Greguš, Michal, Jr.

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

Mathematica SlovacaMichal Greguš, Jr.The existence of a solution of a nonlinear boundary value problemMathematica Slovaca, Vol. 30 (1980), No. 2, 127--132Persistent URL: http://dml.cz/dmlcz/136235Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1980Institute 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.czMath. Slovaca 30,1980, No. 2,127—132 THE EXISTENCE OF A SOLUTION OF A NONLINEAR BOUNDARY VALUE PROBLEM MICHAL GREGUS jr. In this paper a nonlinear singular boundary value problem with the third order A T - ± dx'1'~Xdx-differential operator DL2 is studied, where D =—, L =x -j—. Hence the equation is of the form (1) x2y'" + 3xy" + y'=f(x,y,y'), X6(0,l ) and the conditions which are considered are either (I) lirn*y(*) = 0, sup |xy'(x)|<oo, y(l) = y'(l) = 0, *—° 1 6 ( 0 , 1> OГ (П) lim y(x) <oo, sup \y'(x)\<™, y(í) = 0. *e(0, 1> This problem is a generalization of a problem studied in [5]. The existence of a solution is investigated with the help of the Green function, constructed on the basis of the theory and results given in [4], Tichonoff fixed point theorem [1] and Ascoli-Arzela's theorem [3]. Part I. Let us consider the problem (1)—(I). The Green function of this problem is . . _ f-! ln2 t + lntlnjt 0 < x ^ t ^ l , 0 ( * ' ° - l |ln2jc 0 < t ^ J t ^ l . The function G(JC, t) has the following properties: G(x, t)^0 foreach (JC, t), G(l, t) = G(jc, 1) = 0, limjc G(JC, t) = 0, x—*0 Gx(x,t)^0 for each (JC, t), G,(l, t) = Gx(jc, 1) = 0, lini_jt Gx(jc,t) = lnt, and for xi=t: Gxx(x,t)^0, G^C*, 1) = 0, lini_ JC2 Gxx(x, t)= -Int. 127 m When g is a real continuous function on (0, 1) such that \g(x)\ = j , x e (0, 1), m > 0 , then the function y(jc), y(x)= \ G(x,t)g(t)dt satisfies the inequalities Jo |jcy(jc)| = m, |jcy'(jc)| = m for JC e(0, 1) and is a solution of the linear differential equation DL2y = #(*)• Moreover, y(jc) satisfies the conditions (I). Suppose that the function f(x,u,v) is continuous and bounded by m/3 on (0, 1) x 91 x gi and consider the space 9£x of all real continuous functions with continuous first derivatives on (0,1) and with the finite norm | |y| |*= sup X 6 ( 0 , 1> {|jcy(jc)|, |jcy'(jc)|}, ye%?x. On an arbitrary compact interval ^c=(0, 1) we can define the seminormp^(y) = sup {|jcy(x)|, |jcy'(x)|},y e$?i.The convergence in 9CX xe<€ with the topology defined by these seminorms is the uniform convergence on each compact set <#. The system {p^n}"=1, m>n = /—, 1V n e JV is a countable family of seminorms on (0, 1), satisfying Hausdorffs axiom of separation [3]. The system {{y e%?\\p€n(y)<e}}n=i is a subbase of neighbourhoods of the point zero (i.e. of y = 0 on (0, 1)). d£x with this topology is a complete space. Let us take a closed ball SF with a radius R^m, i.e.: ZF= {y e$?x\ \\y ||* = R } . The set :F is closed, bounded and convex in the topology defined by the system of seminorms {P^n}r=i. It is convenient to consider the operator T: :jF—>5F deter-mined by Ty(x)= ľ Jo G(x,t)f(t,y(t),y'(í))át, yeSe,. T is continuous if for any y0, each e > 0 and neJf, there exists <50 > 0 and n0 e N, n0 = n such that p^n0(y - y0) < d0 implies p^n (Ty - Ty0) < e. The function / is uniformly continuous on any compact set (g„0 x J>k x$k, where $k = {— k,k), keJf. Therefore if we choose 6 > 0 sufficiently small and | y ( 0 - y o < 0 l < « J y ' ( 0 - ^ t e ^n o. For 6 there exists d0>0 such that if the functions y, y0 satisfy p« (y - y0) < 8 60, where5 0 = 5 min { J C } = — , then \y(t)-y0(t)\<8, |y ' (0->'o(0l<<5- But then xe€n0 n0 for xe<€no = <€„ we have x|Ty(jc)-Ty0(jc)| =E and Jc|(Ty(jc))' - (Ty0(jc))'| = E . Hence T is continuous. To show the relative compactness of T(!F) we use Ascoli-Arzela's theorem. Since for JC e (0, 1) 0 = [ G(jc, l)d l<oo and 0 = [ Gx(x, t) dt>-™, Jo Jo 128 the sets T(^) and [T(:F)]' are equibounded on %>n. Equicontinuity follows from the fact that / is bounded on <6n and G(JC, t) and G*(JC, t) are uniformly continuous functions on <£„. Hence there follows from Tichonoff theorem the existence of y e 2F such that Ty =y. Since y satisfies the conditions (I), we have Theorem 1. Let f(x,u,v) be bounded and continuous on (0, 1) X 01 x 01. Then there exists a solution y(x) of the boundary value problem jc2y"' + 3jcy" + y '=/( jc ,y,y ' ) , jce(0, 1> y(l) = y '( l) = 0, Jinljcy(jc) = 0 such that sup |jcy'(jc)|<o°. * 6 ( 0 , 1> Remark 1. If we consider more closely the solution y(jc) of problem (1)—(I), we can see that on every compact set ^ cz (0,1) y (JC), together with its derivatives y\ y'\y'" is a bounded function, and lim JC y'(JC) = 0 and sup |jc2y"(jc)| < o°. X^° *e(0, 1> If we considered the space 3f2 of functions with continuous second derivatives and the finite norm |M|*= sup {\x'y(x)\,\x'y'(x)\,\xY(x)\}, X 6 ( 0 , 1 > i = 1 or 2,/ = 1 or 2, and with the system of seminorms on compact subsets of (0, 1) defined in a similar way as before, respectively, we could prove the existence of the solution of equation (1) with the right-hand side equal to f(x,y,y',y"), f continuous and bounded. Then the proof proceeds as follows: (a) For m > 0 and |^f(jc)|^^ we have |jc2y"(jc)|^m. (b) O f G^x^O t^^Koo toт xe(0,1>. Jo X (c) In the proof of the equicontinuity of [T(^0]" we cannot proceed as in the proof of Theorem 1, because G^(JC, t) has a jump for JC = t. However, it is possible to prove it otherwise: Let us take an arbitrary e > 0 and an arbitrary function y"e[T(^)]". For x1^x2e($n we have \y,f(x1)-y,,(x2)\^ IГ (*r2(l - In*,) - * i 2 ( l - lnx2))f(t, y(t), y'(t), y"(t)) dt + f\x72 lnt)f(t, y(t), y'(t), y"(t)) dt -J xi f V ( l -\nx2)f(t, y(t), y'(t), y"(t)) dt + J xi 129 where f (-*r2+*í2)(in/)/(/, y(t), y'(t), y(t)) d/|_i — \xT2(í -lnXi)-X22(í -lnx2)\xt - — xí2\x2 \nx2-x2-x1\nx1+x1\ + ~\x-22(\-\nx2)\\x2-xl\ + '~\x-22-x-l2\\l-x2 \nx2 + x2\ ž™ iP.fo) - ?1(x2)\+jn2\P2(x1) - P2(x2)\ + ^n2(l + n)\P3(xi)-P3(x2)\ + j(2 + n)\P4(x1)-P.(x2)\, P,(x) = ̂ -2(1-Inx), P2(x) = Jt-(ln*-l), P3(x) = x, P4(x) = x' From the uniform continuity of P., / = 1,2, 3,4 on ^ there follows the existence of n~2 4 a d > 0 such that for |JC1-JC2 |<6 we have |Pf-(jCi) — P*(JC2)I < T T £> ' = 1- 2, 3, 4. Now it is easy to show that for |JCI - JC2| <5, xt <x2 e ^n |y"0ci) -y"(x2)\ <e is true. This result can be formulated in the form of the following existence theorem : Theorem 2. Letf(x, u,v,w)be bounded and continuous on (0, 1) x <R x <3l x 01. Then there exists a solution y(x) of the boundary value problem x2y'" + 3xy" + y'=f(x,y,y',y"), xe(0, 1) y(l) = y'(l) = 0, linljcy(jc) = 0, limx2y'(x) = 0 x-+0 x-*0 such that sup |JCV'(JC)|<O°. * € ( 0 , 1> Part II. Let us consider now the problem (1)—(II). Its Green function is Н (*'°-{-1пНп2л- + (1п x^t, x x^t. Note that H(JC, t) = -G( t , JC). Other properties of H(x, t) are: H(jc,t)^0, Um>H(-r,r)=-Jln2/, and H X (JC,0-^0, \w^Hx(x,t) = 0. The solution of the differential equation DL2y^h(x), where h(x) is a real continuous function on (0, 1), is y(x)=fн(д.,/)Ä(/)d/. J0 130 If \h(x)\<j9 m positive, x e(0, 1), then \y(x)\^m, \y'(x)\^m and y satisfies the boundary condition (II), too. Let f(x, u, v) be continuous and bounded by - j on the set (0,1) x (-K, K) x (-K, K), K^m. Consider the space ©i (—, 1J of all real functions with the first derivative continuous on (—, 1), n eJf and the norm ||y||rt = sup {|y(x)|, \ " / *6<A,i> \y '(x)\}. The space 3)i (-, 1 j is a Banach space [2]. Let #? be the space of all real functions with continuous first derivatives on (0, 1) and the finite norm ||y||« = sup {\y(x)\, |y'(x)|}, ye%. Take a ball % = {y e%\ \\y\\^K} and define the -6(0.1) operator S: <53-»S3 by Sy(x)= f H(x, t) f(t, y(t), y'(t)) dt, ye%. S is well-def-Jo ined, S is continuous (this follows from the uniform continuity of / on (—, l) x (~K,K) x (-K, K)) and S(39) is relatively compact (this follows from Ascoli-Arzela's theorem, as well as the fact that 0.S I H(JC, t) dt = -x> - <», Jo I Hx(x,t)dt = l and that H(x, t) and Hx(x,t) are uniformly continuous on Jo (—, 1) . Therefore we have Theorem 3. Let f(x, u,v) be a bounded and continuous function (0, 1) x <% x 91. Then there exists a solution y(x) of the nonlinear boundary value problem x2y'" + 3xy" + y'=f(x,y,y'), JCG(0,1> limy(x) <oo, sup |y'(^)|<oo, y(l) = 0 X^° X6(0, 1> such that sup |jty"(*)|<°°. X€(0, 1> Remark 2. It is interesting to note that in the case of conditions (II) lim y(x) x—»0 exists and is finite. This fact follows from the boundedness of y '(x). From the mean value theorem we then have that y (JC) is uniformly continuous on (0, 1). Therefore each solution y(x) of (1)—(II) is continuously extendable on (0, 1) and bounded. On the other hand, in problem (1)—(I) difficulties arise. Generally we can say about a solution of that problem only that it is on the left end bounded by the function —, in other words that its "growth" is not arbitrarily large. 131 REFERENCES [1] HARTMAN, P.: Ordinary diffeгential equation, John Wiley and sons, New York—London—Sydn-ey 1964. [2] ŠILOV, G. J.: Matematická analýza, Alfa, Bгatislava 1974. [3] YOSIDA, K.: Functional analysis, Springeг-Verlag, Berlin—Heidelberg—New York 1968. [4] COLLATZ, L.: Pгoblemy chaгakteristických hodnôt s technickými aplikáciami, SVTL, Pгaha 1965. [5] ĎURIKOVIČOVÁ, M.: On the existence of solution of a singular boundaгy value problem, Mat. Čas., 23, 1973, 34—39. Received Novembeг 25, 1977 Katedгa matematickej analýzy Príгodovedeckej fakulty UK Mlynská dolina 816 31 Bгatislava СУЩЕСТВОВАНИЕ РЕШЕНИЯ ОДНОЙ НЕЛИНЕЙНОЙ КРАЕВОЙ ЗАДАЧИ Михал Грегуш, мл. Резюме При помощи теоремы о неподвижной точке доказаны теоремы о существовании решения дифференциального уравнения (1) с краевыми условиями (I) или (II). Решение краевой задачи (1), (I) существует в том случае, когда функция / непрерывна и ограничена (теорема 1,2). Напротив того, решение краевой задачи (1), (II), при условии непрерывности и ограниченности/, ограничено и продолжительно на отрезок (0, 1) (теорема 3). В статье тоже исследованы свойства функции Грина выше упомянутого уравнения (1). 132 

The existence of a solution of a nonlinear boundary value problem

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The existence of a solution of a nonlinear boundary value problem

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