BOUNDED VARIATION SOLUTIONS TO STURM-LIOUVILLE PROBLEMS

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1 Elecronic Journal of Differenial Equaions, Vol. 18 (18, No. 8, pp ISSN: URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu BOUNDED VARIATION SOLUTIONS TO STURM-LIOUVILLE PROBLEMS JACEK GULGOWSKI Communicaed by Pavel Drabek Absrac. In his aricle we consider singular Surm-Liouville problems whose righ-hand side is a funcion of bounded Jordan variaion. We presen necessary and sufficien condiions for all soluions o be of bounded Jordan variaion. 1. Regular and singular Surm-Liouville problems Le I = [, 1] denoe he closed uni inerval. As usual, L p (a, b will denoe he space of all equivalence classes (of almos everywhere equaliy of he real-valued funcions whose p-h power is Lebesgue-inegrable on (a, b. By L p loc (a, b we will denoe he space of funcions belonging o L p (a, b for all [a, b ] (a, b. The space of funcions absoluely coninuous in he closed inerval [a, b] will be denoed by AC[a, b], while he se of all funcions x: (a, b R which resricions belong o all spaces AC[a, b ], for [a, b ] (a, b will be denoed by AC loc (a, b. The Jordan variaion of he funcion x: [a, b] R will be denoed by b a x and he space of all funcions of bounded Jordan variaion in he inerval [a, b] will be denoed by BV ([a, b]. By χ A we will denoe he characerisic funcion of he se A I. Le us now have a look a he classical linear Surm-Liouville problem (p(x ( + q(x( = h( for a.e. (, 1 x( sin η p(x ( cos η =, x(1 sin ζ + p(1x (1 cos ζ =, (1.1 where η, ζ [, π ]. The soluion o his problem is such a funcion x : I R ha boh x ( and (p(x ( exiss a.e. in I, and x, px AC(I and he boundary condiions are saisfied. Assume now ha p C 1 (I, q C(I and p( > for I. In his case we may alk of he soluions o he problem (1.1 in a classical sense (i.e. belonging o C (I when h C(I and o C 1 (I wih x AC(I when h L 1 (I see [3, 7]. Such soluions are, of course, funcions of bounded Jordan variaion. Bu he siuaion may change when he assumpions on coefficiens p and q are released as we will furher observe. 1 Mahemaics Subjec Classificaion. 34B4, 6A45, 45A5. Key words and phrases. Boundary value problem; bounded variaion; Green funcion; Jordan variaion, Surm-Liouville problem. c 18 Texas Sae Universiy. Submied November 6, 17. Published January 6, 18. 1

2 J. GULGOWSKI EJDE-18/8 The more general aiude owards Surm-Liouville boundary value problems allows for such p, q ha 1/p, q L 1 (I, wih he funcion r = 1/p allowing for sign changes and even for achieving on he posiive measure se. In his las case we should hink of x ( = whenever r( =. The assumpions of his kind sill keep us in he orbi of he so-called regular Surm-Liouville problems bu we may also release even hese, leading o he so-called singular Surm-Liouville problems wih Fokker-Planck, Bessel s or Whiaker problems as an example (see [4, Chaper.5]. In his case i may happen ha he soluions x are no defined in he endpoins a {, 1}, hence he boundary condiions should be undersood in a differen way. This issue will be discussed in deails laer. In case of regular Surm-Liouville problems he quesion wheher a soluion x is of bounded Jordan variaion seems o have a rivially posiive answer (since x AC(I, bu acually his may be saed only when x L 1 (I, which is no always he case! We should noe ha in case of singular problems we canno assume ha x is absoluely coninuous in he closed inerval I. Hence here may exis a soluion x of unbounded Jordan variaion. Example 1.1. Le p : I R be given by p( = ( 1 n+1 1 n for [ n 1, n ], where n = 6 n π 1 k, n = 1,,.... Then r = 1/p L 1 (I bu r L 1 ([, a] for each a [, 1. Le us look a he boundary value problem (p(x ( = 1 x( = x(1 =. for a. e. I (1. As i may be easily observed, we have x ( = r(( C, for some consan C R and [, 1. Since r(τ(τ C L 1 (, for any [, 1 we can see ha, having he boundary condiion x( =, x( = τr(τdτ C r(τdτ. Le us denoe R( = r(τdτ and R ( = τr(τdτ, hen we are looking for a funcion x( = R ( CR(. Le us now assume ha = N. We will find he values R( N and R ( N. R( N = N k k 1 ( 1 k+1 kdτ = 6 π N ( 1 k+1 1 k. Since he series + ( 1k+1 1 k converges o ln, i follows ha lim N + R( N = 6 π ln. Similarly, R ( N = N k k 1 ( 1 k+1 kτdτ = 6 π N ( 1 k+1 1 k k 1 + k. Since he sequence ( 1 ( k 1 + k is monoone and bounded we conclude, by k N Abel s convergence es, ha lim N + R ( N = r exiss. We also observe ha

3 EJDE-18/8 STURM-LIOUVILLE PREOBLEMS 3 r < 6 π ln. This is a consequence of he esimae r = 6 + ( 1 π k 1 < 6 + ( 1 π k 1 = 6 π + < 6 + ( 1 π k + k 1 k + k 1 1 k 1 k k + k 1 ( 1 k 1 1 k k 1 1 k 6 = ln. π k 1 + k k + k 1 As we can see in case ( N 1, N he values of R( and R ( lay beween R( N 1 and R( N or R ( N 1 and R ( N respecively, so lim 1 R( = 6 π ln and lim 1 R ( = r. This means ha he lim 1 x( = r C 6 π ln and wih he appropriae value of consan C = C = π r 6 ln we may say ha x(1 =. Le us now esimae he difference x( n x( n 1 = 6 ( 1 n+1 1 n + n 1 π C ( 1 n+1 1 = 6 n + n 1. n n nπ C Since lim n+ n 1 n + = 1 C, i follows ha + n=1 x( n x( n 1 = + which shows ha x is no he funcion of bounded Jordan variaion. In case of he singular Surm-Liouville problems he boundary condiions as given in (1.1 may urn ou o be no reasonable any more. For a deailed review of differen siuaions we sugges he monograph by Zel (see [9]. Also he review paper [6] would be of much help here. The mos general form of he Surm-Liouville equaion is given as he specral problem (p(x ( + q(x( = λw(x(. (1.3 We will refer o his general heory in a special case of w( 1 and λ =. We will also limi ourselves o he real-valued funcions belonging o he domain D, which is defined as he naural domain of he differenial operaor x (p(x ( + q(x(. (1.4 i.e. is given as he se D of all such funcions x : I R ha boh x, px AC loc (I. Acually, we will consider he so-called maximal domain of he differenial operaor (1.4 D max = D L (I (see [9]. This is he very naural assumpion o ake when we are going o look a he problem from he operaor heory perspecive. Hence, we will look a he linear differenial equaion (p(x ( + q(x( = (1.5 in he search for soluions x D max. The ineresing issues are relaed o he behaviour of he differenial operaor near he endpoins of he inerval I, i.e. in he neighbourhood of and 1. There is a sandard classificaion of endpoins (see [9, Definiion 7.3.1], which daes back o 191 and he works of Weyl. Le us assume ha he endpoin a {, 1}. Then we have he following:

4 4 J. GULGOWSKI EJDE-18/8 Regular endpoin if here exiss such an open inerval J I wih a J such ha q, r L 1 (J; Singular endpoin if here exiss such an open inerval J I wih a J, J I such ha q( + r( d = + ; J he singular poin is called Limi-poin (denoed laer by LP when here exiss a leas one soluion x o he problem (1.5 saisfying J x ( d = + for cerain open inerval J I, J I saisfying a J; he singular poin is called Limi-circle (denoed laer by LC when all soluions o he problem (1.5 belong o L (J for an open inerval J I, a J, J I; Remark 1.. Since we look for locally AC soluions he classificaion given above does no depend on he selecion of he inerval J, as long as J I. Remark 1.3. The original definiions of LP and LC aken from [9] or [6] refer o he more general specral problem (1.3 and formally depend on λ. Bu i may acually be proved ha i does no depend on λ (cf. [6, Remark 5.1] and he classificaion given above is he same as he one given in [9]. The mos general seing of boundary condiions referring o (1.3 requires he noion of he Lagrange sesquilinear form (cf. [9, Remark 8..1], which is given by [f, g]( = f((pg ( g((pf ( = p((f(g ( f (g(, for all f, g D max. I may be shown (see [9, Lemma 1..3] ha for any f, g D max boh limis +[f, g]( and lim [f, g]( lim exis and are finie. From now on, when wriing [f, g]( and [f, g](1 we will refer o he appropriae limi. The appropriae selecion of funcion g D max gives rise o he boundary condiion depending on [g, x](1 or [g, x]( =. The deails depend here on he endpoin classificaion. One should, firs of all, observe ha in case of he LP endpoin a here is no need o specify he boundary condiion a all, since for all funcions g, x D max here is [g, x](a = (see [9, Lemma 1.4.1]. In case of one regular or LC and one LP endpoin he boundary value problem may be given as 1 (p(x ( + q(x( = h( for a.e. (, 1 [g, x](a =, where a {, 1} is no a LP endpoin and g D max. When boh endpoins are regular or LC, (p(x ( + q(x( = h( for a.e. (, 1 [g 1, x]( [g, x]( =, [g 1, x](1 [g, x](1 =, (1.6 (1.7 for cerain selecion of funcions g 1, g D max. From now on we refer o he mos general version of Surm-Liouville boundary value problem as he one given by (p(x ( + q(x( = h( for a.e. (, 1 [g 1, x]( [g, x]( = if is no an LP endpoin, [g 1, x](1 [g, x](1 = if 1 is no an LP endpoin, (1.8

5 EJDE-18/8 STURM-LIOUVILLE PREOBLEMS 5 wihin he convenion ha if here is only one LP endpoin hen we ake g =. I may be shown (cf. [9, Chaper 1, Secion 4.1] ha, in case of he regular endpoin, his general form of boundary condiions covers he case of classical boundary condiions (1.1. Remark 1.4. We should noe ha he boundary condiions in he form [g 1, x](a [g, x](a =, where a {, 1} may also ake he form [g i, x](a =, where i {1, }. This is he case when he funcion g j for j i will saisfy g j ( = in some open neighbourhood of a.. Green funcion The main ool used when solving he problem (1.1 is he Green funcion, i.e. such G : (, 1 (, 1 R ha x is a soluion o (1.1 if and only if x( = G(s, h(sds. (.1 This requires, of course, some care, a leas in specifying he space which funcion h belongs o, as well as he assumpion ha is no he eigenvalue of he problem (1.1. We are no going o be very sric abou i a he momen and we will reurn o his issue laer. In he classical heory he Green funcion (see [7, Chaper XI, Exercise.1] for he problem (1.1 wih q C(I, p C 1 (I, p >, is given as { c 1 x 1 (sx ( s 1 G(s, = c 1 (. x 1 (x (s s 1, where x 1, x : I R are linearly independen soluions of (1.1 saisfying he iniial condiions x 1 ( sin η p(x 1( cos η =, x (1 sin ζ + p(1x (1 cos ζ = and c is he appropriae consan, c. Similar formula (see [8] may be given for he problem (1.1 wih r = 1/p L 1 (I, p C 1 ((, 1, (, + and q =. Moreover, in he paper [5, Theorem 3.1] he Green funcion G : (, 1 (, 1 R for he singular problem (1.1 saisfying r = 1/p L 1 (, c and r = 1/p L 1 (c, 1 for some c (, 1, and such q L 1 loc (, 1 ha q( r(sds L1 (, 1 is given by (., wih he appropriae selecion of he basic soluions x 1 and x. General heorems on he exisence of he Green funcion for he problem (1.8, wih r, q L 1 loc (I, are given for example in [9, Theorems 9.4., 1.1.1]. I is worh noing ha here are, in general, no assumpions on he sign of p and q. The heorem given below is acually he reformulaion of facs given in [9], bu we presen i here, wih a proof, for he sake of he clariy. Theorem.1. Assume he funcion x = is he only soluion o (1.8 wih h =. Le x 1, x D max be wo linearly independen soluions of he equaion (1.5 saisfying: (a exacly wo LP endpoins: no furher assumpions; (b exacly one non LP (i.e. regular or LC endpoin a, he oher endpoin being LP : [g 1, x 1 ](a = ; (.3

6 6 J. GULGOWSKI EJDE-18/8 Then (c boh endpoins being regular or LC: [g 1, x 1 ]( [g, x 1 ]( =, (.4 [g 1, x ](1 [g, x ](1 =, (.5 (i he funcion (, 1 (p(x 1(x ( (p(x (x 1 ( = c is a nonzero consan; (ii he funcion G : (, 1 (, 1 R given by { c 1 x 1 (sx ( < s < 1 G(s, = c 1 (.6 x 1 (x (s < s < 1 is he Green funcion for problem (1.8. This means ha for any h L (I here exiss a unique soluion x D max o problem (1.8 given by x( = = c 1 x ( G(s, h(sds x 1 (sh(sds + c 1 x 1 ( x (sh(sds. (.7 Proof. Propery (i is a sandard Wronskian-ype argumen, checked by a direc calculaion. Since he funcion (p(x 1(x ( (p(x (x 1 ( is locally absoluely coninuous i is enough o check ha is derivaive equals almos everywhere. To prove propery (ii i is sufficien o observe ha subsiuion of x given by (.7 ino our differenial operaor gives (p(x ( + q(x( = h(. We should now check ha he funcion x saisfies he boundary condiions. We will check if [g 1, x](a [g, x](a = for endpoin a which is no LP. Firs, le us now observe ha for i = 1, we have [g i, x]( = p(g i (x ( p(g i(x( ( = c 1 p(g i ( x ( x 1 (sh(sds + x 1( ( c 1 p(g i( x ( = c 1 [g i, x ]( x 1 (sh(sds + x 1 ( x 1 (sh(sds + c 1 [g i, x 1 ]( x (sh(sds x (sh(sds x (sh(sds. We should noe ha, since he funcions x 1, x, h belong o L (I, all inegrals in he formula above are finie. We also know ha he values [g i, x j ]( and [g i, x j ](1 are well-defined and finie (for i, j = 1, cf. [9, Secion 1.4]. Therefore, [g i, x](1 = c 1 [g i, x ](1 [g i, x]( = c 1 [g i, x 1 ]( x 1 (sh(sds, x (sh(sds, [g 1, x]( [g, x]( = c 1 x (sh(sds ( [g 1, x 1 ]( [g, x 1 ]( =,

7 EJDE-18/8 STURM-LIOUVILLE PREOBLEMS 7 [g 1, x](1 [g, x](1 = c 1 x 1 (sh(sds ( [g 1, x ](1 [g, x ](1 =, which follows from assumpions (.4 and (.5. The uniqueness is he consequence of he assumpion ha guaranees ha is he only soluion o he problem (1.8 wih he righ-hand side equal o. Remark.. In case boh endpoins are regular we ake g 1 and g as funcions saisfying g 1 ( = cos η, p(g 1( = sin η, g 1 (1 =, p(1g 1(1 = and g (1 = cos ζ, p(1g (1 = sin ζ, g ( =, p(g ( = (see [9, Lemma 1.4.4] and furher discussion here. This acually recreaes he classical boundary condiions given for problem ( Bounded variaion soluions As menioned in he previous secion, having he Green funcion, we may ranslae he problem (1.8 ino he Hammersein equaion (.1 wih G : (, 1 (, 1 R given by (.6 and x 1, x saisfying assumpions of Theorem.1. Then, he value of he inegral operaor K : L (, 1 L (, 1 is given by x( = (Kh( = G(s, h(sds (3.1 is he unique soluion o he problem (1.8 wih he righ-hand side h BV (I. Now, he quesion wheher he problem (1.8 admis he soluions of unbounded variaion may be ranslaed ino he properies of he kernel of he inegral operaor, for he Hammersein equaion (i.e. he Green funcion. We should refer o some of he recen resuls on he subjec. Firs of all, in [, Theorem 4] we may find he characerizaion of he kernels ha induce he coninuous linear map from BV (I o BV (I, i.e. he assumpions ha guaranee ha for a BV righ-hand side h of he problem (1.8 here exis only BV soluions: Theorem 3.1. Le K be a linear inegral operaor generaed by he kernel k : I I R, ha is, K is given by he formula Kx( = k(, sx(sds, I. (3. The operaor K maps coninuously he space BV (I ino iself if and only if he following condiions are saisfiedp: (H1 for every I, he funcion s k(, s is Lebesgue inegrable on I; (H here exiss a posiive consan M such ha sup I ( k(, sds M. Moreover, for some kernels k : I I R he inegral operaor K given by (3. maps he space L (I of all essenially bounded funcions ino he space BV (I of funcions of bounded variaion (see [, Proposiion 4]. Tha would mean ha in such case, for any righ-hand side h belonging o L (I, we have only BV soluions o problem (1.8. This may acually be easily generalized o:

8 8 J. GULGOWSKI EJDE-18/8 Theorem 3.. Assume p [1, + ] is fixed and he kernel k : I I R is such ha (a for every I he funcion s k(, s is Lebesgue measurable; (b he funcion s k(, s belongs o L p (I; (c 1 k(, s m(s for a.e. s I, where m Lp (I,. Then he operaor K, generaed by he kernel k, coninuously maps he space L q (I ino he space BV (I, where 1/p + 1/q = 1. Based on he observaion given above we prove he following heorem. Theorem 3.3. Assume here exis funcions x 1, x : I R saisfying assumpions of Theorem.1 and x 1, x BV (I. Then for any h L 1 (I he soluion x o he problem (1.8 is of bounded variaion. Proof. We are going o show ha he Green funcion G(s, given by (.6 saisfies he condiions (a (c of Theorem 3. for a funcion m L (I. Since funcions belonging o BV (I are measurable and bounded, hen condiions (a and (b are obvious. Now, we are going o check he condiion (c. Le us fix s I. Then and k(, s = G(s, = c 1 x 1 (x (sχ [,s] ( + c 1 x 1 (sx (χ [s,1] ( ( s k(, s c 1 x (s x 1 + x 1 (s + c 1 x 1 (s ( c 1 x 1 (s ( 1 s x + x (s x + x (s x 1 + x 1 (s x (s. ( Wih m(s := c 1 x 1 (s 1 x + x (s 1 x 1+ x 1 (s x (s m L (I. This complees he proof. here is, of course, Now, he quesion appears wha if x 1 or x are no of bounded Jordan variaion. Does i imply ha here exiss he unbounded variaion soluion o he problem (1.8, wih he righ-hand side belonging o he space BV (I? We are going o show ha i is no ha easy. The firs observaion is ha, since boh funcions are locally AC, hen he one wih unbounded variaion mus have infinie variaion in he neighbourhood of or 1. To characerize he Green funcions G ha induce he inegral operaor ransferring funcions of bounded Jordan variaion ino he funcions of bounded Jordan variaion, we will refer o he characerizaion given in Theorem 3.1. Firs, he crucial role of he funcions ϕ : I R, given by ϕ ( = G(s, χ [,] (sds (3.3 for fixed [, 1], should be noed. This is because he assumpion (H of Theorem 3.1 requires he variance of all funcions ϕ be uniformly bounded. Acually, as i may be easily observed, we may replace he funcions ϕ, by funcions ψ : I R, given by ψ ( = G(s, χ [,1] (sds. (3.4

9 EJDE-18/8 STURM-LIOUVILLE PREOBLEMS 9 In wha follows we will also refer o he funcion α: I R given by α( = G(s, ds = c 1 x ( x 1 (sds + c 1 x 1 ( x (sds. (3.5 Lemma 3.4. Le [, 1], ϕ, ψ, α: I R be given by (3.3, (3.4 and (3.5 respecively. Then ϕ = c 1 ] x 1 (sds x + [α c 1 x (sds x 1 and ψ = c 1 x (sds x 1 + [α c 1 x 1 (sds x ]. Proof. Le us fix [, 1]. As we may see { c 1 x ( ϕ ( = x 1(sds for (, 1] c 1 x ( x 1(sds + c 1 x 1 ( x (sds for [, ]. Similarly we have { c 1 x 1 ( ψ ( = x (sds for [, ] c 1 x ( x 1(sds + c 1 x 1 ( x (sds for (, 1]. Le us also observe ha for [, ], ϕ ( = c 1 x ( and for (, 1] we have ψ ( = c 1 x ( This complees he proof. x 1 (sds + c 1 x 1 ( x 1 (sds + c 1 x 1 ( x (sds = α( c 1 x 1 ( x (sds = α( c 1 x ( (3.6 (3.7 x (sds x 1 (sds. Lemma 3.5. Le [, 1], ϕ, ψ, α: I R be given by (3.3, (3.4 and (3.5 respecively. Then if here exiss such (, 1 ha x 1 = + hen 1 ψ = +. Similarly, if 1 x = + for some (, 1, hen 1 ϕ = +. Proof. Assume now ha x 1 = + holds. If his is he case, hen i does no depend on he selecion of (, 1. Then, since x, we may pick such (, 1 ha x (sds = c. Then c 1 1 x (sds x 1 = +, meaning (by Lemma 3.4 ha 1 ψ = +. Similarly we handle he case 1 x = +, concluding ha 1 ϕ = +. Lemma 3.6. Le [, 1], ϕ, ψ : I R be given by (3.3 and (3.4 respecively. Le y 1, y : I R be given by y 1 ( = x 1 ( x (sds and y ( = x ( x 1(sds. Then, if here exiss such (, 1 ha x 1 < + and y = + hen 1 ϕ = +. Similarly, if 1 x < + and 1 y 1 = + for some (, 1, hen 1 ψ = +.

10 1 J. GULGOWSKI EJDE-18/8 Proof. Assume now ha x 1 < + and y = +. We show ha [ ϕ = c 1 x ( x 1 (sds + c 1 x 1 ( x (sds ] = +. As we may observe: [ ϕ = c 1 y ( + c 1 x 1 ( x (sds ] The funcion [, ] x 1 ( x (sds is he produc of he bounded variaion and absoluely coninuous funcions, hence is of bounded variaion. Bu he sum of funcion from BV (I and he one wih unbounded Jordan variaion (i.e. funcion y gives he funcion of unbounded Jordan variaion. This complees he firs par of he proof. The second conclusion may be proved he same way. The following wo proposiions are direc consequences of Lemma 3.4. Proposiion 3.7. Assume ha 1 x < + for each (, 1. Then if lim sup x 1 (sds x ( = +, (3.8 hen sup I 1 ϕ = +. + Proposiion 3.8. Assume ha x 1 < + for each (, 1. Then if lim sup x (sds x 1 ( = +, (3.9 hen sup I 1 ψ = +. 1 Now, we are ready o presen he condiions for funcions x 1 and x, so he corresponding Green funcion is he kernel of he inegral operaor (3.1 which maps he space of funcions of bounded variaion ino iself. Theorem 3.9. Assume he map G : (, 1 (, 1 R is given as in Theorem.1. Then he inegral operaor K : L (I L (I given by (3.1 maps he space BV (I ino iself if and only if all of he following condiions are saisfied: (1 here exiss (, 1 such ha 1 x < + ; ( here exiss (, 1 such ha x 1 < + ; (3 here exiss (, 1 such ha 1 y 1 < +, where y 1 ( = x 1 ( x (sds; (4 here exiss (, 1 such ha y < +, where y ( = x ( x 1(sds; (5 lim sup + x 1(sds 1 x = M < + ; (6 lim sup 1 x (sds x 1 = M 1 < +. Remark 3.1. Because x 1, x are locally absoluely coninuous, he quanifier (, 1 appearing in condiions (1 (4 can be equivalenly replaced wih (, 1. In wha follows we should also remember ha x 1, x L (I, so also x 1, x L 1 (I and he funcions x 1(sds and x (sds are absoluely coninuous in he enire closed inerval I.

11 EJDE-18/8 STURM-LIOUVILLE PREOBLEMS 11 Proof of Theorem 3.9. We observe ha if all of he condiions (1 (6 are saisfied, hen ϕ < +. sup I Firs, we observe ha 1 α < +. Le us esimae ( α = c 1 x ( x 1 (sds + c 1 x 1 ( ( c 1 x ( 1/ ( c 1 x ( 1/ ( + c 1 x 1 ( x 1 (sds + x 1 (sds + 1/ c 1 y + c 1 1/ + c 1 x 1 ( c 1 x 1 ( ( c 1 x ( 1/ x (sds + x (sds ( c 1 x 1 ( 1/ x x 1 (s ds 1/ x (s ds + c 1 y 1. 1/ x (sds x 1 (sds x (sds The esimaes of he firs and fourh erm are finie by (3 and (4. The second and hird erms are esimaed by (1 and ( because b a xy x BV y BV (see [1, Proposiion 1.1] and b a c x(sds = b x(s ds for any [a, b] [, 1]. a Le us now ake a closer look a he condiions (5 and (6. Firs, we should noe ha from (5 i may be concluded ha here exiss such (, 1, ha for all (, x 1 (sds x M + 1. On he oher hand, for all [, 1 we esimae x 1 (sds x x 1 (s ds x. Le us denoe M := x 1(s ds 1 x. By (1 M is finie. Hence we know ha sup x 1 (sds x M + M + 1 := K < +. (,1 Similarly, from (6 and (, we may conclude ha sup (,1 x (sds x 1 := K 1 < +

12 1 J. GULGOWSKI EJDE-18/8 From Lemma 3.4 we infer ha ϕ = c 1 [ x 1 (sds x + α( c 1 x 1 ( and we esimae ( ϕ α + c 1 x 1 (sds x + α + K + K 1 < +, ] x (sds x (sds x 1 and he esimae is valid for any (, 1. This proves ha he condiion (H is saisfied and hence ha, by Theorem 3.1, he inegral operaor K maps BV (I o BV (I. Now, i is ime o noe ha each of he assumpions (1 (6 is a necessary condiion for having all soluions o he problem (1.8 wih righ-hand side h BV (I of bounded variaion. Condiions (1 and ( are necessary by Lemma 3.5. Condiions (3 and (4 are necessary by Lemma 3.6. Condiions (5 and (6 are necessary by Proposiions 3.7 and 3.8 and Lemma 3.5. Example 3.11 (Legendre equaion [6, Secion 19]. Le us consider he equaion ((1 x ( = (, 1 (3.1 wih he boundary condiions x( = and [g, x](1 =, where g( = 1. Here is he regular endpoin, while 1 is he LC endpoin. The wo basic soluions x 1 ( = 1 1+ ln 1 and x ( = 1 saisfy he assumpions of Theorem.1. I is no difficul o check ha he funcions x 1, x saisfy all assumpions (1 (6 of heorem 3.9, even if x 1 is of unbounded Jordan variaion. Hence here does no exis a soluion o he Legendre equaion which is a funcion of unbounded variaion, when he righ-hand side h belongs o BV (I. Example 3.1 (Legendre equaion, again. Le us again look a he Lengendre equaion (3.1 bu now wih he differen boundary condiions x ( = and [g, x](1 = wih g( = 1 1+ ln 1. As we can easily check he wo basic soluions x 1 ( = 1 and x ( = 1 1+ ln 1. Since he funcion x is unbounded in he neighbourhood of 1 we have 1 1/ x = +. Since (1 is no saisfied we know ha here exiss he righ-hand side of (3.1 of bounded variaion wih he corresponding soluion being funcion of unbounded Jordan variaion. References [1] Appell, J.; Banaś, J.; Merenes, N.; Bounded variaion and around, De Gruyer Sudies in Nonlinear Analysis and Applicaions, no. 17, De Gruyer, Berlin, 14. [] Bugajewski, D.; Gulgowski, J.; Kasprzak, P.; On inegral operaors and nonlinear inegral equaions in he spaces of funcions of bounded variaion, J. Mah. Anal. Appl. 444 (16, 3 5. [3] Coddingon, E. A.; Levinson, N.; Theory of Ordinary Differenial Equaions, McGraw-Hill, New York, [4] D. G. Duffy; Green s funcions wih applicaions, Applied Mahemaics, CRC Press, 1. [5] M. Duhoux; Green funcions and he propery of posiiviy for singular second order boundary value problems, Tara M. Mah. Publ. 19 (, 1.

13 EJDE-18/8 STURM-LIOUVILLE PREOBLEMS 13 [6] W. N. Everi; A caalogue of Surm-Liouville differenial equaions, (5, [7] P. Harman; Ordinary differenial equaions, Birkhauser, Boson-Basel- Sugar, 198. [8] Sun, Yan; Liu, L.; Cho, Y. L.; Posiive soluions of singular nonlinear Surm-Liouville boundary value problems, ANZIAM J. 45 (4, [9] A. Zel; Surm-Liouville heory, AMS, 5. Jacek Gulgowski Insiue of Mahemaics, Faculy of Mahemaics, Physics and Informaics, Universiy of Gdańsk, 8-38 Gdańsk, Poland address: dzak@ma.ug.edu.pl