(x,y))Dyjzi(x, y) = fi(x, y, z(x,y)),i = 1, . . . , n, (x, y) ∈ Ia0 , with nonlocal condition(2) z(0,y) +r∑k=1(hk)(0,y)z(ak,y) = ϕ(0,y), y ∈ Rm,508 J. Turowhere ak, k = 1, . . . , r, are finite numbers such that 0 < a1 < a2 < · · · <ar ≤ a0.The nonlocal condition (2) may be also written in the form(3) z(x, y) +r∑k=1hk(x, y)z(ak + x, y) = ϕ(x, y), (x, y) ∈ I.For r = n, τ = 0 and hkij = hijδki (δki is the Kronecker sym-bol) nonlocal boundary condition (2) reduces to the nonlocal condition“a´ la Cesari” [8], [1]. If hkij = δkiδij then (2) reduces to the Nicoletticondition [10], [12]. Furthermore, if all ak = 0, k = 1, . . . , r then we getthe usual Cauchy condition.Nonlocal condition was considered for parabolic problems in [4], [5],[8], and for hyperbolic problems in [2], [3], [6], [9]. Mixed problems forsystem (1) in two independent variables were investigated in [10].System (1) contains as particular cases the system of differentialequations with a retarded argument, differential-integral systems anddifferential-functional equations with operators of the Volterra type (seeSection 4).In this paper, we consider the local existence and uniqueness of gen-eralized solutions of nonlocal problem (1), (2). The method used in thepaper is based on characteristics theory and the fixed point theorem.2. Assumptions and LemmaFor η = (η1, . . . , ηk) ∈ Rk we write |η|k = max{|ηi| : 1 ≤ i ≤ k}.For the matrix U = [uij ], i = 1, . . . , n, j = 1, . . . ,m, we define ||U || =max{∑mj=1 |uij | : 1 ≤ i ≤ n}. Let ||v|| denote the supremum norm ofv ∈ C(D,Rn) and C(D,Rn; p) = {v ∈ C(D,Rn) : ||v|| ≤ p}, p ∈ R+.Let L([α, β],R) be the set of all integrable functions l : [α, β] → R. Wedenote by CL(D,Rn) the class of all functions v ∈ C(D,Rn) satisfyingthe condition(4) |v(s, t)− v(s¯, t¯)|n ≤∣∣∣∣∫ s¯sω(ξ) dξ∣∣∣∣ + q|t− t¯|m, (s, t), (s¯, t¯) ∈ D,where ω ∈ L([−τ, 0],R+), q ∈ R+ (ω and q depend on v). For v ∈CL(D,Rn) we define ||v||L = ||v||∗ + ||v||, where||v||∗ = inf{q +∫ 0−τω(ξ) dξ : q and ω satisfy (4)}.Functional partial differential equations 509Let CL(D,Rn; p) = {v ∈ CL(D,Rn) : ||v||L ≤ p}, p ∈ R+. Denoteby Λ the set of all functions λ : [0, a0] × R+ → R+ such that λ(·, t) ∈L([0, a0],R+) for each t ∈ R+ and λ(s, ·) is nondecreasing on R+ foralmost every (a.e.) s ∈ [0, a0].Assumption H1. Suppose that1. the matrix-valued function (·, y, v) : [0, a0] → Rnm is measurablefor every (y, v) ∈ Rm × C(D,Rn) and (x, ·) : Rm × C(D,Rn) →Rnm is continuous for a.e. x ∈ [0, a0];2. there exists d ∈ Λ such that||(x, y, v)|| ≤ d(x, p)for all (y, v) ∈ Rm × C(D,Rn; p), a.e. x ∈ [0, a0];3. there exists l ∈ Λ such that||(x, y, v)− (x, y¯, v¯)|| ≤ l(x, p)[|y − y¯|m + ||v − v¯||]for all (y, v), (y¯, v¯) ∈ Rm × CL(D,Rn; p) and a.e. x ∈ [0, a0].Assumption H2. Suppose that1. there exist constants p¯ ∈ (0, 12], q¯0 ∈ R+ and a function ω¯0 ∈L([−τ, 0],R+) such thatr∑k=1||hk(x, y)|| ≤ p¯,r∑k=1||hk(x, y)− hk(x¯, y¯)|| ≤ |∫ x¯xω¯0(s) ds|+ q¯0|y − y¯|m;2. there exist constants p0, q0∈R+ and a function ω0∈L([−τ, 0],R+)such that|ϕ(x, y)|n ≤ p0,|ϕ(x, y)− ϕ(x¯, y¯)|n ≤∣∣∣∣∫ x¯xω0(s) ds∣∣∣∣ + q0|y − y¯|mfor all (x, y), (x¯, y¯) ∈ I.510 J. TuroFor a ∈ (0, a0] we denote by Cϕ,a[p, ω, q] the set of all functions v ∈C([−τ, a]× Rm,Rn) such that|z(x, y)|n ≤ p,|z(x, y)− z(x¯, y¯)|n ≤∣∣∣∣∫ x¯xω(s) ds∣∣∣∣ + q|y − y¯|mfor (x, y), (x¯, y¯) ∈ Ia and v satisfies condition (3) on I.We consider a Carathe´odory solution of (1), (2). More precisely, afunction z is called a Carathe´odory solution of problem (1), (2) providedthe following conditions hold:(i) z ∈ Cϕ,a[p, ω, q];(ii) z satisfies (1) almost everywhere in Ia and (2) everywhere in Rn.For z ∈ Cϕ,a[p, ω, q] we consider the following problem(5) η′(t) = i(t, η(t), z(t,η(t))), η(x) = y, i = 1, . . . , n.Note that (5) is an ordinary differential equation. If Assumption H1 issatisfied then for every z ∈ Cϕ,a[p, ω, q] the right hand side of (5) satisfiesCarathe´odory assumptions and the following Lipschitz condition|i(t, ξ, z(t,ξ))− i(t, ξ¯, z(t,ξ¯))| ≤ l(t, ra)(1− q)|ξ − ξ¯|m,holds, where ra = p+q+∫ a−τ ω(s) ds. Thus, the existence and uniquenessof the solution gi[z](·;x, y) : [0, a] → R of (5) follows from classicaltheorems.Lemma 1. If Assumption H1 is satisfied and z, z¯ ∈ Cϕ,a[p, ω, q] then(6) |gi[z](t;x, y)− gi[z](t; x¯, y¯)|m≤ exp[(1 + q)∫ xtl(s, ra) ds] [∣∣∣∣∫ x¯xd(s, p) ds∣∣∣∣ + |y − y¯|m]t ∈ [0,min(x, x¯)], i = 1, . . . , n, and(7) |gi[z](t;x, y)− gi[z¯](t;x, y)|m≤∫ xtl(s, ra) ds exp[(1 + q)∫ xtl(s, ra) ds]||z − z¯||a, t ∈ [0, x],Functional partial differential equations 511where || · ||a denotes the supremum norm in the space C(Ia,Rn).Proof: We will consider the case where x ≤ x¯. We have, by Assump-tion H1,|gi[z](t;x, y)− gi[z](t; x¯, y¯)|m ≤ |y − y¯|m +∫ x¯xd(s, p) ds+∣∣∣∣∫ txl(s, ra)(1 + q)|gi[z](s;x, y)− gi[z](s; x¯, y¯)|m ds∣∣∣∣ , t ∈ [0, x].Hence,by the above inequality and by Gronwall’s inequality we get (6).We consider the case x > x¯ analogously to the case x ≤ x¯.It follows, from Assumption H1, that|gi[z](t;x, y)−gi[z¯](t;x, y)|m≤∣∣∣∣∫ txl(s, ra)[|gi[z](s;x, y)−gi[z¯](s;x, y)∣∣∣∣m+ ||z(s,gi[z](s;x,y)) − z¯(s,gi[z¯](s;x,y))||] ds|, t ∈ [0, x].Since||z(s,gi[z](s;x,y)) − z¯(s,gi[z¯](s;x,y))||≤ ||z − z¯||a + q|gi[z](s;x, y)− gi[z¯](s;x, y)|m,we get|gi[z](t;x, y)− gi[z¯](t;x, y)|m ≤∣∣∣∣∫ txl(s, ra) ds∣∣∣∣ ||z − z¯||a+∣∣∣∣∫ txl(s, ra)(1 + q)|gi[z](s;x, y)− gi[z¯](s;x, y)| ds∣∣∣∣ .Utilizing the Gronwall’s inequality, we obtain (7). The proof ofLemma 1 is complete.Assumption H3. Suppose that1. the vector-valued function f(·, y, v) : [0, a0] → Rn is measurablefor every (y, v) ∈ Rm × C(D,Rn) and f(x, ·) : Rm × C(D,Rn) →Rn is continuous for a.e. x ∈ [0, a0];2. there exists d1 ∈ Λ such that|f(x, y, v)|n ≤ d1(x, p)for all (y, v) ∈ Rm × C(D,Rn; p), a.e. x ∈ [0, a0];3. there exists l1 ∈ Λ such that|f(x, y, v)− f(x, y¯, v¯)|| ≤ l1(x, p)[|y − y¯|m + ||v − v¯||]for all (y, v), (y¯, v¯) ∈ Rm × CL(D,Rn; p) and a.e. x ∈ [0, a0].512 J. Turo3. The main theoremTheorem 1. Suppose that Assumptions H1-H3 are satisfied. Thenthere exist a ∈ (0, a0], p, q ∈ R+ and ω ∈ L([−τ, a],R+) such that prob-lem (1), (2) has a unique solution z in the class Cϕ,a[p, ω, q].Proof: First, we take a so small that(8)∫ a0d1(t, p) dt ≤ 1, Ka = exp[La(1 + q)] ≤ 2,L1a(1 + q) ≤ 1, RaLaKa + L1a < 12 ,where La =∫ a0l(t, p) dt, L1a =∫ a0l1(t, p) dt and Ra = q0 + q¯0p + p¯q +L1a(1 + q).Let us choose constants p, q with(9)p ≥ (1− p¯)(1 + p0),q ≥ 2(1− 2p¯)−1(q0 + q¯0p+ 1)and function(10) ω(t) = max{qd(t, p) + d1(t, p), (1− p¯)−1[ω0(t) + pω¯(t)]}.We define the following operator(Tz)i(x, y) = ϕi(0, gi(0;x, y))−r∑k=1hki(0, gi(0;x, y))zi(ak, gi(0;x, y))+∫ x0fi(t, gi(t;x, y), z(t,gi(t;x,y))) dt, (x, y) ∈ Ia,(Tz)i(x, y) = ϕi(x, y)−r∑k=1hki(x, y)zi(ak + x, y), (x, y) ∈ I.From Assumptions H2, H3 and inequalities (8), (9), we obtain|(Tz)i(x, y)| ≤ |ϕi(0, gi(0;x, y))|+∣∣∣∣∣r∑k=1hki(0, gi(0;x, y))zi(ak, gi(0;x, y))∣∣∣∣∣+∣∣∣∣∫ x0fi(t, gi(t;x, y), z(t,gi(t;x,y))) dt∣∣∣∣≤ p0 + p¯p+∫ a0d1(t, p)dt ≤ p, (x, y) ∈ Ia,Functional partial differential equations 513and|(Tz)i(x, y)| ≤ |ϕi(x, y)|+∣∣∣∣∣r∑k=1hki(x, y)zi(ak + x, y)∣∣∣∣∣≤ p0 + p¯p ≤ p, (x, y) ∈ I.For any (x, y), (x¯, y¯) ∈ Ia, from Assumptions H1-H3 and Lemma 1, weget|(Tz)i(x, y)−(Tz)i(x¯, y¯)| ≤ Ra|gi(t;x, y)−gi(t; x¯, y¯)|+∣∣∣∣∫ x¯xd1(t, p) dt∣∣∣∣≤ RaKa∣∣∣∣∫ x¯xd(t, p) dt∣∣∣∣ +∣∣∣∣∫ x¯xd1(t, p) dt∣∣∣∣ +RaKa|y − y¯|m.For (x, y), (x¯, y¯) ∈ I, we have|(Tz)i(x, y)− (Tz)i(x¯, y¯)| ≤∣∣∣∣∫ x¯xω0(t) dt∣∣∣∣ + p¯∣∣∣∣∫ x¯xω(t) dt∣∣∣∣+ p∣∣∣∣∫ x¯xω¯0(t) dt∣∣∣∣ + (q0 + p¯q + pq¯0)|y − y¯|m.Thus, from (8)-(10), we obtain|(Tz)i(x, y)− (Tz)i(x¯, y¯)| ≤∣∣∣∣∫ x¯xω(t) dt∣∣∣∣ + q|y − y¯|m, (x, y) ∈ Ia.We see that Tz ∈ Cϕ,a[p, ω, q].Now, we prove that T is a contraction. Indeed, for any z, z¯∈Cϕ,a[p, ω, q],we have|(Tz)i(x, y)− (T z¯)i(x, y)| ≤ |ϕi(0, g[z](0;x, y))− ϕi(0, g[z¯](0;x, y))|+∣∣∣∣∣r∑k=1[hki(0, gi[z](0;x, y))− hki(0, gi[z¯](0;x, y))]zi(ak, gi[z](0;x, y))∣∣∣∣∣+∣∣∣∣∣r∑k=1[hki(0, gi[z¯](0;x, y))[zi(ak, gi[z](0;x, y))− z¯i(ak, gi[z¯](0;x, y))]∣∣∣∣∣+∣∣∣∣∫ x0fi(t, gi[z](t;x, y), z(t,gi[z](t;x,y)))−fi(t, gi[z¯](t;x, y), z¯(t,gi[z¯](t;x,y)))] dt∣∣∣∣≤ (RaLaKa + L1a + p¯)||z − z¯||a, (x, y) ∈ Ia,514 J. Turoand|(Tz)i(x, y)−(T z¯)i(x, y)| ≤∣∣∣∣∣r∑k=1hki(x, y)[zi(ak + x, y)− z¯i(ak + x, y)∣∣∣∣∣≤ p¯||z − z¯||a, (x, y) ∈ I.Thus, it follows from (8) that T is a contraction.It remains to prove that the fixed point z of the operator T is theCarathe´odory solution of (1), (2). By taking y = gi(x; 0, η) in the equa-tion zi(x, y) = (Tz)i(x, y), we get(11) zi(x, gi(x; 0, η)) = ϕi(0, η)−r∑k=1hki(0, η)zi(ak, η)+∫ x0fi(t, gi(t; 0, η), z(t,gi(t;0,η))) dt,since the solution gi of (5) satisfies the following group propertygi(t′; t, gi(t;x, y)) = gi(t′;x, y). By differentiation of (11) with respectto x, using the chain rule differentiation Lemma (4.ii) of [7], and byputting again y = gi(x; 0, η), we obtain that z satisfies (1) almost ev-erywhere in Ia. It follows immediately that z satisfies (2). The proof ofTheorem 1 is complete.4. Special cases of system (1)We list below a few examples of problems which can be derived from(1) by specializing the functions  and f .1) Suppose that ¯ : Ia0 ×Rn×Rn → Rnm and f¯ : Ia0 ×Rn×Rn → Rnare given functions. Let(x, y, v) = ¯(x, y, v(0, 0),∫Dv(s, t) ds dt),f(x, y, v) = f¯(x, y, v(0, 0),∫Dv(s, t) ds dt), (x, y, v) ∈ Ia×C(D,Rn).Then system (1) reduces to the differential-integral systemDxzi(x, y) +m∑j=1¯ij(x, y, z(x, y),∫Dz(x+ s, y + t) ds dt)Dyjzi(x, y)= f¯i(x, y, z(x, y),∫Dz(x+ s, y + t) ds dt), i = 1, . . . , n.Functional partial differential equations 5152) Suppose that α = (α0, α′), β = (β0, β′) : Ia0 × C(D,Rn) → R1+m,¯ : Ia0×Rn×Rn → Rnm and f¯ : Ia0×Rn×Rn → Rn are given. Assumethat(α0(x, y, v)− x, α′(x, y, v)− y) ∈ D,(β0(x, y, v)− x, β′(x, y, v)− y) ∈ Dfor (x, y, v) ∈ Ia0 × C(D,Rn).Put(x, y, v) = ¯(x, y, v(0, 0), v(α0(x, y, v)− x, α′(x, y, v)− y)),f(x, y, v) = f¯(x, y, v(0, 0), v(β0(x, y, v)− x, β′(x, y, v)− y)).Then system (1) reduces to the differential system with a retarded argu-mentDxzi(x, y)+m∑j=1¯ij(x, y, z(x, y), z(α0(x, y, z(x,y)),α′(x, y, z(x,y))))Dyjzi(x, y)= f¯i(x, y, z(x, y), z(β0(x, y, z(x,y)), β′(x, y, z(x,y)))), i = 1, . . . , n.The functions α and β depend on the functional argument. Therefore,we cannot apply existence theorems from [13], [14] to the above system.3) If we take(x, y, v) = ¯(x, y, v(0, 0), (V (I(x,y)v))(x, y)),f(x, y, v) = f¯(x, y, v(0, 0), (V (I(x,y)v))(x, y)),where (I(x,y)v)(s, t) = v(s − x, t − y) then system (1) reduces to thesystem of differential-functional equations ([13], [14])Dxzi(x, y) +m∑j=1¯ij(x, y, z(x, y), (V z)(x, y))Dyjzi(x, y)= f¯i(x, y, z(x, y), (V z)(x, y)), i = 1, . . . , n.References1. P. Bassanini, The problem of Graffi-Cesari, Proc. Inter. Conf. Non-linear Phenomena Math. Sci. Arlington, Acad. Press, USA (1982),87–101.516 J. Turo2. L. Byszewski, Existence and uniqueness of solutions of nonlocalproblems for hyperbolic equation uxt = F (x, t, u, ux), J. Appl. Math.Stochastic Anal. 3 (1990), 163–168.3. L. Byszewski, Theorem about existence and uniqueness of contin-uous solution of nonlocal problem for nonlinear hyperbolic equation,Appl. Anal. 40 (1991), 173–180.4. L. Byszewski, Existence of a solution of a Fourier nonlocal quasi-linear parabolic problem, J. Appl. Math. Stochastic Anal. 5 (1992),43–68.5. L. Byszewski, Monotone iterative method for a system of nonlo-cal initial-boundary parabolic problems, J. Math. Anal. Appl. 177(1993), 445–458.6. L. Byszewski and V. Lakshmikantham, Monotone iterativetechnique for nonlocal hyperbolic differential problem, J. Math.Phys. Sci. 26 (1992), 346–359.7. L. Cesari, A boundary value problem for quasilinear hyperbolicsystem, Riv. Mat. Univ. Parma 3 (1974), 107–131.8. J. Chabrowski, On non-local problems for parabolic equations,Nagoya Math. J. 93 (1984), 109–131.9. T. Cz!lapin´ski, On existence and uniquenes of solutions ofnonlocal problems for hyperbolic differential-functional equations,Preprint.10. Z. Kamont and K. Topolski, Mixed problems for quasilinear hy-perbolic differential-functional systems, Math. Balkanica 6 (1992),313–324.11. A. Pli´s, Generalization of the Cauchy problem for a system of par-tial differential equations, Bull. Acad. Polon. Sci. Cl. III 4 (1956),741–744.12. J. Szarski, Generalized Cauchy problem for differential-functionalequations with first order partial derivatives, Bull. Acad. Polon. Sci.Ser. Sci. Math. Astr. Phys. 24 (1976), 575–580.13. J. Turo, Local generalized solutions of mixed problems for quasilin-ear hyperbolic systems of functional partial differential equations intwo independent variables, Ann. Polon. Math. 49 (1989), 259–278.Functional partial differential equations 51714. J. Turo, Generalized solutions to functional partial differentialequations of the first order,, Zeszyty Nauk. Politech. Gdan´skiej Mat.14 (1988), 1-99.Department of MathematicsTechnical University of Gdan´sk80-952 Gdan´skNarutowicza 11/12POLANDe-mail: turo@mif.pg.gda.plPrimera versio´ rebuda el 30 de Maig de 1996,darrera versio´ rebuda el 17 de Marc¸ de 1997

Nonlocal problems for quasilinear functional partial differential equations of first order

University of Groningen

Heterozygosity for the Cys282Tyr mutation in the HFE gene and the risk of colorectal cancer (Netherlands)

Background &amp; Aims: Heterozygosity for the Cys282Tyr transition in the HFE-gene is associated with slightly increased iron levels and may therefore be a potential risk factor for colorectal cancer.Methods: We studied the relationship between Cys282Tyr-heterozygosity and colorectal cancer using a case-control design. The 240 colorectal cancer cases and 635 controls in our study were derived from a prospective cohort study of 12,242 postmenopausal women, who were invited for an experimental breast cancer screening program in Utrecht, the Netherlands. The women were age 51-69 at time of inclusion and were followed for a period of 20 years. HFE genotyping was performed by PCR and allele-specific oligonucleotide (ASO) hybridization.Results: The risk of colorectal cancer was higher for women who were heterozygous for the Cys282Tyr mutation, than for those who were Cys282Tyr-wildtypes, although this was not statistically significant (Age-adjusted OR = 1.2, 95% CI: 0.6-2.2). Cys282Tyr-heterozygotes who smoked seemed to be at higher risk of colorectal cancer, although the p-value for interaction was not significant (p-value 0.42).Conclusions: The Cys282Tyr mutation is not associated with an increased risk for colorectal cancer in postmenopausal women, although in combination with smoking a slightly increased risk cannot be excluded.</p

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van der A, D.L.

van der Hel, O.L.

Roest, M.

van der Schouw, Y.T.

van Gils, C.H.

Marx, J.J.M.

van Noord, P.A.H.

Peeters, P.H.M.

Utrecht University Repository

Heterozygosity for the Cys282Tyr mutation in the HFE gene and the risk of colorectal cancer

J. Turo

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Proceedings - University of Groningen

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Background & Aims: Heterozygosity for the Cys282Tyr transition in the HFE-gene is associated with slightly increased iron levels and may therefore be a potential risk factor for colorectal cancer. Methods: We studied the relationship between Cys282Tyr-heterozygosity and colorectal cancer using a case-control design. The 240 colorectal cancer cases and 635 controls in our study were derived from a prospective cohort study of 12,242 postmenopausal women, who were invited for an experimental breast cancer screening program in Utrecht, the Netherlands. The women were age 51-69 at time of inclusion and were followed for a period of 20 years. HFE genotyping was performed by PCR and allele-specific oligonucleotide (ASO) hybridization. Results: The risk of colorectal cancer was higher for women who were heterozygous for the Cys282Tyr mutation, than for those who were Cys282Tyr-wildtypes, although this was not statistically significant (Age-adjusted OR = 1.2, 95% CI: 0.6-2.2). Cys282Tyr-heterozygotes who smoked seemed to be at higher risk of colorectal cancer, although the p-value for interaction was not significant (p-value 0.42). Conclusions: The Cys282Tyr mutation is not associated with an increased risk for colorectal cancer in postmenopausal women, although in combination with smoking a slightly increased risk cannot be excluded

Nonlocal problems for quasilinear functional partial differential equations of first order

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Dissertations of the University of Groningen

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Proceedings - University of Groningen

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Proceedings - University of Groningen

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