EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER ITERATIVE BOUNDARY-VALUE PROBLEM

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1 Elecronic Journl of Differenil Equions, Vol. 208 (208), No. 50, pp. 6. ISSN: URL: hp://ejde.mh.xse.edu or hp://ejde.mh.un.edu EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER ITERATIVE BOUNDARY-VALUE PROBLEM ERIC R. KAUFMANN Absrc. We consider he exisence nd uniqueness of soluions o he secondorder ierive boundry-vlue problem x () = f(, x(), x [2] ()), b, where x [2] () = x(x()), wih soluions sisfying one of he boundry condiions x() =, x(b) = b or x() = b, x(b) =. The min ool employed o esblish our resuls is he Schuder fixed poin heorem.. Inroducion The sudy of ierive differenil equions cn be rced bck o ppers by Peuhov [9] nd Eder [4]. In 965 Peuhov [9] considered he exisence of soluions o he funcionl differenil equion x = λx(x()) under he condiion h x() mps he inervl [ T, T ] ino iself nd h x(0) = x(t ) = α. He obined condiions on λ nd α for he exisence nd uniqueness of soluions. In 984, Eder [4] sudied soluions of he firs order equion x () = x(x()). The uhor proved h every soluion eiher vnishes ideniclly or is sricly monoonic. The uhor esblished condiions for he exisence, uniqueness, nlyiciy, nd nlyic dependence of soluions on iniil d. In 990, using Schuder s fixed poin heorem Wng [0] obined soluion of x = f(x(x())), x() =, where is one endpoin of he inervl of exisence. In 993, Fečkn showed he exisence of locl soluions vi he Conrcion Mpping Principle for he iniil vlue problem for he ierive differenil equion x () = f(x(x())), x(0) = 0. For more on ierive differenil equions see he ppers [, 2] [5]-[8], []-[4] nd references herein. In his pper we consider he exisence nd uniqueness of soluions o he secondorder ierive boundry-vlue problem x () = f(, x(), x [2] ()), < < b, (.) where x [2] () = x(x()), wih soluions sisfying one of he following boundry condiions: x() =, x(b) = b; (.2) x() = b, x(b) =. (.3) 200 Mhemics Subjec Clssificion. 34B5, 34K0, 39B05. Key words nd phrses. Ierive differenil equion; Schuder fixed poin heorem; conrcion mpping principle. c 208 Texs Se Universiy. Submied Sepember Published Augus 8, 208.

2 2 E. R. KAUFMANN EJDE-208/50 We ssume hroughou h f : [, b] R R R is coninuous. Due o he ierive erm x [2] (), in order for soluions o be well-defined, we require h he imge of x be in he inervl [, b]; h is, we need x() b for ll [, b]. In Secion 2, we firs rewrie (.), (.2) s n inegrl equion nd hen se condiion under which soluions of he inegrl equion will be soluions of he boundry vlue problem. We lso se properies of he kernel h will be needed in he sequel. In Secion 3, we se nd prove heorems on he exisence nd uniqueness of soluions for he boundry vlue problems (.), (.2) nd (.), (.3). We provide n exmple o demonsre our resuls. 2. Preliminries Our gols in his secion re o conver he boundry vlue (.), (.2) o fixed poin problem nd o se heorems we will need o prove he exisence nd uniqueness. To his end, le x C 2 [, b] be soluion of x () = f(, x(), x [2] ()), < < b, x() =, x(b) = b. We begin by inegring he equion x () = f(, x(), x [2] ()) wice. x() = + x ()( ) + ( s)f(s, x(s), x [2] (s)) ds. (2.) Afer pplying he boundry condiion x(b) = b, we cn solve for x () o obin, x () = b Now subsiue his expression for x () ino (2.). x() = ( ) b (b s)f(s, x(s), x [2] (s)) ds + We cn rewrie his equion in he form x() = b b (b s)f(s, x(s), x [2] (s)) ds. ( )(b s)f(s, x(s), x [2] (x)) ds ( )(b s)f(s, x(s), x [2] (s)) ds ( s)f(s, x(s), x [2] (s)) ds. + b ( s)f(s, x(s), x[2] (s)) ds. Finlly, we combine he ls wo inegrls nd simplify he inegrnd. x() = + b + b Thus, if x C 2 [, b] is soluion of ( )(s b)f(s, x(s), x [2] (x)) ds ( b)(s )f(s, x(s), x [2] (s)) ds. x () = f(, x(), x [2] ()), < < b, x() =, x(b) = b,

3 EJDE-208/50 SECOND ORDER ITERATIVE BVP 3 hen x C[, b] mus sisfy he inegrl equion where x() = + G(, s)f(s, x(s), x [2] (s)) ds, b, (2.2) { G(, s) = ( b)(s ), s b, b ( )(s b), s b. Define he operor T : C[, b] C[, b] by (T x)() = + Noe h (T x)() = nd (T x)(b) = b. Also, G(, s)f(s, x(s), x [2] (s)) ds. nd (T x) () = + b b (s )f(s, x(s), x [2] (s)) ds (b s)f(s, x(s), x [2] (s)) ds, (T x) () = f(s, x(s), x [2] (s)). Recll h in order for he soluion of (.), (.2) o be well-defined we need x() b, for ll b. As such, if x C[, b] is fixed poin of T such h (T x)() b for ll [, b], hen x is soluion of (.), (.2). We hve he following lemm. Lemm 2.. The funcion x is soluion of (.), (.2) if nd only if (T x)() b nd x is fixed poin of T. To esblish our uniqueness resuls we will need he following resuls concerning he kernel of (2.2). The proof of his lemm is srigh forwrd nd hence omied. Lemm 2.2. The funcion sisfies G(, s) = b G(, s) G(s, s), { ( b)(s ), s b, ( )(s b), s b, s [, b] [, b], G(s, s) ds = 6 (b )2. We conclude his secion wih Schuder s fixed poin heorem [3]. Theorem 2.3 (Schuder). Le A be nonempy compc convex subse of Bnch spce nd le T : A A be coninuous. Then T hs fixed poin in A.

4 4 E. R. KAUFMANN EJDE-208/50 3. Exisence nd uniqueness of soluions We presen our min resuls in his secion. From Lemm 2. we noe h we need (T x)() b for ll [, b]. The following condiion will be used o conrol he rnge of T x. (H) There exiss consns K, L > 0 such h K f(, u, v) L for ll [, b], u, v R nd b 2 (K + L) > 0. We re now redy o se our firs resul. Theorem 3.. Suppose h condiion (H) holds. The here exiss soluion of he boundry-vlue problem (.), (.2). Proof. Consider he Bnch spce Φ = (C[, b], ) wih he norm defined by x = mx [,b] x(). Le m = mx{, b } nd le Φ m = {x Φ : x m}. Since (H) holds, (T x) () = + b b K b (s )f(s, x(s), x [2] (s)) ds (b s)f(s, x(s), x [2] (s)) ds (s ) ds L b b (K + L) > 0. 2 (b s) ds Consequenly T x is incresing. Since (T x)() = nd (T x)(b) = b, hen (T x)() b for ll [, b]. An pplicion of Schuder s heorem yields fixed poin x of T nd he proof is complee. By Lemm 2. he funcion x is soluion of (.), (.2). Using he sme echnique s in Secion 2, we cn show h he boundry-vlue problem (.), (.3) is equivlen o he inegrl equion (T 2 x)() = (b + ) + provided (T 2 x)() b. G(, s)f(s, x(s), x [2] (s)) ds Theorem 3.2. Suppose h condiion (H) holds. The here exiss soluion of he boundry-vlue problem (.), (.3). Proof. As in he proof of Theorem 3., we firs show h T 2 x is monoone. From condiion (H) we hve (T 2 x) () = + b b (s )f(s, x(s), x [2] (s)) ds (b s)f(s, x(s), x [2] (s)) ds + b (K + L) < 0. 2 The res of he proof is he sme s in Theorem 3..

5 EJDE-208/50 SECOND ORDER ITERATIVE BVP 5 Exmple 3.3. Consider he following boundry-vlue problem wih prmeer k. x () = k cos(x [2] ()) (3.) x(0) = 0, x(π) = π. (3.2) Here, f(, u, v) = k cos v. Since k k cos v k, hen b 2 (K +L) = π 2 k. By Theorem 3. here exiss soluion of (3.), (3.2) for ll vlues of k such h k < 2 π. We now consider uniqueness of soluions of (.), (.2) nd (.), (.3). To his end, we need he following condiion. (H2) There exiss M, N > 0 such h f(, u, v ) f(, u 2, v 2 ) M u u 2 + N v v 2 for ll [, b], u, u 2, v, v 2 R. Theorem 3.4. Suppose h (H) nd (H2) hold. Assume h 6 (M + N)(b )2 <. Then here exiss unique soluion of (.), (.2). Proof. Since (H) holds, hen here exiss fixed poin x of T. Suppose h x nd x 2 re wo disinc fixed poins of T. Then for ll [, b] we hve, x () x 2 () = (T x )() (T x 2 )() = G(, s) ( f(s, x (s), x [2] (s)) f(s, x 2(s), x [2] 2 (s))) ds G(, s) f(s, x (s), x [2] (s)) f(s, x 2(s), x [2] 2 (s)) ds ( G(s, s) M x (s) x 2 (s) + N x [2] 6 (M + N)(b )2 x x 2 < x x 2. ) (s) x[2] 2 (s) Thus, x x 2 < x x 2 nd we hve conrdicion. Consequenly, he fixed poin x of T is unique. By Lemm 2. x is he unique soluion of (.), (.2) nd he proof is complee. In similr mnner we cn prove he following heorem. Theorem 3.5. Suppose h (H) nd (H2) hold. Assume h 6 (M + N)(b )2 <. Then here exiss unique soluion of (.), (.3). Exmple 3.6. We gin consider he boundry-vlue problem (3.), (3.2), x () = k cos(x [2] ()) x(0) = 0, x(π) = π. By he Men Vlue Theorem we know here exiss ξ [0, π] such h k cos v k cos v 2 = k sin ξ v v 2 k v v 2.

6 6 E. R. KAUFMANN EJDE-208/50 We hve 6 (M + N)(b )2 = k π 2 /6. By Theorem 3.4 here exiss unique soluion of (3.), (3.2) for ll vlues of k such h k < 6/π 2. Noe h he resuls in his pper cn be exended o boundry-vlue problems of he form x = f (, x(), x [2] (),..., x [n] () ), x() =, x(b) = b, s well s boundry-vlue problems of he form x = f (, x(), x [2] (),..., x [n] () ), x() = b, x(b) =. References [] Andrzej, P.; On some ierive differenil equions I, Zeszyy Nukowe Uniwersyeu Jgiellonskiego, Prce Memyczne, 2 (968), [2] Berinde, V.; Exisence nd pproximion of soluions of some firs order ierive differenil equions, Miskolc Mh. Noes, () (200), [3] Buron, T. A.; Sbiliy by fixed poin heory for funcionl differenil equions. Dover Publicions, Inc., Mineol, NY, [4] Eder, E.; The funcionl-differenil equion x () = x(x()), J. Differenil Equions, 54 (984), no. 3, [5] Fečkn, M.; On cerin ype of funcionl-differenil equions, Mh. Slovc, 43 (993), no., [6] Ge, W.; Liu, Z.; Yu, Y.; On he periodic soluions of ype of differenil-ierive equions, Chin. Sci. Bull., 43 (3) (998), [7] Liu, H. Z.; Li, W.R.; Discussion on he nlyic soluions of he second-order iered differenil equion, Bull. Koren Mh. Soc., 43 (2006), no. 4, [8] Liu, X. P.; Ji, M.; Iniil vlue problem for second order non-uonomous funcionldifferenil ierive equion, (Chinese) Ac Mh. Sinic (Chin. Ser.), 45 (2002), no. 4, [9] Peuhov, V. R.; On boundry vlue problem, (Russin. English summry), Trudy Sem. Teor. Differencil. Urvneniĭs Oklon. Argumenom Univ. Dru zby Nrodov Pris Limumby, 3 (965), [0] Wng, K.; On he equion x () = f(x(x())), Funkcil. Ekvc. 33 (990), [] Zhng, P.; Anlyic soluions of firs order funcionl differenil equion wih se derivive dependen dely. Elecron. J. Differenil Equions, 2009 (2009), No. 5, 8 pp. [2] Zhng, P.; Anlyic soluions for ierive funcionl differenil equions, Elecron. J. Differenil Equions, 202 (202), No. 80, 7 pp. [3] Zhng, P.; Gong, X.; Exisence of soluions for ierive differenil equions, Elecron. J. Differenil Equions, 204 (204), No. 07, 0 pp. [4] Zho, H.; Smooh soluions of clss of ierive funcionl differenil equions, Absr. Appl. Anl., 202, Ar. ID , 3 pp. Eric R. Kufmnn Deprmen of Mhemics & Sisics, Universiy of Arknss Lile Rock, Lile Rock, AR 72204, USA E-mil ddress: erkufmnn@ulr.edu