Existence and Uniqueness of Positive Solution for Singular BVPs on Time Scales

Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 728484, 12 pages doi:10.1155/2009/728484 Research Article Existence and Uniqueness of Positive Solution for Singular BVPs on Time Scales Ana G ´ omez Gonz´ alez1and Victoria Otero-Espinar2 1Departamento de Matem´ atica Aplicada, Facultade de Matem´ aticas, Universidade de Santiago de Compostela, 15782 Galicia, Spain 2Departamento de An´ alise Matem´ atica, Facultade de Matem´ aticas, Universidade de Santiago de Compostela, 15782 Galicia, Spain Correspondence should be addressed to Victoria Otero-Espinar, [email protected] Received 27 March 2009; Accepted 12 May 2009 Recommended by Alberto Cabada This paper is devoted to derive some sufficient conditions for the existence and uniqueness of positive solutions to a singular second-order dynamic equation with Dirichlet boundary conditions. Copyright q2009 A. G´ omez Gonz´ alez and V. Otero-Espinar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Hilger 1introduced the notion of time scale in 1990 in order to unify the theory of continuous and discrete calculus. The field of dynamic equations on time scale contains, links, and extends the classical theory of differential and difference equations, besides many others. There are more time scales than just Rcorresponding to the continuous caseand Ndiscrete caseand hence many more classes of dynamic equations. By time scale we mean a closed subset of the real numbers. Let Tbe an arbitrary time scale. We assume that Thas the topology that it inherits from the standard topology on R. Assume that a<bare points in Tand define the time scale interval a, bT{t∈T:a≤t≤ b}. For t∈T, define the forward jump operator σ:T→Tby σtinf{s∈T:s>t}and the backward jump operator ρ:T→Tby ρtsup{s∈T:s<t}. In this definition we put σttif Tattains a maximum tand ρttif Tattains a minimum t.Ifσt>t,tis said to be right-scattered and if σtt, t is said to be right-dense. If ρt<t,tis said to be left-scattered and if ρtt, t is said to be left-dense. A function f:T→Ris said to be rd-continuous provided it is continuous at all rightdense points of T, and its left-sided limit exists at left-dense points of T. For t∈T,thedelta 2 Advances in Difference Equations derivative fΔtof fat tis defined to be the number if existssuch that for given >0, there exists a neighborhood Uof tsuch that fσt−fs−fΔtσt−s≤|σt−s|,∀s∈U. 1.1 See 2for general theory about time scales. The problem we will consider in this work is of the type −uΔΔ gt, uσt, uauσ2b0. 1.2 Under this general form it included the Emden-Fowler equation, which arises in several fields, such as the following: iAstrophysics: related to the stellar structure gaseous dynamics. In this case the fundamental problem is to investigate the equilibrium configuration of the mass of spherical clouds of gas. iiGas dynamics and fluid mechanics. The solutions of physical interest in this context are bounded nonoscillatory and possess a positive zero. iiiRelativistic mechanics. ivNuclear physics. vChemically reacting systems: in the theory of diffusion and reaction this equation appears as governing the concentration uof a substance which disappears by an isothermal reaction at each point of a slab of catalyst. We refer to Wong 3, for a general historical overview about this equation. Many works on this equation have been written in the continuous case, and we can cite among others, 4,5or 6. On the discrete case we find the book 7which studies the oscillation properties of the solutions of different difference equations. For the specific problem uΔΔtptuγσt  0, where p≥0andγquotient of odd positive numbers, also oscillation properties were studied in 8. On time scales some results on existence and uniqueness of solutions in the sense of distribution for this equation can be found in the article 9. Considering classical solutions, oscillation properties have also been studied, in works such as 10with delayor 11. In the present paper we present some results on time scales considering classical solutions which generalize the ones from the continuous case. 2. Lower and Upper Solutions Method Let a, b ∈T, such that a<ρb. Let us put Ja, σ2bT,J κa, σbT,andJκo a, σbTif a/ σaand Ja, σ2bT,J κa, σbT,andJκoa, σbTif aσa. Advances in Difference Equations 3 We consider the second-order dynamic equation with Dirichlet boundary conditions, −uΔΔtft, uσt,t∈Jκo, uaA, uσ2bB, P where f:Jκo×A → R,A⊂R, satisfies the following condition. H1iFor every x∈A,f·,x∈CrdJκo, iift, ·is continuous on Auniformly in t∈Jκo. For convenience, we denote Eh∈CrdJκo,R:σb a σs−aσ2b−shsΔs<∞.2.1 We say that fsatisfies the condition H2on B⊂Jκo×Aif there exists a function h∈E such that H2 ft, x≤ht,∀t, x∈B.2.2 Definition 2.1. A solution of Pis a function u∈C2 rda, bTsuch that ut∈A, for all t∈a, σ2bT, which satisfies the equalities on Pfor each t∈Jκo, where C2 rda, bTy∈Ca, σ2bT,y ΔΔ :Jκo−→ R,y ΔΔ ∈Crda, bT.2.3 Definition 2.2. We say that α∈C2 rda, bTis a lower solution of Pif αt∈A, for all t∈a, σ2bTand −αΔΔt≤ft, ασt,t∈Jκo, αa≤A, ασ2b≤B. 2.4 An upper solution β∈C2 rda, bTof Pis defined similarly by reversing the previous inequalities. We have the following result. Theorem 2.3. Let αand βbe, respectively, a lower and upper solution for problem P, such that α≤βon a, σ2bT.Iffsatisfies H1and the conditions H2on Dβ αt, x∈Jκo×R:ασt≤x≤βσt,2.5 then problem Phas at least one solution usuch that α≤u≤βon a, σ2bT. 4 Advances in Difference Equations Proof. We consider the following modified problem: −uΔΔtf∗t, uσt,t∈Jκo, uaA, uσ2bB. Pm with f∗t, x ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ft, ασt ασt−x 1ασt−x,if x≤ασt, ft, x,if ασt≤x≤βσt, ft, βσtx−βσt 1x−βσt,if βσt≤x. 2.6 Due to the hypothesis it can be easily checked that H1and H2are satisfied by the function f∗. Note that, if uis a solution of Pmsuch that α≤u≤βon a, σ2bT, then uis a solution of P, also satisfying α≤u≤βon a, σ2bT. To show that any solution uof Pmis between αand β,letvtαt−ut,and suppose that there exists t∗∈a, σ2bTsuch that vt∗>0. As va≤0andvσ2b ≤0, then there exists t0∈a, σ2bTwith vt0maxvt,t∈a, σ2bT>0,2.7 and vt<vt0for t∈t0,σ2bT. The point t0is not simultaneously left-dense and right-scattered see 12, Theorem 2.1 this implies that σ◦ρt0t0, and we have that vΔΔρt0 ≤0see 12,so −vΔΔρt0uΔΔt0−αΔΔt0 ≤−f∗ρt0,u σρt0fρt0,α σρt0 −αt0−ut0 1αt0−ut0 <0. 2.8 So vΔΔρt0 >0, that is a contradiction. And so we have proved that vt≤0,∀t∈ a, σ2bT. Analogously it can be proved that ut≤βt,∀t∈a, σ2bT. We only need to prove that problem Pmhas at least one solution. Consider now the operator N:Ca, σ2bT→Ca, σ2bT, defined by Nutφtσb a Gt, sf∗s, uσsΔs, 2.9 Advances in Difference Equations 5 for each t∈a, σ2bT, where see 2 Gt, s1 σ2b−a⎧ ⎨ ⎩ t−aσ2b−σs,t≤s, σs−aσ2b−t,σ s≤t 2.10 is Green’s function of the problem −xΔΔ 0, xaxσ2b0, 2.11 and for t∈a, σ2bT φtAB−A σ2b−at−a2.12 is the solution of −xΔΔ 0 such that φaAand φσ2b  B. Clearly, Gt, s>0ona, σ2bT×a, σ2bT,Gt, ·is rd-continuous on a, σbT and G·,sis continuous on a, σ2bT. The function Nu defined by 2.9belongs to Ca, σ2bTbecause f∗checks the conditions H1and H2on Jκo×Rand Gt, s≤s1−s, for each t, s ∈a, σ2bT. It is obvious that u∈Ca, σ2bTis a solution of Pmif and only if uNu.Sothe problem now is ensuring the existence of fixed-points of N. First of all, Nis well defined, is continuous, and NCa, σ2bT is a bounded set. The existence of a fixed-point of Nfollows from the Schauder fixed-point theorem, once we have checked that NCa, σ2bT is relatively compact, that using the Ascoli-Arzela theorem is equivalent to proving that NCa, σ2bT is an equicontinuous family. Let h∗∈Ebe the function related to f∗by condition H2. We compute the first derivative of Nu using 2, Theorem 1.117 NuΔt1 σ2b−aB−A−t a σs−af∗s, uσsΔs σb tσ2b−σsf∗s, uσsΔs ≤1 σ2b−a|B−A|t a σs−ah∗sΔsσb tσ2b−σsh∗sΔs :1 σ2b−a|B−A|λt. 2.13 6 Advances in Difference Equations Finally it is enough to check that λ∈L1Jκo, using integration by parts we obtain σb a |λs|Δsσb a λsΔs lim r→σb−r at a σs−ah∗sΔsΔt lim r→aσb rσb tσ2b−σsh∗sΔsΔt 2σb a σs−aσ2b−σsh∗sΔs −lim r→σb−σ2b−rr a σs−ah∗sΔs −lim r→ar−aσb rσ2b−σsh∗sΔs<∞, 2.14 due to h∗∈E, and the fact σ2b−rr a σs−ah∗sΔs≤σb a σs−aσ2b−σsh∗sΔs, r−aσb rσ2b−σsh∗sΔs≤σb a σs−aσ2b−σsh∗sΔs. 2.15 And so the result is proved. 3. Existence and Uniqueness of Positive Solution Let g:Jκo×0,∞→Rin the condition H1, where A0,∞, and consider the problem −uΔΔtgt, uσt,t∈Jκo, uauσ2b0. Q We will deduce the existence of solution to Qby supposing that the following hypothesis holds. H3There exists a constant L>0 such that for any compact set D⊂Jκo, there is εεD>0: gt, x>L, ∀t, x∈D×0,ε.3.1 Advances in Difference Equations 7 Theorem 3.1. Suppose that H1and H3hold. If, for any δ>0,gsatisfies the condition H2on Jκo×δ, ∞, then problem Qhas at least one solution. Proof. Let’s consider {an}n≥1,{bn}n≥1⊂Jκoas two sequences such that {an}n≥1⊂ a, aσb/2Tis strictly decreasing to aif aσa,andanafor all n≥1ifa<σa,and {bn}n≥1⊂aσb/2,σbTis strictly increasing to σbif ρσb  σb,b nρσb for all n≥1ifρσb <σb. We denote as Dn:an,b nT⊂Jκo,n≥1. Due to the first hypothesis, we can then ensure the existence of εn>0 such that gt, x>L, for all t, x∈Dn×0,ε n. We can suppose, without restriction that {εn}is a decreasing sequence and limn→∞εn0. Consider the function γ:a, a1T→R, such that γa0, and if a/ a1then γt ε2for t∈a2,a 1,γtεn, for all t∈Dn\Dn−1,withn≥3. Since γis nondecreasing, we obtain that for t∈a, a1 γ1t:1 σ2b−at a γsΔs≤γt3.2 is continuous and increasing. Repeating this argument twice, for i2,3 we define γit:1 σ2b−at a γi−1sΔs≤γt.3.3 So γ3∈C2a, a1Tis a convex function verifying γ3t≤εn,∀t∈Dn\Dn−1,n≥2. Analogously, we can consider γ:b1,σ2bT→R, such that γσ2b  0,γσb  0,γtε2, for all t∈b1,b 2and γtεn, for all t∈Dn\Dn−1,withn≥3. Taking γ0γ,we obtain, for i1,2,3, that for t∈b1,σ2b γit:1 σ2b−aσ2b tγi−1sΔs≤γt3.4 is continuous and decreasing, and γ3∈C2b1,σ2bTa convex function such that γ3t≤ εn,∀t∈Dn\Dn−1,n≥2. We now define αt⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ γ3t,t∈a, a1T, α∗t,t∈a1,b 1T, γ3t,t∈b1,σ2bT, 3.5 with α∗ta convenient function, so that α∈C2a, σ2bT,and0<αt≤ε1for all t∈ a1,b 1T. So α∈C2a, σ2bTis a function such that αaασ2b  0,αt>0fort∈ a, σbT,αt≤ε1for t∈D1,andαt≤εn,fort∈Dn\Dn−1,n≥2. In this way, we note that gt, x>L, ∀t, x∈Jκo×{x∈0,∞:0<x≤αt}.3.6 8 Advances in Difference Equations Let m0min{1,L/αΔΔ 1}. Let Ft, x≥gt, x, for all t, x∈Jκo×0,∞,withFin the conditions H1.We will prove that if v∈C2 rda, σ2bTis any solution of −vΔΔ Ft, vσt,3.7 with va≥0,vσ2b ≥0andvt>0 for all t∈Jκo, then, vt≥m0αt,∀t∈a, σ2bT.3.8 Suppose there exists t∈a, σ2bT, such that vt−m0αt<0. Then, we can assure, using arguments analogous to the ones in the proof of Theorem 2.3, that there exists t∗∈a, σ2bT verifying vΔΔρt∗≥m0αΔΔρt∗,3.9 which implies that L<F ρt∗,vσρt∗−vΔΔρt∗≤−m0αΔΔρt∗<m 0  αΔΔ  1≤L, 3.10 which is a contradiction. We define now, for each n∈N gnt, x:maxgτnt,x,gt, x,3.11 where, for each t∈Jκo τntmax{an,min{t, bn}}.3.12 For each nwe have that gnis a function verifying H1on Jκo×0,∞,andgnt, x≥gt, x, for all t, x∈J×0,∞and gnt, xgt, x,fort∈Dn. The sequence {gn}nconverges to guniformly in every set of the form D×0,∞, where D⊂Jκois a compact set. Defining, by induction, g1t, xg1t, xand for n>1 gn1t, xmingnt, x,gn1t, x,3.13 we have that, for each n, the function gnsatisfies the condition H1on Jκo×0,∞.As well g1≥g2≥···≥gn≥···≥g, and {gn}nconverges to guniformly, in every set of the form D×0,∞, where D⊂Jκois a compact set. It is also verified that gnt, xgt, x,t∈an,b n,x∈0,∞. Advances in Difference Equations 9 Now we define the following problems: uΔΔtgnt, uσt 0,t∈Jκo, uauσ2bεn. Qn We will prove that for any c∈0,ε n, the constant function αn≡cis a strictlower solution for Qn. It is obvious that cαna≤εn,andcαnσ2b ≤εn. Now we have to prove that gnt, c>0,t∈Jκo,c∈0,ε n.3.14 For n1, let c∈0,ε 1such that g1t, cg1t, cmaxgτ1t,c,gt, c≥gτ1t,c>L. 3.15 Suppose now that gnt, c>0,t∈Jκo,c∈0,ε n,foragivenn≥1, and we will check that gn1t, c>0,t∈Jκo.Letc∈0,ε n1such that gn1t, cmingnt, c,gn1t, c≥mingnt, c,gτn1t,c>0.3.16 Thus the assertion is proved. Moreover, as gn≥gn1on Jκo×0,∞, it can be easily checked that any solution un of Qnis an upper solution for Qn1. To show that problem Q1has at least one solution. We fix a constant M≥ε1.From the assumption imposed, there exists a function hM∈CrdJκo,Rsuch that gt, x≤hMt,∀t∈Jκo,x≥M. 3.17 Note that gτ1t,x≤hMτ1t ≤R, ∀t∈Jκo,∀x≥M, 3.18 with R>0 is a suitable constant. Set qt:hMtR. We have that q∈CrdJκo,0,∞,and g1t, x≤qt,∀t∈J, x ≥M. 3.19 Let β∈C2 rda, bTbe a solution of the boundary value problem: uΔΔtqt0, uauσ2bM. 3.20