# Unbounded Solutions for Functional Problems on the Half-Line

Authors: Hugo Carrasco; Feliz Minhós
Journal: Abstract and Applied Analysis
 (2016)

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## Abstract

This paper presents an existence and localization result of unbounded solutions for a second-order differential equation on the half-line with functional boundary conditions. By applying unbounded upper and lower solutions, Green’s functions, and Schauder fixed point theorem, the existence of at least one solution is shown for the above problem. One example and one application to an Emden-Fowler equation are shown to illustrate our results.


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## Body

## 1. Introduction

The authors consider the following boundary value problem composed by the differential equation:(1)u′′t=ft,ut,u′t,t≥0,where f:[0,+∞[×R2→R is continuous and bounded by some L1 function, and the functional boundary conditions on the half-line are as follows:(2)Lu,u0,u′0=0,u′+∞=B,with B∈R and L:C[0,+∞[×R2→R a continuous function verifying some monotone assumption:(3)u′+∞≔limt→+∞⁡u′t.Boundary value problems on the half-line arise naturally in the study of radially symmetric solutions of nonlinear elliptic equations, and many works have been done in this area; see [1]. The functional dependence on the boundary conditions allows that problem (1), (2) covers a huge variety of boundary value problems such as separated, multipoint, nonlocal, integrodifferential, with maximum or minimum arguments, as it can be seen, for instance, in [2–11] and the references therein. However, to the best of our knowledge, it is the first time where this type of functional boundary conditions are applied to the half-line.Lower and upper solutions method is a very adequate technique to deal with boundary value problems as it provides not only the existence of bounded or unbounded solutions but also their localization and, from that, some qualitative data about solutions, their variation and behavior (see [12–14]). Some results are concerned with the existence of bounded or positive solutions, as in [15, 16], and the references therein. For problem (1), (2) we prove the existence of two types of solution, depending on B: if B≠0 the solution is unbounded and if B=0 the solution is bounded. In this way, we gather different strands of boundary value problems and types of solutions in a single method.The paper is organized as follows. In Section2 some auxiliary results are defined such as the adequate space functions, some weighted norms, a criterion to overcome the lack of compactness, and the definition of lower and upper solutions. Section 3 contains the main result: an existence and localization theorem, which proof combines lower and upper solution technique with the fixed-point theory. Finally, last two sections contain, to illustrate our results, an example and an application to some problem composed by a discontinuous Emden-Fowler-type equation with a infinite multipoint conditions, which are not covered by the existent literature.


## 2. Definitions and Auxiliary Results

Consider the space(4)X=x∈C10,+∞:limt→+∞⁡xt1+t∈R,limt→+∞⁡x′t∈Rwith the norm xX=maxx0,x′1, where (5)ω0≔sup0≤t<+∞⁡ωt1+t,ω′1≔sup0≤t<+∞⁡ω′t.In this way (X,·X) is a Banach space.Solutions of the linear problem associated to (1) and usual boundary conditions are defined with Green’s function, which can be obtained by standard calculus.Lemma 1.
Letth, h∈L1[0,+∞[. Then the linear boundary value problem composed by(6)u′′t=ht,t≥0,u0=A,u′+∞=B,for A,B∈R, has a unique solution in X, given by (7)ut=A+Bt+∫0+∞Gt,shsds,where(8)Gt,s=-s,0≤s≤t-t,t≤s<+∞.Proof.
Ifu is a solution of problem (6), then the general solution for the differential equation is (9)ut=c1+c2t+∫0tt-shsds,where c1, c2 are constants. Since u(t) should satisfy the boundary conditions, we get (10)c1=A,c2=B-∫0+∞hsds.The solution becomes (11)ut=A+Bt-t∫0+∞hsds+∫0tt-shsds.And by computation (12)ut=A+Bt+∫0+∞Gt,shsds,with G given by (8).
Conversely, ifu is a solution of (7), it is easy to show that it satisfies the differential equation in (6). Also u(0)=A and u′(+∞)=B.The lack of compactness ofX is overcome by the following lemma which gives a general criterion for relative compactness, referred to in [1].Lemma 2.
A setM⊂X is relatively compact if the following conditions hold:(1)
all functions fromM are uniformly bounded;(2)
all functions fromM are equicontinuous on any compact interval of [0,+∞[;(3)
all functions fromM are equiconvergent at infinity; that is, for any given ϵ>0, there exists a tϵ>0 such that (13)xt1+t-limt→+∞⁡xt1+t<ϵ,x′t-limt→+∞⁡x′t<ϵ∀t>tϵ,x∈M.The existence tool will be Schauder’s fixed point theorem.Theorem 3 (see [17]).
LetY be a nonempty, closed, bounded, and convex subset of a Banach space X, and suppose that P:Y→Y is a compact operator. Then P is at least one fixed point in Y.The functions considered as lower and upper solutions for the initial problem are defined as follows.Definition 4.
GivenB∈R, a function α∈X is a lower solution of problem (1), (2) if (14)α′′t≥ft,αt,α′t,t≥0,Lα,α0,α′0≥0,α′+∞<B.
A functionβ is an upper solution if it satisfies the reverse inequalities.


## 3. Existence and Localization Results

In this section we prove the existence of at least one solution for the problem (1), (2), and, moreover, some localization data.Theorem 5.
Letf:[0,+∞[×R2→R be a continuous function, verifying that, for each ρ>0, there exists a positive function φρ with φρ,tφρ∈L1[0,+∞[ such that for (x(t),y(t))∈R2 with sup0≤t<+∞|x(t)|/(1+t),y(t)<ρ,(15)ft,x,y≤φρt,t≥0.Moreover, if L(x1,x2,x3) is nondecreasing on x1 and x3 and there are α, β, lower and upper solutions of (1), (2), respectively, such that(16)αt≤βt,∀t≥0,then problem (1), (2) has at least one solution u∈X, with α(t)≤u(t)≤β(t), for t≥0.Proof.
Letα, β be, respectively, lower and upper solutions of (1), (2) verifying (16). Consider the modified problem(17)u′′t=ft,δt,ut,u′t+11+t3ut-δt,ut1+ut-δt,ut,t≥0,u0=δ0,u0+Lu,u0,u′0,u′+∞=B,where δ:[0,+∞[×R→R is given by (18)δt,x=βt,x>βtx,αt≤x≤βtαt,x<αt.
For clearness, the proof will follow several steps.
Step
1 (if
u
is a solution of (17), then
α
(
t
)
≤
u
(
t
)
≤
β
(
t
)
,
∀
t
≥
0
). Let u be a solution of the modified problem (17) and suppose, by contradiction, that there exists t≥0 such that α(t)>u(t). Therefore (19)inf0≤t<+∞⁡ut-αt<0.
If there ist∗∈]0,+∞[ such that (20)min0≤t<+∞⁡ut-αt≔ut∗-αt∗<0,we have u′(t∗)=α′(t∗) and u′′(t∗)-α′′(t∗)≥0. By Definition 4 we get the contradiction (21)0≤u′′t∗-α′′t∗=ft∗,δt∗,ut∗,u′t∗+11+t∗3ut∗-δt∗,ut∗1+ut∗-δt∗,ut∗-α′′t∗=ft∗,αt∗,α′t∗+11+t∗3ut∗-αt∗1+ut∗-αt∗-α′′t∗≤ut∗-αt∗1+ut∗-αt∗<0.
Sou(t)≥α(t), ∀t>0.
If the infimum is attained att=0 then (22)min0≤t<+∞⁡ut-αt≔u0-α0<0.As u is solution of (17), by the definition of δ, the following contradiction is achieved (23)0>u0-α0=δ0,u0+Lu,u0,u′0-α0≥α0-α0=0.
If(24)inf0≤t<+∞⁡ut-αt≔u+∞-α+∞<0,then u′(+∞)-α′(+∞)≤0. As u is solution of (17), by Definition 4, this contradiction holds (25)0≥u′+∞-α′+∞=B-α′+∞>0.
Thereforeu(t)≤α(t), ∀t≥0.
In a similar way we can prove thatu(t)≥β(t), ∀t≥0.
Step
2 (problem (17) has at least one solution). Let u∈X and define the operator T:X→X(26)Tut=Δ+Bt+∫0+∞Gt,sFusds,with (27)Fus≔fs,δs,us,u′s+11+s3us-δs,us1+us-δs,us,Δ:=δ(0,u(0)+L(u,u(0),u′(0))), and G is the Green function given by (8).
Therefore, problem (17) becomes(28)u′′t=Fut,t≥0,u0=Δ,u′+∞=B,and if tFu(t), Fu(t)∈L1[0,+∞[, by Lemma 1 it is enough to prove that T has a fixed point.
Step
2.1 (
T
is well defined). As f is a continuous function, Tu∈C1[0,+∞[ and, by (15), for any u∈X with ρ>maxuX,αX,βX(29)∫0+∞Fusds≤∫0+∞ϕρs+11+s3ds<+∞.That is Fu(t) and tFu(t)∈L1[0,+∞[. By Lebesgue Dominated Convergence Theorem, (30)limt→+∞⁡Tut1+t=limt→+∞⁡Δ+Bt1+t+∫0+∞limt→+∞⁡Gt,s1+tFusds≤B+∫0+∞ϕρs+11+s3ds<+∞,and analogously for (31)limt→+∞⁡Tu′t=B-limt→+∞⁡∫t+∞Fusds=B<+∞.
ThereforeTu∈X.
Step
2.2 (
T
is continuous). Consider a convergent sequence un→u in X; there exists ρ1>0 such that maxsupnunX,αX,βX<ρ1.
WithM:=sup0≤t<+∞|G(t,s)|/(1+t), we have (32)Tun-TuX=max⁡Tun-Tu0,Tun′-Tu′1≤∫0+∞MFuns-Fusds+∫t+∞Funs-Fusds⟶0,as n→+∞.
Step
2.3 (
T
is compact). Let B⊂X be any bounded subset. Therefore there is r>0 such that uX<r, ∀u∈B.
For eachu∈B, and for maxr,αX,βX<r1, (33)Tu0=sup0≤t<+∞⁡Tut1+t≤sup0≤t<+∞⁡Δ+Bt1+t+∫0+∞sup0≤t<+∞⁡Gt,s1+tFusds≤sup0≤t<+∞⁡Δ+Bt1+t+∫0+∞Mϕr1s+11+s3ds<+∞,Tu′1=sup0≤t<+∞⁡Tu′t≤B+∫t+∞Fusds≤B+∫t+∞ϕr1s+11+s3ds<+∞.
SoTuX=maxTu0,(Tu)′1<+∞; that is, TB is uniformly bounded in X.
T
B is equicontinuous, because, for L>0 and t1,t2∈[0,L], we have, as t1→t2,(34)Tut11+t1-Tut21+t2≤Δ+Bt11+t1-Δ+Bt21+t2+∫0+∞Gt1,s1+t1-Gt2,s1+t2Fusds≤Δ+Bt11+t1-Δ+Bt21+t2+∫0+∞Gt1,s1+t1-Gt2,s1+t2ϕr1s+11+s3ds⟶0,Tu′t1-Tu′t2=∫t1+∞Fusds-∫t2+∞Fusds≤∫t1t2Fusds≤∫t1t2ϕr1s+11+s3ds⟶0.
SoTB is equicontinuous.
MoreoverTB is equiconvergent at infinity, because, as t→+∞,(35)Tut1+t-limt→+∞⁡Tut1+t≤Δ+Bt1+t-B+∫0+∞Gt,s1+t+1Fusds≤Δ+Bt1+t-B+∫0+∞Gt,s1+t+1ϕρ1+11+s3ds⟶0,Tu′t-limt→+∞⁡Tu′t=∫t+∞Fusds≤∫t+∞ϕρ1+11+s3ds⟶0,ast⟶+∞.
So, by Lemma2, TB is relatively compact.
Step
2.4. Let D⊂X be a nonempty, closed, bounded, and convex subset. Then TD⊂D.
LetD⊂X defined by (36)D≔u∈X:uX≤ρ2with (37)ρ2≔max⁡ρ1,β0+B+∫0+∞Mϕρ1s+11+s3ds,B+∫t+∞ϕρ1s+11+s3ds,with ρ1 given by Step 2.1.
Foru∈D and t∈[0,+∞[, we get (38)Tu0≤sup0≤t<+∞β0+Bt1+t+∫0+∞Mϕρ1s+11+s3ds≤β0+B+M∫0+∞ϕρ1s+11+s3ds≤ρ2,(39)Tu′1≤B+∫t+∞ϕρ1s+11+s3ds≤ρ2.
ThenTuX≤ρ2; that is, TD⊂D.
Then, by Schauder’s Fixed Point Theorem,T has at least one fixed point u1∈X.
Step
3 (
u
1
is a solution of (1), (2)). By Step 1, as u1 is a solution of (17) then α(t)≤u1(t)≤β(t), ∀t∈[0,+∞[. So, the differential equation (1) is obtained. It remains to prove that α(0)≤u1(0)+L(u1,u1(0),u1′(0))≤β(0).
Suppose, by contradiction, thatα(0)>u1(0)+L(u1,u1(0),u1′(0)). Then (40)u10=δ0,u10+Lu1,u10,u1′0=α0and by the monotony of L and Definition 4, the following contradiction holds (41)0>u10+Lu1,u10,u1′0-α0=Lu1,α0,u1′0≥Lα,α0,α′0≥0.
Soα(0)≤u1(0)+L(u1,u1(0),u1′(0)) and in a similar way we can prove that u1(0)+L(u1,u1(0),u1′(0))≤β(0).
Therefore,u1 is a solution of (1), (2).A similar result can be obtained iff is a L1-Carathéodory function and(42)u′′t=ft,ut,u′t,a.e.t≥0.Definition 6.
A functionf:[0,+∞[×R2→R is said to be L1-Carathéodory if it verifies the following:(1)
for each(x,y)∈R2, t↦f(t,x,y) is measurable on [0,+∞[;(2)
for almost everyt∈[0,+∞[, (x,y)↦f(t,x,y) is continuous in R2;(3)
for eachρ>0, there exists a positive function φρ with φρ,tφρ∈L1[0,+∞[ such that, for (x(t),y(t))∈R2 with sup0≤t<+∞|x(t)|/(1+t),y(t)<ρ, (43)ft,x,y≤φρt,a.e.t∈0,+∞.However in this case an extra assumption onf must be assumed.Theorem 7.
Letf:[0,+∞[×R2→R be a L1-Carathéodory function such that f(t,x,y) is monotone on y.
If there areα, β, lower and upper solutions of (42), (2), respectively, such that (44)αt≤βt,∀t≥0,and L(x1,x2,x3) is nondecreasing on x1 and x3, then problem (42), (2) has at least one solution u∈X with α(t)≤u(t)≤β(t), ∀t≥0.Proof.
The proof is similar to Theorem5 except the first step.
Letu(t) be a solution of the modified problem composed by (45)u′′t=ft,δt,ut,u′t+11+t3ut-δt,ut1+ut-δt,ut,a.e.t>0,and the boundary conditions (46)u0=δ0,u0+Lu,u0,u′0,u′+∞=B.If, by contradiction, there is t∗∈]0,+∞[ such that (47)min0≤t<+∞⁡ut-αt≔ut∗-αt∗<0,then u′(t∗)=α′(t∗), u′′(t∗)-α′′(t∗)≥0, and there exists an interval I-:=]t-,t∗[ where u(t)<α(t), u′(t)≤α′(t), ∀t∈I-.
By Definition4 and if f(t,x,y) is nondecreasing on y, this contradiction holds for t∈I-: (48)0≤u′′t-α′′t=ft,δt,ut,u′t+11+t3ut-δt,ut1+ut-δt,ut-α′′t≤ft,αt,α′t+11+t3ut-αt1+ut-αt-α′′t≤ut-αt1+ut-αt<0.
The same remains valid iff is nonincreasing, considering an interval I+:=]t∗,t+[ where u(t)<α(t), u′(t)≥α′(t), ∀t∈I+.
So in both casesu(t)≥α(t), ∀t∈[0,+∞[.
The remaining steps are identical to the proof of Theorem5, and we omit them.


## 4. Example

Consider the second-order problem in the half-line with one functional boundary condition:(49)u′′t=sin⁡ut+1+u′t3+ute-t1+t3,t>0,4u20+min0≤t<+∞⁡ut+u′0-2=0,u′+∞=0,5.Remark that the above problem is a particular case of (1), (2) with (50)ft,x,y=sin⁡x+1+y3+xe-t1+t3,B=0,5,La,b,c=4b2+min0≤t<+∞⁡at+c-2.f is continuous in [0,+∞[, and, for u∈X, assumption (15) holds with φρ=k/1+t3, for some k>0 and ρ>1.AsL(a,b,c) is not decreasing in a and c, and the functions α(t)≡-1 and β(t)=t are lower and upper solutions for (49), respectively, then, by Theorem 5, there is at least an unbounded solution u of (49) such that (51)-1≤ut≤t,∀t∈0,+∞.


## 5. Application

Emden-Fowler-types equations (see [18]) can model, for example, the heat diffusion perpendicular to parallel planes by (52)∂2ux,t∂x2+αx∂ux,t∂x+afx,tgu+hx,t=∂ux,t∂t,0<x<t,where f(x,t)g(u)+h(x,t) means the nonlinear heat source and u(x,t) is the temperature.In the steady-state case, and withh(x,t)≡0, last equation becomes(53)u′′x+αxu′x+afxgu=0,x≥0.Iff(x)≡1 and g(u)=un, (53) is called the Lane-Emden equation of the first kind, whereas in the second kind one has g(u)=eu. Both cases are used in the study of thermal explosions. For more details see [19].In the literature, Emden-Fowler-types equations are associated to Dirichlet or Neumann boundary conditions (see [20, 21]).To the best of our knowledge, it is the first time where some Emden-Fowler is considered together with functional boundary conditions on the half-line.Consider that we are looking for nonnegative solutions for the problem composed by the discontinuous differential equation(54)u′′x=u′x1+x3+u4xex,a.e.x>0,coupled with the infinite multipoint conditions(55)∑n=1+∞anuηn-u0+u′0=0,u′+∞=δ,0<δ<1,where an and ηn are nonnegative sequences such that a1η1≥a2η2≥⋯≥anηn≥⋯, ∑n=1+∞anu(ηn), and ∑n=1+∞anηn are convergent with ∑n=1+∞anηn+k≤1-k, (0<k<1).This is a particular case of (42), (2), where (56)fx,y,z=z1+x3+y4ex,B=δ,Lv,y,z=∑n=1+∞anvηn-y+z.fx,y,z≤k11+x3+k2ex≔φrx,k1,k2>0,r>1.Asφr(x),xφr(x)∈L1[0,+∞[ thus f is L1-Carathéodory. Also f is monotone on z; more precisely f is nondecreasing on z. As L(v,y,z) is not decreasing in v and z, and functions α(x)≡0 and β(x)=x+k are lower and upper solutions for problem (54), (55), respectively, then, by Theorem 7, there is at least an unbounded and nonnegative solution u of (54), (55) such that (57)0≤ux≤x+k,∀x∈0,+∞.



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