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Existence of a solution for a three-point boundary value problem for a second-order differential equation at resonance

Abstract

We present a new existence result for a second-order nonlinear ordinary differential equation with a three-point boundary value problem when the linear part is noninvertible.

MSC:34B10, 34B15.

1 Introduction

The study of multi-point boundary value problems for linear second-order ordinary differential equations goes back to the method of separation of variables [1]. Also, some questions in the theory of elastic stability are related to multi-point problems [2]. In 1987, Il’in and Moiseev [3, 4] studied some nonlocal boundary value problems. Then, for example, Gupta [5] considered a three-point nonlinear boundary value problem. For some recent works on nonlocal boundary value problems, we refer, for example, to [615] and references therein.

As indicated in [16], there has been enormous interest in nonlinear perturbations of linear equations at resonance since the seminal paper of Landesman and Lazer [17]; see [18] for further details.

Here we study the following nonlinear ordinary differential equation of second order subject to the three-point boundary condition:

u ( t ) = f ( t , u ( t ) ) , t [ 0 , T ] , u ( 0 ) = 0 , α u ( η ) = u ( T ) ,
(1)

where T>0, f:[0,T]×RR is a continuous function αR and η(0,T).

In this paper we consider the resonance case αη=T to obtain a new existence result. Although this situation has already been considered in the literature [19], we point out that our approach and methodology is different.

2 Linear problem

Consider the linear second-order three-point boundary value problem

u ( t ) = σ ( t ) , t [ 0 , T ] , u ( 0 ) = 0 , α u ( η ) = u ( T )
(2)

for a given function σC[0,T].

The general solution is

u(t)= c 1 + c 2 t 0 t (ts)σ(s)ds

with c 1 , c 2 arbitrary constants.

From u(0)=0, we get c 1 =0. From the second boundary condition, we have

(Tαη) c 2 = 0 T (Ts)σ(s)dsα 0 η (ηs)σ(s)ds.
(3)

2.1 Nonresonance case

If αηT, then

c 2 = 1 T α η [ 0 T ( T s ) σ ( s ) d s α 0 η ( η s ) σ ( s ) d s ] ,

and the linear problem (2) has a unique solution for any σC[0,T]. In this case, we say that (2) is a nonresonant problem since the homogeneous problem has only the trivial solution as a solution, i.e., when σ=0, c 1 = c 2 =0 and u=0. Note that the solution is given by

u(t)= 0 T g(t,s)σ(s)ds
(4)

with

g(t,s)={ t ( T s ) T α η t α ( η s ) T α η ( t s ) , 0 s < min ( η , t ) , t ( T s ) T α η t α ( η s ) T α η , 0 t < s < η < T , t ( T s ) T α η ( t s ) , 0 η < s < t T , t ( T s ) T α η , max ( η , t ) < s T .

For T=1 this is precisely the function given in Lemma 2.3 of [20] or in Remark 12 of [21].

2.2 Resonance case

If T=αη, then (3) is solvable if and only if

0 T (Ts)σ(s)ds=α 0 η (ηs)σ(s)ds,
(5)

and then (2) has a solution if and only if (5) holds. In such a case, (2) has an infinite number of solutions given by

u(t)=ct 0 t (ts)σ(s)ds,cR.

In particular ct, cR is a solution of the homogeneous linear equation

u (t)=0,t[0,T]

satisfying the boundary conditions

u(0)=0,αu(η)=u(T).

Note that

u(T)u(η)= c 2 T 0 T (Ts)σ(s)ds c 2 η+ 0 η (ηs)σ(s)ds,

and then

c 2 = 1 T η [ u ( T ) u ( η ) + 0 T ( T s ) σ ( s ) d s 0 η ( η s ) σ ( s ) d s ] .

We now use that u(T)= T η u(η) to get

1 T η [ u ( T ) u ( η ) ] = 1 T u(T)

and

c 2 = 1 T η [ 0 T ( T s ) σ ( s ) d s 0 η ( η s ) σ ( s ) d s ] + 1 T u(T).

Hence the solution of (2) is given, implicitly, as

u(t)= 0 T t ( T s ) T η σ(s)ds 0 η t ( η s ) T η σ(s)ds 0 t (ts)σ(s)ds+ t T u(T)

or, equivalently,

u(t)= 0 T k(t,s)σ(s)ds+ t T u(T),
(6)

where

k(t,s)={ s , 0 s < min ( η , t ) , t , 0 t < s < η T , t ( T s ) T η ( t s ) , 0 η < s < t T , t ( T s ) T η , max ( η , t ) < s T .

We note that kC([0,T]×[0,T],R) and k(t,s)0 for every (t,s)[0,T]×[0,T].

3 Nonlinear problem

Defining the operators:

F : C [ 0 , T ] C [ 0 , T ] , [ F u ] ( t ) = f ( t , u ( t ) ) , u C [ 0 , T ] , t [ 0 , T ] , K : C [ 0 , T ] C [ 0 , T ] , [ K σ ] ( t ) = 0 T k ( t , s ) σ ( s ) d s , σ C [ 0 , T ] , t [ 0 , T ] , L : C [ 0 , T ] C [ 0 , T ] , [ L u ] ( t ) = t T u ( T ) , u C [ 0 , T ] , t [ 0 , T ] ,

the nonlinear problem is equivalent to

u=Nu,

where N=KF+L.

We note that (6) can be written as

u(t) t T u(T)= 0 T k(t,s)σ(s)ds

and the nonlinear problem (1) as

u(t) t T u(T)= 0 T k(t,s)f ( s , u ( s ) ) ds.

This suggests to introduce the new function v(t)=u(t) t T u(T). To find a solution u, we have to find v and u(T).

For every constant cR, we solve

v(t)= 0 T k(t,s)f ( s , v ( s ) + s T c ) ds
(7)

and let φ(c) be the set of solutions of (7). This set may be empty (no solution), a singleton (unique solution) or with more than one element (multiple solutions). For every v c φ(c), we consider

u c (t)= v c (t)+ t T c,

and hence

u c (t)= 0 T k(t,s)f ( s , u c ( s ) ) ds+ t T c.

If c= u c (T), then u c is a solution of the nonlinear problem (1). We then look for fixed points of the map

cR u c (T)R.

For cR fixed, we try to solve the integral equation (7).

Assume that there exist a,bC[0,T] and α[0,1) such that

| f ( t , u ) | a(t)+b(t) | u | α
(8)

for every t[0,T], uR.

For vC[0,T], define F c vC[0,T] as

[ F c v](t)=f ( t , v ( t ) + t T c ) .

Thus, a solution of (7) is precisely a fixed point of K F c = K c . Note that K c is a compact operator. For vC[0,T], let v= sup t [ 0 , T ] |v(t)|.

For λ(0,1), if v=λ K c (v) we have

v(t)=λ 0 T k(t,s)f ( s , v ( s ) + s T c ) ds,

and

| v ( t ) | k 0 T f ( s , v ( s ) + s T c ) dskT [ a + b ( v + c ) α ] .

Hence there exist constants a 0 , b 0 such that

v a 0 + b 0 ( v + c ) α
(9)

for any vC[0,T] and λ(0,1) solution of v=λ K c (v). This implies that v is bounded independently of λ(0,1), and hence by Schaefer’s fixed point theorem (Theorem 4.3.2 of [22]), K c has at least a fixed point, i.e., for given c, equation (7) is solvable.

Now suppose f is Lipschitz continuous.

Then there exists l>0 such that

| f ( t , x ) f ( t , y ) | l|xy|
(10)

for every t[0,T] and x,yR.

Then, for v,wC[0,T], we have

| [ K c v ] ( t ) [ K c w ] ( t ) | 0 T k(t,s)l | v ( s ) w ( s ) | ds

and

K c v K c wklTvw.

Thus, for l>0 small, equation (7) has a unique solution in view of the classical Banach contraction fixed point theorem.

Now, under conditions (8) and (10), set

cR v c C[0,T],

where v c is the unique solution of (7), and as a consequence of the contraction principle, this map is continuous.

Define the map

φ : R R , φ ( c ) = v c ( T ) .

If there exists cR such that φ(c)=0, then for that c we have v c (T), and the function

u c (t)= v c (t)+ t T c

is such that u c (T)=c, and therefore u c is a solution of the original nonlinear problem (1).

Now, assume that

lim u ± f(t,u)=±
(11)

uniformly on t[0,T].

Then the growth of v is sublinear in view of estimate (9). However, c growths linearly. Hence the norm of the function

v c (s)+ s T c

growths asymptotically as c.

This implies that lim c ± φ(c)=±, and there exists cR with φ(c)=0.

We have the following result.

Theorem 3.1 Suppose that f satisfies the growth conditions (8) and (10). If (11) holds, then (1) is solvable for l sufficiently small.

Note that condition (11) is crucial since for f(t,u)=σ(t) and, in view of (5), the problem (1) may have no solution.

References

  1. Gregus M, Neumann F, Arscott FM: Three-point boundary value problems for differential equations. J. Lond. Math. Soc. 1971, 3: 429–436.

    Article  MATH  Google Scholar 

  2. Timoshenko S: Theory of Elastic Stability. McGraw-Hill, New York; 1961.

    Google Scholar 

  3. Il’in VA, Moiseev EI: Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Differ. Equ. 1987, 23: 803–810.

    MATH  Google Scholar 

  4. Il’in VA, Moiseev EI: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator. Differ. Equ. 1987, 23: 979–987.

    MATH  Google Scholar 

  5. Gupta CP: Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation. J. Math. Anal. Appl. 1992, 1683: 540–551.

    Article  Google Scholar 

  6. An Y: Existence of solutions for a three-point boundary value problem at resonance. Nonlinear Anal. 2006, 65: 1633–1643. 10.1016/j.na.2005.10.044

    Article  MATH  MathSciNet  Google Scholar 

  7. Franco D, Infante G, Zima M: Second order nonlocal boundary value problems at resonance. Math. Nachr. 2011, 284: 875–884. 10.1002/mana.200810841

    Article  MATH  MathSciNet  Google Scholar 

  8. Han X, Gao H, Xu J: Existence of positive solutions for nonlocal fourth-order boundary value problem with variable parameter. Fixed Point Theory Appl. 2011., 2011: Article ID 604046

    Google Scholar 

  9. Infante G, Zima M: Positive solutions of multi-point boundary value problems at resonance. Nonlinear Anal. 2008, 69: 2458–2465. 10.1016/j.na.2007.08.024

    Article  MATH  MathSciNet  Google Scholar 

  10. Pietramala P: A note on a beam equation with nonlinear boundary conditions. Bound. Value Probl. 2011., 2011: Article ID 376782

    Google Scholar 

  11. Ma R: A survey on nonlocal boundary value problems. Appl. Math. E-Notes 2007, 7: 257–279.

    MATH  MathSciNet  Google Scholar 

  12. Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc. 2006, 74: 673–693. 10.1112/S0024610706023179

    Article  MATH  MathSciNet  Google Scholar 

  13. Webb JRL, Zima M: Multiple positive solutions of resonant and non-resonant nonlocal boundary value problems. Nonlinear Anal. 2009, 71: 1369–1378. 10.1016/j.na.2008.12.010

    Article  MATH  MathSciNet  Google Scholar 

  14. Zhang P: Existence of positive solutions for nonlocal second-order boundary value problem with variable parameter in Banach spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 43

    Google Scholar 

  15. Zhang H-E, Sun J-P: Positive solutions of third-order nonlocal boundary value problems at resonance. Bound. Value Probl. 2012., 2012: Article ID 102

    Google Scholar 

  16. Korman P: Nonlinear perturbations of linear elliptic systems at resonance. Proc. Am. Math. Soc. 2012, 140: 2447–2451. 10.1090/S0002-9939-2011-11288-7

    Article  MATH  MathSciNet  Google Scholar 

  17. Landesman EM, Lazer AC: Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech. 1970, 19: 609–623.

    MATH  MathSciNet  Google Scholar 

  18. Ambrosetti A, Prodi G: A Primer of Nonlinear Analysis. Cambridge University Press, Cambridge; 1993.

    Google Scholar 

  19. Ma R: Multiplicity results for a three-point boundary value at resonance. Nonlinear Anal. 2003, 53: 777–789. 10.1016/S0362-546X(03)00033-6

    Article  MATH  MathSciNet  Google Scholar 

  20. Lin B, Lin L, Wu Y: Positive solutions for singular second order three-point boundary value problems. Nonlinear Anal. 2007, 66: 2756–2766. 10.1016/j.na.2006.04.005

    Article  MathSciNet  Google Scholar 

  21. Webb JRL: Positive solutions of some three-point boundary value problems via fixed point index theory. Nonlinear Anal. 2001, 47: 4319–4332. 10.1016/S0362-546X(01)00547-8

    Article  MATH  MathSciNet  Google Scholar 

  22. Smart DR: Fixed Point Theorems. Cambridge University Press, Cambridge; 1974.

    MATH  Google Scholar 

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Acknowledgements

This research has been partially supported by Ministerio de Economía y Competitividad (Spain), project MTM2010-15314, and co-financed by the European Community fund FEDER. The author is thankful to the referees for their useful suggestions.

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Nieto, J.J. Existence of a solution for a three-point boundary value problem for a second-order differential equation at resonance. Bound Value Probl 2013, 130 (2013). https://doi.org/10.1186/1687-2770-2013-130

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