EXISTENCE AND ITERATION OF MONOTONE POSITIVE POLUTIONS FOR MULTI-POINT BVPS OF DIFFERENTIAL EQUATIONS

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1 U.P.B. Sci. Bull., Series A, Vol. 72, Iss. 3, 2 ISSN EXISTENCE AND ITERATION OF MONOTONE POSITIVE POLUTIONS FOR MULTI-POINT BVPS OF DIFFERENTIAL EQUATIONS Yuji Liu By applying monoone ieraive meho, we obain no only he exisence of monoone posiive soluions for a kind of muli-poin boundary value problems, bu also esablish ieraive schemes for approximaing he soluions. A boundary value problem ha our resuls can readily apply, whereas he known resuls in he curren lieraure do no cover, is presened a he end of he paper. Keywor: Muli-poin boundary-value problem; p-laplacian; half-line; posiive soluions; exisence; uniqueness. MSC2: primary 49J4, 49K2; secondary 58E7, 65K, 53C65.. Inroducion Recenly an increasing ineres has been observed in invesigaing he exisence of posiive soluions of boundary-value problems. This ineres comes from siuaions involving nonlinear ellipic problems in annular regions. Erbe and Tang [9] noed ha, if he boundary-value problem wih u = F x, u in R < x < ˆR u = for x = R, u = for x = ˆR; or u u = for x = R, x = for x = ˆR; or = for x = R, u = for x = ˆR u x is radially symmeric, hen i can be ransformed ino he so called wo-poin Surm- Liouville problem x = f, x,, αx βx =, γx δx =. where α, β, γ, δ are posiive consans. Paper [9] may be he firs one concerned wih he exisence of posiive soluions of a boundary value problem. In he laes en years, muli-poin boundary-value problems BVPs for shor for second order differenial equaions wih p Laplacian have gained much aenion, see papers [- 8]. Prof., Dr., Guangdong Universiy of Business Sudies, Guangzhou, Guangdong, 532, P. R. China, liuyuji888@sohu.com 99

2 Yuji Liu Ma in [,2] sudied he following BVP [px ] qx f, x =,,, αx βpx = m a ixξ i, γx δpx = m b ixξ i, where < ξ < < ξ m <, α, β, γ, δ, a i, b i wih ρ = γβ αγ αδ >. By using Green s funcions which complicae he sudies of BVP and Guo- Krasnoselskii fixed poin heorem [7-], he exisence and mulipliciy of posiive soluions for BVP were given. In paper [2], he exisence of posiive soluions for he m-poin boundaryvalue problem y = f, y, y,, αy βy =, y = m 2 α iyξ i and y = f, y, y,, αy βy =, y = m 2 α iyξ i was considered, where α >, β >, he funcion f is coninuous, and f, y, y, for all [, ], y y R. The presence of he hird variable y in he funcion f, y, y causes some considerable difficulies, especially, in he case where an approach relies on a fixed poin heorem on cones and he growh rae of f, y, y is sublinear or superlinear. The approach used in [2] is based on an analysis of he corresponding vecor field on he y, y face-plane and on Kneser s propery for he soluion s funnel. Recenly, many auhors sudied he exisence of muliple posiive soluions of he following BVP consising of he second order differenial equaion [φx ] qf, x, x =,, 2 and one of he following boundary condiions x = a i xξ i, x = and x = x = x = a i xξ i, x = a i x ξ i, x = a i x ξ i, x = b i xξ i b i x ξ i, b i xξ i, b i x ξ i, where < ξ < < ξ m <, a i, b i for all i =,, m, f is defined on [, ] [, R, φ is called p Laplacian, see papers [4-6] and [3-8,2].

3 Exisence and ieraion of monoone posiive poluions for muli-poin BVPs of differenial equaions To he auhor s knowledge, here has been no paper concerning wih he compuaional meho of he following boundary value problems [pφx ] qf, x, x =,,, αx βx = m a ixξ i, x = 3 m b ix ξ i where < ξ < < ξ m <, α, β, a i, b i for all i =,, m, f is defined on [, ] [, [,, p C [, ] and q C [, ], φ is called p Laplacian, φx = x r 2 x wih r >, is inverse funcion is denoed by φ x wih φ x = x s 2 x wih /r /s =. The purpose of his paper is o invesigae he ieraion and exisence of posiive soluions for BVP3. By applying monoone ieraive echniques, we will consruc some successive ieraive schemes for approximaing he soluions in his paper. The sequel of his paper is organized as follows: he main resul is presened in Secion 2, and some examples are given in Secion Main Resuls In his secion, we firs presen some background definiions in Banach spaces. Definiion 2.. Le X be a real Banach space. The nonempy convex closed subse P of X is called a cone in X if ax P and x y P for all x, y P and a and x X and x X imply x =. Definiion 2.2. Le X be a real Banach space and P a cone in X. A map ψ : P [, is a nonnegaive coninuous concave funcional map provided ψ is nonnegaive, coninuous and saisfies ψx y ψx ψy, for all x, y P and [, ]. Definiion 2.3. Le X be a real Banach space. An operaor T ; X X is compleely coninuous if i is coninuous and maps bounded ses ino pre-compac ses. Choose X = C [, ]. We call x y for x, y X if x y and x y for all [, ]. We define he norm x = max{ max x, max [,] [,] x } for x X. I is easy o see ha X is a semi-ordered real Banach space. Define he cone in X by P = {x X : x and is concave and increasing on [, ]}. For a posiive number H, denoe he subse P H by P H = {x P : x < H} and P H = {x P : x H}. Suppose ha A α, β, a i, b i for all i =,, m wih m a i < α, and m b iφ pξ i p < ; A2 p C [, ],, ; A3 f : [, ] [, [, [,, q : [, ] [, are coninuous wih qf,, on each sub-inerval of [,], f, x, y f, x 2, y 2 for all [, ], x x 2 and y y 2.

4 2 Yuji Liu Lemma 2.. Suppose ha A, A2 and A3 hold and x saisfies x C [, ] wih [pφx ] on [, ]. Then x is concave. Proof. Suppose x = max [,] x. If <, for,, since x, we have p φx. Then pφx for all, ]. Le φ ps τ =. φ ps Then τ C[, ], [, ] and is sricly increasing on [, ] since i is easy o see ha τ = and τ =. Thus implies ha dx dτ = dτ d = φ p φ ps dx d = dx dτ dτ d = dx dτ >, φ p φ ps φ φ pφx. ps Since [pφx ] = qf, x, x, we ge ha φ pφx is decreasing as increases. Since is increasing as τ increases, we ge ha dx dτ is decreasing as τ increases. Then x is concave on [, ]. If =, similarly o above discussion, we ge ha x is concave. The proof is compleed. Throughou he paper, δ, a, b and d are defined by δ = φ, m b iφ pξ i p m a ξi δ i φ δps qudu a = a, i βφ δ δp qudu b = a, i d = φ δ qudu. δ min [,] p Lemma 2.2. Suppose ha A A3 hold. If x X is a soluion of BVP3, hen x = B x φ pφa x s qufu, xu, x udu 4 ps ps

5 Exisence and ieraion of monoone posiive poluions for muli-poin BVPs of differenial equaions 3 where saisfies A x = and B x saisfies [ A x, φ qufu, xu, ] x udu δp b i φ p pξ i φa x B x = α m [ m a i a i ξi φ p ps φa x βφ p p φa x ξ i qufu, xu, x udu pξ i s qufu,xu,x udu ps qufu,xu,x udu p ] Then Proof. Since x is soluion of 3, we ge pφx = pφx x = x qufu, xu, x udu, [, ]. φ p ps φy s qufu, xu, x udu. ps The BCs in 3 imply ha αx βφ p p φx qufu, xu, x udu p = x and a i ξi a i x = φ p ps φy b i φ p pξ i φx I follows ha x = a i [ m ξi a i φ p ps φy βφ p p φx s qufu, xu, x udu ps ξ i qufu, xu, x udu pξ i s qufu, xu, x udu ps ] qufu, xu, x udu p.. 8

6 4 Yuji Liu Le If we ge Gc = c b i φ p pξ i φc ξ i qufu, xu, x udu pξ i ξ i qufu, xu, x udu = for each i =,, m, Gc = b i φ pξ i p c. Then Gc = implies ha c =. If b i = for all i =,, m, hen Gc = implies ha c =. If here exiss i {,, m} such ha ξ i qufu, xu, x udu and here exiss j {,, m} such ha b j, i is easy o see ha G and Gc p = b i φ c pξ i qufu, xu, x udu φc pξ i and Gc/c is coninuous, increasing on, and on,, respecively. One sees from A ha Gc lim = b i φ c c pξ i p Gc >, lim =. c c Hence Gc < for all c,. On he oher hand, i follows from and ξ i Gc lim = c G φ qufu,xu,x udu δp φ qufu,xu,x udu δp = = b i φ pξ i p δp qufu, xu, x udu pξ i δ b i φ pξ i p ξ i qufu, xu, x udu.

7 Exisence and ieraion of monoone posiive poluions for muli-poin BVPs of differenial equaions 5 ha here exiss unique consan [ c, φ qufu, xu, ] x udu δp 9 such ha Gc =. Togeher wih 8, we ge ha c = x = A x. Hence we ge ha here exis consans A x saisfying 5 and 6, and B x saisfying 7 such ha x saisfies 4. The proof is compleed. Lemma 2.3. Suppose ha A, A2 and A3 hold. If x X is a soluion of BVP3, hen x > for all,. Proof. Suppose x saisfies 3. I follows from he assumpions ha pφx is decreasing on [, ]. I follows from Lemma 2. ha x is concave on [,]. Then x is decreasing on [,]. I follows from Lemma 2.2 ha x. I follows ha x is increasing on [,]. Then x αx = a i xξ i a i x. I follows ha α a i x βx. We ge ha x since A and x. Hence x > for, ]. The proof is complee. Define he nonlinear operaor T : P X by T x = B x φ p ps φa s x qufu, xu, x udu ps for x P, where A x saisfies 6, and B x saisfies 7. Lemma 2.4. Suppose ha A A3 hold. I is easy o show ha i T x saisfies [pφt x ] qf, x, x =,,, αt x βt x = m a it xξ i, T x = m b it x ξ i ; ii T y P for each y P ; iii x is a soluion of BVP3 if and only if x is a soluion of he operaor equaion x = T x in P ; iv T : P P is compleely coninuous; v T x T x 2 for all x, x 2 P wih x x 2. Proof. The proofs of i, ii and iii are simple. To prove iv, i suffices o prove ha T is coninuous on P and T is relaive compac. I is similar o ha of he proof of Lemma 2.9 in [8] or Lemmas in [6] and are omied.

8 6 Yuji Liu To prove v, i is easy o see ha A x is increasing in x and consequenly, using A3 and considering B x as a funcion of A x and f we ge he monooniciy of B x. Suppose x x 2, we ge ha x x 2 and x x 2 for all [, ]. Then one fin ha T x T x 2, T x T x 2, [, ]. I follows ha T x T x 2. The proof is compleed. Theorem 2.. Suppose ha A A3 hold. Furhermore, suppose ha here exiss a consan A > such ha f saisfies max f, 2A, 2A φm, [, ], [,] where { A M = min, a b A d }. Then BVP3 has a leas one posiive soluion x P wih x = lim n u n or x = lim n v n, where u =, v = A A and u n = T u n, v n = T v n. Proof: We firs prove ha T : P 2A P 2A. For x P 2A, one has ha x 2A, x 2A, [, ]. Then implies ha f, x, x f, 2A, 2A max f, 2A, 2A φm. [,]

9 Exisence and ieraion of monoone posiive poluions for muli-poin BVPs of differenial equaions 7 Le A x and B x saisfy 6 and 7 respecively. I follows from Lemma 2.2 ha A x saisfies 5. By he definiion of T x, we ge ha T x = B x = φ p ps φa x s qufu, xu, x udu ps a i [ m ξi a i φ p ps φa s x qufu, xu, x udu ps βφ p pφa x qufu, xu, ] x udu p φ ps pφa s x qufu, xu, x udu ps a i [ m ξi a i φ s quφmdu quφmdu δps ps βφ quφmdu quφmdu ] δp p φ s quφmdu quφmdu δps ps M m a ξi δ i φ δps qudu a i βφ δ δp qudu a i = Ma b Md 2A, φ δ δ min [,] p qudu

10 8 Yuji Liu and T x p = φ p φa x p φ quφandu δp p Mφ δ qudu δ min [,] p = Md A 2A. I follows ha T x P 2A. By he definiions of u and v, we have qufu, xu, x udu u v, u v, [, ]. quφmdu Then u v. Then Lemma 2.4v implies ha u n v n for all n =, 2, 3,. Now, we prove ha u n u n. I suffices o prove ha u u. Firs, one has = u = T u m a ξi p i φ ps φa u ps s qufu, a u, adu a i βφ p pφa u p qufu,, du Second, we have a i φ ps pφa u ps = u. u = T u = φ p pφa u p = u. s qufu,, du qufu,, du I follows ha u u and u u for all [, ]. Then u u. Hence one has ha u u u 2 u n. 2 Now, we prove ha v n v n. I suffices o prove ha v v. Firs, 9 implies ha A v φ qufu, v u, v udu. δp Sice max [,] f, 2A, 2A φm, [, ], we ge f, v, v = f, A A, A f, 2A, 2A φm, [, ].

11 Exisence and ieraion of monoone posiive poluions for muli-poin BVPs of differenial equaions 9 Then = v = T v m a ξi p i φ ps φa v ps βφ p pφa v p a i s qufu, bu a, bdu qufu, bu a, bdu a i φ ps pφa v ps m a i ξi φ δ δps s qufu, bu a, bdu qufu, v u, v udu a i βφ δ δp qufu, v u, v udu M a i δ φ δps m a ξi δ i φ δps qufu, v u, v udu a i βφ δ δp qudu a i M qudu φ δ δps m a ξi δ i φ δps qudu a i βφ δ δp qudu a i = Ma b Md Second, we have A A = v. v = T v = φ p pφa v p φ δ δ min [,] p = Mφ δ δ min [,] p = Md A = v. φ δ δ min [,] p qudu qudu qufu, v u, v udu quφmdu qudu

12 Yuji Liu I follows ha v v and v v for all [, ]. Then v v. Hence one has ha v v v 2 v n. 3 I follows from 2 and 3 ha = u u u n v n v v = A A. Since u n is a uniformly increasing sequence, i is easy o show ha u n is equiconinuous. Then lim n u n = u P saisfies u = a u b a, T u = u. Then x = u is a soluion of BVP3. Similarly o above discussion, saisfies lim v n = v P n u = a v b a, T v = v. Then x = v is a soluion of BVP3. I is easy o see ha BVP3 has unique soluion x in {x P : x A A} if u = v. BVP3 has muliple soluions if u v. The proof is complee. Remark 2. The quaniy A x is given implicily, as a roo of equaion 6, i mus be deermined in every sep of ieraion. Theorem 2.2. Suppose ha A A3 hold. Furhermore, suppose ha f saisfies { f, 2A, 2A lim sup sup < φ min, }. 4 x φa a b d [,] Then BVP3 has a leas one posiive soluion x P. Proof. I follows from 4 ha here exiss a consan A > such ha { f, 2A, 2A φaφ min, }. a b d The remainder of he proof is similar o ha of he proof of Theorem 2. and is omied. 3. An example Now, we presen a boundary value problem o which our resuls can readily apply, whereas he known resuls in he curren lieraure do no cover. Example 3.. Consider he following BVP [x 3 ] f, x, x =,,, x x = 2 x/2, x = 4 x /2. Corresponding o BVP3, one sees ha φx = x 3 wih φ x = x 3, α =, β =, ξ = /2, a = /2, b = /4, q, [, ], p =, f, u, v = is nonnegaive and coninuous. 24 x3 24 y3 5

13 Exisence and ieraion of monoone posiive poluions for muli-poin BVPs of differenial equaions One fin ha δ = φ = 63, m b iφ pξ i p m a ξi δ i φ δps qudu a = b = a i βφ δ δp qudu a i d = φ δ δ min [,] p = , qudu = , = , and { } A A 3 63A M = min 2 3 8, 4 = I is easy o check ha here exiss a consan A > such ha f, 2A, 2A φa = 3 A3 3 A3 A 3 63, [, ]. 64 One can check ha A, A2, A3 hold. Then Theorem 2. implies ha BVP5 has a leas one posiive soluion x wih x = lim n u n or x = lim n v n, where u =, v = A A and u n = T u n, v n = T v n, where T is defined by and A x saisfies T x = 2 [ 2 A x = 4 /2 A 3 x A 3 x A 3 x A 3 x s fu, xu, x udu /3 /3 ] fu, xu, x udu s /2 fu, xu, x udu /3 fu, xu, x udu /3 for x C [, ] wih x, [, ]. Remark 3.. BVP5 can no be solved by heorems obained in [-2]. Acknowledgemens: The auhor is graeful o he reviewers and he edior for heir helpful commens and suggesions which make he paper easy o read. The presen research is suppored by Naural Science Foundaion of Guangdong province No: and Naural Science Foundaion of Hunan province, P. R. ChinaNo: 6JJ58.

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