A singular Cauchy-Nicoletti problem for a system of three
ordinary differential equations is considered. An approach which combines topological method of T. Ważewski and Schauder\u27s principle is used.
Theorem concerning the existence of a solution of this problem (a graph of which lies in a given domain) is proved. Moreover, an estimation of its coordinates is obtained

J. Baštinec

J. Diblík

A singular Cauchy-Nicoletti problem for a system of three

ordinary differential equations is considered. An approach which combines topological method of T. Ważewski and Schauder's principle is used.

Theorem concerning the existence of a solution of this problem (a graph of which lies in a given domain) is proved. Moreover, an estimation of its coordinates is obtained

Baštinec, J.

Diblík, J.

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Mathematical Communications 2(1997), 211-218 211.212 Three-point singular problemThree-point singular boundary-value problem for asystem of three differential equations∗Jarom´ir Basˇtinec†and Josef Dibl´ik‡Abstract. A singular Cauchy-Nicoletti problem for a system ofthree ordinary differential equations is considered. An approach whichcombines topological method of T. Waz˙ewski and Schauder’s principleis used. Theorem concerning the existence of a solution of this prob-lem (a graph of which lies in a given domain) is proved. Moreover, anestimation of its coordinates is obtained.Key words: Three-point boundary value problem, singular Cauchy-Nicoletti problem, Waz˙ewski’s principleAMS subject classifications: 34B10, 34B15Received May 25, 1998, Accepted November 19, 19981. IntroductionIn the presented paper the following Cauchy-Nicoletti problemy′i(x) = ωi(x)yi + fi(x, y1, y2, y3), i = 1, 2, 3, (1)y1(x+1 ) = A1, y2(x±2 ) = A2, y3(x−3 ) = A3, (2)where x ∈ I = [a, b], a = x1 < x2 < x3 = b and Ai, i = 1, 2, 3 are real constants,is considered. Denote Ii = I \ {xi}, i = 1, 2, 3 and J = I1 ∩ I2 ∩ I3. We shallsuppose that ωi ∈ C (Ii,R) , i = 1, 2, 3 and fi ∈ C(Ωi,R), i = 1, 2, 3 whereΩi ⊂ Ii × R3, Ωi ∩ {x = x∗} 6= ∅ for x∗ ∈ Ii. Note that continuity of the functionsωi and fi is not required at point xi, i = 1, 2, 3. A solution of the problem (1), (2)is defined in the following sense:Definition 1. A vector-function y(x) = (y1(x), y2(x), y3(x)) ∈ C(I) where yi(x) ∈C1(Ii), i = 1, 2, 3, is said to be a solution of the problem (1), (2) if it satisfies thesystem (1) on J and, moreover, y1(x+1 ) = A1, y2(x±2 ) = A2, y3(x−3 ) = A3.∗This work has been supported by the Grant No 201/96/0410 of Czech Grant Agency (Prague)†Department of Mathematics, Faculty of Electrical Engineering and Computer Sci-ence,Technical University (VUT), Technick 8, 616 00 Brno, Czech Republic; e-mail:bastinec@dmat.fee.vutbr.cz‡Department of Mathematics, Faculty of Electrical Engineering and Computer Sci-ence,Technical University (VUT), Technick 8, 616 00 Brno, Czech Republic; e-mail:diblik@dmat.fee.vutbr.czThree-point singular problem 213Although singular boundary value problems were widely considered by using ofvarious methods (see e.g. [1] – [3], [5] – [7]) the method used here is based on a newapproach – a combination of topological method of T. Waz˙ewski and Schauder’sprinciple. (Note that method of T. Waz˙ewski was used to investigation variousasymptotic and singular problems, e.g., in [2], [3], [8] — [10].) Each equation ofthe system (1) is considered separately (as scalar equation) under supposition thatnondiagonal variables are changed by functions taken from a given set of functionsM . For every scalar equation (together with corresponding Cauchy initial conditionwhich follows from (2)) it is shown, with the aid of Waz˙ewski’s principle, thatthere is its solution with the same properties which are supposed for coordinates ofcorresponding functions from M . By this way an operator T is defined. Stationarypoint of operator T is a solution of the problem (1), (2).2. Existence of solutuins of problem (1), (2)Let us suppose that ωi ∈ C (Ii,R) , i = 1, 2, 3, function fi ∈ C(Ωi,R), i = 1, 2, 3whereΩi = {(x, y1, y2, y3) : x ∈ Ii, (x, y1, y2, y3) ∈ Ω},Ω = {(x, y1, y2, y3) : x ∈ I, αi(x)− ε∗ ≤ yi ≤ βi(x) + ε∗, i = 1, 2, 3},ε∗ is a small positive number and αi(x), βi(x), i = 1, 2, 3 are real functions such thatαi(x), βi(x) ∈ C1 (I) , αi(x) < βi(x), for x ∈ Ii, α1(x1) = β1(x1) = A1, α2(x2) =β2(x2) = A2, α3(x3) = β3(x3) = A3. Involve (for i, j = 1, 2, 3 and i < j) the setΩij = {(x, yi, yj) : x ∈ Ik, k ∈ {1, 2, 3}, k 6= i, j, αs(x) ≤ ys ≤ βs(x), s = i, j}.Let us define auxiliary functionsFi(x, y1, y2, y3) ≡ ωi(x)yi − y′i + fi(x, y1, y2, y3), i = 1, 2, 3.The result of the paper is given in the following theorem.Theorem 1. Assume that F1(x, α1(x), y2, y3)·F1(x, β1(x), y2, y3) < 0 if (x, y2, y3) ∈Ω23,F2(x, y1, α2(x), y3) · F2(x, y1, β2(x), y3) < 0 if (x, y1, y3) ∈ Ω13 andF3(x, y1, y2, α3(x)) · F3(x, y1, y2, β3(x)) < 0 if (x, y1, y2) ∈ Ω12. (3)Let, moreover,|fi(x, y1, y2, y3)−fi(x, z1, z2, z3)| ≤Mi(x)|y1−z1|+Ni(x)|y2−z2|+Pi(x)|y3−z3| (4)for any (x, y1, y2, y3), (x, z1, z2, z3) ∈ Ωi where Mi(x), Ni(x), Pi(x) are functions,continuous on Ii, i = 1, 2, 3, such that|ωi(x)| > Mi(x) +Ni(x) + Pi(x), i = 1, 2, 3, x ∈ Ii (5)and ω1(x)F1(x, β1(x), y2, y3) > 0 if (x, y2, y3) ∈ Ω23,ω2(x)F2(x, y1, β2(x), y3) > 0 if (x, y1, y3) ∈ Ω13,ω3(x)F3(x, y1, y2, β3(x)) > 0 if (x, y1, y2) ∈ Ω12.(6)Then there is at least one solution y(x) = (y1(x), y2(x), y3(x)) of the problem (1), (2)such that αi(x) < yi(x) < βi(x) where x ∈ Ii, i = 1, 2, 3.214 J. Basˇtinec and J. Dibl´ikProof. The idea of the proof is as follows. At first, with the aid of (1), (2), anoperator T is constructed, such that T (M) ⊂M whereM = {(ϕ1(x), ϕ2(x), ϕ3(x)) : x ∈ I, ϕi(x) ∈ C(I), αi(x) ≤ ϕi(x) ≤ βi(x), i = 1, 2, 3}.The next step is verification of conditions of Schauder principle for this operator.The stationary point ϕ(x) ∈ M of T will be a solution of the problem (1), (2).The construction of operator T uses the topological principle of T. Waz˙ewski. (Thedetails of the application of this principle can be found, e.g., in [3], [4], [8] — [10]and therefore will be omitted.)The proof is divided into two parts (construction of the operator T and verificationof Schauder’s principle).I. Construction of operator T . Let us consider the system of three equationsy′1 = ω1(x)y1 + f1(x, y1, ϕ2(x), ϕ3(x)), (7)y′2 = ω2(x)y2 + f2(x, ϕ1(x), y2, ϕ3(x)), (8)y′3 = ω3(x)y3 + f3(x, ϕ1(x), ϕ2(x), y3), (9)where (ϕ1(x), ϕ2(x), ϕ3(x)) ∈M. This system consists of separated scalar equations.Therefore we shall consider equations of this system separately in the sequel. Define(for i = 1, 2, 3): wi(x, yi) ≡ (yi − αi(x))(yi − βi(x)),Ni = {(x, yi) : x ∈ Ii, wi(x, yi) = 0},Di = {(x, yi) : x ∈ int Ii, wi(x, yi) < 0}. a) Let us investigate the equation (7).Next we prove that there is at least one solution y1 = y1(x) ∈ C(I) ∩ C1(I1)satisfying the following properties:y1(x+1 ) = A1; α1(x) < y1(x) < β1(x), x ∈ I1. (10)Let us evaluate the derivative of w1(x, y1) along the trajectories of the equation (7)if (x, y1) ∈ N1. Then either y1 = β1(x) or y1 = α1(x). In the first case we havedw1(x, y1)dx∣∣∣∣y1=β1(x)= (β1(x)− α1(x)) · F1(x, β1(x), ϕ2(x), ϕ3(x))and in the second onedw1(x, y1)dx∣∣∣∣y1=α1(x)= F1(x, α1(x), ϕ2(x), ϕ3(x)) · (α1(x)− β1(x)).In view of condition (1.) we have eitherF1(x, α1(x), ϕ2(x), ϕ3(x)) < 0 (11)orF1(x, α1(x), ϕ2(x), ϕ3(x)) > 0. (12)If (11) holds, thendw1(x, y1)dx∣∣∣∣(x,y1)∈N1> 0Three-point singular problem 215and all points of the set N1 for x ∈ (x1, x3) are the points of strict egress forD1with respect to (7). Then each point (x3, y∗1) where y∗1 ∈ (α1(x3), β1(x3)) definesa solution of equation (7) such that (10) holds. In the sequel we will take the uniquesolution y1(x) which satisfies condition y1(x−3 ) = y∗1 =12 (α1(x3) + β1(x3)).If (12) holds, thendw1(x, y1)dx∣∣∣∣(x,y1)∈N1< 0and all points of the set N1 for x ∈ (x1, x3) are the points of strict ingress forD1 with respect to (7). Then from the Waz˙ewski principle follows that there is atleast one solution y1 = y1(x) such that w1(x, y1(x)) < 0 if x ∈ I1, i.e. (10) holds.Suppose that a set Y1 consists of all such solutions and denote y∗∗1 = min{y1(x3) :y1(x) ∈ Y1}. The value y∗∗1 exists in view of elementary properties of solutionsof differential equations. In the sequel we will take the unique solution y1(x) ofequation (7) which satisfies condition y1(x−3 ) = y∗∗1 . Therefore for both cases (11)or (12), we have defined, by an unique manner, a solution y1(x) of equation (7)with property (10).b) Now consider the equation (9). We prove (similarly as in the part a)) that thereis at least one solution y3 = y3(x) ∈ C(I) ∩ C1(I3) satisfying the conditionsy3(x−3 ) = A3; α3(x) < y3(x) < β3(x), x ∈ I3. (13)We evaluate the derivative of w3(x, y3) along the trajectories (9) if (x, y3) ∈ N3. Byanalogy with the previous computations we (in view of (3)) getdw3(x, y3)dx∣∣∣∣(x,y3)∈N3> 0 (14)ifF3(x, ϕ1(x), ϕ2(x), α3(x)) < 0 (15)anddw3(x, y3)dx∣∣∣∣(x,y3)∈N3< 0 (16)ifF3(x, ϕ1(x), ϕ2(x), α3(x)) > 0. (17)Therefore, as in the part a), for both cases (14) or (16) (with the aid of the setsN3,D3), we can define, by an unique manner, a solution y3(x) of eguation (9) withproperty (13).c) Let us consider the equation (8). We prove that there is at least one solutiony2 = y2(x) ∈ C(I) ∩ C1(I2) such thaty2(x±2 ) = A2; α2(x) < y2(x) < β2(x), x ∈ I2. (18)By condition (1.) there are possible following four cases:dw2(x, y2)dx∣∣∣∣(x,y2)∈N∈> 0, ordw2(x, y2)dx∣∣∣∣(x,y2)∈N∈< 0,216 J. Basˇtinec and J. Dibl´ikordw2(x, y2)dx∣∣∣∣ (x, y2) ∈ N∈x ∈ [x1, x2)> 0,dw2(x, y2)dx∣∣∣∣ (x, y2) ∈ N∈x ∈ (x2, x3]< 0,ordw2(x, y2)dx∣∣∣∣ (x, y2) ∈ N∈x ∈ [x1, x2)< 0,dw2(x, y2)dx∣∣∣∣ (x, y2) ∈ N∈x ∈ (x2, x3]> 0.By analogy, as in part a) above we define (with the aid of the sets N2,D2), byan unique manner, a solution y2(x) of equation (8) which satisfies conditions (18).d) From parts a) – c) above it follows that for each function ϕ(x) = (ϕ1(x), ϕ2(x),ϕ3(x)) ∈M there can be chosen, by indicated rule, a unique function y(x) = (y1(x),y2(x), y3(x)) ∈ M (here we put y1(x1) = y1(x+1 ), y2(x2) = y2(x±2 ) and y3(x3) =y3(x−3 )). This correspondence defines mentioned operator T on M , i.e. for eachϕ(x) ∈M we have Tϕ(x) ∈M and, therefore, T (M) ⊂M .II.Verification of Schauder’s assumptions. Let us consider the Banach space Λ offunctions λ(x) = (λ1(x), λ2(x), λ3(x)) , continuous on I, with the norm ‖λ(x)‖ =maxi=1,2,3{maxI|λi(x)|}. Obviously M ⊂ Λ and, as it follows from the properties ofthe functions αi(x), βi(x), i = 1, 2, 3, M is a closed, bounded and convex set.It remains to prove that T is a continuous mapping such that T (M) is a relativelycompact subset. With respect to relatively compactness of T (M) it is sufficient toprove by Arczela–Ascoli Theorem that T (M) is uniformly bounded and equicontin-uous on I.α) The uniform boudnedness follows from inequality ‖ϕ‖ ≤ L where L = maxI{|αi(x)|,|βi(x)|, i = 1, 2, 3} which holds for every ϕ ∈M .β) Let us prove the equicontinuity of each function ϕ(x) ∈ T (M). On I1 the firstcoordinate ϕ1(x) of ϕ(x) satisfies an equation of the typeϕ′1(x) = ω1(x)ϕ1(x) + f1(x, ϕ1(x), ν2(x), ν3(x)) (19)where (ϕ1(x), ν2(x), ν3(x)) ∈ M. Since ω1(x) ∈ C(I1,R) and f1 ∈ C(Ω1,R), from(19) we get |ϕ′1(x)| < Kδ, x ∈ [x1 + δ, x3], x1 + δ < x3, 0 < δ = const, where theconstant Kδ exists and depends on δ. Let us put δ1 = min(δ/2, ε/Kδ/2) where ε isan arbitrary positive number and δ is so small thatmax[x1,x1+δ]|β1(x)−A1| < ε/2, max[x1,x1+δ]|α1(x)−A1| < ε/2.Let us suppose that |z1 − z2| < δ1, z1, z2 ∈ [x1, x3]. Then either z1, z2 ∈ [x1, x1 + δ]or z1, z2 ∈ [x1 + δ/2, x3]. In the first case|ϕ1(z1)− ϕ1(z2)| ≤ |ϕ1(z1)−A1|+ |ϕ1(z2)−A1| < ε/2 + ε/2 = εand in the second case (by Lagrange’s theorem) |ϕ1(z1)− ϕ1(z2)| ≤ Kδ/2 |z1 − z2| <ε. So, for each positive ε there is a δ1 > 0 such that |ϕ1(z1)− ϕ1(z2)| < ε for|z1 − z2| < δ1 and each function of the type of ϕ1(x) is equicontinuous. By analogywe can show that the functions of the type ϕ2(x) or ϕ3(x) are equicontinuous too.Finally, for |z1− z2| < δ1, we get ‖ϕ(z1)−ϕ(z2)‖ < ε and the equicontinuity of theset T (M) is proved.Three-point singular problem 217γ) Continuity of the operator T . Let us suppose that y0(x) ∈M, y˜(x) ∈M andY 0(x) = Ty0(x), Y˜ (x) = T y˜(x).In the sequel we prove that the operator T is continuous, i.e. that‖Y 0(x)− Y˜ (x)‖ < ε if ‖y0(x)− y˜(x)‖ < δ ≤ ε. (20)The last inequality (in which ε is an arbitrary sufficiently small positive number)will be supposed in the sequel. Consider the identityY 0i′(x) ≡ ωi(x)Y 0i (x) + fi(x, η01(x), η02(x), η03(x)),where i = 1, 2, 3, η0i (x) = Y0i (x), η0j (x) ≡ y0j (x), j 6= i, (x, η01(x), η02(x), η03(x)) ∈ Ωiand the equation (which has a solution Y˜i = Yi(x))Y˜ ′i = ωi(x)Y˜i + fi(x, η˜1, η˜2, η˜3), (21)where i = 1, 2, 3, η˜i = Y˜i, η˜j = η˜j(x) ≡ y˜j(x), j 6= i, (x, η˜1, η˜2, η˜3) ∈ Ωi. Define (fori = 1, 2, 3): Wi(x, Y˜i) = (Y˜i − Y 0i (x))2 − ε2, ε = const, 0 < ε < ε∗,Pi = {(x, Y˜i) : x ∈ Ii,Wi(x, Y˜i) = 0}. γ1) Let us evaluate the derivative ofW1(x, Y˜1)along the trajektories of equation (21) for i = 1 if (x, Y˜1) ∈ P1. Then eitherY˜1 = Y 01 (x) + ε or Y˜1 = Y01 (x)− ε. ThereforedW1(x, Y˜1)dx∣∣∣∣∣Y˜1=Y 01 ±ε== ±2ε [ω1(x)(±ε) + f1(x, Y 01 (x)± ε, y˜2(x), y˜3(x))− f1(x, Y 01 (x), y02(x), y03(x))] .According to (4) and (5)|f1(x, Y 01 (x)± ε, y˜2(x), y˜3(x))− f1(x, Y 01 (x), y02(x), y03(x))| ≤≤ (M1(x) +N1(x) + P1(x)) ε < |ω1(x)|ε.ThereforedW1(x, Y˜1)dx∣∣∣∣∣(x,Y˜1)∈P1> 0 if ω1(x) > 0 on I1 (22)anddW1(x, Y˜1)dx∣∣∣∣∣(x,Y˜1)∈P1< 0 if ω1(x) < 0 on I1. (23)If (22) and (11) hold simultaneously, then all points of the set ∂Q1 where Q1 ={(x, Y˜1) : x ∈ (x1, x3), w1(x, Y˜1) < 0,W1(x, Y˜1) < 0} are, for x ∈ (x1, x3), the pointsof strict egress for Q∞ with respect to (21) where i = 1 (since this equation is atthe same time an equation of the type (7)). Since Y 01 (x+1 ) = Y˜1(x+1 ) and (in view ofconstruction of operator T ) Y 01 (x−3 ) = Y˜1(x−3 ), then |Y 01 (x)− Y˜1(x)| < ε. Indeed, ifthis inequality does not hold, then there is a x∗ ∈ I1 such that |Y 01 (x∗)−Y˜1(x∗)| = εand by (22) |Y 01 (x) − Y˜1(x)| > ε on (x∗, x3]. This is impossible. If (23) and (12)218 J. Basˇtinec and J. Dibl´ikhold, then all points of the set ∂Q1 are, for x ∈ (x1, x3), the points of strict ingressfor Q∞ with respect to (21) where i = 1. If inequality |Y 01 (x)− Y˜1(x)| < ε does nothold, then there is a x∗ ∈ I1 such that |Y 01 (x∗)−Y˜1(x∗)| = ε and |Y 01 (x)−Y˜1(x)| < εon (x1, x∗). This is impossible. In both considered cases |Y 01 (x)− Y˜1(x)| < ε on I1and, consequently, on I too. We conclude that in cases (22), (11) and (23), (12)|Y˜1(x)− Y 01 (x)| < ε on I if ‖y˜(x)− y0(x)‖ < δ.Cases (22), (12) and (23), (11) are impossible according to (6).γ2) Let us evaluate the derivative of W3(x, Y˜3) along the trajectories of equation(21) for i = 3 if (x, Y˜3) ∈ P3. We getdW3(x, Y˜3)dx∣∣∣∣∣(x,Y˜3)∈P3> 0 if ω3(x) > 0 on I3, (24)ordW3(x, Y˜3)dx∣∣∣∣∣(x,Y˜3)∈P3< 0 if ω3 < 0 on I3. (25)In both of these cases we can prove, as in the part γ1), that |Y 03 (x) − Y˜ 03 (x)| < εon I if ‖y˜(x) − y0(x)‖ < δ. Cases (24), (17) and (25), (15) are impossible in viewof (6).γ3) Let us evaluate the derivative of W2 along the trajectories of equation (21) fori = 2 if (x, Y˜2) ∈ P∈. We getdW2(x, Y˜2)dx∣∣∣∣∣(x,Y˜2)∈P2> 0 if ω2 > 0 on I2, ordW2(x, Y˜2)dx∣∣∣∣∣(x,Y˜2)∈P2< 0 if ω2(x) < 0 on I2,or dW2(x, Y˜2)dx∣∣∣∣∣ (x, Y˜2) ∈ P∈x ∈ [x1, x2)> 0 if ω2(x) > 0 on [x1, x2),dW2(x, Y˜2)dx∣∣∣∣∣ (x, Y˜2) ∈ P∈x ∈ (x2, x3]< 0 if ω2(x) < 0 on (x2, x3],or dW2(x, Y˜2)dx∣∣∣∣∣ (x, Y˜2) ∈ P∈x ∈ [x1, x2)< 0 if ω2(x) < 0 on [x1, x2),dW2(x, Y˜2)dx∣∣∣∣∣ (x, Y˜2) ∈ P∈x ∈ (x2, x3]> 0 if ω2(x) > 0 on (x2, x3].Each of the these cases can be considered as above in the parts γ1) and γ2) and,therefore, |Y 03 (x)− Y˜3(x)| < ε on I if ‖y˜(x)−y0(x)‖ < δ. Other cases are impossiblein view of (6).Three-point singular problem 219Connecting parts γ1) — γ3), we conclude that (20) holds and, consequently,operator T is continuous. All conditions of Schauder’s principle are valid and,therefore, operator T has a fixed point, i.e. has a solution of problem (1), (2) withindicated properties which follow from the form of the setM . The proof is complete.3. ExamplesExample 1. Let us consider singular problem:xy′1 = y1 + x(y1 − x)(y2 − y3) exp (−2/x),(x− 1/2)y′2 = 2y2 +(y2 − (x− 1/2)2)(y1 − y3) exp(− (x− 1/2)−2),(x− 1)y′3 = y3 + (x− 1) (x− 1− y3) (y1 + y2) exp (2/(x− 1)),y1(0+) = 0, y2(0.5±)= 0, y3(1−) = 0.This problem has trivial solution. Moreover, if rewrite this system in the form(1), all conditions of Theorem 1 are valid for α1(x) = x/2, β1(x) = 2x, α2(x) =(x− 1/2)2 /2, β2(x) = 2 (x− 1/2)2 , α3(x) = (1− x) /2 and β3(x) = 2 (1− x) .Consequently, there is at least one nontrivial solution of this problem y(x) =(y1(x), y2(x), y3(x)) such that x/2 < y1(x) < 2x on (0, 1], (x− 1/2)2 /2 < y2(x) <2 (x− 1/2)2 on [0, 1/2) ∪ (1/2, 1], (1− x) /2 < y3(x) < 2 (1− x) on [0, 1).Example 2. Let us consider singular problem:y′1 = −y1x2+y210+ 1,y′2 = −5y2(x− 1)2 +y310+ 1,y′3 =5y3(x− 2)2 −y110− 1,y1(0+) = y2(1±) = y3(2−) = 0.All conditions of Theorem 1 are valid for α1(x) = 0.1x2, β1(x) = 2x2, α2(x) =0.1 (x− 1)2 , β2(x) = 2 (x− 1)2 , α3(x) = 0.1 (x− 2)2 and β3(x) = 2 (x− 2)2 . Con-sequently, there is at least one solution of this problem y(x) = (y1(x), y2(x), y3(x))such that αi(x) < yi < βi(x), x ∈ Ii, i = 1, 2, 3.References[1] V.A. Chechyk, Investigations of systems of ordinary differential equationswith singularities, Trudy Moskovskovo Mat. Obchestva, 8 (1959) 155 – 198.(In Russian)[2] J. Dibl´ık, On existence of 0-curves of a certain singular system of differentialequations, Math. Nachr., 122 (1985) 247 - 258. (In Russian)220 J. Basˇtinec and J. Dibl´ik[3] J. Dibl´ık, A multimensional singular boundary value problem of the Cauchy-Nicoletti type. Georgian Math. J., 4, (1997) 303 - 312.[4] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964.[5] I.T. Kiguradze, Some singular boundary-value problems for ordinary differ-ential equations, Izd. Tbilis. Univ., Tbilisi, 1975. (In Russian)[6] N.B. Konyuchova, Singular Cauchy problem for systems of ordinary differ-ential equations, Zh. Vych. Matematiki i Mat. Fiziki, 23, (1983) 629 – 645. (InRussian)[7] B. Pa, On a singular boundary-value problem for system of ordinary differen-tial equations, Arch. Math. (Brno), 13, (1977) 207 – 226. (In Russian)[8] B. Vrdoljak, On behaviour of solutions of system of linear differential equa-tions, Mathematical Communications, 2 (1997) 47 – 57.[9] B. Vrdoljak, On solutions of the Lagerstrom equation, Arch. Math. (Brno),24, (1988) 111 – 222.[10] T. Waz˙ewski, Sur un principe topologique de l’examen de l’allure asympto-tique des inte´grales des e´quations diffe´rentielles, Ann. Soc. Math. Polon., 20(1947) 279-313.

Three-point singular boundary-value problem for a system of three differential equations

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