Šeda, V.

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

EQUADIFF 3Valter ŠedaOn a nonlinear boundary value problem of higher orderIn: Miloš Ráb and Jaromír Vosmanský (eds.): Proceedings of Equadiff III, 3rd CzechoslovakConference on Differential Equations and Their Applications. Brno, Czechoslovakia, August 28 -September 1, 1972. Univ. J. E. Purkyně - Přírodovědecká fakulta, Brno, 1973. Folia FacultatisScientiarum Naturalium Universitatis Purkynianae Brunensis. Seria Monographia, Tomus I.pp. 145--153.Persistent URL: http://dml.cz/dmlcz/700087Terms of use:© Masaryk University, 1973Institute 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.czON A NONLINEAR BOUNDARY VALUE PROBLEM OF HIGHER ORDER by V. §EDA 1. The theory of boundary value problems (in short BVP) for ordinary differential (d.) equations has been largely developed as one can see from a comprehensive and carefully prepared report [1]. It is characterized by a great number of problems and a variety of methods attacking them. In the last years two methods, the subfunction method and the method based on the funnel theorem of Kneser, became successful in guaranteeing the existence of solutions to BVP. Both methods have been applied to solving second-order BVP of different types in more than 15 papers. Here only [2], [3] and [4] will be mentioned. As to the BVP of higher order, the subfunction method has been used only in [5] and some BVP of the third order were solved by means of the funnel technique in [6]. Another application of the subfunction theory to anon-linear BVP of a higher order will be presented now. The problem arose in the theory of semiconductors and it has been solved by reducing it to a set of second-order ones. The details can be found in [7]. 2. Suppose a, b, /, a, /?, y, S, A, B, C, D, Fare positive numbers. The problem in question is yw - a[y" + b(y'y'" + y"2)] = 0, (0 = x = I), / = - ^ , etc., (1) j(0) = a{Fy'(P) - By'(0) [C + Dy"(0)]} y'(0) = P{Fy"'(0) - By'(0) [C + Dy"(0)]} y(l) = A - y{Fy'"(l) - By'(l) [C + Dy\lj\) y'(l) = 5{Fy'"(l) - By'(l) [C + Dy"(I)]} (2) A solution y of that problem represents potential and the function z defined through yby z = Fym - By\C + Dy") (3) means current in a semiconductor. We shall be interested to prove the existence, if possible the uniqueness, of the solution y to (1), (2) as well as to find some properties of the associated function z given by (3). By the double integration of (1) and assuming that Fab - BD = 0, (4) the BVP (1), (2) can be reduced to four different types of BVP. 145 10 Equadiff HI When 1. P(Fa - BC) - 1 = &(Fa ~ BC) - I = 0, then (1), (2) is equivalent to b y" = aly+ —/'] +c2, (5') jy'(o)-y(o) = o, (6) JLy'(i) + y(i)-A = o 1 , and the associated function z = ~o y • If 2. P(Fa - BC) - 1 = 0 ^ 'HI7'3 - BC) "~ 1> t n e n (*)' (2) reduces to the BVP (5'), 5 - / ( 0 ) - y ( 0 ) = 0, (7) >>(/) = A, y'(l) = 0. The associated function z = - r - / . In the case 3. jS(Fa - BC) - 1 * 0 = <5(Fa - AC) - 1 instead of (1), (2) we have (5'), j,(0) = y'(0) = 0 (8) -fy(0 + K0-^ = o and z = —y'. o When 4. j8(Fa - BC) - 1 -̂  0 # <5(Fa - BC) - 1 is assumed, the BVP (1), (2) means the BVP y" = aly + үУ' ) + cгx + c2, (5) , л aҒc. ,ínЛ ßFct * 0 ) в l-ß(Fa-BC)> y{0)=l-ß(Fa-BC)> Ю - Л - У F h fíгҳ, y V ) - Í Л -1 - 8(Fa - BC)9 J w 1 - c5(Fa - BC) and the function z = (Fa — BC) y' + Fc1. In all mentioned cases ct, c2 G -̂  a r e arbitrary. Since the BVP (5), (6) contains all the others as special cases, first this problem will be studied. Nevertheless, the problem can be further generalized and the existence, uniqueness and a comparison property of the solution to the more general BVP will be established in the next section. 146 3, Consider the d. equation y"=/(*,y,y') (J°) with / being denned and continuous on the set S = {(x, y, y'): 0 _ x _ /, | y | + + | y' | < +00}. The notions of a lower (an upper) solution of (10) as well as of a Nagumo condition for f have the usual meaning and they are defined in [3], p. 460. Further, if cp is a lower and \j/ an upper solution of (10) and cp(0) < \j/(0) (cp(l) < \j/(l)), then by H1(H2) will be denoted the class of all continuous functions g = g(y, y') (h = h(y, y')) defined on (cp(0), \j/(0)y x R «<p(/), \j/(l)} x R) which are nondecreas-ing in y' and satisfy g[<p(0), <p'(0)] ^ 0, g^(0), ^'(0)] _ 0 (h[_cp(l), cp'(l)] _ 0, h[*K/), *A'(0] _ 0). H3(H4) will mean the class of all continuous functions g = = g(y, y') (h = h(y, y')) defined on R2 which are nonincreasing in y (nondecreasing in y) and nondecreasing in y'. The following existence theorem has been proved as Theorem 3.6 in [3], p. 464. Theorem 1. Let <p0/0 be a lower (an upper) solution of the d. equation (10) with cp _ ij/ on <0, /> and let <p(0) < ij/(0), cp(l) < \j/(l). Let f satisfy a Nagumo condition with respect to the pair cp, \j/ and let ge Hl9 he H2. Then there is a solution y of the BVP (10), g[y(0)9yf(0)-] = o = h[y(i),yXi)2 (11) which satisfies cp _ y _ \j/ on <0, 1>. With a help of this theorem another existence as well as a comparison theorem will be proved. Theorem 2. Suppose ft = ft(x, y, y'), f2 = f2(x, y, y') are continuous and fx S _/2( / i =f2) on S. Suppose lim f2(x,y, y') = -00 ( lim f2(x,y,y') = +00) uniform-y-* — 00 y~* + CXD ly in x, y' on any compact subset of <0, /> xR. Assume further f2 satisfies a Nagumo condition with respect to any pair x9coe Co«0, / » , x _ °>- Suppose g = g(y, y') e H3 andh = h(y, y') e H4. Then, if there is a solution yx of the problem (11), y"=/i(^,y,y'), (12) there also exists a solution y2 of the BVP (11), y"=/2(*,y,y') (13) and a number c > 0 such that yx — c _ y2 _ yx (yt _ y2 _ yx + c) on <0, />. Proof. Only the case/i ^ / 2 will be considered. Since fx ^f2, j i is an upper solution of (13). By the assumption off2, there is a y0 such that for all xe <0, /> and all y _ y0 f2(x,y,y[(x)) _ m = min fx(x, y^(x), y'i(x)). Then for any c > 0 *e<0,i> with yx(x) — c _ y0 (x e <0, / » the function yx — c is a lower solution of (13). g and 147 h being from H3 and H4 respectively, g e Hx and he H2. The result follows on the basis of Theorem 1. The following theorem generalizes a uniqueness result in [8], p. 108. Theorem 3. Let f = f(x9 y9 y') be defined and increasing in y on S. Let g = g(y, y') (h = h(y9 y'j) be decreasing in y (increasing in y) and nondecreasing in y' on R2. Then there exists at most one solution of the BVP (10), (11). Proof. Suppose there are two solutions yl9 y2 of the considered BVP. By the properties of g and h, yt(0) > y2(0) (yt(0) < y2(0)) implies that y[(0) > y2(0) (yi(0) < ya(0)). Similarly, yt(l) > y2(l) (yx(l) < y2(l)) can stand only with y[(l) < < y2(0 (yi(0 > y'lO))- All four cases lead to the existence of either a positive local maximum or a negative minimum of yx — y2. This cannot happen, with respect to the assumption on / Thus, y^O) = y2(0), yx(l) = y2(l) and yi = y2 on <0, />. Remark. Theorems 1 up to 3 are valid also in the case when one or both of the functions g9 h show inverse monotonic properties. 4, The above stated theorems can be applied to the BVP (5), (6). By Theorem 3, there exists at most one solution of that problem. On the basis of Theorem 1, the existence of a solution to that BVP can be assured by finding suitable lower and upper solutions to (5). Taking linear functions for lower and upper solutions we get Lemma 1. There exists a unique solution y to the problem (5), (6). This solution satisfies the inequalities where ~^-x + q = y(x) S ~~x + qx (0 = x = I), (14) «-* [j(4^T^')^-f} ™[TK>^(H--V-£} The mentioned solution will be denoted as y(c1, c2). Some of its properties are given in the following lemmas. Lemma 2. If c[x + c2 > cxx + c2 on (0, /> and c = max —[cix + c2 — (ctx + c2)], (15) .xe<0,.'> a then y(ci ,c2) - c = y(c\, c2) < y(ct, c2) on <0, />. (16) 148 Proof . By Theorem 2 and by the uniqueness of y(c[, c2), there is a c > 0 such that y(ct, c2) - c ^ y(c\, c2) ^ y(c1, c2) on <0, />, A direct calculation gives (15). The strict inequality in (16) can be easily shown. Lemma 3. For any two numbers cx, d there exist uniquely determined numbers c2, c"2 such that y(ct, c2) (0) = d = y(ct, c"2) (I). Proof. (16) implies that the functions y(c1, c2) (0), y(cl, c2) (I) of the variable c2 are continuous and decreasing in c2. With respect to this and to the comparison properties of y(cx, c2) only the unboudedness of y(0, c2) (0), y(0, c2) (I) has to be proved. Using some properties of y(0, c2) we get the following statements. First, (14) implies the inequalities y(0, c2) > 0 for all c2 < 0, and y(0, c2) < A for all c2 > 0, on <0,/>. (17) Secondly, if one of the functions y(0, c2) (0), y(0, c2) (I) is bounded, there exists a linear function M + Nx such that | y(0, c2) (x) | < M + Nx (xe <0, / » (18) for all c2 sufficiently great in absolute value. At the same time y'2(0,c2)<M1 on <0,/> (19) is true since on both intervals of monotonicity of y(0, c2) the function v(0, c2) = dv = / 2 ( 0 5 c2)/2 satisfies the d. equation -̂ — = abv + (ay + c2). When e.g. c2 < 0 is sufficiently great in absolute value, then — Y < 0 a n ^ hence v(0, c2) is smaller dy than v[y(0, c2) (0)] or v[y(0, c2) (/)] which are bounded. From (18) and (19) it follows that for any K < 0 there is a'c2 < 0 such that y"(0, c2) < K on <0, />. This gives a contradiction with (17). Lemma 4. When y"(cx, c2) ^ 0 in <0, />, then y(ci,c2)(0)ikjy(c,,c2)(\)/(i + ^ , y(c,, c2) (/) ?£ j - y(Cl, c2) (o)/(i + l)j + A (i + j ) '+*B*]/[(i+'+*X,+*)]-If y"(cv, c2)^0 in <0, />, then v ( c 1 ^ 2 ) ( 0 ) ^ - | X c 1 , c 2 ) ( Z ) / ( z + ^ 149 X and X c 1 ) c 2 ) ( O è | Я c 1 , c 2 ) ( 0 ) / ( / + | ) . Proof. The corresponding homogeneous BVP to (5), (6) has only the trivial solution, therefore there exists its Green's function G -= G(x, t). A direct calculation yields that G(0, t) = jG(l, t)/(l + j \ and G(l, t) = -|- G(0, t)/(l + - U for 0 = t = /. From these two inequalities the lemma follows. 5. By means of the foregoing lemmas the BVP (1), (2) in the first case is solved in Theorem 4. Let Fab — BD = 0, ft = S, Fa — BC = — . Then for each number y0 there exists a unique solution yyo of the BVP (I), (2) such that yyo(0) = y0. The solutions yyo possess the following properties: When y0 < yl9 then yyo < yyi on <0, /> and yyo continuously depend on y0. Further there is a unique c0, 0 < c0 < A, such that for y0 > c0 or for y0 < 0 there exists a number d0 = d0(yo), 0 < d0 < I, for which the following is true: When y0 > c0, then yyo(x) > 0 for 0 ^ x < d0 and yyo(x) < 0 for d0 < x = I. I/*yo < 0, then y'yo(x) < Ofor 0 ^ x < d0 as well as yyo(x) > Ofor d0 < x ^ /. When 0 ^ y0 = c0, then y'yo(x) _• 0 0n <0, /> and there is at most one zero point °fy'yo-Proof. The first part of the theorem, concerning the existence of yyo, its monotony and continuity property follows from Lemmas 1, 2 and 3. By (14), 0 < y(0, 0) (0) ^ = y(0, 0) (x) ^ y(0, 0) (I) < A, 0 < yf(0, 0) (x), 0 < y"(0, 0) (x) on <0, />. Hence yy0 > yo < 0, behaves according to the statement of the theorem. Further there is a c20 > 0 such that y0 = y(0, c20) and y"(0, c20) > 0, y'(0 c20) = 0 on <0, /> with the only zero-point at 0. By Lemma 3, there is a c21 < 0 such thaty(0, c21) (/) = = A, and by the properties of y(0, c2) it follows that y"(0, c21) < 0, y'(0, c2l) = 0 on <0, />, the only zero-point being at /. When c0 = y(0, c21) is put, yyo, y0> c0, shows the properties mentioned in the theorem. In the ease c21 < c2 < c20 y(0, c2) has no local extrema and thus, it is increasing in <0, />. In the case 2 (3) each solution to (5'), (7) ((5'), (8)) also satisfies the BVP (5'), (6) and the solutions y(0, c21), y(0, c20) are the unique solutions of the mentioned problems, so we have the following Corollary. Let (4) be satisfied, p ^ S, Fa - BC = j (Fa - BC = y V Then there exists one and only one solution y of the BVP (1), (2). y satisfies the inequalities y^0,y'^0, y" = 0 (y = 0, y = 0, y = 0) on <0, />, the only zero-point of y' being at I (at 0). 150 Consider now the last case. By Lemma 3, for any cx there is a unique c2 = c2(ci) such that ^••^'•BW-rr^rBg- <*» y(ci» ^2(̂ 1)) is continuous on K x <0, /> and it satisfies the d. equation (5) with ci = ^2(ci)> and the first half of the conditions (9). The second half will be satisfied (for a cx) iff c! is a root of the equation S(cO = F(ci), where S(cx) = y(ct, c2(c0) (/) ( - oo < cx < + oo) (21) and T^ = A~ T^W^c) (— < ^ < + -)• (--> The function S is continuous. Its further properties are given by Lemma 5. When 1 - p(Fa - BC) < 0, S is decreasing. If 1 - j3(Fa - BC) > 0 and l < ^L (23) = 1 - P(Fa - BC) ' K } then S is increasing. Proof. Suppose cx > c\. The difference u = y(ct, c2(c,)) — y(c\, c2(c\)) satisfies u" - p(x) u' - au = H(x) (24) H<°>=T=m^te>-c'^ u'^=T^whsc)^-'•>• (25) X-u'(l) + u(l) = 0 where p(x) = ^- [y'(cl9 c1(c1)) (x) + / ( c i , c2(c[)) (x)], H(x) = (ct - c\) x + + c2(ct) - c2(cl), (x e <0, / » . When 1 - P(Fa - BC) < 0, by (25), u must attain a negative local minimum at £, 0 < £ < /, and by (24) it has no local maximum in (£, />. Thus u(/) < 0. If 1 — P(Fa — BC) > 0, then there is a positive local maximum of u at a point X e (0, /). Since au(X) + H(X) = 0, by (23), we get H(l) = 0 and thus H = 0 in <0, />. If w(ju) = 0, X < p < /, then there exists a non-positive local minimum of u at v, p < v < I. But au(v) + H(v) < 0, what gives a contradiction. When u(l) = 0, and hence u'(l) = 0, the comparing u with the solution v of the BVP v" - p(x) v' - av = 0 (x e <0, / » (26) 1,(0) = u(0), v(l) = u(l) 151 gives that u _ v in <0, /> and hence v'(l) > 0, from where the existence of two zeros of v follows. This contradicts the disconjugacy of (26). The proof is complete. From the last lemma the next theorem follows immediately. Theorem 5. Let Fab - BD = 0 and let either a) 1 - P(Fa - BC) > 0, 1 -- S(Fa - BC) > 0, / = aaF/[l - p(Fa - BC)] or b) 1 - P(Fa - BC) < 0, 1 -— 8(Fa — BC) < 0. Then there exists a unique solution y of the BVP (1), (2). In the case a) the corresponding associated function z to that solution is positive, while in the case b) z is either positive or its only local extremum is a local minimum. When 1 is sufficiently small, this minimum is positive. Proof. In the proof of Theorem 4 the inequalities 0 < 5(0) < A were shown. Since S and Tare continuous, F(0) = A, Lemma 5 implies the first part of the theorem. The associated function z = (Fa — BC) y' + Fcx of the established solutions sat­isfies the BVP n abFcx , ab , n^ ,._. 2 ""-T^rt2+T^Bc"-BC"- ( 2 7 > (28) imply that z(0) > 0, z(l) > 0. In the case a) from (27) it follows that z cannot have a non-positive local minimum and hence, z > 0 in <0, />. In the case b) z may have a local minimum. Carrying on some considerations we get that this minimum is positive, when / is sufficiently small. Theorem 6. Let Fab - BD = 0,1 - p(Fa - BC) < 0, 1 - d(Fa - BC) > 0. Let, further y 1 - d(Fa - BC) or -{^VjT-ßiła-BC) (29) I < _ 1 _ a (30) 1 - d(Fa -BC) SI- P(Fa - BC) y ' K } + T Then there exists at least one solution y of the BVP (1), (2). In the former case (the latter case) the associated function z shows the properties: z(0) < 0, z(l) > 0, z'(x) ^ 0 for all x in <0, />(z(0) > 0, z(l) < 0, z'(x) ^ 0 in <0, /». Proof. When c± > 0, the investigation of y(cl9 c2(c$) as well as of y'(cl9 c2(ci)) gives that y"(c1, c2(c1)) ^ 0 in <0, /> and hence, Lemma 4 can be applied. Thus, when (29) is true, there is a c± > 0 such that S(ci) > T(cx) what establishes the exist­ence of a solution to the BVP (1), (2). (28) guarantee the inequalities z(0) < 0, z(l) > 0 and z' = (Fa - BC) y" ^ 0. If ct < 0 is sufficiently great in absolute value, 152 one can prove that y"(c1, c2(c1)) = 0 in <0, /> and again, with a help of Lemma 4, on the basis of (30), the result can be proved. When 1 - p(Fa - BC) > 0, 1 - S(Fa - BC) < 0, then the situation is a little more complicated. Still the methods from the last proof are efficient to prove Theorem 7. Let Fab - BD = 0, 1 - P(Fa - BC) > 0, 1 - <5(Fa - BC) < 0. Let, further, y ( . Jfi\ a 1 - S(Fa -BC)< \ a / 1 - P(Fa - BC) ' or > 1 - S(Fa - BC) > max{-|a/[(/ + | ) [ 1 - KFa - BC)]], ~} % _ fj(/a _ BC)}• Then there exists at least one solution of the BVP (1), (2). In the former (the latter) case the associated function z to the mentioned solution shows the properties z(0) < 0, z(l) > 0 and either z' ^ 0 on <0, /> or there is a subinterval <0, /x> such that z' = 0 on <0, /x> and z' = 0 in </x, /> (z(0) > 0, z(l) < 0, and either z' = 0 on <0, /> or ^hcre is a subinterval <0, /x> such that z' ^ 0 I/J <0, /x> and z' _ 0 in </x, />). REFERENCES [1] R. CONTI: Recent trends in the theory of boundary value problems for ordinary differential equations. Boll. U. M. I. 22 (1967), 135-178. [2] L. K. JACKSON: Subfunctions and Second Order Differential Inequalities. Advan. Math. 2 (1968), 307-363. [3] L. H. ERBE: Nonlinear Boundary Value Problems for Second Order Differential Equations. J. Diff. Eqs. 7 (1970), 459-472. [4] J. W. BEBERNES, R. WILHELMSEN: A General Boundary Value Problem Technique. J. Diff. Eqs. 8 (1970), 404-415. [5] L. JACKSON and K. SCHRADER: Subfunctions and Third Order Differential Inequalities. J. Diff. Eqs. 8 (1970), 180-194. [6] L. JACKSON and K. SCHRADER: Existence and Uniqueness of Solutions of Boundary Value Problems for Third Order Differential Equations. J. Diff. Eqs. 9 (1971), 46-54 . [7] V. SEDA: On a nonlinear boundary value problem. Acta F. R. N. U. C (to appear). [8] G. SANSONE: Equazioni differenziali nel campo reale, Parte Seconda (Russian Translation). Moscow: Izdat. Inostr. Lit. 1954. Author's address: Valter Seda, Department of Mathematical Analysis, Comenius University, Mlynskd dolina, Bratislava 16, Czechoslovakia. 153 

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On a nonlinear boundary value problem of higher order

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On a nonlinear boundary value problem of higher order

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

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