STURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES

STURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES 1. Regulr Sturm-Liouville Problem The method of seprtion of vribles to solve boundry vlue problems leds to ordinry differentil equtions on intervls with conditions t the endpoints of the intervls. For exmple het propgtion in rod of length L whose end points re kept t temperture leds to the ODE problem X (x) + λx(x) =, X() =, X(L) =. If the endpoints re insulted, the ODE problem becomes X (x) + λx(x) =, X () =, X (L) =. These re exmples of Sturm-Liouville problems. A regulr Sturm-Liouville problem (SL problem for short) on n intervl [, b] is second order ODE problem, with endpoints conditions, of the form (1) where (p(x)y ) + [λr(x) q(x)] y = α 1 y() + α y () = β 1 y(b) + β y (b) = p, q nd r re continuous functions on [, b] such tht p(x) > nd r(x) > x [, b] α 1, α, β 1, nd β re constnts such tht α 1, α re not both zero, nd β 1, β re not both zero. λ R is prmeter Remrk 1. The second order ODE of the SL-problem (1) is in self-djoint form (suitble for certin mnipultions). Most second order liner ODEs cn be trnsformed into self-djoint form. Consider the second order ODE A(x)y + B(x)y + C(x)y = y + B A y + C A y = with A(x) > on n intervl I. Let p(x) be defined by ( ) B(x) p(x) = exp A(x) dx. Then p = (B/A)p nd py + p(b/a)y = (py ). Hence, if we multiply the lst ODE by p we obtin py + p(b/a)y + p(c/a)y = (py ) + p(c/a)y = This lst ODE is in self-djoint form. Dte: Mrch, 16. 1

STURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES Note tht y is solution of the SL-Problem (1). It is the trivil solution. For most vlues of the prmeter λ, problem (1) hs only the trivil solution. An eigenvlue of the the SL-problem (1) is vlue of λ for which nontrivil solution exist. The nontrivil solution is clled n eigenfunction. Note tht if y(x) solves (1), then so does ny multiple cy(x), where c is constnt. Thus if y 1 is n eigenfunction of (1) with eigenvlue λ 1, then ny function cy 1 (x) is lso n eigenfunction with eigenvlue λ 1. In fct the set of ll eigenfunctions, corresponding to n eigenvlue λ, together with the zero function forms vector spce: the eigenspce of the eigenvlue. Exmple 1. For the SL-problem y (x) + λy(x) =, y() = y(l) = we hve p(x) = 1, r(x) = 1, q(x) =, α 1 = β 1 = 1, nd α = β =. This problem, tht we hve encountered severl times lredy, hs infinitely mny eigenvlues λ n = (nπ/l), nd for ech n Z +, the eigenspce is generted by the function y n (x) = sin(nπx/l). Exmple. The eigenvlues nd eigenfunction of the SL-problem y (x) + λy(x) =, y () = y (L) = (here we hve α 1 = β 1 = nd α = β = 1) re: λ = with y (x) = 1, nd for n Z +, λ n = (nπ/l) with y n (x) = cos(nπx/l). Exmple 3. Consider the SL-problem y (x) + λy(x) =, y() =, y (L) =, (this time α 1 = 1, α =, β 1 =, nd β = 1). To find the eigenvlues, we consider three cses depending on the vlues of λ. If λ <, set λ = ν with ν >. In this cse the generl solution of the ODE is y(x) = C 1 sinh(νx) + C cosh(νx). In order for y() =, we need to hve C 1 =, thus y = C 1 sinh(νx) nd y = νc 1 cosh(νx). To get y (L) =, we need νc 1 cosh(νl) =. Since cosh x > ( x R)nd since ν >, then necessrily C 1 = nd then y. Thus λ < cnnot be n eigenvlue. If λ =, then y(x) = A + Bx is the generl solution of the ODE nd the boundry conditions give A = B =, nd consequently λ = is not n eigenvlue. If λ >, we set λ = ν with ν >. The generl solution of the ODE is y(x) = C 1 sin(νx) + C cos(νx). The condition y() = implies C =. Then y = C 1 sin(νx) gives y = νc 1 cos(νx) nd the condition y (L) = gives νc 1 cos(νl) =. In order to obtin nontrivil solution, we need C 1. Consequently, cos(νl) =. Hence νl = π (j + 1)π + jπ =, j =, 1,, The eigenvlues nd eigenfunctions of the SL-problem re therefore ( ) (j + 1)π (j + 1)πx λ j =, y j (x) = sin, j =, 1,, This is the typicl sitution for the generl SL-problem (1). The set of eigenvlues forms n incresing sequence nd for ech eigenvlue, the corresponding eigenspce is generted by one function. More precisely, we hve the following Theorem,

Theorem 1. sequence STURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES 3 The set of eigenvlues of the SL-problem (1) forms n incresing λ 1 < λ < λ 3 <, with lim j λ j = nd for ech j Z +, the eigenspce of λ j is one-dimensionl: it is generted by one eigenfunction y j (x) The proofs of the Theorems of this Note will not given here but cn be found in either Ordinry Differentil Equtions by G. Birkhoff nd G. Rott; Methods of Mthemticl Physics by R. Cournt nd D. Hilbert; A First Course in PDE by H.F. Weinberger. Exmple 4. Let us find the eigenvlues nd eigenfunctions of the SL-problem (x y ) + λy = 1 < x < y(1) = y() = Note tht ODE cn be written s x y + xy + λy = which Cuchy-Euler eqution. Its chrcteristic eqution is m + m + λ =. The chrcteristic roots re m 1 = 1 1 4λ nd m = 1 + 1 4λ. We distinguish three cses depending on the sign of 1 4λ. If λ < 1/4, then 1 4λ >. Both chrcteristic roots m 1 nd m re distinct nd rel. The generl solution of the ODE is The endpoints conditions re y = C 1 x m 1 + C x m C 1 + C =, C 1 m 1 + C m =. Since m 1 m, the only solution is C 1 = C =. Thus λ < 1/4 cnnot be n eigenvlue. If λ = 1/4, then m 1 = m = 1/. The generl solution of the ODE is y = Ax 1/ + Bx 1/ ln x. The endpoints conditions led to A = nd then B 1/ ln =. We hve gin y nd λ = 1/4 is not n eigenvlue. If λ > 1/4, set 1 4λ = 4ν with ν >. In this cse the chrcteristic roots re m 1, = 1 ± iν. Two independent solutions of the ODE re xm1 nd x m. These re however (conjugte) complex-vlued functions. We need to tke their rel nd imginry prt. For this recll tht x +ib = x x ib = x e ib ln x = x cos(b ln x) + x sin(b ln x). The generl solution of the ODE is therefore y(x) = x 1/ (C 1 cos(ν ln x) + C sin(ν ln x)) = C 1 cos(ν ln x) + C sin(ν ln x) x. The endpoints conditions give y(1) = C 1 =, y() = C 1 cos(ν ln ) + C sin(ν ln ) =.

4 STURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES Hence, C sin(ν ln ) =. To obtin nontrivil solution y, we need to hve sin(ν ln ) =. Thus, ν = nπ ln with n Z+. From 1 4λ = 4ν, we get the eigenvlues λ n = 1 4 + ν n = 1 ( nπ ) 4 +, n Z + ln nd the corresponding eigenfunctions y n (x) = 1 nπ ln x sin. x ln. Orthogonlity of Eigenfunctions In view of pplying eigenfunctions of SL-problems (p(x)y ) + [λr(x) q(x)] y = α 1 y() + α y () = β 1 y(b) + β y (b) = to expnd functions nd solve BVP, we introduce the following inner product in the spce Cp[, b] of piecewise continuous functions on the intervl [, b]. We define the inner product with weight r(x) s follows: for f, g Cp[, b], < f, g > r = f(x)g(x)r(x)dx. The norm f r of function f is defined s f r = ( 1/ b < f, f > r = f(x) r(x)). Two function f, g C p[, b] re sid to be orthogonl if < f, g > r =. We re going to show tht for given SL-problem, eigenfunctions corresponding to distinct eigenvlues re orthogonl. More precisely, we hve the following Theorem. Theorem. Let y(x) nd z(x) be two eigenfunctions of the SL-problem corresponding to two distinct eigenvlues λ nd µ. Then y nd z re orthogonl with respect to the inner product <, > r. Tht is, < y, z > r = Proof. By using µrz = (pz ) + qz, we get µ y(x)z(x)r(x)dx = = y(x)z(x)r(x)dx =. y(x) [ (p(x)z (x)) + q(x)z(x)] dx y(x)(p(x)z (x)) dx + q(x)y(x)z(x)dx We use integrtion by prts for the first integrl in the right hnd side y(x)(p(x)z (x)) dx = [y(x)p(x)z (x)] b p(x)y (x)z (x)dx = [p(x)(y(x)z (x) y (x)z(x))] b + + (p(x)y (x)) z(x)dx

STURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES 5 Now we use the endpoints conditions to get Similrly, β 1 (y(b)z (b) y (b)z(b)) = (β 1 y(b))z (b)) y (b)(β 1 z(b)) = β y (b)z (b) + β y (b)z (b) = β (y(b)z (b) y (b)z(b)) =. Since β 1 nd β re not both zero, then y(b)z (b) y (b)z(b) =. An nlogous rgument gives y()z () y ()z() =. This imply tht Therefore, nd then µ y(x)z(x)r(x)dx Since λ µ, then necessrily () [p(x)(y(x)z (x) y (x)z(x))] b =. y(x)(p(x)z (x)) dx = = = = λ b Consider the regulr SL-problem (p(x)y (x)) z(x)dx (p(x)y (x)) z(x)dx + [ (p(x)y (x)) + q(x)y(x)] z(x)dx y(x)z(x)r(x)dx. y(x)z(x)r(x)dx =. 3. Generlized Fourier Series (p(x)y ) + [λr(x) q(x)] y = α 1 y() + α y () = β 1 y(b) + β y (b) = q(x)y(x)z(x)dx We know tht the eigenvlues λ j form n incresing sequence nd lim j λ j = nd for ech j, the eigenspce hs dimension 1 (Theorem 1). Let y j (x) be n eigenfunction corresponding to the eigenvlue λ j. By nlogy with Fourier series, we re going to ssocites to ech piecewise continuous function f on [, b] series in the eigenfunctions y j : f(x) c j y j (x) where the coefficients c j re given by (3) c j = < f, y j > r y j r = f(x)y j(x)r(x)dx y j (x)r(x)dx The following Theorem sys tht the system of eigenfunctions {y j } j is complete in the spce C p[, b] mening tht the bove generlized Fourier series is representtion of f.

6 STURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES Theorem 3. Let f be piecewise smooth function on the intervl [, b] nd let {y j } j Z + be the system of eigenfunctions of the SL-problem (). Then we hve the following representtion of the function f f v (x) = c j y j (x), where the coefficients c j re given by formul (3). In prticulr, t ll points x t which f is continuous, we hve f(x) = c j y j (x), Exmple 1. Consider the SL-problems y (x) + λy(x) = < x < 1 y() = y(1) y (1) = It cn be verified (I leve it s n exercise) tht λ < cnnot be n eigenvlue. For λ =, the generl solution of the ODE is y(x) = A + Bx. The first condition y() = gives A =. For y = Bx, the second condition becomes is trivilly stisfied (B B = ). Hence λ = is n eigenvlue with eigenfunction y = x. For λ >, we set λ = ν, the generl solution of the ODE is y(x) = C 1 cos(νx)+ C sin(νx). The condition y() = gives C 1 =. With y = C sin(νx) the second condition y(1) y (1) = gives C sin(ν) C ν cos(ν) =. In order to hve nontrivil solution (so C ), the prmeter ν > must stisfy the eqution sin(ν) ν cos(ν) = tn(ν) = ν. This eqution hs unique solution in ech intervl (jπ, jπ + (π/)). Indeed, Positive roots of tnν =ν ν 1 ν ν 3 ν 4 consider the function g(ν) = sin ν ν cos ν. We hve g(jπ) = (jπ)( 1) j nd

STURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES 7 g(jπ + (π/)) = ( 1) j. Thus g(jπ)g(jπ + (π/)) = jπ <. The intermedite vlue theorem implies tht g hs zero between jπ nd jπ + (π/). Since the derivtive g (ν) = ν sin ν does not chnge sign in the intervl [jπ, jπ + (π/)], then g hs unique zero in the intervl. The sme rgument shows tht g hs no zeros in the intervl [jπ + (π/), jπ + π]. The set of eigenvlues nd eigenfunctions of the SL-problem consists therefore of λ =, y (x) = x, nd for j Z + λ j = ν j, y j (x) = sin( x), where tn =, (jπ, jπ + (π/)) The pproximte vlues of the first eight vlues of nd λ j re listed in the following tble j 1 3 4 5 6 7 8 4.493 7.75 1.94 14.66 17..371 3.519 6.666 λ j. 59.7 118.9 197.9 96.6 415. 553. 711.1 Exmple. Let us find the representtion of the function f(x) = 1 on [, 1] in terms of the eigenfunctions x, sin( x), j = 1,, 3, of the SL-problem of Exmple 1. Tht is, 1 = c x + c j sin( x), x (, 1). The coefficients re given by c = < 1, x > x, c j = < 1, sin(x), j = 1,, 3, sin( x) Note tht the weight in this exmple is r(x) 1. We hve, nd < 1, x >= < 1, sin( x) >= 1 1 For the norms of the eigenfunctions, we hve nd (recll tht sin = cos ) 1 x = xdx = 1 sin( x)dx = 1 cos. 1 x dx = 1 3 1 sin( x) = sin ( x)dx = 1 (1 cos( x))dx = 1 [ x sin(ν ] 1 jx) = 1 ( 1 sin(ν ) j) = 1 ( 1 sin ν ) j cos = 1 cos = sin.

8 STURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES We hve therefore, c = 1/ 1/3 = 3, nd c j = (1 cos )/ (1 cos )/ = (1 + cos ), j 1. The expnsion of f(x) = 1 is 1 = 3 x + (1 + cos ) sin(x) x (, 1). By using the pproximte vlues of the given in the tble, we get 1 = x +.569 sin(4.493x) +.9 sin(7.75x)+ 3 +. sin(1.94x) +.133 sin(14.67x) + 4. Applictions to Boundry Vlue Problems We illustrte the use of SL problems in solving boundry vlue problems. Exmple 1. Consider the problem deling with the one-dimensionl het propgtion. The temperture u(x, t) stisfies the problem u t = ku xx < x < L, t > u(, t) =, u(l, t) = u x (L, t) t > u(x, ) = f(x) < x < L where is positive constnt. Suppose tht < L. Let u(x, t) = X(x)T (t) be nontrivil solution of the homogeneous prt (u t = ku xx, u(, t) =, nd u(l, t) = u x (L, t)). The seprtion of vribles leds to the following ODE problems for X nd T : { X (x) + λx(x) = X() =, X(L) = X nd T (t) + kλt (t) =. (L) The X-problem is regulr SL-problem. It is nlogous to the problem of the exmple of the previous section. This time however λ = is not n eigenvlue. Indeed, for λ =, the generl solution of the ODE is X(x) = A + Bx, the first condition X() = gives A =, nd the second implies tht BL = B. Since < L, then B = nd X. The eigenvlues nd eigenfunctions re λ j = ν j, X j (x) = sin( x), j Z + where is the j-th positive root of the eqution sin(νl) = ν cos(νl) tn(νl) = ν. For ech j, the corresponding solution solution of the homogeneous prt is u j (x, t) = exp( kνj t) sin(x). The superposition principle gives the generl series solution s u(x, t) = c j e k t sin( x). In order for such solution to stisfy the nonhomogeneous condition, we need to hve u(x, ) = f(x) = c j sin( x).

STURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES 9 Therefore, c j = < f(x), sin(ν L jx) > sin( x) = f(x) sin(x)dx L sin ( x)dx The norms (squred) of the eigenfunctions sin( x) cn be clculted s in the previous section: L L sin( x) = sin ( x)dx = 1 (1 cos( x))dx = 1 [ x sin(ν ] L jx) = 1 ( L sin(ν ) jl) = 1 ( L sin(ν ) jl) cos( L) = L cos ( L). If for exmple f(x) = x, then L [ ] L x cos(νj x) L < f(x), sin( x) > = x sin( x)dx = The coefficients c j re = L cos(l) ( L) cos(l) c j = L cos ( L) The solution of the BVP is u(x, t) = ( L) + sin(l) ν j = L cos(l) + cos( L) ν j = ( L) cos(l) + cos( L) = ( L) (L cos ( L)). cos( x) cos( L) (L cos ( L)) exp( k t) sin( x). Exmple. The following problem models the wve propgtion in nonhomogeneous string (the mss density of the string is not constnt). u tt = (1 + x) u tt, < x < 1, t >, u(, t) =, u(1, t) =, t >, u(x, ) = f(x), < x < 1, u t (x, ) = g(x), < x < 1. The mss density of the string t the point x is 1/(1 + x). The seprtion of vribles u(x, t) = X(x)T (t) for the homogeneous prt leds to the following ODE problems X λ + (1 + x) X = nd T + λt =. X() = X(1) = The X-problem is regulr SL-problem with weight r(x) = 1/(1 + x). dx

1 STURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES Now we need to find the eigenvlues nd eigenfunctions of the X-problem. The ODE cn be written s (1 + x) X + λx =. It is Cuchy-Euler type eqution in the vrible (1 + x). We seek then solutions of the form (1 + x) m. It leds to the chrcteristic eqution m m + λ =. The chrcteristic roots re therefore m 1, = 1 ± 1 4 λ Depending on the prmeter λ, we consider three cses. If λ < 1/4, then m 1 nd m re rel nd distinct. The generl solution of the ODE is X(x) = C 1 (1 + x) m1 + C (1 + x) m. The boundry conditions give X() = C 1 + C =, X(1) = C 1 m 1 + C m =. The only solution is C 1 = C = nd such λ cnnot be n eigenvlue. If λ = 1/4, then m 1 = m = 1/. The generl solution of the ODE is X(x) = A 1 + x + B 1 + x ln(1 + x). In this cse gin the boundry conditions imply tht A = B = nd λ = 1/4 cnnot be n eigenvlue. If λ > 1/4, then m 1 nd m re complex conjugte. Set λ = ν + (1/4), with ν >, so tht the chrcteristic roots re m 1 = 1 + iν, m = 1 iν. The C-vlued independent solutions of the ODE re (1 + x) m1, = (1 + x) 1/ (1 + x) ±iν = 1 + xe ±iν ln(1+x). The generl R-vlued solution is therefore X(x) = A 1 + x cos(ν ln(1 + x)) + B 1 + x sin(ν ln(1 + x)). The condition X() = gives A =, nd then the condition X(1) = gives B sin(ν ln ) =. In order to obtin nontrivil solution (B ), the prmeter ν needs to stisfy sin(ν ln ) = ν ln = jπ, j Z +. The eigenvlues nd eigenfunctions of the X-problem re therefore λ j = 1 4 +, X j (x) = 1 + x sin( ln(1 + x)), where = jπ ln. For ech j Z +, the corresponding independent solutions of the T -problem re T 1 j (t) = cos( λ j t), nd T j (t) = sin( λ j t). The generl series solution of the homogeneous prt of the problem is [ u(x, t) = A j cos( λ j t) + B j sin( 1 λ j t)] + x sin(νj ln(1 + x)) In order for such series to stisfy the nonhomogeneous conditions, we need u(x, ) = f(x) = A j 1 + x sin(νj ln(1 + x)) u t (x, ) = g(x) = λj B j 1 + x sin(νj ln(1 + x))

STURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES 11 The coefficients A j nd B j re given by A j B j = < f(x), 1 + x sin( ln(1 + x)) > r, 1 + x sin( ln(1 + x)) r = < g(x), 1 + x sin( ln(1 + x)) > r λj 1 + x sin( ln(1 + x)) r 5. Other Sturm-Liouville Problems Other SL-problems importnt in pplictions include the periodic Sturm-Liouville problem: (p(x)y ) + [λr(x) q(x)] y = y() = y(b) y () = y (b) where the coefficient p stisfies p() = p(b). The set of eigenvlues is s in Theorem 1 but this time for ech eigenvlue λ j, there re two independent eigenfunctions y 1 j (x) nd y j (x). Exmple 1. The following periodic SL-problem ws encountered in connection with BVPs in polr coordintes Θ (θ) + λθ(θ) =, Θ() = Θ(π), Θ () = Θ (π) The eigenvlues re λ j = j nd the eigenfunctions re Θ 1 j (θ) = cos(jθ) nd Θ j (θ) = sin(jθ). The hypotheses on the coefficients of the ODE in regulr SL-problem includes p(x) > nd r(x) > on [, b]. If ny one these hypotheses is wekened, it led to the singulr Sturm-Liouville problem. For some singulr problems, the boundry conditions cn lso be wekened or re not needed. The following singulr SLproblem will be studied in more detils. It is relted to Bessel functions. ) (xy ) + ( m x + λx y =, < x < L, y(l) = Here p(x) = x nd r(x) = x, both functions vnish t x =. The boundry condition t x = is not needed. This eqution is relted to boundry vlue problems in cylindricl coordintes. Another singulr problem of interest is the Legendre eqution ( (1 x )y ) + n(n 1)y = 1 < x < 1 This time r(x) = 1 nd p(x) = 1 x vnishes t both ends ±1 of the intervl. No boundry conditions re needed. This problem ppers in connection to boundry vlue problems in sphericl coordintes. 6. Exercises For exercises 1 to 4: () find the eigenvlues nd eigenfunctions of the Sturm- Liouville problems; (b) find the generlized Fourier series of the functions f(x) = 1 nd g(x) = x. Exercise 1. y + λy =, < x < 1, y() = nd y (1) =

1 STURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES Exercise. y + λy =, 1 < x < 1, y( 1) = y(1) nd y ( 1) = y (1) (periodic SL problem) Exercise 3. y + λy =, < x < 1, y() = nd y(1) + y (1) = Exercise 4. y + λy =, < x < 1, y() = y () nd y(1) = y (1) Exercise 5. Consider the problem x y + xy + λy =, 1 < x < L, y(1) =, y(l) =, with L > 1. (1) Put the ODE in djoint form: (py ) +(q +λr)y = (Hint: multiply by 1/x). () Wht is the inner product relted to this problem? (3) Find the eigenvlues nd eigenfunctions (note: the ODE is Cuchy-Euler). (4) Find the generlized Fourier series of the function f(x) = 1 (Hint: when computing the Fourier coefficients c j, you cn use the substitution t = ln x in the integrl). (5) Sme question for the function g(x) = x. Exercise 6. Sme questions s in Exercise 5 for the SL-problem x y + xy + λy =, 1 < x < L, y (1) =, y (L) =, Exercise 7. Solve the BVP Exercise 8. Solve the BVP Exercise 9. Solve the BVP Exercise 1. Solve the BVP where u t = u xx < x < π, t >, u(, t) = t > u(π, t) + u x (π, t) = t > u(x, ) = sin x < x < π u t = u xx < x < π, t >, u x (, t) = t > u(π, t) = u x (π, t) t > u(x, ) = 1 < x < π u tt = c u xx < x < π, t >, u(, t) = t > u(π, t) u x (π, t) = t > u(x, ) = sin x < x < π u t (x, ) = < x < π u tt = c u xx < x < π, t >, u(, t) = t > u(π, t) u x (π, t) = t > u(x, ) = < x < π u t (x, ) = f(x) < x < π f(x) = { if < x < (π/), 1 if (π/) < x < π.

STURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES 13 Exercise 1. Solve the BVP u tt = u xx < x < π, t >, u x (, t) = u(, t) t > u x (π, t) = t > u(x, ) = < x < π u t (x, ) = 1 < x < π Exercise 13. Solve the BVP u t = (1 + x) u xx < x < 1, t >, u(, t) = u(1, t) = t > u(x, ) = x(1 x) 1 + x < x < 1