DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi Publishing Corporation97640610.1155/2009/976406976406Research ArticleSolutions for m-Point BVP with Sign Changing NonlinearitySuHuaZhangBinggenSchool of Statistics and MathematicShandong University of FinanceJinan, Shandong 250014Chinasdfi.edu.cn200909022009200924102008310120092009Copyright © 2009This 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 study the existence of positive solutions for the following nonlinear m-point boundary value problem for an increasing homeomorphism and homomorphism with sign changing nonlinearity: {(ϕ(u(t)))+a(t)f(t,u(t))=0, 0<t<1, u(0)=i=1m2aiu(ξi), u(1)=i=1kbiu(ξi)i=k+1sbiu(ξi)i=s+1m2biu(ξi), where ϕ:RR is an increasing homeomorphism and homomorphism and ϕ(0)=0. The nonlinear term f may change sign. As an application, an example to demonstrate our results is given. The conclusions in this paper essentially extend and improve the known results.

1. Introduction

In this paper, we study the existence of positive solutions of the following nonlinear m-point boundary value problem with sign changing nonlinearity:(ϕ(u(t)))+a(t)f(t,u(t))=0,0<t<1,u(0)=i=1m2aiu(ξi),u(1)=i=1kbiu(ξi)i=k+1sbiu(ξi)i=s+1m2biu(ξi), where ϕ:RR is an increasing homeomorphism and homomorphism and ϕ(0)=0; ξi(0,1) with 0<ξ1<ξ2<<ξm2<1 and ai,bi,a,f satisfy

ai,bi[0,+), 0<i=1kbii=k+1sbi<1,0<i=1m2ai<1;

a(t):(0,1)[0,+) does not vanish identically on any subinterval of [0,1] and satisfies0<01a(t)dt<;

fC([0,1]×[0,+),(,+)),f(t,0)0andf(t,0)0.

Definition 1.1.

A projection ϕ:RR is called an increasing homeomorphism and homomorphism, if the following conditions are satisfied:

if xy, then ϕ(x)ϕ(y), for all x,yR;

ϕ is a continuous bijection and its inverse mapping is also continuous;

ϕ(xy)=ϕ(x)ϕ(y), for all x,yR.

The study of multipoint boundary value problems for linear second-order ordinary differential equations was initiated by Il'in and Moiseev [1, 2]. Motivated by the study of [1, 2], Gupta  studied certain three-point boundary value problems for nonlinear ordinary differential equations. Since then, more general nonlinear multipoint boundary value problems have been studied by several authors. We refer the reader to  for some references along this line. Multipoint boundary value problems describe many phenomena in the applied mathematical sciences. For example, the vibrations of a guy wire of a uniform cross-section and composed of N parts of different densities can be set up as a multipoint boundary value problems (see Moshinsky ); many problems in the theory of elastic stability can be handled by the method of multipoint boundary value problems (see Timoshenko ).

In 2001, Ma  studied m-point boundary value problem (BVP):u(t)+h(t)f(u)=0,0t1,u(0)=0,u(1)=i=1m2αiu(ξi),where αi>0(i=1,2,,m2),i=1m2αi<1,0<ξ1<ξ2<<ξm2<1, and fC([0,+),[0,+)), hC([0,1],[0,+)). Author established the existence of positive solutions under the condition that f is either superlinear or sublinear.

In , we considered the existence of positive solutions for the following nonlinear four-point singular boundary value problem with p-Laplacian:(ϕp(u(t)))+a(t)f(u(t))=0,0<t<1,αϕp(u(0))βϕp(u(ξ))=0,γϕp(u(1))+δϕp(u(η))=0,where ϕp(s)=|s|p2s,p>1,ϕq=(ϕp)1,1/p+1/q=1,α>0,β0,γ>0,δ0,ξ,η(0,1),ξ<η,a:(0,1)[0,). By using the fixed-point theorem of cone, the existence of positive solution and many positive solutions for nonlinear singular boundary value problem p-Laplacian is obtained.

Recently, Ma et al.  used the monotone iterative technique in cones to prove the existence of at least one positive solution for m-point boundary value problem (BVP):(ϕp(u(t)))+a(t)f(t,u(t))=0,0<t<1,u(0)=i=1m2aiu(ξi),u(1)=i=1m2biu(ξi),where 0<i=1m2bi<1,0<i=1m2ai<1,0<ξ1<ξ2<<ξm2<1, a(t)L1[0,1], fC([0,1]×[0,+),[0,+)).

In , Wang and Hou investigated the following m-point BVP:(ϕp(u(t)))+f(t,u(t))=0,t(0,1),ϕp(u(0))=i=1n2aiϕp(u(ξi)),u(1)=i=1n2biu(ξi),where ϕp(u)=|u|p2u, p>1, ξi(0,1) with 0<ξ1<ξ2<<ξn2<1 and ai,bi satisfy ai,bi[0,+), 0<i=1n2ai<1,0<i=1n2bi<1.

However, in all the above-mentioned paper, the authors discuss the boundary value problem (BVP) under the key conditions that the nonlinear term is positive continuous function. Motivated by the results mentioned above, in this paper we study the existence of positive solutions of m-point boundary value problem (1.1) for an increasing homeomorphism and homomorphism with sign changing nonlinearity. We generalize the results in .

By a positive solution of BVP (1.1), we understand a function u which is positive on (0,1) and satisfies the differential equation as well as the boundary conditions in BVP (1.1).

2. The Preliminary Lemmas

In this section, we present some lemmas which are important to our main results.

Lemma 2.1.

Let (H1) and (H2) hold. Then for u0C[0,1], the problem(ϕ(u(t)))+a(t)f(t,u(t))=0,0<t<1,u(0)=i=1m2aiu(ξi),u(1)=i=1kbiu(ξi)i=k+1sbiu(ξi)i=s+1m2biu(ξi)has a unique solution u(t) if and only if u(t) can be express as the following equation:u(t)=t1ωf(s)ds+B,where A,B satisfyϕ1(A)=i=1m2aiϕ1(A0ξia(s)f(s,u(s))ds),B=11i=1kbi+i=k+1sbi[i=1kbiξi1ωf(s)dsi=k+1sbiξi1ωf(s)ds+i=s+1m2biϕ1(A0ξia(s)f(s,u(s))ds)],whereωf(s)=ϕ1(0sa(r)f(r,u(r))dr+A).Define l=ϕ(i=1m2ai)/(1ϕ(i=1m2ai))(0,1), then there exists a unique A[l01a(s)f(s,u(s))ds,0] satisfying (2.3).

Proof.

The method of the proof is similar to [5, Lemma 2.1], we omit the details.

Lemma 2.2.

Let (H1) and (H2) hold. If uC+[0,1], the unique solution of the problem (2.1) satisfiesu(t)0,t[0,1].

Proof.

According to Lemma 2.1, we first haveA+0sa(r)f(t,u(r))dr0.Sou(1)=B=11i=1kbi+i=k+1sbi[i=1kbiξi1ωf(s)dsi=k+1sbiξi1ωf(s)ds+i=s+1m2biϕ1(A0ξia(s)f(t,u(s))ds)]=11i=1kbi+i=k+1sbi[i=1kbiξi1ωf(s)dsi=k+1sbiξi1ωf(s)ds+i=s+1m2biϕ1(A+0ξia(s)f(t,u(s))ds)]11i=1kbi+i=k+1sbi[i=1kbiξk1ωf(s)dsi=k+1sbiξk1ωf(s)ds]=(i=1kbii=k+1sbi)ξk1ωf(s)ds1i=1kbi+i=k+1sbi0.If t[0,1), we haveu(t)=Bt1ϕ1(A0sa(r)f(r,u(r))dr)ds=u(1)+t1ϕ1(A+0sa(r)f(r,u(r))dr)dsu(1)0.So u(t)0,t[0,1]. The proof of Lemma 2.2 is completed.

Lemma 2.3.

Let (H1) and (H2) hold. If uC+[0,1], the unique solution of the problem (2.1) satisfiesinft[0,1]u(t)γu,where γ=(i=1kbii=k+1sbi)(1ξk)/(1i=1kbiξk+i=k+1sbiξk)(0,1),u=maxt[0,1]|u(t)|.

Proof.

Clearlyu(t)=ϕ1(A0ta(s)f(s,u(s))ds)=ϕ1(A+0ta(s)f(s,u(s))ds)0.This implies thatu=u(0),mint[0,1]u(t)=u(1).It is easy to see that u(t2)u(t1), for any t1,t2[0,1] with t1t2. Hence u(t) is a decreasing function on [0,1]. This means that the graph of u(t) is concave down on (0,1). So we haveu(ξk)u(1)ξk(1ξk)u(0).Together with u(1)=i=1kbiu(ξi)i=k+1sbiu(ξi)i=s+1m2biu(ξi) and u(t)0 on [0,1], we getu(0)i=1kbiu(ξk)u(1)i=1kbiξki=k+1sbiu(ξk)+u(1)i=k+1sbiξk(i=1kbii=k+1sbi)(1ξk)i=1kbiu(ξi)u(1)i=1kbiξki=k+1sbiu(ξi)+u(1)i=k+1sbiξk(i=1kbii=k+1sbi)(1ξk)u(1)(1i=1kbiξk+i=k+1sbiξk)(i=1kbii=k+1sbi)(1ξk)=u(1)γ.The proof of Lemma 2.3 is completed.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B10">8</xref>]).

Let K be a cone in a Banach space X. Let D be an open bounded subset of X with DK=DKϕ and DK¯K. Assume that A:DK¯K is a compact map such that xAK for xDK. Then the following results hold.

If Axx, xDK, then i(A,DK,K)=1.

If there exists x0K{θ} such that xAx+λx0, for all xDK and all x>0, then i(A,DK,K)=0.

Let U be open in X such that U¯DK. If i(A,DK,K)=1 and i(A,DK,K)=0, then A has a fixed point in DKU¯K. The same results hold, if i(A,DK,K)=0 and i(A,DK,K)=1.

Let E=C[0,1], then E is Banach space, with respect to the norm u=supt[0,1]|u(t)|. DenoteK={uuC[0,1],u(t)0,inft[0,1]u(t)γu},where γ is the same as in Lemma 2.3. It is obvious that K is a cone in C[0,1].

We define φ(t)=min{t,1t},t(0,1) andKρ={u(t)K:u<ρ},Kρ*={u(t)K:ρφ(t)<u(t)<ρ},Ωρ={u(t)K:minξm2t1u(t)<γρ}={u(t)E:u0,γuminξm2t1u(t)<γρ}.

Lemma 2.5 (see [<xref ref-type="bibr" rid="B7">13</xref>]).

Ωρ defined above has the following properties:

KγρΩρKρ;

Ωρ is open relative to K;

XΩρ if and only if minξm2t1cx(t)=γρ;

If xΩρ, then γρx(t)ρ for t[ξm2,1].

Now, for the convenience, one introduces the following notations:fγρρ=min{minξm2t1f(t,u)ϕ(ρ):u[γρ,ρ]},f0ρ=max{max0t1f(t,u)ϕ(ρ):u[0,ρ]},fφ(t)ρρ=max{max0t1f(t,u)ϕ(ρ):u[φ(t)ρ,ρ]},fα=limuαsupmax0t1f(t,u)ϕ(u),fα=limuαinfminξm2t1f(t,u)ϕ(u),(α:=or0+),m={(1+i=k+1sbi+i=s+1m2bi)ϕ1((l+1)01a(s)ds)1i=1kbi+i=k+1sbi}1,M={i=1kbii=k+1sbi1i=1kbi+i=k+1sbiξk1ϕ1(0sa(r)dr)ds}1.

3. The Main Result

In the rest of the section, we also assume the following conditions.

There exist ρ1,ρ2(0,+) with ρ1<γρ2 such that(1)f(t,u)>0,t[0,1],u[ρ1φ(t),+),(2)fφ(t)ρ1ρ1ϕ(m),fγρ2ρ2ϕ(Mγ).

There exist ρ1,ρ2(0,+) with ρ1<ρ2 such that(3)f(t,u)>0,t[0,1],u[min{γρ1,ρ2φ(t)},+),(4)fγρ1ρ1ϕ(Mγ),fφ(t)ρ2ρ2ϕ(m).

There exist ρ1,ρ2,ρ3(0,+) with ρ1<γρ2 and ρ2<ρ3 such that(1)f(t,u)>0,t[0,1],u[ρ1φ(t),+),(2)fφ(t)ρ1ρ1ϕ(m),fγρ2ρ2ϕ(Mγ),fφ(t)ρ3ρ3ϕ(m).

There exist ρ1,ρ2,ρ3(0,+) with ρ1<ρ2<γρ3 such that(3)f(t,u)>0,t[0,1],u[min{γρ1,ρ2φ(t)},+),(4)fγρ1ρ1ϕ(Mγ),fφ(t)ρ2ρ2ϕ(m),fγρ3ρ3ϕ(Mγ).

There exist ρ,ρ(0,+) with ρ<γρ such that(1)f(t,u)>0,t[0,1],u[ρφ(t),+),(2)fφ(t)ρρϕ(m),fγρρϕ(Mγ),0f<ϕ(m).

There exist ρ,ρ(0,+) with ρ<ρ such that(3)f(t,u)>0,t[0,1],u[min{γρ,ρφ(t)},+),(4)fγρρϕ(Mγ),fφ(t)ρρϕ(m),ϕ(M)<f.

Our main results are the following theorems.

Theorem 3.1.

Assume that (H1),(H2),(H3),(A3) hold. Then BVP (1.1) has at least three positive solutions.

Theorem 3.2.

Assume that (H1),(H2),(H3),(A4) hold. Then BVP (1.1) has at least two positive solutions.

Theorem 3.3.

Assume that (H1),(H2),(H3) hold and also assume that (A1) or (A2) hold. Then BVP (1.1) has at least a positive solution.

Theorem 3.4.

Assume that (H1),(H2),(H3) hold and also assume that (A5) or (A6) hold. Then BVP (1.1) has at least two positive solutions.

Proof of Theorem <xref ref-type="statement" rid="thm3.1">3.1</xref>.

Without loss of generality, we suppose that (A3) hold. Denotef*(t,u)={f(t,u),uρ1φ(t),f(t,ρ1φ(t)),0u<ρ1φ(t),it is easy to check that f*(t,u)C([0,1]×[0,+),(0,+)).

Now define an operator T:KC[0,1] by setting(Tu)(t)=11i=1kbi+i=k+1sbi[i=1kbiξi1ω(s)dsi=k+1sbiξi1ω(s)ds+i=s+1m2biϕ1(A0ξia(s)f*(s,u(s))ds)]t1ω(s)ds,whereω(s)=ϕ1(0sa(r)f*(τ,u(r))dr+A).By Lemma 2.3, we have T(K)K. So by applying Arzela-Ascoli's theorem, we can obtain that T(K) is relatively compact. In view of Lebesgue's dominated convergence theorem, it is easy to prove that T is continuous. Hence, T:KK is completely continuous.

Now, we consider the following modified BVP (1.1):(ϕ(u))+a(t)f*(t,u(t))=0,0<t<1,u(0)=i=1m2aiu(ξi),u(1)=i=1kbiu(ξi)i=k+1sbiu(ξi)i=s+1m2biu(ξi).Obviously, BVP (3.10) has a solution u(t) if and only if u is a fixed point of the operator T. From the condition (A3)(2), we havefφ(t)ρ1*ρ1ϕ(m),fγρ2*ρ2ϕ(Mγ),fφ(t)ρ3*ρ3ϕ(m).

Next, we will show that i(T,Kρ1*,K)=1.

In fact, by fφ(t)ρ1*ρ1ϕ(m), for uKρ1*, we have(Tu)(t)=11i=1kbi+i=k+1sbi×(i=1kbiξi1ω(s)dsi=k+1sbiξi1ω(s)ds+i=s+1m2biϕ1(A0ξia(s)f*(s,u(s))ds))t1ω(s)ds11i=1kbi+i=k+1sbi(i=1kbi01ϕ1((l+1)01a(r)f*(r,u(r))dr)ds+i=s+1m2biϕ1((l+1)01a(s)f*(s,u(s))ds))+01ϕ1((l+1)01a(r)f*(r,u(r))dr)ds(1+i=k+1sbi+i=s+1m2bi)ϕ1((l+1)01a(s)ds)1i=1kbi+i=k+1sbiϕ1(ϕ(ρ1)ϕ(m))=ρ1=u.This implies that Tuu for uKρ*. By Lemma 2.4(1), we havei(T,Kρ1*,K)=1.

Furthermore, we will show that i(T,Kρ2,K)=1.

Let e(t)1, for t[0,1], then eK1. We claim thatuTu+λe,uΩρ2,λ>0.

In fact, if not, there exist u0Ω2 and λ0>0 such that u0=Tu0+λ0e.

By (A3) and Lemma 2.1, we have for t[0,1],0sa(τ)f*(τ,u(τ))dτ+Aϕ(ρ2)ϕ(Mγ)(0sa(τ)dτ),so thatω(s)=ϕ1(0sa(τ)f*(τ,u(τ))dτ+A)ρ2Mγϕ1[0sa(τ)dτ].Then, we have thatu0(t)=Tu0(t)+λ0e(t)11i=1kbi+i=k+1sbii=1k(biξk1(ω(s))dsi=k+1sbiξk1(ω(s))ds)+λ0i=1kbii=k+1sbi1i=1kbi+i=k+1sbiρ2Mγξk1ϕ1(0sa(r)dr)ds+λ0=γρ2+λ0.This implies that γρ2γρ2+λ0, this is a contradiction. Hence, by Lemma 2.4(2), it follows thati(T,Ωρ2,K)=0.

Finally, similar to the proof of i(T,Kρ1*,K)=1, we can show that i(T,Kρ3*,K)=1.

By Lemma 2.5(a) and ρ1<γρ2 and ρ2<ρ3, we have K¯ρ1Kγρ2Ωρ2Kρ2Kρ3. It follows from Lemma 2.4(3) that T has three positive fixed points u1,u2,u3 in Kρ1*,Ωρ2K*¯ρ1,Kρ3*, respectively. Therefore, BVP (3.10) has three positive solutions u1,u2,u3 in Kρ1*,Ωρ2K*¯ρ1,Kρ3*, respectively.

Then, BVP (3.10) has three positive solutions u1,u2,u3[ρ1φ(t),), which means that u1,u2,u3 are also the positive solutions of BVP (1.1).

Proof of Theorem <xref ref-type="statement" rid="thm3.2">3.2</xref>.

The proof of Theorem 3.2 is similar to that of Theorem 3.1, and so we omit it here. The proof of Theorem 3.2 is completed.

Proof of Theorem <xref ref-type="statement" rid="thm3.3">3.3</xref>.

Theorem 3.3 is corollary of Theorem 3.1. The proof of Theorem 3.3 is completed.

Proof of Theorem <xref ref-type="statement" rid="thm3.4">3.4</xref>.

We show that condition (A5) implies condition (A3). Let k(f,ϕ(m)), then there exists r>ρ such that maxt[0,1]f(t,u)kϕ(u),u[r,) since 0f<ϕ(m). Denoteβ=max{maxt[0,1]f(t,u):ρφ(t)ur},ρ3>max{ϕ1(βϕ(m)k),ρ}.Then we havemaxt[0,1]f(t,u)kϕ(u)+βkϕ(ρ3)+βϕ(m)ϕ(ρ3),u[ρφ(t),).This implies that fφ(t)ρ3ρ3ϕ(m) and (A3) holds. Similarly condition (A6) implies condition (A4).

By an argument similar to that Theorem 3.1, we can obtain the result of Theorem 3.4. The proof of Theorem 3.4 is completed.

4. ExamplesExample 4.1.

Consider the following five-point boundary value problem with p-Laplacian:(ϕ(u))+f(t,u)=0,0<t<1,u(0)=1128u(14)+1256u(12)+164u(34),u(1)=18u(14)164u(12),where a1=1/128,a2=1/256,a3=1/64,b1=1/8,b2=1/64,b3=0,ξ1=1/4,ξ2=1,/2,ξ3=3/4:ϕ(u)={u2,u0,u2,u>0,f(t,u)={15(1+t)(u(t)φ(t)2)30,(t,u)[0,1]×(0,2],15(1+t)(2φ(t)2)30,(t,u)[0,1]×(2,+).

It is easy to check that f:[0,1]×[0,+)[0,+) is continuous. It follows from a direct calculation thatm={(1+i=k+1sbi+i=s+1m2bi)ϕ1((l+1)01a(s)ds)1i=1kbi+i=k+1sbi}1=0.96,γ=(i=1kbii=k+1sbi)(1ξk)1i=1kbiξk+i=k+1sbiξk=21250,M={i=1kbii=k+1sbi1i=1kbi+i=k+1sbiξk1ϕ1(0sa(r)dr)ds}1=0.76.

Choose ρ1=1,ρ2=250, it is easy to check that ρ1<γρ2 andf(t,u)>0,t[0,1],u[φ(t),+),fρ1φ(t)ρ1=max{max0t1(1/5)(1+t)(u(t)φ(t)/2)3012}=(1/5)(1+1)13012=25<ϕ(m)=m2=0.92,t[0,1],u[φ(t)ρ1,ρ1],fγρ2ρ2=min{min3/4t1(1/5)(1+t)(2φ(t)/2)302502}=(1/5)(1+3/4)(21/2)302502=1.0742>ϕ(Mγ)=(Mγ)2=0.004,t[34,1],u[γρ2,ρ2]. It follows that f satisfies the condition (A1) of Theorem 3.3, then problems (1.1) have at least two positive solutions.

Remark 4.2.

Let φ(u)=u, the problem is second-order m-point boundary value problem.

Remark 4.3.

Let ϕp(s)=|s|p2s,p>1, the problem is boundary value problem with p-Laplacian operators.

Hence our results generalize boundary value problem with p-Laplacian operators.

Acknowledgment

This research is supported by the Doctor of Scientific Startup Foundation of Shandong University of Finance of China (08BSJJ32).

Il'inV. A.MoiseevE. I.Nonlocal boundary value problem of the second kind for a Sturm-Liouville operatorDifferential Equations1987238979987Il'inV. A.MoiseevE. I.Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspectsDifferential Equations1987237803810GuptaC. P.Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equationJournal of Mathematical Analysis and Applications1992168254055110.1016/0022-247X(92)90179-HMR1176010ZBL0763.34009SuH.WangB.WeiZ.ZhangX.Positive solutions of four-point boundary value problems for higher-order p-Laplacian operatorJournal of Mathematical Analysis and Applications2007330283685110.1016/j.jmaa.2006.07.017MR2308411ZBL1127.34013MaD.-X.DuZ.-J.GeW.-G.Existence and iteration of monotone positive solutions for multipoint boundary value problem with p-Laplacian operatorComputers & Mathematics with Applications2005505-672973910.1016/j.camwa.2005.04.016MR2165635ZBL1095.34009MaR.Positive solutions for a nonlinear m-point boundary value problemComputers & Mathematics with Applications2001426-7755765ZBL0987.34018MR184618410.1016/S0898-1221(01)00195-XSuH.WeiZ.XuF.The existence of countably many positive solutions for a system of nonlinear singular boundary value problems with the p-Laplacian operatorJournal of Mathematical Analysis and Applications2007325131933210.1016/j.jmaa.2006.01.064MR2273527ZBL1108.34015GuoD. J.LakshmikanthamV.Nonlinear Problems in Abstract Cones19885Boston, Mass, USAAcademic Pressviii+275Notes and Reports in Mathematics in Science and EngineeringMR959889ZBL0661.47045WangY.HouC.Existence of multiple positive solutions for one-dimensional p-LaplacianJournal of Mathematical Analysis and Applications2006315114415310.1016/j.jmaa.2005.09.085MR2196536ZBL1098.34017SuH.WeiZ.XuF.The existence of positive solutions for nonlinear singular boundary value system with p-LaplacianApplied Mathematics and Computation2006181282683610.1016/j.amc.2006.02.017MR2269962ZBL1111.34020SuH.WeiZ.WangB.The existence of positive solutions for a nonlinear four-point singular boundary value problem with a p-Laplacian operatorNonlinear Analysis: Theory, Methods & Applications200766102204221710.1016/j.na.2006.03.009MR2311023ZBL1126.34017SuH.Positive solutions for n-order m-point p-Laplacian operator singular boundary value problemsApplied Mathematics and Computation2008199112213210.1016/j.amc.2007.09.043ZBL1148.34019MR2415806MoshinskyM.Sobre los problems de condiciones a la frontiera en una dimension de carac- teristicas discontinuasBoletín de la Sociedad Matemática Mexicana19507125TimoshenkoS. P.Theory of Elastic Stability19612ndNew York, NY, USAMcGraw-Hillxvi+541MR0134026