Iteratioal Mathematical Forum 2 2007 o. 66 3241-3267 A Aalysis of a Certai Liear First Order Partial Differetial Equatio + f ( x y) = 0 z x by Meas of Topology z y T. Oepomo Sciece Egieerig ad Mathematics Divisio LA Ageles Harbor & West LA College 945 South Idaho Uits: 134 La Habra Ca 90631 USA Oepomots@lahc.edu Oepomot@wlac.edu or oepomotj@lattc.edu Abstract We shall cosider oly real fuctios of real variables. By a solutio of z + x z f ( x y) = 0 y i a ope regio R we shall mea a fuctio I(xy) which is of class C ad satisfy the equatio i R. By a solutio of the partial differetial equatio i a closed regio R we shall mea a fuctio I(xy) which at each poit of R is defied ad cotiuous ad at each iterior poit of R is of class C ad satisfies the subject
3242 T. Oepomo partial differetial equatio. It is the ited of this paper to emphasize why it was ecessary i Kamke s theorem to specify that the fiite limit poits of the regio g lie i G. Mathematics Subject Classificatios: 35M99; 51N005 Keywords: Partial Differetial Equatio Topology Itroductio: The liear ad first order partial differetial equatio z + x z f ( x y) = 0 y [KA] (1) was studied ad proved by Kamke [KA] i his article i the Aale of 1928. We shall re-cosider re-write ad refie his aalysis i more precise method. 1. A review of Kamke s proof. I this sectio we shall review Kamke s proof [KA] oce more i more detail. 1.1 Kamke s Proof: The author has proved the followig: Theorem: If f(xy) ad f y (xy) are defied ad are cotiuous i a ope regio G ad if g is a ope simply coected regio lyig etirely i * G i which f(xy) is bouded; the there exists a fuctio Ψ ( x y) such that i g: a) Ψ ( x y) is defied ad is of class C with respect to x ad y; b) Ψ ( x y) is costat alog each solutio curve ** of y = f(xy) (2)
Aalysis of PDE by meas of topology 3243 c) Ψy ( x y) > 0 (3) * A ope regio g is said to lie etirely i a ope regio G if all of the poits of g ad also all of the fiite limit poits of g are poits of G ** A solutio of curve (2) shall be called a characteristic of (1) d) Ψ ( x y) satisfies equatio (1) For this paper it is ecessary to review Kamke s method of proof i detail. The below metioed paragraph follows a codesed quotatio of parts of the article: Topological Part of the Proof: 1. Here we use specifically oly that through each poit of g there passes exactly oe characteristic of (1) that these characteristics deped cotiuously o the iitial poit ad that i both directios of the x-axis they either have ubouded ordiates or approach arbitrarily close to the boudary of g. 2. Deote by s a bouded vertical ope segmet i g ad by o(s) the set of poits i g which belog to characteristics of (1) lyig i g ad passig through s. Each o(s) is a ope regio (i fact simply coected eve if g were ot simply coected). 3. If P ad Q are two poits of o(s) o a vertical segmet ad if the whole segmet PQ is i g the PQ is cotaied i o(s). 4. Amog the set of ope regios o(s) there exists a coutable subset o 1 o 2. With the properties: α ) o 1 + o 2 + ------ = g;
3244 T. Oepomo β ) o o is cotaied etirely i g -1 o 1 + o 2 + ------ + o ; γ ) each o has at least oe poit i commo with g -1. 5. Each of the regios g = o 1 + o 2 + -------- +o is a ope regio. 6. If P ad Q are o a vertical segmet ad are both i g ad if PQ is cotaied i g the PQ is cotaied i g. 7. For the costructio of Ψ ( x y) it is expediet to alter the set o somewhat: there exists a set of regios q 1 q 2 with the properties: α ) each q q(t ) is the set of poits i g which lie o characteristics of (1) lyig i g ad passig through a ope fiite vertical segmet t i g; β ) each h q 1 + q 2 + ---------+q is a ope regio; γ ) q 1 + q 2 + ---------= g; δ ) the commo poits of t ad h -1 form exactly oe ope segmet ad t projects out of h -1 i exactly oe directio. Coclusio of the Proof: 1. Deote by x υ the abscissa of υ t ad by y = ϑ ( ξ of (1) passig through the poit ( ξ of G. The ( ξ possesses for h 1 rather tha g (ad with a) b) c ad d) of Kamke s theorem. x the characteristics Ψ ϑ ( x ξ ξ η istead of x y) the properties 2. To prove the existece of such a fuctio Ψ ( ξ for the whole regio g assume it has already bee costructed for h -1 ad that for each υ < 1
Aalysis of PDE by meas of topology 3245 there are two real costats aυ bυ >0 such that i hυ -hυ -1 Ψ( ξ aυ + bυ ϑ( x ν ξ. It must ow be show that there is a real umber pair a b > 0 such that the fuctio defied by Ψ( ξ aυ + bυ ϑ( ν ξ x i hυ - hυ -1 for υ = 1 2 3 has the properties of a) b) c) ad d) of Kamka s Theorem i the regio h. 3. Just oe eds of t say the upper projects out of h -1. Let V W be the coordiate of the ed poits of this half-ope segmet of t. There exists a smallest (1 <) such that for some v < V the ope segmet (v V ) of t belogs to q ad also to h -h 1. The characteristics of (1) through the ope iterval (v V ) therefore ru through t ad cut out o this fiite segmet a ope iterval (u U -1). U is either a iterior or a boudary poit of g i either case a iterior poit of G. Through this poit therefore there passes a characteristic of (1)
3246 T. Oepomo W t the curve (4) t U V u v Figure 1 ( x x U ) y = ϑ (4) which is defied for x i a certai eighborhood of x. 4. But the curve (4) is defied at least for x x x. Assume for simplicity that x < x η. If (4) were ot defied i the etire closed w iterval x x the (4) would as far as it were defied i this iterval either ot be bouded or would be bouded but would come arbitrarily close to the boudary of G. That either of these coditios ca obtai we see from the cotiuous depedece of characteristics o the iitial poit.
Aalysis of PDE by meas of topology 3247 5. Exted the half-ope iterval ( u U) upward to form a ope segmet s that belogs etirely to G. The fuctio ϑ( ξ x is defied i o(s) ad has there cotiuous partial derivatives with respect to ξ ad η. Sice the curve (4) is i o(s) these partial derivatives exist i particular i the poits of the curve (4). Therefore we ca defie: ( x x V ) b b (5). ϑ η a a + bu bv (6) ( ξ ( ξ a + b ϑ( x ξ Ψ i h 1 Ψ is thus well defied i h ad i h 1 which b > 0. From (7) ( ) ξη h (7) h is a liear fuctio of ϑ( η ξ x i Ψ has the same value for all poits ( ξ of the same characteristic of (1). The proof of the theorem will be complete whe it is show the fuctio Ψ ( ξ is of class C ad satisfies (1) i h istead of g. We will be cocered oly with the boudary poits of h -1 belogig to h h 1. These poits are obtaied from curve (4). Use the fuctios ( x ξ a + i o(s) (8) b ϑ ( x ξ a + bϑ i h (9) Each is defied i a regio which cotais the poits of the curve (4) i questio ad each possesses i its regio of defiitio cotiuous partial derivatives with respect to
3248 T. Oepomo ξ ad η. I the poits of the curve (4) i questio the fuctios (8) ad (9) have the value a + b U ad a + bv respectively. From (6) these values coicide. It is ow easily show that Ψ ( ξ ) ad ( ξ ξ Ψ exists ad are cotiuous ad that Ψ ( x.y) satisfies (1) i the poits of curve (4) i questio. We are ow i a positio to see the value to Kamke s method of proof of requirig that the fiite limit poits of g be cotaied i the ope regio G. Kamke makes use of this coditio i his paragraph 3.3 to show that the curve (4) is defied at least for x i a eighborhood of x. Agai i his paragraph 3.4 to prove η that the curve (4) is defied for x x x ; agai i his paragraph 3.5 to imbed the characteristic (4) i a ope regio o(s) made up of characteristics through a ope segmet s belogig etirely to G. This is used to show the existece ad cotiuity of the partial derivatives ϑ ( x ξ ad ϑ ( ξ ξ x i o(s) ad therefore i particular i η the poits of the curve (4). Kamke is thus eabled to defie b a ad ( ξ ψ i h h 1 ad thus to complete his iductio. If this existece of the ope regio G were ot postulated i Kamke s theorem ad if o further hypotheses were made cocerig the fuctio f(xy) ad the regio g the the method used by Kamke to prove his theorem could ot be carried out for uder these coditios the existece ad cotiuity of the limits of the partial derivatives ϑ ( x ξ ad ϑ ( ξ ξ Bedixso x alog the curve (4) are ot certai. By a theorem of η
Aalysis of PDE by meas of topology 3249 ϑ ( ξ x = η x f y ξ e [ x ϑ ( x ξ η )] dx i q( t ) (10) hece without further hypotheses o f(xy) ad o g we caot assert the existece ad cotiuity of the limits of the partial derivatives ϑ ( x ξ ad ϑ ( ξ ξ x alog the η curve (4). T. Wazewski [WA] has costructed a example based o these cosideratios to show that without the eclosig regio G the hypotheses of Kamke s theorem are ot sufficiet to isure the existece of a solutio (1) havig the properties described. 1.2 The required ifiite sequece of regio. The ifiite sequece of the regio q( t ) is eeded i Kamke s method of proof. We see at oce that if the regio g is ubouded there may be eed for a ifiite umber of regios q( t ) sice each of the ope segmet t is fiite. Eve if g is bouded a fiite umber of regios q( t ) may ot be sufficiet for Kamke s method of proof. This is show by the followig example: Assume that g is bouded ad that C is a characteristic of (1) which has oly two poits i commo with the boudary of g. C divides g ito two sub-regios oe of which is above C ad the other below C. Cosider the ope simply coected regio g eclosed by a simple closed curve which above C coicides with the boudary of g but below C lies i the lower subregio of g ad has a ifiite umber of loops each of which is taget to C.
3250 T. Oepomo C Figure 2 Boudary of G boudary of g BodaryBoudary of g Clearly usig Kamke s method with a ifiite sequece of regios q( t ) we ca defie a solutio of (1) havig the properties describes i Kamke s Theorem clear to the boudary of g. But we caot usig his method defie such a solutio of (1) i g by meas oly a fiite umber of regios q( t ). The segmets of characteristics of (1) lyig i the sub-regios of g bouded by C ad the part of the boudary of g below C caot be trasverse by less tha a ifiite umber of vertical segmets t. 1.3 The geerality of Kamke s method of proof
Aalysis of PDE by meas of topology 3251 It has bee show that eve if the regio g is bouded a fiite umber of regio o(s) is ot sufficiet to isure the geerality of Kamke s method of proof. However if the regio g is bouded ad a differet kid of regio o(s) is used it ca be show that a fiite umber of such regios are sufficiet to costruct a solutio of (1) havig the properties described i Kamke s Theorem. Let us assume the hypotheses of Kamke s Theorem ad assume that the regio g is bouded. There exists a bouded ope simply coected regio R such that g lies etirely i R ad R lies etirely i G. Deote by s a bouded vertical segmet i R ad by o(s) the set of poits i R which belog to characteristics of (1) lyig i R ad passig through s. Each of the sets o(s) is a ope simply coected regio. Each of poit of the closure of g is cotaied i at least oe of the regios o(s). Hece by the Borel Coverig Theorem there exists a fiite subset of this set of ope regios o o o. o 1 2 3... m such that every poits of the closure of g is cotaied i at least oe of the regio o 1 o2 o3.... o m. Clearly there is a orderig of this fiite set such that each g o 1 + o2 + o3 +... + o is a ope regio. By splittig of some of the regios with the cosequet fiite icrease i the umber of regios it ca be brought about that the regios g ad the segmets s are such that the commo poits of s ad g -1 form exactly oe ope segmet ad that s projects out of g -1 i exactly oe directio. Hece we may state Lemma 1: There exists a fiite set of ope regios q... 1 q2 qr with the properties as follows:
3252 T. Oepomo α ). each q q1 + q2 +... + q (=12..r) is the set of poits i R which lie o characteristics of (1) lyig i R ad passig through a ope fiite vertical segmet t i R; β ). each h q1 + q2 +... + q (=123.r) is a ope regio; γ ). g lies etirely i the regio h q + q +... q ; r 1 2 r δ ). the commo poits of t ad h -1 (=2.r) form exactly oe ope segmet ad t projects out of h -1 i exactly oe directio. Thus i a fiite umber of steps we ca defie a fuctio ( x y) Ψ possessig the properties described i Kamke s Theorem i the regio h r ad therefore i the regio g. The fuctio ( x y) of g. Ψ is evidetly defied ad cotiuous i every poit of the closure
Aalysis of PDE by meas of topology 3253 Boudary of G Boudary of G Boudary of R Figure: 3 Boudary of g 2. A Variatio of Kamke s method of Proof 2.1 A Review of Kamke s method of Proof Lemma 2: Assume that f(x) is a fuctio such that a) f(x) is defied ad is of class C over rage α<x<a b<x<β a<b; b) f(x) ad f (x) have defiite fiite left-had limits at x = a ad defiite righthad limits at x = b. Deote these limits by f(a) f (a) f(b) f (b) respectively; c) f(a) < f(b) f (a) >0 ad f (b) >0. The there exists a fuctio F(x) such that: a) F(x) is defied ad is of class C i α<x<β;
3254 T. Oepomo b) F (x)>0 i a x b; c) F(x) f(x) i α<x<a b<x<β. As depicted i Figure 4 costruct a rectagle amed MHNK ad divide it ito 4 quarters umbered I II III IV. Exted a lie segmet with slope equal to f (a) from M to a poit A iside III ad exted a lie segmet with slope equal to f (b) from N to a poit B i side I. Choose a poit P i the iterior of MA ad a poit Q iside III ad i the iterior of AB such that AP = AQ. Costruct the circle 0 taget to MA at P ad to AB at Q. Similarly choose poits S ad R i I ad costruct the circle0 taget to AB at R ad to BN at S. Y=f(x) Y=f(x) Figure 4 α a b β
Aalysis of PDE by meas of topology 3255 K N [bf(b)] II B I S R O O III P A Q IV M[af(a)] Figure 5 H Y=f(x) The curve composed of the segmet MP the arc PQ of the circle O the segmet QR the arc RS of the O ad the segmet SN is cotiuous ad has a cotiuously turig taget. The slope of this curve is everywhere positive ad the limits of the slope at M ad N coicide with those of the slope of the curve y = f(x). For a x b defie F(x) to be the ordiate of the poit o the above-described curve whose abscissa is x. For α < x < b b < x < β defie F(x) f(x). The fuctio F(x) thus defied i α < x < β possesses the properties a) b) ad c) as described i the coclusio of Lemma 2. 2.2 Proof to Kamke s Theorem by Iductio The theorem of Kamke quoted i paragraph 1 ca be proved usig the sequece of ope regios q(t ) but a differet method of fittig.
3256 T. Oepomo Assume that a fuctio Ψ ( ξ as described i the theorem has already bee costructed for h -1 ad that for υ < it is i each q(tυ ) a fuctio of ( ξ ϑ x υ of class C amely w [ ϑ( x ξ ] where (y) is of class C o υ w υ ' t ad ( y) υ w υ has defiite fiite positive limits at the edpoits of t υ. We shall use paragraphs 3.3 ad 3.4 of Kamke s article that is summarized o pages 3 through 7 of this paper. Exted the half-ope iterval ( u U ) (see Figure 1) upward to form a ope segmet s that belogs etirely to G. The fuctio ϑ( x ξ is defied i o(s) ad has there cotiuous partial derivatives with respect to ξ ad η. Deote the ordiates of the upper ad lower edpoits of s by s 2 ad s 1 respectively. By our assumptio w ( y) has a defiitio fiite limit at y = U ad w ' ( y) has a defiite fiite positive limit at y = U. Hece usig a modificatio of Lemma 2 we ca defie a fuctio w(y) such that a) w(y) is defied ad is cotiuous o s 1 y s 2 ad is of class C o s 1 < y < s 2 ; b) w (y) > 0 o s 1 < y < s 2 ad w (y) has defiite fiite positive limits at y=s 1 ad y=s 2 ; c) w(y) w y) Ψ( x y) s y < U ( o 1. Defie a fuctio ( ξ λ ( ξ w[ ϑ( x ξ ] λ i o(s) as follows: (11)
Aalysis of PDE by meas of topology 3257 The fuctio ( ξ a) λ ( ξ Ψ( ξ b) ( ξ 0 λ is of class C i o(s) ad also i the part of o(s) which is i h -1 ; λ η > i o(s). From this follows that ( ξ Ψ Ψ ( ξ ad ( ξ ξ Ψ have defiite fiite η cotiuous limits o the curve (4) (see page 5) ad that the limit ( ξ the curve (4) is positive. Ψ o η The iductio will complete whe we have defied a fuctio w (y) such that a) w (y) is of class C o t ; ' b) w ( y) > 0 o t ad it has defiite fiite positive limits at the edpoits of t ; c) The fuctio Ψ ( ξ defied for = q by υ 1 2. i each ( ) w [ ϑ( x ξ ] possesses the properties described i the coclusio of υ υ Kamke s Theorem for h istead of g. For this usig the fact that as ( ) ξη Ψ ( ξ ad ( ξ ξ Ψ is already defied i h -1 Ψ ( ξ Ψ all posses defiite fiite cotiuous limits the last η positive o the curve (4) ad usig Lemma 2 defie a fuctio w (y) such that a) w (y) is of class C o w < y < W ; t υ
3258 T. Oepomo ' ' b) w ( y) > 0 o w <y<w ad w ( y) has defiite fiite positive limits y = w ad y = W [If M ad ' M are ay real umbers such that M > w (V ) ad ' M > 0 we ca specify that lim y w η w ( y) = M ad that ' ' lim w ( y) = M ]; y w η c) w (y) ( x y) ψ o w < y < V. Cosider the fuctio ( ξ Ψ defied i q(t ) η η ( ξ w [ ϑ( x ξ ] This fuctio is of class C i q(t ) ad also ' a) Ψ ( ξ ) = ϑ( x ξ > 0 η Ψ (12) w i q(t ); b) i that part of q(t ) which is cotaied i h -1 Ψ ( ξ [ ϑ( x ξ ] Ψ( ξ c) i q(t ) Ψ ξ w Ψ[ x ϑ( x ξ ] (13) ' ( ξ + ( ξ Ψ ( ξ = w ϑ ( x ξ + f ( ξ ϑ ( x ξ ) so that ( ξ η { } = 0 f (14) Ψ is a solutio of (1) i q(t ). ξ Thus fuctio Ψ ( ξ defied for υ = 1 2 3 i each ( ) w [ ϑ( x ξ ] possesses i h istead of g (ad with ( υ υ q by t υ ξ istead of (xy)) the properties described i the coclusio of Kamke s Theorem ad the iductio is complete. 2.3 Upper ad Lower Boud of a fuctio as described i Kamke s Theorem
Aalysis of PDE by meas of topology 3259 We have see that if the upper ed of t projects out of h -1 so that W lies above V (see Figure 1) ad if M is ay real umber such that M > w (V ) we ca defie the fuctio w (y) o the whole of t so that ( y) w y w lim = M. Similarly if the lower ed of t projects out of h -1 so that W lies below V ad if P is ay real umber such that P < w (V ) we ca defie the fuctio w (y) o the whole of t so that lim ( y) = P. It is clear that we ca choose w y w the umbers M h ad P k so that these coditios are satisfied ad so that p < M h < M for every M h ad P < Pk < M for every P k where M ad P are fiite real umbers. If this is doe the i g ( x y) M P < Ψ < (15) Further if a> 0 ad b are arbitrary real umbers ad if Ψ ( xu) is a solutio of (1) havig the properties described i Kamke s Theorem the ( x y) aψ( x y) + b Ψ (16) is also a solutio of (1) havig these properties. Hece we ca make the followig Remark: If L ad U are arbitrary fiite real umbers such that L<U the the fuctio ( x y) Ψ accordig to Kamke s Theorem ca be costructed by the method as described i the preset paragraph so that i g we will have L< Ψ ( x y) <U.
3260 T. Oepomo 3. The Value of f(xy) ad Its First Derivative Cotiuous Limits O The Boudary 3.1 The Cotiuous Limits of f(xy) ad f y (xy) O the Boudary of Regio g Let us assume that g is a bouded ope simply coected regio ad that f(xy) ad f y (xy) have defiite cotiuous limits o the boudary of g. The through each poit of g there passes exactly oe characteristic of (1) which lies i g. These characteristics deped cotiuously o the iitial poit ad i both directios of the x-axis they approach arbitrary close to the boudary of g. Deote by s a vertical ope segmet i g ad by o(s) the set of poits i g that belog to characteristics of (1) lyig i g ad passig through s. I a maer exactly similar to that Kamke it ca be show that these poits sets o(s) posses i g the properties described with referece to the regio g i paragraph 2 of Kamke s article (see pages 3 through 4 of this paper). I particular Kamke s paragraph 2.7 (see pages 3 through 4) is valid uder the preset coditios.
Aalysis of PDE by meas of topology 3261 g Figure 6 O(s) S Assume as i paragraph 2.2 that a fuctio Ψ ( ξ with the properties described i Kamke s Theorem has already bee costructed for h -1 ad that for υ < it is i each q ( t υ ) a fuctio ϑ( x υ ξ of class C amely υ w [ ϑ( x υ ξ ] where ( y) w υ is of ' ' class C o t υ w υ ( y) > 0 o t υ ad ( y) has defiite fiite positive limits at the edpoits of t υ. w υ The segmet t projects out of h -1 i just oe directio assume for defiiteess the upper. Let v ; w deote the ordiates of the edpoits of the half-ope segmet of t projectig outside h -1. The lower segmet of t belogs clear to its lower edpoit w to h -1. Hece there exists a smallest ( 1 < ) such that for some v v < V the
3262 T. Oepomo ope segmet ( v V ) of t belogs to qu ad also to h h 1. The characteristics of (1) through the ope iteral (v V ) of t belog to q ad also to h h 1. The characteristics of (1) through ope iterval (v V ) therefore ru through t ad cut out o this segmet a ope iterval ( u U ). U is ether a iterior or a boudary poit of g. The characteristics of (1) y = ϑ( x x (17) through the iterval u U ) are the class C their slopes are bouded ad they satisfy ( (2). Hece η varies mootoically from u to uiformly to a curve y Φ( x y) U the characteristics (17) coverge = (18) which is cotiuous ad sigle-valued. The curve (18) exteds at least from x = x to x = x. Every poit of the curve (18) lies either i g or o the boudary of g. Further sice f (x y) has a defiite fiite cotiuous limit o the boudary of g ad sice the characteristics (17) satisfy (2) the curve (18) is itself a solutio curve of (2) that is to say a characteristic of (1). It will be show that ϑ( x ξ ϑξ ( x ξ ad ϑη ( x ξ have defiite fiite cotiuous limits the last positive o the curve (18). It is evidet that the limit of ϑ( x ξ o the curve (18) is U.
Aalysis of PDE by meas of topology 3263 The fuctio ϑ( x ξ has q( t ) for its exact regio defiitio is costat for all poits of oe ad the same characteristics of (1) ad (by a theorem of Bedixso) has cotiuous partial derivatives with respect to ξ ad η. Further ϑη ( x ξ = x f y [ x ξ e ϕ ( x ξ ] dx >0 ad (19) ϑξ ( x ξ +f( ξ η ) ϑη ( x ξ 0 i q( t ) (20) Let the poit ( ξ approach a poit P(xy) o the curve (18) from withi h -1. ( ξ ϑ x is of class C with respect to ξ ad η ad it is evidet that as (ξ approach (xy) the fuctio ϑ( ξ x approaches a limit Φ (x) which is at least cotiues. The fuctio f y (xy) has a defiite fiite cotiuous limit o the curve (18). The covergece to these limits is uiform for x x x. As the poit (ξ η approaches (xy) we see from (19) that ϑη ( x ξ approaches a defiite fiite cotiuous limit. This limit is positive sice the itegrad i the expoet i (19) is bouded. It is clear from (20) that ϑ ( ξ x has e defiite fiite cotiuous limit o ξ the curve (18). From our assumptios the fuctio Ψ ( ξ has already bee costructed for h -1 ad [ ] Ψ ( ξ w ϑ( x ξ i q( t ) (21)
3264 T. Oepomo It is clear from (21) that Ψ ( ξ has a defiite fiite cotiuous limit alog the curve (18). As the poit (ξ approaches (xy) o the curve (18) from withi q( t ) the fuctio ( ξ Ψ approaches η hece ( ξ η { lim w '( y) y U } { lim ϑ ( x η ξ ) }; (22) ( 18) othecurve Ψ has a defiite fiite cotiuous limit o the curve (18). This limit is positive sice each limit i (22) is positive. Similarly we see that ( ξ defiite fiite cotiuous limit alog curve (18). Ψ has a We are ow i a positio to complete the itroductio exactly as i Paragraph 2.2. Sice also 2.3 ca be applied to the preset case we may state Theorem 1: Assume that g is a bouded ope simply coected plae regio ad f(xy) is a fuctio such that a) f(xy) ad f y (xy) are defied ad are cotiuous i g; ξ b) f(xy) ad f y (xy) have defiite fiite cotiuous limits o the boudary of g. Assume also that L ad U are fiite real members such that L<U. The there exists a fuctio ( x y) a) ( x y) Ψ such that i g: Ψ is defied ad is of class C with respect to x ad y; b) ( x y) Ψ is costat alog each characteristics of (1);
Aalysis of PDE by meas of topology 3265 Boudary of g W η t t η Figure 7 U The curve (18) V η v η u w η c) ( x y) Ψ >0; (3) y d) ( x y) Ψ satisfies (1); e) L< Ψ ( x y) <U. 3.2 Review of Wazewski s Proof I referece [WA] pages 103-116 it has bee proved by Wazewski the followig Theorem: There exists a ope simply coected regio G ad a fuctio f(xy) of class costat. ( ) C i G such that every solutio of (1) which is defied i the whole of G is From this theorem it is clear that if i Theorem 1 the coditio that f(xy) ad f y (xy) posses defiite fiite cotiuous limits o the boudary of g is omitted the remaiig
3266 T. Oepomo hypothesis do ot ecessarily imply the existece of a solutio Ψ( x y) that Ψ ( x y) > 0 y i g. of (1) i g such However the coditio that f(xy) ad f y (xy) posses defiite fiite cotiuous limits o the boudary of g is ot ecessary i order that there exist a solutio of Ψ( x y) g such that Ψ ( x y) > 0 y i g. Cosider the partial differetial equatio of (1) i z 1 z + x y y = 0 (23) Let g be the ope simply coected regio which lies i the first quadrat ad is eclosed by the x-axis the parabola 2 The fuctio Ψ( y) y 2x + 4 y 2 = 2x ad the ordiate x=2. x (24) is defied i g is of class C with respect to x ad y i g ad also Ψ x ( x y) 1 Ψ( x y) + y y = 1 Ψ 1 + y x ( x y) y Ψ( y) = 0 i g (25) so that ( x y) Ψ is a solutio of (23) i g. Further Ψ ( x y) y y = > 0 i g (26) Ψ ( x y) Equatio (23) is of the form (1) with f(xy) = y 1. The fuctios f(xy) = y 1 ad f y 1 = are defied ad are cotiuous i g but either possesses a defiite fiite 2 y cotiuous limit o the etire boudary of g.
Aalysis of PDE by meas of topology 3267 Y y 2 =2x Figure 8 2 1 g x=2 0 1 2 X Refereces [KA] E. Kamke Zur Theorie der Differetialgleichuge. Mathematische Aale. 99 (1928) 602-615. [WA] T. Wazewski Sur u Probleme de Caractere Itegral Relatif a L Equatio z + x z Q(xy) = 0. Mathematica 8 (1934) 103-116. y Received: February 12 2007