JFSA Journal of Function Spaces and Applications 1758-4965 0972-6802 Hindawi Publishing Corporation 641617 10.1155/2013/641617 641617 Research Article Existence Results for a Fully Fourth-Order Boundary Value Problem Li Yongxiang Liang Qiuyan Ma To Department of Mathematics, Northwest Normal University Lanzhou 730070 China nwnu.edu.cn 2013 22 7 2013 2013 21 12 2012 27 06 2013 2013 Copyright © 2013 Yongxiang Li and Qiuyan Liang. 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.

We discuss the existence of solution for the fully fourth-order boundary value problem u(4)=f(t,u,u,u′′,u′′′), 0t1, u(0)=u(1)=u′′(0)=u′′(1)=0. A growth condition on f guaranteeing the existence of solution is presented. The discussion is based on the Fourier analysis method and Leray-Schauder fixed point theorem.

1. Introduction and Main Results

In this paper we deal with the existence of solution for the fully fourth-order ordinary differential equation boundary value problem (BVP) (1)u(4)(t)=f(t,u(t),u(t),u′′(t),u′′′(t)),0t1,u(0)=u(1)=u′′(0)=u′′(1)=0, where f:[0,1]×4 is continuous. This problem models deformations of an elastic beam whose two ends are simply supported in equilibrium state, and its research has important significance in mechanics.

For the special case of BVP(1) that f does not contain derivative terms u and u′′′, namely, simply fourth-order boundary value problem (2)u(4)(t)=f(t,u(t),u′′(t)),0t1,u(0)=u(1)=u′′(0)=u′′(1)=0,

the existence of solution has been studied by many authors; see . In , Aftabizadeh showed the existence of a solution to PBV(2) under the restriction that f is a bounded function. In [2, Theorem 1], Yang extended Aftabizadeh’s result and showed the existence for BVP(2) under the growth condition of the form (3)|f(t,u,v)|a|u|+b|v|+c, where a, b, and c are positive constants such that (4)a  π4+b  π2<1.

In , under a more general linear growth condition of two-parameter nonresonance, del Pino and Manásevich also discussed the existence of BVP(2) and the result of Yang was further extended. For more results involving two-parameter nonresonance condition see [4, 7]. All these works are based on Leray-Schauder degree theory. In [5, 6], the upper and lower solutions method is applied to discuss the existence of BVP(2). Recently, in  the fixed point index theory in cones is employed to BVP(2) and some existence results of positive are obtained, where f may be super-linear growth.

For the more simple case of BVP(1) that f does not contain any derivative terms, the following fourth-order boundary value problem (5)u(4)(t)=f(t,u(t)),0t1,u(0)=u(1)=u′′(0)=u′′(1)=0,

has been studied by more researchers, and various theorems and methods of nonlinear analysis have been applied; see  and reference therein.

However, few researchers consider the fully fourth-order boundary value problem BVP(1). The purpose of this paper is to discuss the existence of solution of BVP(1). We will extend the Yang’s result previously mentioned from BVP(2) to the general BVP(1). Our results are as follows.

Theorem 1.

Assume that fC([0,1]×4,) and it satisfies the growth condition (6)|f(t,x0,x1,x2,x3)|c0|x0|+c1|x1|+c2|x2|+c3|x3|+M,

for all t[0,1] and (x0,x1,x2,x3)4, where c0,c1,c2,c30 and M>0 are constants and c0,c1,c2,c3 satisfy the restriction (7)c0π4+c1π3+c2π2+c3π<1.

Then the BVP(1) possesses at least one solution.

Theorem 1 is a directly extension of Yang’s result previously mentioned. In Theorem 1, the condition (7) is optimal. If the condition (7) does not hold, the existence of solution of BVP(1) cannot be guaranteed. Strengthening the condition (6) of Theorem 1, we can obtain the following uniqueness result.

Theorem 2.

Assume that fC([0,1]×4,) and it satisfies the Lipschitz-type condition (8)|f(t,x0,x1,x2,x3)-f(t,y0,y1,y2,y3)|i=03  ci|xi-yi|,

for any (t,x0,x1,x2,x3) and (t,y0,y1,y2,y3)[0,1]×4, where c0, c1, c2, c30 are constants and satisfy (7). Then BVP(1) has a unique solution.

If the partial derivatives fx0, fx1, fx2, and fx3 exist, then from Theorem 2 and the theorem of differential mean value, we have the following.

Corollary 3.

Let fC([0,1]×4,) and the partial derivatives fx0, fx1, fx2, and fx3 exist. If there exist positive constants c0,c1,c2,c3 such that (9)|fxi(t,x0,x1,x2,x3)|ci,i=0,1,2,3,

and the constants c0,c1,c2,c3 satisfy (7), then BVP(1) has one unique solution.

The proofs of Theorems 1 and 2 are based on the Fourier analysis method and Leray-Schauder fixed point theorem, which will be given in Section 2.

2. Proof of the Main Results

Let I=[0,1] and H=L2(I) be the usual Hilbert space with the interior product (u,v)=01u(t)v(t)dt and the norm u2=(01|u(t)|2dt)1/2. For m, let Wm,2(I) be the usual Sobolev space with the norm um,2=i=0mu(i)22. uWm,2(I) means that uCm-1(I), u(m-1)(t) is absolutely continuous on I and u(m)L2(I).

Given hL2(I), we consider the linear fourth-order boundary value problem (LBVP) (10)u(4)(t)=h(t),tI,u(0)=u(1)=u′′(0)=u′′(1)=0.

Let G(t,s) be the Green’s function to the second-order linear boundary value problem (11)-u′′=0,u(0)=u(1)=0,

which is explicitly expressed by (12)G(t,s)={t(1-s),0ts1,s(1-t),0st1.

For every given hL2(I), it is easy to verify that the LBVP(10) has a unique solution uW4,2(I) in Carathéodory sense, which is given by (13)u(t)=01G(t,τ)G(τ,s)h(s)dsdτ:=Sh(t).

If hC(I), the solution is in C4(I) and is a classical solution. Moreover, the solution operator of LBVP(10), S:L2(I)W4,2(I) is a linearly bounded operator. By the compactness of the Sobolev embedding W4,2(I)C3(I) and the continuity of embedding C3(I)W3,2(I), we see that S maps L2(I) into W3,2(I) and S:L2(I)W3,2(I) is a completely continuous operator.

Choose a subspace of W3,2(I) by (14)D={uW3,2(I)u(0)=u(1)=0,u′′(0)=u′′(1)=0}.

Clearly, D is a closed subspace, and hence D is a Banach space by the norm u3,2 of W3,2(I). Define another norm on D by (15)uX=u′′′2,uD.

One easily verifies that uX is equivalent to u3,2. Hereafter, we use X to denote the Banach space D endowed the norm uX, namely, (16)X=(D,·X).

By the boundary condition of LBVP(10), the solution operator S maps H into D. Hence S:HX is completely continuous.

Lemma 4.

For LBVP(10), the following two conclusions hold.

The norm of the solution operator of LBVP(10) S:HX satisfies S(H,X)1/π.

For every hH, the unique solution of LBVP(10) uW4,2(I) satisfies the inequalities (17)u21π3u′′′2,u21π2u′′′2,u′′21πu′′′2.

Proof.

Since sine system {sinkπtk} is a complete orthogonal system of L2(I), every hL2(I) can be expressed by the Fourier series expansion (18)h(t)=k=1  hksinkπt, where hk=201h(s)sinkπsds, k=1,2,, and the Parseval equality (19)h22=12k=1  |hk|2,

holds. Let u=Sh; then uW4,2(I) is the unique solution of LBVP(10), and u, u′′, and u(4) can be expressed by the Fourier series expansion of the sine system. Since u(4)=h, by the integral formula of Fourier coefficient, we obtain that (20)u(t)=k=1  hkk4π4sinkπt,u′′(t)=-k=1  hkk2π2sinkπt.

On the other hand, since cosine system {coskπtk=0,1,2,} is another complete orthogonal system of L2(I), every vL2(I) can be expressed by the cosine series expansion (21)v(t)=a02+k=1  akcoskπt, where ak=201h(s)coskπsds, k=0,1,2,. For the above u=Sh, by the integral formula of the coefficient of cosine series, we obtain the cosine series expansions of u and u′′′: (22)u(t)=k=1  hkk3π3coskπt,(23)u′′′(t)=-k=1  hkkπcoskπt.

Now from (23), (19), and Parseval equality, it follows that (24)ShX2=u′′′22=12k=1  |hkkπ|212π2k=-  |hk|2=1π2h22.

This means that S(H,X)1/π, namely, (a) holds.

By (20)–(22) and Paserval equality, we have that (25)u22=12k=1  |hkk4π4|212π6k=1  |hkkπ|2=1π6  u′′′22,(26)u22=12k=1  |hkk3π3|212π4k=1  |hkkπ|2=1π4  u′′′22,(27)u′′22=12k=1  |hkk2π2|212π2k=1  |hkkπ|2=1π2  u′′′22.

This shows that the conclusion (b) holds.

Proof of Theorem <xref ref-type="statement" rid="thm1">1</xref>.

We define a mapping F:XH by (28)F(u)(t):=f(t,u(t),u(t),u′′(t),u′′′(t)),uX.

From the assumption (6) and the property of Carathéodory mapping it follows that F:XH is continuous and it maps every bounded set of X into a bounded set of H. Hence, the composite mapping SF:XX is completely continuous. We use the Leray-Schauder fixed-point theorem to show that SF has at least one fixed-point. For this, we consider the homotopic family of the operator equations: (29)u=λ(SF)(u),0<λ<1.

We need to prove that the set of the solutions of (29) is bounded in X. See .

Let uX be a solution of an equation of (29) for λ(0,1). Set h=λF(u); then by the definition of S, u=ShW4,2(I) is the unique solution of LBVP(10). By (a) of Lemma 4, we have (30)uX=ShXS(X,H)h21πh21πF(u)2.

From (28), (6), and (b) of Lemma 4, it follows that (31)F(u)2c0u2+c1u2+c2u′′2+c3u′′′2+M(c0π3+c1π2+c2π+c3)u′′′2+M=π(c0π4+c1π3+c2π2+c3π)uX+M.

Combining this inequality with (30), we obtain that (32)uXM1-(c0/π4+c1/π3+c2/π2+c3/π):=C0.

This means that the set of the solutions for (29) is bounded in X. Therefore, by the Leray-Schauder fixed-point theorem , SF has a fixed-point u0X. Let h0=F(u0). By the definition of S, u0=Sh0W4,2(I) is a solution of LBVP(10) for h=h0. Since W4,2(I)C3(I), from (28) it follows that h0C(I). Hence u0C4(I) is a classical solution of LBVP(10), and by (28) u0 is also a solution of BVP(1).

The proof of Theorem 1 is completed.

Proof of Theorem <xref ref-type="statement" rid="thm2">2</xref>.

Let M=max{|f(t,0,0,0,0)|:tI}+1. From condition (8) of Theorem 2 we easily see that Condition (6) of Theorem 1 holds. By Theorem 1, the BVP(1) has at least one solution.

Now, let u1,u2C4(I) be two solutions of BVP(1); then ui=S(F(ui)), i=1,2. From (8) and (28), we obtain that (33)|F(u2)(t)-F(u1)(t)|+i=03ci|u2(i)(t)-u1(i)(t)|,

for tI. Since u2-u1 is the solution of LBVP(10) for h=F(u2)-F(u1), by (33) and (b) of Lemma 4, we have (34)F(u2)-F(u1)2i=03ciu2(i)-u1(i)2(c0π3+c1π2+c2π+c3)u2′′′-u1′′′2=π(c0π4+c1π3+c2π2+c3π)u2-u1X.

From this and (a) of Lemma 4, it follows that (35)u2-u1X=S(F(u2)-F(u1))XS(X,H)F(u2)-F(u1)2(c0π4+c1π3+c2π2+c3π)u2-u1X.

Since c0/π4+c1/π3+c2/π2+c3/π<1, from (35) we see that u2-u1X=0, that is u2=u1. Therefore, BVP(1) has only one solution.

The proof of Theorem 2 is completed.

Example 5.

Consider the following fully linear fourth-order boundary value problem (36)u(4)(t)=a0(t)u(t)+a1(t)u(t)+a2(t)u′′(t)+a3(t)u′′′(t)+h(t),tI,u(0)=u(1)=u′′(0)=u′′(1)=0, where the coefficient functions a0,a1,a2,a3C(I) and the inhomogeneous term hC(I). All the known results of  are not applicable to this equation. Let (37)f(t,x0,x1,x2,x3)=a0(t)x0+a1(t)x1+a2(t)x2+a3(t)x3+h(t),ci=maxtI|ai(t)|,i=0,1,2,3.

It is easy to see that the partial derivatives fx0, fx1, fx2, and fx3 exist and (38)|fxi(t,x0,x1,x2,x3)|=|ai(t)|ci,i=0,1,2,3.

Assume that the constants c0,c1,c2,c3 satisfy (7). Then by Corollary 3, (36) has a unique solution.

Example 6.

Consider the following nonlinear fourth-order boundary value problem (39)u(4)(t)=i=03  bi(t)|u(i)(t)|αi+sinπt,tI,u(0)=u(1)=u′′(0)=u′′(1)=0, where biC(I), αi(0,1),  i=0,1,2,3. Let (40)f(t,x0,x1,x2,x3)=i=03  bi(t)|xi|αi+sinπt.

Then fC([0,1]×4,) and it satisfies that (41)lim|x0|+|x1|+|x2|+|x3|maxtI  f(t,x0,x1,x2,x3)  |x0|+|x1|+|x2|+|x3|  =0.

From this one easily proves that there exists a positive constant M>0 such that (42)|f(t,x0,x1,x2,x3)||x0|+|x1|+|x2|+|x3|+M.

Since (7) holds for the constants c0=c1=c2=c3=1, by (42) f satisfies the conditions of Theorem 1. Hence by Theorem 1, (39) has at least one solution. This conclusion cannot be obtained from the results in .

Acknowledgments

This research is supported by NNSFs of China (11261053, 11061031) and the NFS of Gansu province (1208RJZA129).

Aftabizadeh A. R. Existence and uniqueness theorems for fourth-order boundary value problems Journal of Mathematical Analysis and Applications 1986 116 2 415 426 10.1016/S0022-247X(86)80006-3 MR842808 ZBL0634.34009 Yang Y. S. Fourth-order two-point boundary value problems Proceedings of the American Mathematical Society 1988 104 1 175 180 10.2307/2047481 MR958062 ZBL0671.34016 del Pino M. A. Manásevich R. F. Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition Proceedings of the American Mathematical Society 1991 112 1 81 86 10.2307/2048482 MR1043407 ZBL0725.34020 De Coster C. Fabry C. Munyamarere F. Nonresonance conditions for fourth order nonlinear boundary value problems International Journal of Mathematics and Mathematical Sciences 1994 17 4 725 740 10.1155/S0161171294001031 MR1298797 ZBL0810.34017 Ruyun M. Jihui Z. Shengmao F. The method of lower and upper solutions for fourth-order two-point boundary value problems Journal of Mathematical Analysis and Applications 1997 215 2 415 422 10.1006/jmaa.1997.5639 MR1490759 ZBL0892.34009 Bai Z. The method of lower and upper solutions for a bending of an elastic beam equation Journal of Mathematical Analysis and Applications 2000 248 1 195 202 10.1006/jmaa.2000.6887 MR1772591 ZBL1016.34010 Li Y. Two-parameter nonresonance condition for the existence of fourth-order boundary value problems Journal of Mathematical Analysis and Applications 2005 308 1 121 128 10.1016/j.jmaa.2004.11.021 MR2141607 ZBL1071.34016 Li Y. On the existence of positive solutions for the bending elastic beam equations Applied Mathematics and Computation 2007 189 1 821 827 10.1016/j.amc.2006.11.144 MR2330259 ZBL1118.74032 Gupta C. P. Existence and uniqueness results for the bending of an elastic beam equation at resonance Journal of Mathematical Analysis and Applications 1988 135 1 208 225 10.1016/0022-247X(88)90149-7 MR960814 ZBL0655.73001 Agarwal R. P. On fourth order boundary value problems arising in beam analysis Differential and Integral Equations 1989 2 1 91 110 MR960017 ZBL0715.34032 Bai Z. Wang H. On positive solutions of some nonlinear fourth-order beam equations Journal of Mathematical Analysis and Applications 2002 270 2 357 368 10.1016/S0022-247X(02)00071-9 MR1915704 ZBL1006.34023 Li F. Zhang Q. Liang Z. Existence and multiplicity of solutions of a kind of fourth-order boundary value problem Nonlinear Analysis. Theory, Methods & Applications A 2005 62 5 803 816 10.1016/j.na.2005.03.054 MR2153213 ZBL1076.34015 Han G. Li F. Multiple solutions of some fourth-order boundary value problems Nonlinear Analysis. Theory, Methods & Applications A 2007 66 11 2591 2603 10.1016/j.na.2006.03.042 MR2312608 ZBL1126.34013 Deimling K. Nonlinear Functional Analysis 1985 New York, NY, USA Springer xiv+450 MR787404