PŮŽA, AND I. P. STAVROULAKIS

Transcription

1 Georgin Mthemticl Journl Volume , Number 4, ON SINGULAR BOUNDARY VALUE PROBLEMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS OF HIGHER ORDER I. KIGURADZE, B. PŮŽA, AND I. P. STAVROULAKIS Dedicted to the memory of Professor N. Muskhelishvili on the occsion of his 110th birthdy Abstrct. Sufficient conditions re estblished for the solvbility of the boundry vlue problem x n t = fxt, h i x = 0 i = 1,..., n, where f is n opertor h i i = 1,..., n re opertors cting from some subspce of the spce of n 1-times differentible on the intervl ], b[ m-dimensionl vector functions into the spce of loclly integrble on ], b[ m-dimensionl vector functions into the spce R m Mthemtics Subject Clssifiction: 34K10. Key words nd phrses: singulr functionl differentil eqution, boundry vlue problem, Fredholm property, priori boundedness principle. 1. Formultion of the Min Results 1.1. Formultion of the problem nd brief survey of literture. Consider the functionl differentil eqution of n-th order with the boundry conditions x n t = fxt 1.1 h i x = 0 i = 1,..., n. 1.2 When the opertors f : C [, b]; R m L[, b]; R m nd h i : C [, b]; R m R m i = 1,..., n re continuous, problem 1.1, 1.2 is clled regulr. If the opertor f opertors h i i = 1,..., n cts from some subspce of the spce C ], b[ ; R m into the spce L loc ], b[ ; R m into the spce R m, problem 1.1, 1.2 is clled singulr. The bsic principles of the theory of wide enough clss of regulr problems of form 1.1, 1.2 re constructed in the monogrphs [4], [5], [43]. Optiml sufficient conditions for such problems to be solvble nd uniquely solvble re given in [7], [8], [10] [12], [22], [24], [26] [28], [39]. ISSN X / $8.00 / c Heldermnn Verlg

2 792 I. KIGURADZE, B. PŮŽA, AND I. P. STAVROULAKIS As to singulr problems of form 1.1, 1.2, they hve been studied with sufficient completeness in the cse with the opertor f hving the form fxt = g t, xt,..., x t see [1], [2], [14] [21], [32] [35], [37], [45] nd the references cited therein. For the singulr functionl differentil eqution 1.1, the weighted initil problem is studied in [30], [31], two-point problems in [3], [6], [23], [36], [38], [40] [42], wheres the multi-point Vllée-Poussin problem in [25]. In the generl cse the singulr problem 1.1, 1.2 remins studied but little. An ttempt is mde in this pper to fill up this gp to some extent. Throughout the pper the following nottion will be used. R = ], + [, R + = [0, + [. R m is the spce of m-dimensionl column vectors x = x i m with the components x i R i = 1,..., m nd the norm m x = x i. R m + = {x = x i m : x i R + i = 1,..., m}. R m m is the spce of m m mtrices X = x ik m i,k=1 with the components x ik R i, k = 1,..., m nd the norm X = m i,k=1 x ik. If x = x i m R m nd X = x ik m i,k=1 R m m, then x = x i m nd X = x ik m i,k=1. R m m + = {X = x ik m i,k=1 : x ik R + i, k = 1,..., m}. rx is the spectrl rdius of the mtrix X R m m. Inequlities between mtrices nd vectors re understood componentwise, i.e., for x = x i m, y = y i m, X = x ik m i,k=1 nd Y = y ik m i,k=1 we hve nd x y x i y i i = 1,..., m X Y x ik y ik If k is nturl number nd ε ]0, 1[, then k k ε! = i ε. i, k = 1,..., m. If m nd n re nturl numbers, < < b < +, α R nd β R, then C ], b[ ; Rm is the Bnch spce of -times continuously differentible vector functions x : ], b[ R m hving limits limt α i x i 1 t, t limb t β i x i 1 t i = 1,..., n, 1.3 t b

3 ON SINGULAR BOUNDARY VALUE PROBLEMS 793 where α i = α + i n + α + i n 2, β i = β + i n + β + i n 2 i = 1,..., n. 1.4 The norm of n rbitrry element x of this spce is defined by the equlity { n } = sup t α i b t β i x i 1 t : < t < b. k=1 C ], b[ ; Rm is the set of x C ], b[ ; Rm for which x is loclly bsolutely continuous on ], b[, i.e., bsolutely continuous on [ + ε, b ε] for rbitrrily smll positive ε. L ], b[ ; R m nd L ], b[ ; R m m re respectively the Bnch spce of vector functions y : ], b[ R m nd the Bnch spce of mtrix functions Y : ], b[ R m m whose components re summble with weight t α b t β. The norms in these spces re defined by the equlities y L = b t α b t β yt dt, Y L = b t α b t β Y t dt. L ], b[ ; R m + = {y L ], b[ ; R m : yt R m + for t ], b[ }. L ], b[ ; R m m + = {Y L ], b[ ; R m m : Y t R m m + for t ], b[ }. In the sequel it will lwys be ssumed tht < < b < +, α [0, n 1], β [0, n 1], 1.5 wheres f : C ], b[ ; Rm L ], b[ ; R m nd h i : C ], b[ ; Rm R m i = 1,..., n re continuous opertors which, for ech ρ ]0, + [, stisfy the conditions sup { fx : sup { h i x : ρ } L ], b[ ; R +, 1.6 ρ } < + i = 1,..., n. 1.7 By solution of the functionl differentil eqution 1.1 is understood vector function x C ], b[ ; Rm stisfying 1.1 lmost everywhere on ], b[. A solution of 1.1 stisfying 1.2 is clled solution of problem 1.1, Theorem on the Fredholm property of liner boundry vlue problem. We begin by introducing Definition 1.1. A liner opertor p : C ], b[ ; Rm R m is clled strongly bounded if there exists ζ L ], b[ ; R + such tht pxt ζt for < t < b, x C ], b[ ; Rm. 1.8

4 794 I. KIGURADZE, B. PŮŽA, AND I. P. STAVROULAKIS Consider the boundry vlue problem x n t = pxt + qt, 1.9 l i x = c 0i i = 1,..., n, 1.10 where p : C ], b[ ; Rm L ], b[ ; R m is liner, strongly bounded opertor, l i : C ], b[ ; Rm R m i = 1,..., m re liner bounded opertors, q L ], b[ ; R m, c 0i R m i = 1,..., m. Theorem 1.1. For problem 1.9, 1.10 to be uniquely solvble it is necessry nd sufficient tht the corresponding homogeneous problem x n t = pxt, l i x = 0 i = 1,..., n hve only trivil solution. Moreover, if problem hs only trivil solution, then there exists positive constnt γ such tht for ny q L ], b[ ; R m nd c 0i R m i = 1,..., m, solution x of problem 1.9, 1.10 dmits the estimte n γ c 0i + q L The vector differentil eqution with deviting rguments x n t = P i tx i 1 τ i t + qt, 1.12 where τ i : [, b] [, b] i = 1,..., n re mesurble functions, P i : ], b[ R m m i = 1,..., n re mtrix functions with mesurble components nd q L ], b[ ; R m, is prticulr cse of eqution 1.9. Along with 1.12, consider the corresponding homogeneous eqution From Theorem 1.1 follows x n t = P i tx i 1 τ i t Corollry 1.1. Let lmost everywhere on ], b[ the inequlities be fulfilled. Moreover, τ i t > for i > n α, τ j t < b for j > n β 1.13 t α b t β τ i t α i b τi t β i P i t dt < + i = 1,..., n Here nd in the sequel it will be ssumed tht if α i = 0 β i = 0, then τ i t α i 1 τ i t b βi 1.

5 ON SINGULAR BOUNDARY VALUE PROBLEMS 795 Then for problem 1.12, 1.2 to be uniquely solvble, it is necessry nd sufficient tht the corresponding homogeneous problem , hve only trivil solution. Moreover, if problem hs only trivil solution, then there exists positive constnt γ such tht for ny q L ], b[ ; R m nd c 0i R m i = 1,..., m, solution x of problem 1.12, 1.2 dmits estimte A priori boundedness principle for the nonliner problem 1.1, 1.2. To formulte this principle we hve to introduce Definition 1.2. Let γ be positive number. The pir p, l i n of continuous opertors p : C ], b[ ; Rm C ], b[ ; Rm L ], b[ ; R m nd l i n : C ], b[ ; Rm C ], b[ ; Rm R mn is sid to be γ-consistent if: i the opertors px, : C ], b[ ; Rm L ], b[ ; R m nd l i x, : C ], b[ ; Rm R m re liner for ny fixed x C ], b[ ; Rm nd i {1,..., n}; ii for ny x nd y C ], b[ ; Rm nd for lmost ll t ], b[ we hve inequlities px, yt δ t, y C, l i x, y δ 0 x C y C, where δ 0 : R + R + is nondecresing, δ, ρ L ], b[ ; R + for every ρ R +, nd δt, : R + R + is nondecresing for every t ], b[ ; iii for ny x C ], b[ ; Rm, q L ], b[ ; R m nd c i R m i = 1,..., n, n rbitrry solution y of the boundry vlue problem dmits the estimte y n t = px, yt + qt, l i x, y = c i i = 1,..., n 1.15 y C n γ c i + q L Definition 1.2. The pir p, l i n of continuous opertors p : C ], b[ ; Rm C ], b[ ; Rm L ], b[ ; R m nd l i n : C ], b[ ; Rm C ], b[ ; Rm R mn is sid to be consistent if there exists γ > 0 such tht this pir is γ-consistent. Theorem 1.2. Let there exist positive number ρ 0 nd consistent pir p, l i n of continuous opertors p : C ], b[ ; Rm C ], b[ ; Rm R mn nd l i n : C ], b[ ; Rm C ], b[ ; Rm R mn such tht for ny λ ]0, 1[ n rbitrry solution of the problem dmits the estimte x n t = 1 λpx, xt + λfxt, 1.17 λ 1l i x, x = λh i x i = 1,..., n 1.18 ρ

6 796 I. KIGURADZE, B. PŮŽA, AND I. P. STAVROULAKIS Then problem 1.1, 1.2 is solvble. For n = 1 nd α = β = 0, Theorem 1.2 implies Theorem 1 from [27]. Corollry 1.2. Let there exist positive number γ, γ-consistent pir p, l i n of continuous opertors p : C ], b[ ; Rm C ], b[ ; Rm L ], b[ ; R m, l i n : C ], b[ ; Rm C ], b[ ; Rm R mn nd functions η : ], b[ R + R + nd η 0 : R + R + such tht the inequlities fxt px, xt η t,, 1.20 h i x l i x, x η0 x C 1.21 re fulfilled for ny x C ], b[ ; Rm nd lmost ll t ], b[. Moreover, η, ρ L ], b[ ; R + for ρ R + nd lim sup ρ + η0 ρ + 1 ρ ρ Then problem 1.1, 1.2 is solvble. s α b s β ηs, ρ ds < 1 γ As n exmple, in C α,0 ], b[ ; R m consider the boundry vlue problem x n t = g t, xτ 1 t,..., x τ n t, 1.23 lim x i 1 t = c i x i = 1,..., k, t lim x i 1 t = c i x i = k + 1,..., n. t b 1.24 Here k {1,..., n 1}, α [0, n k], τ i : [, b] [, b] i = 1,..., n re mesurble functions, c i : C α,0 ], b[ ; R m R m i = 1,..., m re continuous opertors, nd g : ], b[ R mn R m is vector function such tht g, x 1,..., x n : ], b[ R m is mesurble for ny x i R m i = 1,..., n nd gt,,..., : R mn R m is continuous for lmost ll t ], b[. We will lso suppose tht for i > n α the inequlity holds lmost everywhere on ], b[. The following sttement is vlid. τ i t > Corollry 1.3. Let there exist η 0 : R + R +, P i L α,0 ], b[ ; R mn + i = 1,..., n nd q : ], b[ R + R m + such tht c i x η 0 x C α,0 for x C α,0 ], b[ ; R m 1.25

7 ON SINGULAR BOUNDARY VALUE PROBLEMS 797 nd on ], b[ R mn the inequlity gt, x 1,..., x n τi t α i P i t x i + q t, τi t α i x i 1.26 holds. Let, moreover, q, ρ L α,0 ], b[ ; R m + for every ρ R +, the components of qt, ρ re nondecresing with respect to ρ, nd η0 ρ lim + 1 s α qs, ρ ds = ρ + ρ ρ rp < 1, 1.28 where P = k b n k 1 α+α k+1 s α τ i s k+1 i α k+1 P i s ds n k 1!k + 1 i α k+1! + i=k+1 b n i α+α i n i! Then problem 1.23, 1.24 is solvble. b s α P i s ds. Before pssing to the formultion of the next corollry we introduce Definition 1.3. An opertor p : C ], b[ ; Rm L ], b[ ; R m n opertor l : C ], b[ ; Rm R m is clled positive homogeneous if the equlity pλxt = λpxt lλx = λlx is fulfilled for ll x C ], b[ ; Rm, λ R + nd lmost ll t ], b[. Definition 1.4. A positive homogeneous opertor p : C ], b[ ; Rm L ], b[ ; R m positive homogeneous opertor l : C ], b[ ; Rm R m is clled strongly bounded bounded if there exists function ζ L ], b[ ; R + positive number ζ 0 such tht the inequlity pxt ζt lx ζ0 x C holds for ll x C ], b[ ; Rm nd lmost ll t ], b[. Corollry 1.4. Let there exist liner, strongly bounded opertor p : C ], b[ ; Rm L ], b[ ; R m, positive homogeneous, continuous, strongly bounded opertor p : C ], b[ ; Rm L ], b[ ; R m, liner bounded

8 798 I. KIGURADZE, B. PŮŽA, AND I. P. STAVROULAKIS opertors l i : C ], b[ ; Rm R m i = 1,..., n, positive homogeneous, continuous, bounded opertors l i : C ], b[ ; Rm R m i = 1,..., m, nd functions η :], b[ R + nd η 0 : R + R + such tht the inequlities fxt pxt pxt η t,, 1.29 h i x l i x l i x η 0 x C hold for ny x C ], b[ ; Rm nd for lmost ll t ], b[. η, ρ L ], b[ ; R + for ny ρ R +, lim ρ + η0 ρ + 1 ρ ρ nd for ny λ [0, 1] the problem s α b s β ηs, ρ ds 1.30 Moreover, = x n t = pxt + λpxt, l i x + λl i x = 0 i = 1,..., n 1.32 hs only trivil solution. Then problem 1.1, 1.2 is solvble. As n exmple, for the second order singulr hlf-liner differentil eqution u t = p 1 t ut µ u t 1 µ sgn ut + p 2 tu t + p 0 t 1.33 let us consider the two-point boundry vlue problems nd lim ut = c 1, t lim ut = c 1, t lim ut = c t b lim u t = c t b We re interested in the cse where µ [0, 1] nd p i : ], b[ R i = 0, 1, 2 re mesurble functions stisfying either the conditions or the conditions t b t p i t dt < + i = 0, 1, p 1 t λ 1 [σt] 1+µ, [ p 2 t σ t σt t p 0 t dt < + i = 0, 1, p 1 t λ 1 [σt] 1+µ, p 2 t dt < +, ] sgnt 0 t λ 2 σt for < t < b, p 2 t dt < +, p 2 t σ t σt λ 2σt for < t < b

9 ON SINGULAR BOUNDARY VALUE PROBLEMS 799 Here t 0 ], b[, λ i R + i = 1, 2, nd σ : ], b[ R + is loclly bsolutely continuous function such tht either or σ t sgnt 0 t 0 for < t < b, > µ 2 [ + 0 σs ds + t 0 + σs ds ds λ 1 + λ 2 s + s 1+µ/µ > µ 0 t 0 ds λ 1 + λ 2 s + s 1+µ/µ ] σs ds, σs ds By virtue of Theorems 3.1 nd 3.2 from [9] Corollry 1.4 implies Corollry 1.5. Let conditions 1.35 i, 1.36 i nd 1.37 i be fulfilled for some i {1, 2}. Then problem 1.33, 1.34 i hs t lest one solution. This corollry is generliztion of the clssicl result of Ch. de l Vllée- Poussin [44] for eqution Auxiliry Propositions Lemm 2.1. Let ρ > 0, η L ], b[ ; R +, t 0 ], b[, nd S be the set of n 1-times continuously differentible vector functions x : ], b[ R m stisfying the conditions x i 1 t 0 ρ i = 1,..., n, 2.1 Then S x t x s t s ηξ dξ for < s t < b. 2.2 C ], b[ ; Rm nd S is compct set of the spce C ], b[ ; Rm. Proof. Let x be n rbitrry element of the set S. Then by 2.2 the function x is loclly bsolutely continuous on ], b[ nd Therefore x n t ηt for lmost ll t ], b[. 2.3 t x n L ], b[ ; R m, 2.4 x i 1 t t 0 j i t = x j 1 t 0 j i! j=i + 1 t s n i x n s ds for < t < b i = 1,..., n, 2.5 n i! t 0

10 800 I. KIGURADZE, B. PŮŽA, AND I. P. STAVROULAKIS nd x i 1 t εi t for < t < b i = 1,..., n, 2.6 where ε i t = ρ j=i b j i j i! t 1 + t s n i ηs ds i = 1,..., n. 2.7 n i! t 0 Let Then i 1 = mx{i : α i = 0}, i 2 = mx{i : β i = 0}. n i α, α i = 0 for i i 1, α i = α + i n > 0 for i > i 1, n i β, β i = 0 for i i 2, β i = β + i n > 0 for i > i Therefore ε i t ε i + < + for i i 1, < t t 0, 2.9 t 0 ε 1+i1 s ds < + if i 1 < n 1, 2.10 ε i t ε i b < + for i i 2, t 0 t < b, 2.11 t 0 ε 1+i2 s ds < + if i 2 < n If i > i 1, then, with 2.7 nd tken into ccount, for ny δ ]0, t 0 [ we find [ lim sup t α i ε i t ] = lim sup t t 1 n i! +δ [ t α+i n n i! s α ηs ds. Hence, becuse of the rbitrriness of δ, it follows tht Anlogously, it cn be shown tht +δ t s t n i ηs ds [ lim t α i ε i t ] = 0 for i > i t [ lim b t β i ε i t ] = 0 for i > i t b ]

11 ON SINGULAR BOUNDARY VALUE PROBLEMS 801 If i i 1 if i i 2, then by virtue of conditions 2.3 nd conditions 2.3 nd we hve t 0 s n i x n s ds < + Hence 2.5 implies the existence of the limit lim x i 1 t t t 0 b s n i x n s ds < + lim t b x i 1 t. If however i > i 1 i > i 2, then from 2.6 nd 2.13 from 2.6 nd 2.14 we hve limt α i x i 1 t = 0 limb t β i x i 1 t = 0. t t b We hve thereby proved the existence of limit 1.3. Therefore S C ], b[ ; Rm. By the Arzel Ascoli lemm, from estimtes 2.3, 2.6 nd conditions it follows tht S is compct set of the spce C ], b[ ; Rm. Let p, l i n be γ-consistent pir of continuous opertors p : C ], b[ ; Rm C ], b[ ; Rm L ], b[ ; R m nd l i n : C ], b[ ; Rm C ], b[ ; Rm R mn, nd q : C ], b[ ; Rm L ], b[ ; R m, c 0i : C ], b[ ; Rm R m i = 1,..., n be continuous opertors. For ny x C ], b[ ; Rm, consider the liner boundry vlue problem y n t = px, yt + qxt, l i x, y = c 0i x i = 1,..., n By condition iii of Definition 1.2, the homogeneous problem y n t = px, yt, l i x, y = 0 i = 1,..., n hs only trivil solution. By Theorem 1.1 this fct gurntees the existence of unique solution y of problem We write uxt = yt. Lemm 2.2. u : C ], b[ ; Rm C ], b[ ; Rm is continuous opertor. Proof. Let nd Then x i C ], b[ ; Rm, y i t = ux i t i = 1, 2 yt = y 2 t y 1 t. y n t = p 2 x 2, yt + q 0 x 1, x 2 t, l i x 2, y = c i x 1, x 2 i = 1,..., n,.

12 802 I. KIGURADZE, B. PŮŽA, AND I. P. STAVROULAKIS where q 0 x 1, x 2 t = px 1, y 1 t px 2, y 1 t + qx 2 t qx 1 t, c i x 1, x 2 = l i x 1, x 2 l i x 2, y 1 + c 0i x 2 c 0i x 1 i = 1,..., n. Hence, by condition iii of Definition 1.2 we hve n ux 2 ux 1 γ c i x 1, x 2 + q 0 x 1, x 2 L. C Since the opertors p, q, l i nd c 0i i = 1,..., n re continuous, this estimte implies the continuity of the opertor u. Lemm 2.3. Let k {1,..., n 1}, α [0, n k], nd x Cα,0 ], b[ ; R m be vector function stisfying conditions Then on ], b[ the following inequlities re fulfilled: x i 1 t b j i c j x j=i + 1 n i! b n i α+α i t α i yx i = k + 1,..., n, 2.16 x i 1 t b j i c j x j=i b n k 1 α+α k+1 + n k 1!k + 1 i α k+1! t k+1 i α k+1 yx i = 1,..., k, 2.17 where yx = s α x n s ds Proof. Let x 0 t be polynomil of degree not higher thn n 1 stisfying the conditions Then x i 1 0 = c i x i = 1,..., k, x i 1 0 b = c i x i = k + 1,..., n. x i 1 0 t On the other hnd, b j i c j x for t b i = 1,..., n j=i x i 1 t = x i 1 0 t 1n i s t n i x n s ds 2.20 n i! t i = k + 1,..., n, x i 1 t = c i x + t x i s ds i = 1,..., k. 2.21

13 ON SINGULAR BOUNDARY VALUE PROBLEMS 803 By 1.4 Therefore n i α α i 0 i = 1,..., n. s t n i s n i α+α i s α i s α b n i α+α i t α i s α for t s < b i = 1,..., n. If long with this we tke into ccount inequlity 2.19, then from 2.20 we obtin estimtes It is cler tht α k+1 1, since α n k. If α k+1 < 1, then by virtue of 2.16 nd 2.19, from 2.21 follow estimtes To complete the proof of the lemm it remins to consider the cse where α k+1 = 1. Then α = n k nd thus from we find nd x k 1 t b j k c j x j=k t 1 + s τ n k 1 x n s ds dτ n k 1! τ = b j k c j x j=k [ t ] 1 + t s n k 1 x n s ds + s n k x n s ds n k 1! t b j k 1 c j x + j=k n k 1! yx x i 1 t b j i 1 c j x + j=i n k 1!k i! t k i yx i = 1,..., k. Therefore estimtes 2.17 re vlid. 3. Proof of the Min Results Proof of Theorem 1.1. Let B = C ], b[ ; Rm R mn be Bnch spce with elements u = x; c 1,..., c n, where x C ], b[ ; Rm, c i R m i = 1,..., n, nd the norm u B = u C + c i.

14 804 I. KIGURADZE, B. PŮŽA, AND I. P. STAVROULAKIS Fix rbitrrily t 0 ], b[ nd, for ny u = x; c 1,..., c n, set n t t 0 put i 1 = ci + x i 1 t 0 i 1! + 1 t t s pxs ds; c 1 l 1 x,..., c n l n x, n 1! t 0 t 1 qt = t s qs ds; c 01,..., c 0n. n 1! t 0 Problem 1.9, 1.10 is equivlent to the opertor eqution u = pu + q 3.1 in the spce B since u = x; c 1,..., c n is solution of eqution 3.1 if nd only if c i = 0 i = 1,..., n nd x is solution of problem 1.9, As for the homogeneous eqution u = pu it is equivlent to the homogeneous problem 1.9 0, From condition 1.8 nd Lemm 2.1 it immeditely follows tht the liner opertor p : B B is compct. By this fct nd the Fredholm lterntive for opertor equtions [13], Ch. XIII, 5, Theorem 1, eqution 3.1 is uniquely solvble if nd only if eqution hs only trivil solution. Moreover, if eqution hs only trivil solution, then the opertor I p is invertible nd I p 1 : B B is liner bounded opertor, where I : B B is n identicl opertor. Therefore there exists γ 0 > 0 such tht for ny q B the solution u of eqution 3.1 dmits the estimte u B γ 0 q B. However, q B c 0i + γ 1 q L, where γ 1 > 1 is constnt depending only on α, β,, b, t 0 nd n. Hence n u B γ c 0i + q L, 3.2 where γ = γ 0 γ 1. Since problem 1.9, 1.10 is equivlent to eqution 3.1, it is cler tht problem 1.9, 1.10 is uniquely solvble if nd only if problem 1.9 0, hs only trivil solution. Moreover, if 1.9 0, hs only trivil solution, then by virtue of 3.2 the solution x of problem 1.9, 1.10 dmits estimte 1.11.

15 ON SINGULAR BOUNDARY VALUE PROBLEMS 805 Proof of Corollry 1.1. We set pxt = P i tx i 1 τ i t for ny x C ], b[; Rm. Then equtions 1.12 nd tke respectively forms 1.9 nd On the other hnd, in view of 1.13 nd 1.14 p : C ], b[ ; Rm L ], b[ ; R m is strongly bounded liner opertor. Therefore the conditions of Theorem 1.1 re fulfilled. Proof of Theorem 1.2. Let δ, δ 0 nd γ be the functions nd numbers ppering in Definitions 1.2 nd 1.2. We set By 1.6 nd 1.7 ηt = 2ρ 0 δt, 2ρ 0 + sup { fxt : η 0 = 2ρ 0 δ 0 2ρ 0 + sup { h i x : 2ρ 0 }, 2ρ 0 }, ρ 1 = γ η 0 + η L, η t = δt, ρ 1 ρ 0 + ηt, 3.3 B 0 = { x C ], b[ ; } Rm : ρ 1, for 0 s ρ 0 χs = 2 s/ρ 0 for ρ 0 < s < 2ρ 0, for s 2ρ 0 qxt = χ [ ] fxt px, xt, 3.6 c 0i x = χ [ li x, x h i x ] i = 1,..., n. 3.7 η 0 < +, η L ], b[ ; R +, η L ], b[ ; R + nd for every x C ], b[ ; Rm nd lmost ll t ], b[ we hve the inequlities qxt ηt, c 0i x η Let u : C ], b[ ; Rm C ], b[ ; Rm be n opertor which to every x C ], b[ ; Rm ssigns the solution y of problem By Lemm 2.1, u is continuous opertor. On the other hnd, by conditions ii nd iii of Definition 1.2, nottions 3.3, 3.4 nd inequlities 3.8, the vector function y = ux stisfies, for ech x B 0, the conditions y C ρ 1, y t y s t s η ξ dξ for < s t < b.

16 806 I. KIGURADZE, B. PŮŽA, AND I. P. STAVROULAKIS By Lemm 2.2 this implies tht the opertor u mps the bll B 0 into its own compct subset. Therefore, owing to Schuder s principle, there exists x B 0 such tht xt = uxt for < t < b. By nottions 3.6, 3.7 the function x is solution of problem 1.17, 1.18, where λ = χ. 3.9 Let us show tht x dmits estimte Assume the contrry. Then either or ρ 0 < < 2ρ 0, ρ If condition 3.10 is fulfilled, then by virtue of 3.5 nd 3.9 λ ]0, 1[, which, by one of the conditions of the theorem, gurntees the vlidity of estimte But this contrdicts condition Assume now tht inequlity 3.11 is fulfilled. Then by virtue of 3.5 nd 3.9 λ = 0 nd therefore x is solution of problem Thus xt 0 since problem hs only trivil solution. But this contrdicts inequlity The contrdiction obtined proves the vlidity of estimte By 1.19, nd 3.9, it clerly follows from 1.17, 1.18 tht λ = 1 nd x is solution of problem 1.1, 1.2. Proof of Corollry 1.2. By 1.22 there is ρ 0 > 0 such tht γ η 0 ρ + s α b s β ηs, ρ ds < ρ for ρ > ρ Let x be solution of problem 1.17, 1.18 for some λ ]0, 1[. Then y = x is lso solution of problem 1.15 where qt = λ fxt px, xt, c 0i x = λ l i x, x h i x i = 1,..., n. Assume tht ρ =.

17 ON SINGULAR BOUNDARY VALUE PROBLEMS 807 By the γ-consistency of the pir p, l i n nd inequlities 1.20, 1.21 we hve n ρ γ c 0i x + q L γ η 0 ρ + s α b s β ηs, ρ ds. Hence by 3.12 it follows tht ρ ρ 0. Therefore estimte 1.19 is vlid, which due to Theorem 1.2 gurntees the solvbility of problem 1.1, 1.2. Proof of Corollry 1.3. Problem 1.23, 1.24 is obtined from problem 1.1, 1.2 when fxt g t, xτ 1 t,..., x τ n t, 3.13 h i x = lim t x i 1 t c i x i = 1,..., k, h i x = lim t b x i 1 t c i x i = k + 1,..., n By virtue of the restrictions imposed on g, τ i, c i i = 1,..., n nd the inequlity α n k it is obvious tht f : Cα,0 ], b[ ; R m L α,0 ], b[ ; R m nd h i : Cα,0 ], b[ ; R m R m i = 1,..., m re continuous opertors stisfying conditions 1.6 nd 1.7, where β = 0. Assume for ny x, y Cα,0 ], b[ ; R m nd t ], b[ tht px, yt = 0, l i x, y = lim t y i 1 t i = 1,..., k, l i x, y = lim t b y i 1 t i = k + 1,..., n According to Definition 1.2 nd Theorem 1.2 the pir p, l i n of continuous opertors p : Cα,0 ], b[ ; R m Cα,0 ], b[ ; R m L α,0 ], b[ ; R m, l i n : Cα,0 ], b[ ; R m Cα,0 ], b[ ; R m R mn, is consistent. To prove Corollry 1.3, by Theorem 1.2 it is sufficient to show tht for ech λ ]0, 1[ n rbitrry solution x of problem 1.17, 1.18 dmits the estimte α,0 ρ 0, 3.16 where ρ 0 is non-negtive constnt not depending on λ nd x. By virtue of problem 1.17, 1.18 tkes the form x n t = λg t, xτ 1 t,..., x τ n t, 3.17 lim x i 1 t = λc i x i = 1,..., k, t lim x i 1 t = λc i x i = k + 1,..., n. t b 3.18 Let x be solution of problem 3.17, 3.18 for some λ ]0, 1[. Then by virtue of Lemm 2.3 we conclude tht estimtes 2.16, 2.17, where yx is

18 808 I. KIGURADZE, B. PŮŽA, AND I. P. STAVROULAKIS the vector given by equlity 2.18, re true. On the other hnd, on ccount of 1.26 we hve yx s α τ i s α i P i s x i 1 s ds + s α q s, ds. α,0 If, long with 2.16 nd 2.17, we tke into ccount tht α i = 0 i = 1,..., k, then from the ltter inequlity we obtin nd therefore yx Pyx + y 0 x where E is the unique m m mtrix nd y 0 x = + E Pyx y 0 x, 3.19 τi s α i n P i s ds b j i c j x s α q s, α,0 j=i ds By the nonnegtiveness of the mtrix P nd inequlity 1.28, from 3.19 it follows tht yx E P 1 y 0 x. If long with this we tke into ccount condition 1.25 nd equlity 3.20, then 2.16 nd 2.17 imply tht η 1 x C, 3.21 α,0 α,0 where η 1 ρ = µ η 0 ρ + b s α qs, ρ ds nd µ is positive constnt depending only on α i, P i, τ i i = 1,..., n, nd b. On the other hnd, due to condition 1.27 we hve nd therefore where ρ 0 = inf η 1 ρ lim ρ + ρ = 0 η 1 ρ < ρ for ρ > ρ 0, { ρ > 0 : ηs s } < 1 for s [ρ, + [.,

19 ON SINGULAR BOUNDARY VALUE PROBLEMS 809 Therefore from 3.21 we obtin estimte On the other hnd, it is obvious tht the constnt ρ 0 does not depend on λ nd x. Proof of Corollry 1.4. The strong boundedness of the opertors p nd p nd the boundedness of the opertors l i n nd l i n gurntee the existence of ζ L ], b[ ; R + nd ζ 0 R + such tht the inequlities pxt + pxt ζt, li x + l i x ζ hold for ech x C ], b[ ; Rm nd lmost ll t ], b[. By Theorem 1.1 nd Definition 1.2 the pir p, l i n is consistent since for λ = 0 problem 1.32 hs only trivil solution. Let us consider for rbitrry λ [0, 1], q L ], b[ ; R m nd c 0i R m i = 1,..., n the boundry vlue problem x n t = pxt + λpxt + qt, 3.23 l i x + λl i x = c 0i i = 1,..., n 3.24 nd prove tht every solution x of this problem dmits the estimte n γ c 0i + q L, 3.25 where γ is positive constnt not depending on λ, q, c 0i i = 1,..., n nd x. Assume the contrry tht this is not so. Then for ech nturl k there re λ k [0, 1], q k L ], b[ ; R m, c ki R m i = 1,..., n such tht the problem x n t = pxt + λ k pxt + q k t, l i x = λ k l i x + c ki i = 1,..., n hs solution x k dmitting the estimte def n ρ k = x k C > k c ki + q k L. If we ssume tht then we hve x k t = ρ 1 k x kt, q k t = ρ 1 k q kt, c ki = ρ 1 k c ki i = 1,..., n, q k C x k C = 1, 3.26 < 1 n k, c ki < 1 k, 3.27 x n k t = px kt + λ k px k t + q k t, 3.28

20 810 I. KIGURADZE, B. PŮŽA, AND I. P. STAVROULAKIS Let t 0 = +b 2 where l i x k = λ k l i x k + c ki i = 1,..., n Then 3.28 implies y k t = z k t = By 3.27 we hve t t 0 i 1 i 1! t x k t = y k t + z k t, 3.30 x i 1 k t t s pxk s + λ k px k s ds, 3.31 n 1! t 0 t 1 t s q n 1! k s ds. t 0 lim z k k + C = 0, lim c ki = 0 i = 1,..., n k + On the other hnd, with 3.22 nd 3.26 tken into ccount, from 3.31 we find y i 1 k t 0 ρ i = 1,..., n, y k t y k s t s ζξ dξ for < s < t < b, where ρ is positive constnt not depending on k. By virtue of these inequlities nd Lemm 1.1, we cn ssume without loss of generlity tht the sequence y k + k=1 is converging in the norm of the spce C ], b[ ; Rm. It cn lso be ssumed without loss of generlity tht the sequence λ k + k=1 is converging. Assume tht λ = lim λ k, xt = lim y kt. k + k + Then by we hve nd xt = t t 0 i 1 i 1! lim x k x k + C = lim y k x k + C = l i x = λl i x i = 1,..., n, t x i 1 1 t 0 + t s pxs + λpxs ds. n 1! t 0 Therefore x is solution of problem On the other hnd, from 3.26 nd 3.33 it clerly follows tht = 1.

21 ON SINGULAR BOUNDARY VALUE PROBLEMS 811 But this is impossible becuse for ech λ [0, 1] problem 1.32 hs only trivil solution. The contrdiction obtined proves the existence of positive number γ tht possesses the bove-mentioned property. By condition 1.31 there is ρ 0 > 0 such tht inequlity 3.12 is fulfilled. To prove Corollry 1.4, it is sufficient due to Theorem 1.2 to estblish tht for ech λ ]0, 1[ n rbitrry solution x of the problem x n t = pxt + λ [ fxt pxt ], 3.34 l i x = λ l i x h i x i = 1,..., n 3.35 dmits estimte It is obvious tht ech solution x of problem 3.34, 3.35 is solution of problem 3.23, 3.24, where qt = λ fxt pxt pxt, c 0i = λ l i x + l i x h i x i = 1,..., n According to the bove proof, x dmits estimte 3.25 from which, with 1.29, 1.30 nd 3.36 tken into ccount, we find γ η 0 x C + s α b s β η s, Hence, by virtue of 3.12, we obtin estimte Acknowledgements ds. This work ws supported by the Reserch Grnt of the Greek Ministry of Development in the frmework of Bilterl S&T Coopertion between the Hellenic Republic nd the Republic of Georgi. References 1. R. P. Agrwl nd D. O Regn, Nonliner superliner singulr nd nonsingulr second order boundry vlue problems. J. Differentil Equtions , No. 1, R. P. Agrwl nd D. O Regn, Second-order boundry vlue problems of singulr type. J. Mth. Anl. Appl , N. B. Azbelev, M. J. Alves, nd E. I. Brvyi, On singulr boundry vlue problems for liner functionl differentil equtions of second order. Russin Izv. Vyssh. Uchebn. Zved. Mt , N. V. Azbelev, V. P. Mximov, nd L. F. Rkhmtullin, Introduction to the theory of functionl differentil equtions. Russin Nuk, Moscow, N. V. Azbelev, V. P. Mximov, nd L. F. Rkhmtullin, Methods of the modern theory of liner functionl differentil equtions. Russin R&C Dynmics, Moscow Izhevsk, 2000.

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24 814 I. KIGURADZE, B. PŮŽA, AND I. P. STAVROULAKIS 44. Ch. de l Vllée-Poussin, Sur l éqution différentielle linéire du second ordre. Détermintion d une intégrle pr deux vleurs ssignés. Extension ux équtions d ordre n. J. Mth. Pures Appl , P. J. Y. Wong nd R. P. Agrwl, Singulr differentil equtions with n, p boundry conditions. Mth. Comput. Modelling , No. 1, Authors ddresses: Received I. Kigurdze A. Rzmdze Mthemticl Institute Georgin Acdemy of Sciences 1, M. Aleksidze St., Tbilisi Georgi E-mil: kig@rmi.cnet.ge B. Půž Msryk University Fculty of Science Deprtment of Mthemticl Anlysis Jnáčkovo nám. 2, Brno Czech Republic E-mil: puz@mth.muni.cz I. P. Stvroulkis Deprtment of Mthemtics University of Ionnin Ionnin Greece E-mil: ipstv@cc.uoi.gr