- Open Access
Nonlinear fourth order boundary value problem
Boundary Value Problems volume 2014, Article number: 189 (2014)
In this paper we consider a nonlinear boundary value problem generated by a fourth order differential equation on the semi-infinite interval in which the lim-4 case holds for fourth order differential expression at infinity. Using the well-known Banach and Schauder fixed point theorems we prove the existence and uniqueness theorems for the nonlinear boundary value problem.
MSC: 34A34, 34B15, 34B16, 34G20.
In the literature a kind of first order nonlinear boundary value problems is of the form
where A is a matrix defined on some interval , F is a vector which is continuous on , , T is a bounded linear operator defined on the space of bounded and continuous -valued functions on I and r is a vector in . Existence and uniqueness theorems of the solutions of the problem (1), (2) have been obtained in many papers. These results can be found in  and references therein. Similar nonlinear boundary value problem has been studied by Agarwal et al. on a time scale as
Second order nonlinear boundary value problems have been investigated by many authors. For example, Baxley has considered second order nonlinear boundary value problem on the semi-infinite interval as
as well as on the interval , where  (further see ). We should note that there are several works in the field of the existence and uniqueness theorems of the second order nonlinear boundary value problems. Some of them can be found in, for example, –. In particular, in  Guseinov and Yaslan have investigated the existence and uniqueness of the solutions of the second order nonlinear boundary value problem on the semi-infinite interval
α, β, γ, δ are real numbers satisfying and , are arbitrary real numbers, denotes the Wronskian of the solutions of (7) and u and v are the solutions of (7). Also they have studied the following nonlinear boundary value problem on the infinite interval :
where p, q are real-valued, measurable functions on such that Weyl’s limit-circle case holds for the equation
In fact, Weyl showed that  at least one of the linearly independent solutions of the equation
is in a squarely integrable space on , where λ is a complex parameter. This result follows from the convergence of the corresponding nested circles. These circles either converge to a circle or a point. In the primary case, two linearly independent solutions of (9) and any combinations of them belong to the squarely integrable space and (9) is said to be the limit-circle case. Otherwise (9) is said to be of limit-point case. These Weyl’s results have been generalized to the fourth order case as well as 2n th order case by Everitt – (further see ). Moreover, we should note that limit-point/circle classifications does not depend on the spectral parameter λ. Using these ideas we generalize the results of (5), (6) to the fourth order case as given in (10), (13)-(16). Using Banach and Schauder fixed point theorems we establish the existence and uniqueness theorems for the singular fourth order nonlinear boundary value problem (10), (13)-(16) in the lim-4 case.
2 Nonlinear problem
We consider the fourth order nonlinear differential equation as
where , y is the desired solution, , , , , , are the real-valued, continuous functions and on I. Further we assume that is real-valued and continuous function on , and for , satisfies the following condition:
where and .
It is well known that the r th quasi-derivative of a function y can be defined as follows :
Therefore (10) can be rewritten as
Let denote the Hilbert space consisting of all real-valued functions y such that with the inner product and the norm .
We assume that and the lim-4 case conditions are satisfied for the equation , –. In other words, we assume that four linearly independent solutions of , , belong to . In the literature there are sufficient conditions in which the lim-4 case holds for , –. For example, in 1977, Eastham  proved that the equation
has four linearly independent solutions belonging to if
Consider the set in consisting of all functions such that () is locally absolutely continuous function on I and . Then for arbitrary two functions y and χ in we have the Green’s formula
where and . Green’s formula implies that for two functions , the limit exists and is finite.
Let , , , and be the solutions of the equation
satisfying the conditions 
where , , , and () are real numbers satisfying
Since the lim-4 case holds for , , all the solutions and , , belong to and . It is clear that
where is the Kronecker delta and . This means that for arbitrary , the values , , and exist and are finite.
Using (12) we get
For , let us consider the following boundary conditions:
where and () are defined as above and and are some real numbers.
3 Green’s function
For , let us consider the following differential equation:
denotes the transpose of the vector y, and
where is defined by (20). Note that since the lim-4 case holds for , one finds that is a Hilbert-Schmidt kernel.
where . Hence (21) can be rewritten as
Therefore solving (23) in is equivalent to find the fixed points of ℒ.
Now we recall the well-known fixed point theorem.
Banach fixed point theorem
Let B be a Banach space and S a nonempty closed subset of B. Assumeis a contraction, i.e., there is a λ, , such thatfor all u, v in S. Then A has a unique fixed point in S.
Letsatisfies the condition (11) and the following Lipschitz condition: there is a constantso that
for all. If
If we take the value λ as
then we obtain from (26)
Therefore ℒ is a contraction mapping. Hence from the Banach fixed point theorem the proof is completed. □
Now consider the set K in as follows:
Letsatisfies the condition (11). Further let us assume that for allthe following inequality holds:
whereis a constant and may depend on R. If
This implies that . Since K is a closed subset of , the Banach fixed point theorem can be applied to obtain a unique solution of (21) in K. This completes the proof. □
4 Fixed points on Banach space
Nonlinear boundary value problems may have solutions without uniqueness. To show that the boundary value problem (10), (13)-(16) have solutions may be without uniqueness, we recall the following well-known theorems.
Schauder fixed point theorem
Let B be a Banach space and K a nonempty bounded, convex, and closed subset of B. Assumeis a completely continuous operator. Then L has at least one fixed point in K provided that.
A setis relatively compact if and only if for every, K is bounded, there exists asuch thatfor alland allwith, there exists a numbersuch thatfor all.
Now we can state the following theorem.
Letsatisfies the condition (11). Further we assume that there exists a numbersuch that
It is clear that K is bounded, convex, and closed. Further one can see that ℒ maps K into itself. Hence the proof of Theorem 4.2 will be completed with the next lemma. □
Let . Then for and we have
Since is a Hilbert-Schmidt kernel, we take . Therefore, one immediately gets
Hence we have obtained for and : there exists a such that implies for . This implies that ℒ is continuous.
Now consider the bounded set
Taking into account (30) we get for all
On the other side from (11) we get
This implies that
for all . Therefore is bounded in .
Now for let us consider the inequality
Since is a Hilbert-Schmidt kernel for there exists a () such that
for all and all with .
Moreover, for we get
This implies that for there exists () such that . Consequently is a relatively compact set in . This completes the proof of Lemma 4.3 and therefore Theorem 4.2. □
Kartsatos AG: A boundary value problem on an infinite interval. Proc. Edinb. Math. Soc. 1973, 19: 245-252. 10.1017/S0013091500015510
Agarwal RP, Bohner M, O’Rean D: Time scale systems on infinite intervals. Nonlinear Anal. 2001, 47: 837-848. 10.1016/S0362-546X(01)00227-9
Baxley JV: Existence and uniqueness for nonlinear boundary value problems on infinite intervals. J. Math. Anal. Appl. 1990, 147: 122-133. 10.1016/0022-247X(90)90388-V
Baxley JV: Existence theorems for nonlinear second order boundary value problems. J. Differ. Equ. 1990, 85: 125-150. 10.1016/0022-0396(90)90092-4
Topal SG, Yantır A, Cetin E: Existence of positive solutions of a Sturm-Liouville BVP on an unbounded time scale. J. Differ. Equ. Appl. 2008, 14: 287-293. 10.1080/10236190701596508
Kosmatov N: Second order boundary value problems on an unbounded domain. Nonlinear Anal. 2008, 68: 875-882. 10.1016/j.na.2006.11.043
Andres J, Pavlačková M: Asymptotic boundary value problems for second-order differential systems. Nonlinear Anal. 2009, 71: 1462-1473. 10.1016/j.na.2008.12.013
Lian H, Wang P, Ge W: Unbounded upper and lower solutions method for Sturm-Liouville boundary value problem on infinite intervals. Nonlinear Anal. 2009, 70: 2627-2633. 10.1016/j.na.2008.03.049
Lian H, Ge W: Calculus of variations for a boundary value problem of differential system on the half line. Comput. Math. Appl. 2009, 58: 58-64. 10.1016/j.camwa.2009.03.088
Liu B, Liu L, Wu Y:Unbounded solutions for three-point boundary value problems with nonlinear boundary conditions on . Nonlinear Anal. 2010, 73: 2923-2932. 10.1016/j.na.2010.06.052
Guseinov GS, Yaslan I: Boundary value problems for second order nonlinear differential equations on infinite intervals. J. Math. Anal. Appl. 2004, 290: 620-638. 10.1016/j.jmaa.2003.10.013
Titchmarsh EC: Eigenfunction Expansions Associated with Second Order Differential Equations. Part I. 2nd edition. Oxford University Press, London; 1962.
Naimark MA: Linear Differential Operators. 2nd edition. Nauka, Moscow; 1969.
Coddington EA, Levinson N: Theory of Ordinary Differential Equations. McGraw-Hill, New York; 1955.
Weyl H: Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen wilkürlichen Funktionen. Math. Ann. 1910, 68: 220-269. 10.1007/BF01474161
Everitt WN: Fourth order singular differential equations. Math. Ann. 1963, 149: 320-340. 10.1007/BF01471126
Everitt WN: Singular differential equations I: the even order case. Math. Ann. 1964, 156: 9-24. 10.1007/BF01359977
Everitt WN: Singular differential equations II: some self-adjoint even order cases. Q. J. Math. 1967, 18: 13-32. 10.1093/qmath/18.1.13
Devinatz A: The deficiency index of certain fourth-order ordinary self-adjoint differential operators. Q. J. Math. 1972, 23: 267-286. 10.1093/qmath/23.3.267
Walker PW: Asymptotics for a class of fourth order differential equations. J. Differ. Equ. 1972, 11: 321-334. 10.1016/0022-0396(72)90048-4
Eastham MSP: The limit-4 case of fourth-order self-adjoint differential equations. Proc. R. Soc. Edinb., Sect. A 1977, 79: 51-59.
Becker RI: Limit circle criteria for fourth order differential operators with an oscillatory coefficient. Proc. Edinb. Math. Soc. 1981, 24: 105-117. 10.1017/S0013091500006404
Fulton CT: The Bessel-squared equation in the lim-2, lim-3 and lim-4 cases. Q. J. Math. 1989, 40: 423-456. 10.1093/qmath/40.4.423
Kodaira K: On ordinary differential equations of any even order and the corresponding eigenfunction expansions. Am. J. Math. 1950, 72: 502-544. 10.2307/2372051
Krasnosel’skii MA: Topological Methods in the Theory of Nonlinear Integral Equations. Gostekhteoretizdat, Moscow; 1956.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
About this article
Cite this article
Yardımcı, Ş., Uğurlu, E. Nonlinear fourth order boundary value problem. Bound Value Probl 2014, 189 (2014). https://doi.org/10.1186/s13661-014-0189-0
- nonlinear problem
- fourth order problem
- Banach fixed point theorem
- Schauder fixed point theorem