ON A GENERALIZED STURM-LIOUVILLE PROBLEM
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1 Foli Mthemtic Vol. 17, No. 1, pp Act Universittis Lodziensis c 2010 for University of Łódź Press ON A GENERALIZED STURM-LIOUVILLE PROBLEM GRZEGORZ ANDRZEJCZAK AND TADEUSZ POREDA Abstrct. Bsic results of our pper re devoted to generlized Sturm- Liouville problem for n eqution of the form (p(ty (t +q(ty(t = F (t, y( with conditions { α1 y( + α 2 y ( = 0 β 1 y(b + β 2 y (b = 0 where α α2 2, β2 1 + β2 2 > 0, p(t 0 for t [, b], q C([, b] nd F is continuous trnsformtion from [, b] C([, b] to C([, b]. It is required tht the Green s function ssocited with this problem be nonnegtive Mthemtics Subject Clssifiction: 35J65. Key words nd phrses: Sturm-Liouville problem. 1. Introduction. Let F be continuous trnsformtion from [, b] C([, b] to C([, b] with the supremum norm. The min problem considered in this pper is the existence of solution of the generlized differentil eqution of the form (1 (py + qy = F (, y with boundry conditions { α1 y( + α 2 y ( = 0 (2 β 1 y(b + β 2 y (b = 0 for y C 2 ([, b] where α α2 2, β2 1 + β2 2 > 0, p C1 ([, b], p(t 0 for t [, b], q C([, b]. The modifiction of the Sturm Liouville problem we consider is motivoted by results of Fijłkowski Przerdzki [2] nd Fijłkowski, Przerdzki nd Stńczy [3] on nonlocl elliptic equtions. In our considertions we pply the following clssicl result. Theorem 1. ([1] p.41 Let P be cone in Bnch spce X, i.e. P is closed convex set such tht: (i λp P for λ 0, (ii P ( P = {0},
2 18 GRZEGORZ ANDRZEJCZAK AND TADEUSZ POREDA nd this cone is norml, i.e. there exists positive constnt C such tht v C w, for v, w P, v w. Let, for v, w X, the reltion v w denote tht w v P. Suppose tht mpping T : P X is completely continuous nd nondecresing, i.e. T (v T (w, for v w. If there exist points v 1, v 2 P, v 1 v 2, for which v 1 T (v 1 nd T (v 2 v 2, then the mpping T hs fixed point v 0 P such tht v 1 v 0 v Min results Before presenttion of the min results concerning (1 we need the following lemm: Lemm 1 (On subliner trnsformtion. Let T : C([, b] C([, b] be completely continuous nd monotonic trnsformtion such tht (1 T (0 0, (2 T (y α + βυ(y for some α, β C([, b], α, β 0, where υ is seminorm defined on C([, b] stisfying the condition (3 υ(β < 1. Then there exists function y 0 0 which is fixed point of the trnsformtion T. Proof. We cn notice tht for w 1 = 0 we hve T (w 1 w 1. We re looking for w 2 = c(β + ε, c, ε R +, such tht T (w 2 w 2. By the ssumptions T (w 2 α + βυ(c(β + ε, hence T (w 2 α + cβ(υ(β + ευ(1. On ccount of (1 it is not difficult to choose ε 0 > 0 such tht υ(β < 1 ε 0 υ(1. Then for tht ε 0 the inequlity T (w 2 α + cβ holds. Then we cn tke constnt c 0 big enough to stisfy the inequlity α c 0 ε 0. Thus for w 2 = c 0 (β + ε 0 we hve T (w 2 w 2. We denote by C + ([, b] the set of ll non-negtive functions in C([, b]. Clerly T is completely continuous nd monotonic on C + ([, b]. Moreover T (w 1 w 1 nd T (w 2 w 2 for w 1 = 0 nd w 2 = c 0 (β + ε 0, so the ssumptions of Theorem 1 re fulfiled. Hence there exists function y 0 C + ([, b] such tht w 1 y 0 w 2 nd T (y 0 = y 0. Theorem 2. Let continuous function F : [, b] C([, b] C([, b] stisfy the conditions: (i 0 F (, y 1 F (, y 2 for 0 y 1 y 2, y 1, y 2 C([, b], (ii F (, y f +h υ(y for some nonnegtive functions f, h L 1 ([, b], where υ is seminorm on C([, b].
3 ON A GENERALIZED STURM-LIOUVILLE PROBLEM 19 If the Green s function G ssocited with problem (1-2 is nonnegtive nd υ( G(, th(tdt < 1, then there exists C2 -solution to (1-2. The solution is nonnegtive. Proof. Let us consider n opertor T : C([, b] C([, b] of the form: T (y(s = G(s, t F (t, ydt for y C([, b], s [, b]. We observe tht ny fixed point of T is solution of (1-2. By properties of the function G nd the ssumptions bout the function F, the trnsformtion T is completely continuous. Furthermore T (y G(, tf(tdt + υ(y G(, th(tdt. Using the nottion α = G(, tf(tdt nd β = G(, th(tdt, we cn see tht T stisfies the ssumptions of the lemm on subliner trnsformtion. Consequently, there exists fixed point of T, which gives the existence of solution to problem (1-2. Theorem 3. If the trnsformtion F discribed in the ssumptions of Theorem 2 is of the form F (t, y = f(t+h(tυ(y, nd υ is n dditive seminorm on C + ([, b] such tht ( υ G(, tf(tdt 0, then the condition ( is lso necessry for existence of the solution y 0 to problem (1-2. For ny such solution υ(y 0 0. Proof. Under the bove ssumptions ny solution y 0 to problem (1-2 stisfies the eqution y 0 (x = G(x, t F (t, y 0 dt for x [, b]. In view of the form of F nd properties of the seminorm υ, we hve υ(y 0 = υ G(, tf(tdt + υ G(, th(tdt υ(y 0.
4 20 GRZEGORZ ANDRZEJCZAK AND TADEUSZ POREDA Hence υ(y 0 ( 1 υ G(, th(tdt = υ G(, tf(tdt Since υ( G(, tf(tdt > 0 nd υ(y 0 0, it follows from the bove equlity tht υ( G(, th(tdt < 1 nd υ(y 0 > 0. Corollry 1. Let f, g, h L 1 ([, b] be nonnegtive functions. differentil- integrl problem (3 (py + qy = f + h g y L 1 for y C([, b]. For ny with boundry conditions (2 nd nonnegtive Green s function G let us denote α = G(, tf(tdt, β = G(, th(tdt. The problem hs solution if nd only if one of the following conditions: holds. (i g α = 0.e., or (ii g β L 1 < 1 Proof. Observe tht the function α is the solution to the differentil-only prt of the problem. Theorem 4. Let the function F in (1 stisfy the following conditions: (i 0 F (, y 1 F (, y 2 if only 0 y 1 y 2 for y 1, y 2 C([, b], nd (ii F (, y f + A(, sy(sds, for some functions A C([, b] [, b], f C + ([, b] nd y C([, b]. Let Γ(A(u, s = G(u, ta(t, sdt for u, s [, b]. If either ( there exist p > 1 nd q such tht 1 p + 1 q = 1 nd or (b mx u [,b] Γ(A(u, L 1 < 1, Γ(A(u, du q L p < 1, then problem (1-2 hs nonnegtive solution in C 2 ([, b]. Proof. Let n opertor T : C([, b] C([, b] be defined in the following wy: T (y(u = G(u, tf (t, ydt.
5 Then T (y(u ON A GENERALIZED STURM-LIOUVILLE PROBLEM 21 G(s, tf(tdt + nd thus for p 1 nd suitble q T (y(u G(u, ta(t, sdt G(s, tf(tdt + Γ(A(u, L p y L q. y(sds for u [, b], The trnsformtion T stisfies the ssumptions of Lemm on subliner trnsformtion. Therefore it hs fixed point in C + ([, b], nd so problem (1-2 hs solution in C 2 ([, b]. References [1] D.Guo nd V.Lkshmiknthm, Nonliner Problems in Abstrct Cones, Acd. Press, Orlndo, [2] P.Fijłkowski nd B.Przerdzki, On rdil positive solution to nonlocl elliptic eqution, Topologicl Methods in Nonliner Anlysis Journl of the Juliusz Schuder Center 21 (2003, pp [3] P.Fijłkowski, B.Przerdzki nd R.Stńczy, A nonlocl elliptic eqution in bounded domin, Bnch Center Publictions. 66 (2004, pp Grzegorz Andrzejczk Institute of Mthemtics, Technicl University of Łódź Wólczńsk 215, Łódź, Polnd E-mil: grzegorz.ndrzejczk@p.lodz.pl Tdeusz Pored Institute of Mthemtics, Technicl University of Łódź Wólczńsk 215, Łódź, Polnd E-mil: tdeusz.pored@p.lodz.pl
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