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

Abstract

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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Ana Gómez González 1 and Victoria Otero-Espinar 2

Recommended by Alberto Cabada

1, Departamento de Matemática Aplicada, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Galicia, Spain
2, Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Galicia, Spain

Received 27 March 2009; Accepted 12 May 2009

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 ... (corresponding to the continuous case) and ... (discrete case) and hence many more classes of dynamic equations.

By time scale we mean a closed subset of the real numbers. Let ... be an arbitrary time scale. We assume that ... has the topology that it inherits from the standard topology on ... . Assume that a<b are points in ... and define the time scale interval _[a,b]... ={t∈...:a≤t≤b} . For t∈... , define the forward jump operator σ:...[arrow right]... by σ(t)=inf {s∈...:s>t} and the backward jump operator ρ:...[arrow right]... by ρ(t)=sup {s∈...:s<t} . In this definition we put σ(t)=t if ... attains a maximum t and ρ(t)=t if ... attains a minimum t . If σ(t)>t, t is said to be right-scattered and if σ(t)=t, t is said to be right-dense. If ρ(t)<t, t is said to be left-scattered and if ρ(t)=t, t is said to be left-dense.

A function f:...[arrow right]... is said to be rd-continuous provided it is continuous at all right-dense points of ... , and its left-sided limit exists at left-dense points of ... . For t∈... , the delta derivative ^fΔ (t) of f at t is defined to be the number (if exists) such that for given ...>0 , there exists a neighborhood U of t such that [figure omitted; refer to PDF] See [2] for general theory about time scales.

The problem we will consider in this work is of the type [figure omitted; refer to PDF]

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 u of 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≥0 and γ quotient 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∈... , such that a<ρ(b) . Let us put J=_{[a,^σ2 (b)]...} , ^Jκ =_[a,σ(b)]... , and ^{(Jκ )o} =_[a,σ(b))... if a≠σ(a) and J=_{(a,^σ2 (b)]...} , ^Jκ =_(a,σ(b)]... , and ^{(Jκ )o} =_(a,σ(b))... if a=σ(a) .

We consider the second-order dynamic equation with Dirichlet boundary conditions, [figure omitted; refer to PDF] where f:(^Jκ)o ×...9C;[arrow right]..., ...9C;⊂... , satisfies the following condition.

(_H1 ) : (i) For every x∈...9C; , f(·,x)∈_Crd (^{(Jκ )o} ) ,

: (ii) f(t,·) is continuous on ...9C; uniformly in t∈(^Jκ)o .

For convenience, we denote [figure omitted; refer to PDF] We say that f satisfies the condition (_H2 ) on [Bernoulli]⊂^{(Jκ )o} ×...9C; if there exists a function h∈E such that

(_H2 ) : [figure omitted; refer to PDF]

Definition 2.1.

A solution of (P) is a function u∈^Crd2 (_(a,b)... ) such that u(t)∈...9C; , for all t∈_{[a,^σ2 (b)]...} , which satisfies the equalities on (P) for each t∈(^Jκ)o , where [figure omitted; refer to PDF]

Definition 2.2.

We say that α∈^Crd2 (_(a,b)... ) is a lower solution of (P) if α(t)∈...9C; , for all t∈[a,^σ2 (b)_]... and [figure omitted; refer to PDF] An upper solution β∈^Crd2 (_(a,b)... ) of (P) is defined similarly by reversing the previous inequalities.

We have the following result.

Theorem 2.3.

Let α and β be, respectively, a lower and upper solution for problem (P) , such that α≤β on _{[a,^σ2 (b)]...} . If f satisfies (_H1 ) and the conditions (_H2 ) on [figure omitted; refer to PDF] then problem (P) has at least one solution u... such that α≤u...≤β on [a,^σ2 (b)_]... .

Proof.

We consider the following modified problem: [figure omitted; refer to PDF] with [figure omitted; refer to PDF] Due to the hypothesis it can be easily checked that (_H1 ) and (_H2 ) are satisfied by the function ^f* .

Note that, if u is a solution of (_Pm ) such that α≤u≤β on _{[a,^σ2 (b)]...} , then u is a solution of (P) , also satisfying α≤u≤β on _{[a,^σ2 (b)]...} .

To show that any solution u of (_Pm ) is between α and β , let v(t)=α(t)-u(t) , and suppose that there exists ^t* ∈_{[a,^σ2 (b)]...} such that v(^t* )>0 . As v(a)≤0 and v(^σ2 (b))≤0 , then there exists _t0 ∈_{(a,^σ2 (b))...} with [figure omitted; refer to PDF] and v(t)<v(_t0 ) for t∈_{(t0 ,^σ2 (b)]...} . The point _t0 is not simultaneously left-dense and right-scattered (see [12, Theorem 2.1]) (this implies that (σ[composite function]ρ)(_t0 )=_t0 ), and we have that ^vΔΔ (ρ(_t0 ))≤0 (see [12]), so [figure omitted; refer to PDF] So ^vΔΔ (ρ(_t0 ))>0 , that is a contradiction. And so we have proved that v(t)≤0, ∀t∈_{[a,^σ2 (b)]...} .

Analogously it can be proved that u(t)≤β(t), ∀t∈_{[a,^σ2 (b)]...} .

We only need to prove that problem (_Pm ) has at least one solution.

Consider now the operator N:C(_{[a,^σ2 (b)]...} )[arrow right]C(_{[a,^σ2 (b)]...} ) , defined by [figure omitted; refer to PDF] for each t∈_{[a,^σ2 (b)]...} , where (see [2]) [figure omitted; refer to PDF] is Green's function of the problem [figure omitted; refer to PDF] and for t∈[a,^σ2 (b)_]... [figure omitted; refer to PDF] is the solution of -^xΔΔ =0 such that [varphi](a)=A and [varphi](^σ2 (b))=B .

Clearly, G(t,s)>0 on _{(a,^σ2 (b))...} ×_{(a,^σ2 (b))...} , G(t,·) is rd-continuous on _[a,σ(b)]... and G(·,s) is continuous on _{[a,^σ2 (b)]...} .

The function Nu defined by (2.9) belongs to C([a,^σ2 (b)_]... ) because ^f* checks the conditions (_H1 ) and (_H2 ) on ^{(Jκ )o} ×... and G(t,s)≤s(1-s) , for each t,s∈_{[a,^σ2 (b)]...} .

It is obvious that u∈C(_{[a,^σ2 (b)]...} ) is a solution of (_Pm ) if and only if u=Nu . So the problem now is ensuring the existence of fixed-points of N .

First of all, N is well defined, is continuous, and N(C(_{[a,^σ2 (b)]...} )) is a bounded set. The existence of a fixed-point of N follows from the Schauder fixed-point theorem, once we have checked that N(C(_{[a,^σ2 (b)]...} )) is relatively compact, that using the Ascoli-Arzela theorem is equivalent to proving that N(C(_{[a,^σ2 (b)]...} )) is an equicontinuous family.

Let ^h* ∈E be the function related to ^f* by condition (_H2 ) . We compute the first derivative of Nu using [2, Theorem 1.117] [figure omitted; refer to PDF]

Finally it is enough to check that λ∈^L1 (^{(Jκ )o} ) , using integration by parts we obtain [figure omitted; refer to PDF] due to ^h* ∈E , and the fact [figure omitted; refer to PDF] And so the result is proved.

3. Existence and Uniqueness of Positive Solution

Let g:^{(Jκ )o} ×(0,+∞)[arrow right]... in the condition (_H1 ) , where ...9C;=(0,+∞) , and consider the problem [figure omitted; refer to PDF] 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 [straight epsilon]=_{[straight epsilon]D} >0 : [figure omitted; refer to PDF]

Theorem 3.1.

Suppose that (_H1 ) and (_H3 ) hold. If, for any δ>0, g satisfies 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 )o} as two sequences such that _{{an }n≥1} ⊂_{(a,(a+σ(b))/2)...} is strictly decreasing to a if a=σ(a) , and _an =a for all n≥1 if a<σ(a) , and _{{bn }n≥1} ⊂_{((a+σ(b))/2,σ(b))...} is strictly increasing to σ(b) if ρ(σ(b))=σ(b), _bn =ρ(σ(b)) for all n≥1 if ρ(σ(b))<σ(b) . We denote as _Dn :=_{[an ,bn ]...} ⊂^{(Jκ )o} , n≥1 .

Due to the first hypothesis, we can then ensure the existence of _{[straight epsilon]n} >0 such that g(t,x)>L , for all (t,x)∈_Dn ×(0,_{[straight epsilon]n} ] . We can suppose, without restriction that {_{[straight epsilon]n} } is a decreasing sequence and _{lim n[arrow right]+∞[straight epsilon]n} =0 .

Consider the function γ: [a,_a1]... [arrow right]^...+ , such that γ(a)=0 , and if a≠_a1 then γ(t)=_{[straight epsilon]2} for t∈[_a2 ,_a1 ], γ(t)=_{[straight epsilon]n} , for all t∈_Dn \_Dn-1 , with n≥3 . Since γ is nondecreasing, we obtain that for t∈[a,_a1 ] [figure omitted; refer to PDF] is continuous and increasing. Repeating this argument twice, for i=2,3 we define [figure omitted; refer to PDF] So _γ3 ∈^C2 (_{[a,a1 ]...} ) is a convex function verifying _γ3 (t)≤_{[straight epsilon]n} , ∀t∈_Dn \_Dn-1 , n≥2 .

Analogously, we can considerγ...:[_b1 ,^σ2 (b)_]... [arrow right]^...+ , such that γ...(^σ2 (b))=0, γ...(σ(b))=0, γ...(t)=_{[straight epsilon]2} , for all t∈[_b1 ,_b2 ] and γ...(t)=_{[straight epsilon]n} , for all t∈_Dn \_Dn-1 , with n≥3 . Taking _γ...0 =γ... , we obtain, for i=1,2,3 , that for t∈[_b1 ,^σ2 (b)] [figure omitted; refer to PDF] is continuous and decreasing, and _γ...3 ∈^C2 (_{[b1 ,^σ2 (b)]...} ) a convex function such that _γ...3 (t)≤_{[straight epsilon]n} , ∀t∈_Dn \_Dn-1 , n≥2 .

We now define [figure omitted; refer to PDF] with ^α* (t) a convenient function, so that α∈^C2 (_{[a,^σ2 (b)]...} ) , and 0<α(t)≤_{[straight epsilon]1} for all t∈_{[a1 ,b1 ]...} .

So α∈^C2 (_{[a,^σ2 (b)]...} ) is a function such that α(a)=α(^σ2 (b))=0, α(t)>0 for t∈_(a,σ(b))... , α(t)≤_{[straight epsilon]1} for t∈_D1 , and α(t)≤_{[straight epsilon]n} , for t∈_Dn \_Dn-1 , n≥2 .

In this way, we note that [figure omitted; refer to PDF]

Let _m0 =min {1,L/(^||α||ΔΔ +1)} .

Let F(t,x)≥g(t,x) , for all (t,x)∈^{(Jκ )o} ×(0,+∞) , with F in the conditions (_H1 ) . We will prove that if v∈^Crd2 (_{(a,^σ2 (b))...} ) is any solution of [figure omitted; refer to PDF] with v(a)≥0, v(^σ2 (b))≥0 and v(t)>0 for all t∈^{(Jκ )o} , then, [figure omitted; refer to PDF] Suppose there exists t∈_{[a,^σ2 (b)]...} , such that v(t)-_m0 α(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))...} verifying [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] which is a contradiction.

We define now, for each n∈... [figure omitted; refer to PDF] where, for each t∈^{(Jκ )o} [figure omitted; refer to PDF] For each n we have that _g...n is a function verifying (_H1 ) on ^{(Jκ )o} ×(0,+∞) , and _g...n (t,x)≥g(t,x) , for all (t,x)∈J×(0,+∞) and _g...n (t,x)=g(t,x) , for t∈_Dn .

The sequence _{{g...n }n} converges to g uniformly in every set of the form D×(0,+∞) , where D⊂(^Jκ)o is a compact set.

Defining, by induction, _g1 (t,x)=_g...1 (t,x) and for n>1 [figure omitted; refer to PDF] we have that, for each n , the function _gn satisfies the condition (_H1 ) on ^{(Jκ )o} ×(0,+∞) . As well _g1 ≥_g2 ≥...≥_gn ≥...≥g, and _{{gn }n} converges to g uniformly, in every set of the form D×(0,+∞) , where D⊂(^Jκ)o is a compact set.

It is also verified that _gn (t,x)=g(t,x), t∈[_an ,_bn ], x∈(0,+∞) .

Now we define the following problems: [figure omitted; refer to PDF] We will prove that for any c∈(0,_{[straight epsilon]n} ] , the constant function _αn ≡c is a (strict) lower solution for (Q_)n .

It is obvious that c=_αn (a)≤_{[straight epsilon]n} , and c=_αn (^σ2 (b))≤_{[straight epsilon]n} . Now we have to prove that [figure omitted; refer to PDF]

For n=1 , let c∈(0,_{[straight epsilon]1} ] such that [figure omitted; refer to PDF] Suppose now that _gn (t,c)>0, t∈^{(Jκ )o} , c∈(0,_{[straight epsilon]n} ] , for a given n≥1 , and we will check that _gn+1 (t,c)>0, t∈^{(Jκ )o} . Let c∈(0,_{[straight epsilon]n+1} ] such that [figure omitted; refer to PDF] Thus the assertion is proved.

Moreover, as _gn ≥_gn+1 on ^{(Jκ )o} ×(0,+∞) , it can be easily checked that any solution _un of (Q_)n is an upper solution for (Q_)n+1 .

To show that problem (Q₎₁ has at least one solution. We fix a constant M≥_{[straight epsilon]1} . From the assumption imposed, there exists a function _hM ∈_Crd (^{(Jκ )o} ,^...+ ) such that [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] with R>0 is a suitable constant.

Set q(t):=_hM (t)+R . We have that q∈_Crd (^{(Jκ )o} ,(0,+∞)) , and [figure omitted; refer to PDF] Let β∈^Crd2 (_(a,b)... ) be a solution of the boundary value problem: [figure omitted; refer to PDF] It is easy to check that such a function exists and that β(t)≥M , for all t∈_{[a,^σ2 (b)]...} , and thus [figure omitted; refer to PDF] which implies that β is an upper solution of (Q₎₁ .

Taking _α1 ≡_{[straight epsilon]1} , we have that _α1 and _β1 :=β are, respectively, a lower and upper solutions of (Q₎₁ . Moreover _α1 (t)≤_β1 (t) , for all t∈[a,^σ2 (b)_]... . Then, by Theorem 2.3 there exists a solution _u1 of (Q₎₁ such that [figure omitted; refer to PDF] Proceeding by induction, using _αn ≡_{[straight epsilon]n} , _βn ≡_un-1 , lower and upper solution of _(Q)n , n≥2 , we obtain (via Theorem 2.3) a sequence {_un } of solutions of _(Q)n , such that _{[straight epsilon]n} ≤_un (t)≤_un-1 (t) , for all t∈_{[a,^σ2 (b)]...} and _un (a)=_un (^σ2 (b))=_{[straight epsilon]n} , n≥2 .

Since _gn ≥g on ^{(Jκ )o} ×(0,+∞) , and _un is a solution of (Q_)n with _un (t)>0 for all t∈^{(Jκ )o} , then _un (t)≥_m0 α(t) , for every t∈_{[a,^σ2 (b)]...} .

To end, we will prove that [figure omitted; refer to PDF] is a solution of problem (Q) .

We have that, for any n≥1 , [figure omitted; refer to PDF]

Now let D⊂^{(Jκ )o} be a compact interval. There exists an index ^n* =^n* (D)∈... such that D⊂_Dn for all n≥^n* . Let n≥^n* , if t∈D , then [figure omitted; refer to PDF] Hence, _un verifies the first equality of problem (Q) for all t∈D , and for all n≥^n* . Moreover [figure omitted; refer to PDF]

Then by the Ascoli-Arzelá theorem, we can conclude that u... is a solution of (Q) in D . Since the compact D was arbitrary, we have that u...∈C(_(a,σ(b))... ) , and [figure omitted; refer to PDF] As u...(a)=u...(^σ2 (b))=_{lim n[arrow right]+∞[straight epsilon]n} =0 , we just have to check the continuity of u... in a and ^σ2 (b) .

Let [straight epsilon]>0 . Take _{n[straight epsilon]} in such a way that _{un[straight epsilon]} (a)<[straight epsilon] . From the continuity of _{un[straight epsilon]} (t) at a , it follows that we can find a constant δ=_{δ[straight epsilon]} >0 such that [figure omitted; refer to PDF] So [figure omitted; refer to PDF] Using the same argument the continuity at ^σ2 (b) is proved.

In order to prove the uniqueness of the solution to problem (Q) we have the following result.

Theorem 3.2.

Under the hypotheses of Theorem 3.1, if g(t,·) is strictly decreasing in x , for each t∈^{(Jκ )o} , then problem (Q) has a unique solution.

Proof.

By Theorem 3.1 the problem (Q) has at least one solution.

Suppose that there exist _v1 and _v2 , solutions of (Q) and ^t* ∈^{(Jκ )o} such that _v1 (^t* )>_v2 (^t* ) . If this occurs, setting z(t):=_v1 (t)-_v2 (t) we can find _t0 ∈_{(a,^σ2 (b))...} such that z(_t0 )>0 and ^zΔΔ (ρ(_t0 ))≤0 . Using the decreasing property of g(t,x) in x , we obtain [figure omitted; refer to PDF] that is a contradiction.

Corollary 3.3.

Let _p1 ,_p2 ∈_Crd (^{(Jκ )o} ), let_p1 >0 on ^{(Jκ )o} and _p1 +|_p2 |∈E . If ψ∈C((0,+∞)), ψ>0 is strictly decreasing on (0,+∞) and _{lim x[arrow right]⁰⁺} ψ(x)=+∞ , then problem [figure omitted; refer to PDF] has a unique solution in ^Crd2 (J) .

Proof.

Setting g(t,x):=_p1 (t)ψ(x)-_p2 (t) , and for each δ>0, _hδ (t):=_p1 (t)ψ(δ)+|_p2 (t)| , it is clear that the hypothesis in Theorem 3.2 holds. So, applying the result we have ensured the existence of one unique solution to the problem.

Example 3.4.

Let ...=C be the classical ternary Cantor set, and a=0, b=1 . For every t∈... , let _It ={i∈I:_ti ∈R∩[0,t)} with R=_{{ti }i∈I} the set of all right-scattered points of ... .

If we take _p1 (t)=1/^t1/2 σ(t)(1-t), _p2 (t)=0 and [figure omitted; refer to PDF] then problem (3.31) has a unique solution.

Acknowledgments

This research is partially supported by D.G.I. and F.E.D.E.R. project MTM2007-61724 and by the Xunta of Galicia and F.E.D.E.R. project PGIDIT06PXIB207023PR, Spain.

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