Šeda, V.

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 4Valter ŠedaA partially ordered space connected with the de la Vallée Poussin problemIn: Jiří Fábera (ed.): Equadiff IV, Czechoslovak Conference on Differential Equations and TheirApplications. Proceedings, Prague, August 22-26, 1977. Springer-Verlag, Berlin, 1979. LectureNotes in Mathematics, 703. pp. [374]--383.Persistent URL: http://dml.cz/dmlcz/702238Terms of use:© Springer-Verlag, 1979Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czA PARTIALLY ORDERED SPACE CONNECTED WITH THE DE LA VALLEE POUSSIN PROBLEM V. Seda, Bratislava In the paper the m-point BVP (1) (L(x) = ) XL p.(t)x(n-j) = f t t ^ x S o M ^ H , 0=0 0 (2) x ( i - 1 )(t k) = Ai)k(i=l,.o.,.k, is considered where n = 2, 2 = m = n, 1 = r, (k=l,2,..o,m) are natural numbers such that r, + r0 + ... + r = n, and -°°<a< 1 2 m 7 <t,< ... <t <b<<-*°, A- k(i=l, ...,rk, k=l,2,.oo,m) are real num-bers. First the properties of the Green function corresponding to the homogeneous BVP (1») L(x) = 0, (2>) x(i""1)(tk) = 0 (i=l,...,rk, k=l,2,...,m) and their consequences are studied. The regularity properties deter-mine continuity and compactness of the integrodifferential operator associated to (1), (2). The sign of the Green function decides the monotonicity properties of this operator in a partially ordered space which is described here. Using the fixed point theory in partially ordered spaces we obtain new results for the mentioned BVP connected with the notion of the lower (upper) solution of (1). Throughout the paper the following two assumptions are satisfied: (Aj) p.: (a,b) --> R, p-C L-.(a,b) (j=0,l,...,n), PQ(t) = 1 in (a,b) a.e. and f: (a,b) * R^ —-> R satisfies Carath£odory conditions, i.e. f(.,XQ,X,,..O,X ,) is measurable in (a,b) for each fixed (XQ,X, ,...,X , )GR n, for almost all t Q£ (a,b) f(tQ, •,..., •) is continuous in R11 and sup {If (t,xn,x,, ...,xM ,)l : (xn + x, + ... r> l /o «? ^ u x n—x u x + -*£_!> r ) e L l o c ( a > b ) f or each r > 0 ° (A2) The differential equation (1>) is disconjugate in (a,b)o With respect to Assumption (A,), by a solution (1) on an interval jC(a,b) each function x with x ^L,0c(*j) which satisfies (1) a©eo in j is understood. Under the assumptions (A,), (Ap) for any m, 2 = m = n, any rk (k = 1,2,...,m) with 1 = rk, ri + r? + ##° + rm = n» an(3 a n y a < t 1 ^ t 2 < ••• < tm< b the problem (l1), (2>) has only the trivial solution and there exists a unique function G = G(t,s), the so-cal-led Green function of (1>), (2») defined on [t, ,t 1 * U ft jt/,.,! 375 and such that if rCL(t^t ) and w is the unique solution of the problem ( P ) , (2), then the unique solution y of the problem L(y) = rt =» r(t), (2) is given by y(t) = w(t) +J G(t,s)r(s) ds ([4], p. 137)* Hence the BVP (1), (2) is equivalent to the integrodifferential equation (3) x(t)=w(t)+J G(t,s)f [s,x(s),x'(s),...,x(n"1)(s)J ds (teft-^tj) *1 in the sense that each solution x of (3) with x^n ' £ L(t- ,tm) (n) possesses x '£L(tlft ) and is a solution of (1), (2) and con-versely* Lemma !• The function G has the following properties: 1« j, i = (),•••,n-2, is continuous in ft-jit J x Lt-,,t !• ^d t '% m' p <• < <. 2% —-f is continuous in the domains tn = t = s = t and ^ tn-l 1 m t-, = s = t = t • Further, 1 m * ( 4) l i m *n"1Q(t,8> , lim ^ " ^ ( t a? a l ( t l<s<t). t*s+ t ^ 1 t*s- ^t11"1 X m 3« G(*,s) satisfies (1*) in D^Tf^mJ a#e# an(3 the boundary con-ditions (2*)* 4o The sign of G is determined by the inequality GUjsHt-t-O ~ ( t - t 2 ) 2 . . . ( t - t m ) m = 0, ^ = t = tm , t x = s = tm and G(t,s) t 0 for t j £ < t < t j £ + 1 , ^ < s <tm , k = l , . . . , m - l . 5. lim .liSiiifii- lim ^ ( j . 8 ? = 0 for t , < t < t , s -» t ^ 4>tx s -* tm- 9ta and - f 0, 0 = i = n-2 l i m ЭŁ Q ( t l 8 ) я \ 8 + t 1 + -Эt1 l . l f . в n - 1 f O - i - n - 1 l i m ^ ( M , ) . ŕ s -> t m - "Эt1 [ 0, 0 = i = n-2 1, i = n-1 . 376 Proof. Let u = u(t,s) be the Cauchy function for (1*), ioeo u(o,s) is the solution of (1') determined by the conditions x(i"1)(s) = o (i = l,o..,n-l), x(n"1}(s) = 1. Let v ^ , I = = 1,2, ••o,m, j = l,o*«,r* , be the solution of (1>) satisfying the conditions vj^""1)(tk) = ° (k * 1 » i = 1>###>rk> k = 1i2io»*im) „ ( i - l ) / . % = J .. ( i = l , » « o , r / ) . S •- i s the de l t a Kronecker vj£ U i ; J 1 ^ J 1 symbols Then, by [4] , p . 137, k_ 1L T>J-1u(tj93) (5) G( t , s ) = =1 0=1 & -ъt* t = s m ~t ~4 ^ — vU^t} ÌЗГ—» * = s i Л=k+1 j = l J/C U t J x ^ ^ " Ч к t ^ s ) t f c = s = t k + 1 (k = l , . . . , m - l ) and the equality # - -5- ^J - 1u(t y,s) _ , (6) Z _ Z_ v - . ( t ) --4 = u ( t , s ) for each ( t , s ) £ [ t , , t I i * -.*i»V i s t r ue ( [4], p* 136). By (5 ) , j > i = 0 , 1 , • • • , n - l , i s p i ece -wise continuous in U n i t J * [ t n i ^ J and i t s only d i s con t i nu i t i e s may l i e on the l i n e s s = t and s = t k , k = 2,«««,m-l ( i f m>2)o Since -—*• i s continuous in s and » •, = 0 for j = atJ~x ± Dt^1 i = l,.*.,r, , by (5) we get that lim * G^hs) = lim ^ ^ha) * s->tk+ ^ s->tk" 2 t (t£ [tlft ], t ̂  tk, k = 2,ooo,m-l, 0 = i = n-l)« Further, .: , / 0, 0 = i = n-2 l i m *1Q(tl8) , lim 31QU1e) a / (t 1<t<t m) s->t- -dt1 s->t+ ^t 1 •• l i = n-l follows from (5)> (6)0 This implies the statements 1 and 2 as well as (4)• 3 is a direct consequence of (5K The sign of G has been determined in [5] > PPo 80-81o (5) and (6) also imply the statement 5« Denote now the set of all functions x with x^n ^£L(t-,,tm) (with x(n)eL(t1,tm)) by L^-^U-L.t^) (L{n)(tlftm))o Further, let C ^ = C •,([t^|tm]) be provided with the norm ||x|| = 377 = max {max _lx^ ( t ) l ] . Similarly, the norm II x II n = k = 0 , l , . . o , n - l t £ [ t ^ t j = max _, |x ( t ) l wi l l be used in C = C ( r t 1 f t _ J ) . t G T t 1 f t J -- J- m (n-1) The right-hand side of (3) defines in Lv ( t l f t m ) an operator T which i s given by (7) T(x)(t) = w(t) + J G(t,s)f [sJx(s),x»(s),...,x(n"1)(s)Jds 4 (tє[tlftj). By (A-J and Lemma 1, T: L ^ "1* ^ , ^ ) -* L( n )(tl ftm). Since Ll n^(tl ftm)CC j^CL^1 1"1^^,^), T maps Cn - 1 into itself. Lemma 2. T as a mapping from C -, into itself is continuous and compact. Proof. Suppose that sC Cn - 1 is bounded. Then, by (A-ĵ ), there exists a function mCLttpt ) such that for any xES, If(t,x(t),..., x(n-l)(tj| s m(t) (te[tlftf)« On the basis of (7), this implies that there exist constants A-, j = O,l,o..,n-1, such that T(x) = = y satisfies the inequalities ly J (t)l = A. (j = 0,l,...,n-l, t£[tlft J) and, since y is the solution of L(y) = f [t,x(t), n x»(t)f...fx(n"1)(t)]f (2), |y(n)(t)|= H|p.(t)U n . +m(t>. Thus the sets {y } i J = O,l,..o,n-1, are equicontinuous and uniformly bounded on [t-,,tm]o From this the relative compactness of T(S) in C ! followso Thus T is compact. If ^ -» x in Cn-1, then [T(xn)](j)(t) -» [T(X)] (j)(t) (j = O,lfooofn-1) pointwise in [t-.ft ] and the sequences {[T(Xn)J / ar© equicontinuous on that interval. Hence [ K X ^ J ̂' converge uniformly to [T(X)] ( J ) (j = Oflf...fn-l) on [t;ptm] and T is continuous. Many consequences follow from the sign of the Green function. In order to show them, similarly as in [7] , a partial ordering in C -,, and more generally, in C, will be introduced by the rule. If x, yeC x or in C, then x = y iff (-1) k + 1 °°# m [y(t) - x(t)j =~ 0 for all t € [tfc f tk+1] and all k = l,2,ooo,m-l. By Lemma 1, x = y means that x(t) = y(t) (x(t) = y(t)) on each interval [^i^+iJ where G(t,s) = 0 (G(t,s) = 0 ) . Denote E1 = = {kc{l,...,m-lj : rk+1+...+rm is even} , H2 = {k e{l,...,m-l\ 2 rk+l + #" + rm i s o d d) • 378 Let P = (xCCn-1J x = 0} (PQ = (xGC: x = o) ). Clearly P, PQ are cones in the corresponding spaces and (C if_rO ,(C,P0), res-pectively, ordered Banach spaces with positive cones P and PQ, respectively. While P is not normal, PQ already is. If xQ = yQ in C , and in C, respectively, then by the interval [xo*yo--the set (zeC n - 1: xQ = z = yQ } and [z£C: xQ = z = y Q} , res-pectively, will be understood. Further, xEL u'(t-,t ) will be called a lower solution (y£lj (t-.,t ) an upper solution) of the differential equation (1) if it satisfies the differential inequality L(x) = f(t,x,x*,•••, x(n"1}) (L(y) = f(t,y,y»,...,y(n"1))) a.e. in [t-^tj . Finally for each xCF n'(t,,t ) the solution v of the problem L(v) = 0, v(i""1)(tk) = x ( i " 1 ) ( t k ) (i = !»•<>•,rk. k = l,2,...,m) will be said to be the solution of (!') associated with the function x. With help of these notions the following condition will be expre-ssed which plays an important role in subsequent considerations. (A-~) There exist a lower and an upper solution xQ, yQ, respecti-vely, of the equation (1) such that (8) xQ = yQ and the solutions v__ , v of (1>) associated with the function x0 y0 XQ, yQ, respectively, satisfy (9) v = w = v x0 *0 where w is the solution of the problem (1,)> (2). The inequalities (8), (9) are considered in the partially orde-red space C -, • Theorem !• Let the differential equation (1) satisfy the condi-tions (A1 - (A-J and let (10) L(x0)(t)-f(t,x,x1,ooo,xn-1) = 0 = L(y0(t)-f(t,x,x1,..o,xn-1) be true for a l l ( t ^ x - ^ , • • • , x n _ 1 ) € W where W i s given by W = { ( t , x , x 1 , . . o , x n _ 1 ) : xQ(t) = x = y 0 ( t ) , -<~<x±<c>*>9 i = 1, • • • ,n - l } U {(t9x9x19...9xnmml): yQ(t) = x = x Q ( t ) , - ^ < x i < < < ^ , i = l , . « o , n - l } • 379 Then the BVP (1), (2) has a solution x which lies in the in­terval [x0,y0] . Proof. The interval [x0,yQ] is a closed and convex subset of Cn-1* Ifг *Ç all tєEt-^. _-xO»yo] » t h e n ^,x(t),x'(t),...,x(n"1)(t))GW for and by (9), (10), for all t £ [tk,tk+1], keH^, m T(x)(t) = w(t) + J G(t,s) f[s,x(s),x'(s),.o.,x(n"1)(s)] ds = vv (t) + t tl yo rm + J G(t,s) L(yQ)(s) ds = yQ(t). For kCH 2 we get T(x)(t) = tl - y0(*h te[tk»tk+l-l ° I n a s i m i l a r wayi T(x)(t) = xQ(t) (T(x)(t) = xQ(t)) for te[tk,tk+1], kCH 1(keH 2). Thus we see that (8) - (10) imply that T([xQ,y0]) C [x0,yQ]. By Lemma 2, T is continuous© Now we prove that T([xQ,y0]) is relatively compact in C ,• Let 1 = j = n-1 be a natural number. By the continuity properties of G there exists a constant G->0 such that I ̂  G^i3)|g Q. J I <^tJ I J in ["hfi-nJ * ["kiftnji anc3» o n t n e D a s : L S °f (10), the inequality (11) | [ T ( x ) ] ( j ) ( t ) | = | w ( j ) ( t ) | + G^j ( | L ( x 0 ) ( s ) | + | L ( y 0 ) ( s ) | ) d h HÕ is true for x,-, = ic0 = x = yQo Let HQ = max( max lxQ(t) | , t€-tl»tm-l max lyn^t)!)* Since y = T(x) satisfies the equation L(y) = t € & l i * J = f [ t , x ( t ) , x ( t ) , . o . , x ( n " " 1 ) ( t ) ] > (11) with (A-^ give that n | [T(x) ] ( n ) ( t ) |= II |P j ( t ) l H n- j + m ( t ) whe*e ^ W ^ ' V 1 T n i s proves that the functions {[T(X)] ( J )}, j = 0,l,oo.,n-l, are uni-formly bounded and e qui continuous in L^ii^mJ which means that T([xn,yn]) is relatively compact. The Schauder fixed point theorem ensures a fixed point of T in [x0»y0]° In [7] two sufficient conditions are given for xQ and yQ to satisfy (9K Here they will be stated as Lemma 3o Any of the following conditions is sufficient for the lower and the upper solutions x0»y0i respectively, of (1) to sa-tisfy the inequalities (9)« lo xQ and y0 satisfy the boundary conditions 380 X 0 1 " 1 ) ( t k ) = A i ,k = "̂"̂ V* i = 1 » - " » r k » k = 2,...,m--l ( i f such points e x i s t ) , x o i _ 1 ) < V = A i , _ = yo i _ 1 ) <V' ± - -»—'T1* j = 1 » " n+r,-l_ ( r , - l ) _ _. _. n+r,-l ( i f r l f r m - ? 2 ) , (-1) X [ x - 1 ( t x ) - A , ] = 0 = (-1) (r-,-1) " [У0 X <*l> " A Г l , l l • m' x 0 2. r-, = n-1, rp = 1, m = 2, L(x) = x and the lower and the upper solutions xQ, yQ, respectively, of (120 x( n ) = 0 satisfy the boundary conditions y o1"1 ) ( tl) = Ai,l = ̂ O1"1^ ^ i = lf-tn-1, y0(t2) = A1 2 = xQ(t2)o On the basis of Lemma 3, Theorem 1 generalizes Theorem 3 d from [6], pp. 672-673o In some cases the uniqueness of the solution of the BVP (1), (2) can be ensured by the Kellog uniqueness theorem [3] o Since its proof can be based on the theory of the Leray-Schauder degree in locally convex spaces where the degree is defined also for unbounded sets ([2], po 101), a slightly modified version of that theorem is given here as Lemma 4* Let X be a real Banach space with a convex open (not necessarily bounded) subset D and let F: D —> D be a continuous mapping which is continuously Fr^chet differentiable on D and for which F(D) is compacto Suppose that (a) for each x€D, 1 is not an eigenvalue of F'(x), and (b) for each x e "3D, x 7* F(x)o Then F has a unique fixed point. Theorem 2o Suppose that 1. there exist two functions xQ, y 0€C n( [t-^jt.-,]), a < t 1 < t 2 < b , such that y0(t)<x0(t) (te[t pt 2]) and y ^1 } (t±) <x£ i~ 1 ) (t-^, i = l,ooo,n—1, *-_• f» 2o f, together with , i = 0 ,1 ,000,n-1, i s continuous on the set Wx = { ( t f x f x l f . . . , 3 ^ ^ ) : yQ(t) = x = x Q ( t ) , - ~ < X i < ^ , 381 i = l,..o,n—1, t-j = t = t 2 | , 3. (13) x((n)(t) - f(t,x,x1,.o.,xn-1)<0<yjn)(t) - f(t,x,x1,...,xn--1) on Wx, 4. the BVP (n) V"1 ^^[t,x(t),x'(t),...,x(n-1)(t)] (i) y u i ; = Z_ ~- yv ', i=0 a xi (14>) y(i"1)(t1) = 0 , i = l,.0.,n-l, y(t2) = 0 has only the trivial solution for all x CCn-#1([t1,t2J) such that y0(t)<x(t) <xQ(t), teftp^]. Then for all numbers A. -,, i = l,...,n-l, A-, 2 such that 4 1 - 1 ) ( t l ) < : Ai,l < xO i" 1 ) ( tl )> i = lf»in-lf y o ( t 2 ) < A l , 2 < x O ( t 2 ) there exists a unique solution of the problem (12) x ( n ) = f(t,x,xS...,x(n"1)) (14) x(l~1)(t1) = Ai -_, i = l,0..,n-l, x(t2) = A1 2. Proof. Consider the ordered Banach space ^n-1 = C -,( [t-ptp]) with the norm and ordering similar to those in C -,. Let D = = {x^^n-i5 ^o ( t ) <x("t)<xo(t^i t̂_ = t = t2jo D is open and con-vex in D ^ ] ^ Then D = [x0,yQ] (we remind that x - y in V •* iff x(t) = y(t) in [tlftp])© Lemma 3 gives that the assumptions 1 and 3 of the theorem imply (A-J and (10). (A-,) is ensured by the assumption 2. (A2) is clearly satisfied. Hence, from the proof of Theorem 1 it follows that the mapping T-, which corresponds to the problem (12), (14) maps D into itself, is continuous and T^D) is compact. The Frdchet derivative of T-, at x £ D is t T»(x)(h(t)) = j G-.(t,s) 2_ - ---— . %1 h(i)(s)J ds, (teftpt^) where h £ D n - 1 and G-̂  is the Green function of (12*), (14')• T£(x) continuously depends on x in D. Condition (a) in Lemma 4 means that the linear operator T](x) has no nontrivial fixed point which is equivalent to the assumption 4» The condition (b) follows from T-,(D)CD which is guaranteed by (13)• Thus all assumptions of Lemma 4 being satisfied, by this lemma Theorem 2 is true. In the sequel a special case of (1) will be considered, namely 382 (15) L(x) = f(t,x), where f shows the following monotonicity property© (A.) The function f = f(t,x) is nondecreasing in x for each t £ & k,t k + 1], kCH,, and nonincreasing in x for each t€ [tk, Vi-l]' k e H 2 # With respect to (15), the operator T defined by (7) which now *m has the form T(x)(t) = w(t) +J G(t,a)f [s,x(s)] ds (t£[t,,t ]) *1 can be considered either as a mapping from C , into itself or as a mapping from C into Co In both cases it is continuous and com-pact o The advantage of using C lies in the fact that in this space an interval is a bounded set© The condition (A.) implies that T is isotone which means that if x = y are two comparable elements of C o r cn-lf then T(x) = T(y)o When x and y are a lower and an upper solution of (15), (Aj gives (10), and by Theorem 1 the following lemma is true© Lemma 5» Let the differential equation (15) satisfy the conditi-ons (A-̂ ) - (A.)© Then the BVP (15), (2) has a solution -<e[x0,y0]o Remarks© 1© By Theorem 2, [7] , there exist a least x and a greatest solution y of (15), (2) in the interval [xQ>yn]# They can be obtained by an iterative procedure starting with xQ and y0, respectively© 2© In Theorem 1 and Lemma 5 instead of (Ap) it suffices to assume that (1*) is disconjugate in [t-,,t ]© On the basis of Lemma 7, [l], p© 93, the latter assumption implies that (1*) is disconjugate in a greater interval (a,b) D [t-, ,t ] and thus, the use of (Ap) does not weaken the resultso Theorem 3» Assume that (15) satisfies the conditions (A-̂ ) - (A-J and (16) f ( t ,x) - f ( t ,x) = - p(t)(x-x) (x = x, t e [ t k , t k + J , kGH-̂ ) f ( t ,x) - f ( t ,x) = - p(t)(x-x) (x = x, t e [ t k , t k + 1 ] , k£H2) where peL(t1,tm) and it is such that the differential equation (17) L(x) + p(t)x = 0 is disconjugate in [t-,,t ]© Then the BVP (15), (2) has the least x and the greatest solu-tion y in [x0,yQ]. Proof© Defining the operator L-^x) = L(x) + p(t)x (x €. 383 CL^t tp t^) ) and f-̂ tjX) = f(t,x) + p(t)x ( t C ^ , ^ ] , xc e(-<*?,-*>)) w e can write (15) in the form (15x) Lx(x) = f- jU.x) . By (17) , (16) and Remark 2, (152) satisfies (A-̂ ) - (A.). Hence Lemma 5, together with Remark 1, brings the result. Remarko By Proposition 9, [lj , p© 95, there exists a S > 0 de-pending on (1>) as well as on [t-pt ] such that for |p(t)| < S on [tT_>tmJ the equation (17) is disconjugateo References [l] W0A0 Coppel, Disconjugacy, Springer-Verlag, Berlin-Heidelberg-New York 1971. [2] K© Deimling, Nichtlineare Gleichungen und Abbildungsgrade, Springer-Verlag, Berlin-Heidelberg-New York 1974© [3] RoB. Kellog, Uniqueness in the Schauder fixed point theorem, Proco Amer. Math. Soco 60 (1976), 207-210© [4] I0T0 Kiguradze- Some singular boundary value problems for ordi-nary differential equations. Izdat. Tbilisskogo Univ., Tbilisi 1975 (Russian). L5J A.Yuo Levin, Non-oscillation of solutions of the equation x(n)+P1(t)x(n""1)+oo.+p(t)x=0, Uspehi mat. nauk XXIV, 2(146) (1969), 44-96 (Russian) [6j K. Schmitt, Boundary value problems and comparison theorems for ordinary differential equations, Siam J. Appl. Math., 26 (1974), 670-678. [7] Vo Seda, Two remarks on boundary value problems for ordinary differential equations, Jo Differential Equations (in print). Author's address: Department of Mathematical Analysis, Comenius University, 816 31 Bratislava, Czechoslovakia© 

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A partially ordered space connected with the de la Vallée Poussin problem

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A partially ordered space connected with the de la Vallée Poussin problem

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