AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2016/8987374 8987374 Research Article Unbounded Solutions for Functional Problems on the Half-Line Carrasco Hugo 1 Minhós Feliz 1,2 2 Han Maoan 1 Centro de Investigação em Matemática e Aplicações (CIMA) Universidade de Évora Rua Romão Ramalho 59 7000-671 Évora Portugal uevora.pt 2 Departamento de Matemática Escola de Ciências e Tecnologia Universidade de Évora 7000-671 Évora Portugal uevora.pt 2016 722016 2016 24 11 2015 30 12 2015 722016 2016 Copyright © 2016 Hugo Carrasco and Feliz Minhós. 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.

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.

1. Introduction

The authors consider the following boundary value problem composed by the differential equation:(1)ut=ft,ut,ut,t0,where f:[0,+[×R2R is continuous and bounded by some L1 function, and the functional boundary conditions on the half-line are as follows:(2)Lu,u0,u0=0,u+=B,with BR and L:C[0,+[×R2R a continuous function verifying some monotone assumption:(3)u+limt+ut.

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 . 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  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 ). 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 B0 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 Section 2 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=xC10,+:limt+xt1+tR,limt+xtRwith the norm xX=maxx0,x1, where (5)ω0sup0t<+ωt1+t,ω1sup0t<+ωt.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.

Let th, hL1[0,+[. Then the linear boundary value problem composed by(6)ut=ht,t0,u0=A,u+=B,for A,BR, has a unique solution in X, given by (7)ut=A+Bt+0+Gt,shsds,where(8)Gt,s=-s,0st-t,ts<+.

Proof.

If u 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-t0+hsds+0tt-shsds.And by computation (12)ut=A+Bt+0+Gt,shsds,with G given by (8).

Conversely, if u 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 of X is overcome by the following lemma which gives a general criterion for relative compactness, referred to in .

Lemma 2.

A set MX is relatively compact if the following conditions hold:

all functions from M are uniformly bounded;

all functions from M are equicontinuous on any compact interval of [0,+[;

all functions from M 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,xt-limt+xt<ϵt>tϵ,xM.

The existence tool will be Schauder’s fixed point theorem.

Theorem 3 (see [<xref ref-type="bibr" rid="B20">17</xref>]).

Let Y be a nonempty, closed, bounded, and convex subset of a Banach space X, and suppose that P:YY 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.

Given BR, a function αX is a lower solution of problem (1), (2) if (14)αtft,αt,αt,t0,Lα,α0,α00,α+<B.

A function β 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.

Let f:[0,+[×R2R be a continuous function, verifying that, for each ρ>0, there exists a positive function φρ with φρ,tφρL1[0,+[ such that for (x(t),y(t))R2 with sup0t<+|x(t)|/(1+t),y(t)<ρ,(15)ft,x,yφρt,t0.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)αtβt,t0,then problem (1), (2) has at least one solution uX, with α(t)u(t)β(t), for t0.

Proof.

Let α, β be, respectively, lower and upper solutions of (1), (2) verifying (16). Consider the modified problem(17)ut=ft,δt,ut,ut+11+t3ut-δt,ut1+ut-δt,ut,t0,u0=δ0,u0+Lu,u0,u0,u+=B,where δ:[0,+[×RR is given by (18)δt,x=βt,x>βtx,αtxβtαt,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 t0 such that α(t)>u(t). Therefore (19)inf0t<+ut-αt<0.

If there is t]0,+[ such that (20)min0t<+ut-αtut-αt<0,we have u(t)=α(t) and u(t)-α(t)0. By Definition 4 we get the contradiction (21)0ut-αt=ft,δt,ut,ut+11+t3ut-δt,ut1+ut-δt,ut-αt=ft,αt,αt+11+t3ut-αt1+ut-αt-αtut-αt1+ut-αt<0.

So u(t)α(t), t>0.

If the infimum is attained at t=0 then (22)min0t<+ut-αtu0-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,u0-α0α0-α0=0.

If(24)inf0t<+ut-αtu+-α+<0,then u(+)-α(+)0. As u is solution of (17), by Definition 4, this contradiction holds (25)0u+-α+=B-α+>0.

Therefore u(t)α(t), t0.

In a similar way we can prove that u(t)β(t), t0.

Step 2 (problem (17) has at least one solution). Let uX and define the operator T:XX(26)Tut=Δ+Bt+0+Gt,sFusds,with (27)Fusfs,δs,us,us+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)ut=Fut,t0,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, TuC1[0,+[ and, by (15), for any uX with ρ>maxuX,αX,βX(29)0+Fusds0+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+tFusdsB+0+ϕρs+11+s3ds<+,and analogously for (31)limt+Tut=B-limt+t+Fusds=B<+.

Therefore TuX.

Step 2.2 ( T is continuous). Consider a convergent sequence unu in X; there exists ρ1>0 such that maxsupnunX,αX,βX<ρ1.

With M:=sup0t<+|G(t,s)|/(1+t), we have (32)Tun-TuX=maxTun-Tu0,Tun-Tu10+MFuns-Fusds+t+Funs-Fusds0,as n+.

Step 2.3 ( T is compact). Let BX be any bounded subset. Therefore there is r>0 such that uX<r, uB.

For each uB, and for maxr,αX,βX<r1, (33)Tu0=sup0t<+Tut1+tsup0t<+Δ+Bt1+t+0+sup0t<+Gt,s1+tFusdssup0t<+Δ+Bt1+t+0+Mϕr1s+11+s3ds<+,Tu1=sup0t<+TutB+t+FusdsB+t+ϕr1s+11+s3ds<+.

So TuX=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 t1t2,(34)Tut11+t1-Tut21+t2Δ+Bt11+t1-Δ+Bt21+t2+0+Gt1,s1+t1-Gt2,s1+t2FusdsΔ+Bt11+t1-Δ+Bt21+t2+0+Gt1,s1+t1-Gt2,s1+t2ϕr1s+11+s3ds0,Tut1-Tut2=t1+Fusds-t2+Fusdst1t2Fusdst1t2ϕr1s+11+s3ds0.

So TB is equicontinuous.

Moreover TB is equiconvergent at infinity, because, as t+,(35)Tut1+t-limt+Tut1+tΔ+Bt1+t-B+0+Gt,s1+t+1FusdsΔ+Bt1+t-B+0+Gt,s1+t+1ϕρ1+11+s3ds0,Tut-limt+Tut=t+Fusdst+ϕρ1+11+s3ds0,ast+.

So, by Lemma 2, TB is relatively compact.

Step 2.4. Let DX be a nonempty, closed, bounded, and convex subset. Then TDD.

Let DX defined by (36)DuX:uXρ2with (37)ρ2maxρ1,β0+B+0+Mϕρ1s+11+s3ds,B+t+ϕρ1s+11+s3ds,with ρ1 given by Step 2.1.

For uD and t[0,+[, we get (38)Tu0sup0t<+β0+Bt1+t+0+Mϕρ1s+11+s3dsβ0+B+M0+ϕρ1s+11+s3dsρ2,(39)Tu1B+t+ϕρ1s+11+s3dsρ2.

Then TuXρ2; that is, TDD.

Then, by Schauder’s Fixed Point Theorem, T has at least one fixed point u1X.

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, that α(0)>u1(0)+L(u1,u1(0),u1(0)). Then (40)u10=δ0,u10+Lu1,u10,u10=α0and by the monotony of L and Definition 4, the following contradiction holds (41)0>u10+Lu1,u10,u10-α0=Lu1,α0,u10Lα,α0,α00.

So α(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 if f is a L1-Carathéodory function and(42)ut=ft,ut,ut,a.e.t0.

Definition 6.

A function f:[0,+[×R2R is said to be L1-Carathéodory if it verifies the following:

for each (x,y)R2, tf(t,x,y) is measurable on [0,+[;

for almost every t[0,+[, (x,y)f(t,x,y) is continuous in R2;

for each ρ>0, there exists a positive function φρ with φρ,tφρL1[0,+[ such that, for (x(t),y(t))R2 with sup0t<+|x(t)|/(1+t),y(t)<ρ, (43)ft,x,yφρt,a.e.t0,+.

However in this case an extra assumption on f must be assumed.

Theorem 7.

Let f:[0,+[×R2R be a L1-Carathéodory function such that f(t,x,y) is monotone on y.

If there are α, β, lower and upper solutions of (42), (2), respectively, such that (44)αtβt,t0,and L(x1,x2,x3) is nondecreasing on x1 and x3, then problem (42), (2) has at least one solution uX with α(t)u(t)β(t), t0.

Proof.

The proof is similar to Theorem 5 except the first step.

Let u(t) be a solution of the modified problem composed by (45)ut=ft,δt,ut,ut+11+t3ut-δt,ut1+ut-δt,ut,a.e.t>0,and the boundary conditions (46)u0=δ0,u0+Lu,u0,u0,u+=B.If, by contradiction, there is t]0,+[ such that (47)min0t<+ut-αtut-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), tI-.

By Definition 4 and if f(t,x,y) is nondecreasing on y, this contradiction holds for tI-: (48)0ut-αt=ft,δt,ut,ut+11+t3ut-δt,ut1+ut-δt,ut-αtft,αt,αt+11+t3ut-αt1+ut-αt-αtut-αt1+ut-αt<0.

The same remains valid if f is nonincreasing, considering an interval I+:=]t,t+[ where u(t)<α(t), u(t)α(t), tI+.

So in both cases u(t)α(t), t[0,+[.

The remaining steps are identical to the proof of Theorem 5, and we omit them.

4. Example

Consider the second-order problem in the half-line with one functional boundary condition:(49)ut=sinut+1+ut3+ute-t1+t3,t>0,4u20+min0t<+ut+u0-2=0,u+=0,5.

Remark that the above problem is a particular case of (1), (2) with (50)ft,x,y=sinx+1+y3+xe-t1+t3,B=0,5,La,b,c=4b2+min0t<+at+c-2.

f is continuous in [0,+[, and, for uX, assumption (15) holds with φρ=k/1+t3, for some k>0 and ρ>1.

As L(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)-1utt,t0,+.

5. Application

Emden-Fowler-types equations (see ) can model, for example, the heat diffusion perpendicular to parallel planes by (52)2ux,tx2+αxux,tx+afx,tgu+hx,t=ux,tt,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 with h(x,t)0, last equation becomes(53)ux+αxux+afxgu=0,x0.

If f(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 .

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)ux=ux1+x3+u4xex,a.e.x>0,coupled with the infinite multipoint conditions(55)n=1+anuηn-u0+u0=0,u+=δ,0<δ<1,where an and ηn are nonnegative sequences such that a1η1a2η2anηn, n=1+anu(ηn), and n=1+anηn are convergent with n=1+anηn+k1-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,zk11+x3+k2exφrx,k1,k2>0,r>1.

As φr(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)0uxx+k,x0,+.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by national founds of FCT Fundação para a Ciência e a Tecnologia, in the project UID/MAT/04674/2013 (CIMA).

Agarwal R. P. O'Regan D. Infinite Interval Problems for Differential, Difference and Integral Equations 2001 Glasgow, UK Kluwer Academic 10.1007/978-94-010-0718-4 MR1845855 Boucherif A. Second-order boundary value problems with integral boundary conditions Nonlinear Analysis: Theory, Methods & Applications 2009 70 1 364 371 10.1016/j.na.2007.12.007 MR2468243 2-s2.0-55549118921 Graef J. R. Kong L. Minhós F. M. Higher order boundary value problems with -Laplacian and functional boundary conditions Computers & Mathematics with Applications 2011 61 2 236 249 10.1016/j.camwa.2010.10.044 Grossinho M. R. Minhós F. Santos A. I. A note on a class of problems for a higher-order fully nonlinear equation under one-sided Nagumo-type condition Nonlinear Analysis: Theory, Methods & Applications 2009 70 11 4027 4038 10.1016/j.na.2008.08.011 MR2515319 2-s2.0-62949101139 Jiang J. Liu L. Wu Y. Second-order nonlinear singular Sturm Liouville problems with integral boundary conditions Applied Mathematics and Computation 2009 215 4 1573 1582 10.1016/j.amc.2009.07.024 MR2571646 2-s2.0-77953960751 Lian H. Wang P. Ge W. Unbounded upper and lower solutions method for Sturm-Liouville boundary value problem on infinite intervals Nonlinear Analysis: Theory, Methods & Applications 2009 70 7 2627 2633 10.1016/j.na.2008.03.049 MR2499729 2-s2.0-59849090468 Minhós F. Fialho J. On the solvability of some fourth-order equations with functional boundary conditions Discrete and Continuous Dynamical Systems 2009 supplement 564 573 Sun Y. Sun Y. Debnath L. On the existence of positive solutions for singular boundary value problems on the half-line Applied Mathematics Letter 2009 22 5 806 812 10.1016/j.aml.2008.07.009 MR2514916 2-s2.0-62249222504 Yan B. O'Regan D. Agarwal R. P. Unbounded solutions for singular boundary value problems on the semi-infinite interval: upper and lower solutions and multiplicity Journal of Computational and Applied Mathematics 2006 197 2 365 386 10.1016/j.cam.2005.11.010 MR2260412 2-s2.0-33747374885 Yoruk Deren F. Aykut Hamal N. Second-order boundary-value problems with integral boundary conditions on the real line Electronic Journal of Differential Equations 2014 2014 19 1 13 MR3159428 Zhang X. Ge W. Positive solutions for a class of boundary-value problems with integral boundary conditions Computers and Mathematics with Applications 2009 58 2 203 215 10.1016/j.camwa.2009.04.002 MR2535787 2-s2.0-67349249895 Cabada A. Tomecek J. Nonlinear second-order equations with functional implicit impulses and nonlinear functional boundary conditions Journal of Mathematical Analysis and Applications 2007 328 2 1013 1025 10.1016/j.jmaa.2006.06.011 MR2290029 2-s2.0-33845912694 Graef J. R. Kong L. Minhós F. M. Fialho J. On the lower and upper solution method for higher order functional boundary value problems Applicable Analysis and Discrete Mathematics 2011 5 1 133 146 10.2298/aadm110221010g MR2809041 2-s2.0-79953310556 Minhós F. Location results: an under used tool in higher order boundary value problems 1124 Proceedings of the International Conference on Boundary Value Problems: Mathematical Models in Engineering, Biology and Medicine September 2008 Santiago de Compostela, Spain 244 253 AIP Conference Proceedings 10.1063/1.3142939 Liu L. Wang Z. Wu Y. Multiple positive solutions of the singular boundary value problems for second-order differential equations on the half-line Nonlinear Analysis: Theory, Methods & Applications 2009 71 7-8 2564 2575 10.1016/j.na.2009.01.092 MR2532782 2-s2.0-67349157008 Yan B. O'Regan D. Agarwal R. P. Positive solutions for second order singular boundary value problems with derivative dependence on infinite intervals Acta Applicandae Mathematicae 2008 103 1 19 57 10.1007/s10440-008-9218-2 MR2415171 2-s2.0-45849139760 Zeidler E. Nonlinear Functional Analysis and Its Applications, I: Fixed- Point Theorems 1986 New York, NY, USA Springer Wong J. S. W. On the generalized Emden-Fowler equation SIAM Review 1975 17 2 339 360 10.1137/1017036 MR0367368 2-s2.0-0016495927 Harley C. Momoniat E. First integrals and bifurcations of a Lane-Emden equation of the second kind Journal of Mathematical Analysis and Applications 2008 344 2 757 764 10.1016/j.jmaa.2008.03.014 MR2426306 2-s2.0-43449087611 Habets P. Zanolin F. Upper and lower solutions for a generalized Emden-Fowler equation Journal of Mathematical Analysis and Applications 1994 181 3 684 700 10.1006/jmaa.1994.1052 MR1264540 ZBL0801.34029 2-s2.0-0000911571 Wazwaz A.-M. Adomian decomposition method for a reliable treatment of the Emden–Fowler equation Applied Mathematics and Computation 2005 161 2 543 560 10.1016/j.amc.2003.12.048 MR2112423 2-s2.0-10044247479