Drábek, Pavel

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Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaePavel DrábekA resonance problem for nonlinear Duffing equationCommentationes Mathematicae Universitatis Carolinae, Vol. 29 (1988), No. 2, 205--215Persistent URL: http://dml.cz/dmlcz/106627Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1988Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCOMMENTATIONES MATHEMATICAE UNIVERSTTATIS CAROLINAE 29,2 (1988) A RESONANCE PROBLEM FOR NONLINEAR DUFFING EQUATION Pavel DRABEK Abstract; The purpose of this paper is to study a semilinear periodic problem for the forced Duffing equation at resonance. Nonlinear perturbati-on is allowed to grow linearly. Also some nonuniform nonresonance conditi-ons concerning the nonlinear perturbation are discussed. Key words and phrases; Duffing equation, periodic solutions, nonlinear perturbations with linear growth, Leray-Schauder degree. Classification; 34C25, 34B15, 34C15 1. Introduction. We consider the periodic boundary value problem BVP for an ordinary differential equation of Duffing type (1.1) u"+c a ' + f ( t , u ) = e ( t ) a.e. on r0,T3, (1.2) u(T)=u(0), u'(T)=u'(0). The right hand side e is an element of L (0,T;R), c e R is the damping and f is a nonlinear Caratheodory function. Our aim is to give sufficient conditions for the existence of periodic solutions of (1.1) - (1.2). We can formulate either nonresonance or resonan-ce conditions according to the lower and upper limits of s f(t,s) as s — P - oo . As for the nonresonance case our result is related to Dr^bek, In-vernizzi £.73. The resonance conditions are related to the papers of Landes-man, Lazer 181, Ward [93, Ahmad Cl,23 and DrsSbek L 3J. To prove our results we use essentially the structure of the set of all couples (a,b)%R for which the nonlinear Dirichlet BVP u"+a u+-b u~=0 on [0,T3, u(0)=u(T)=0 has a nontrivial solution. The reader is referred to the Fucfk's monograph {6J in order to see the important role played by the set of such couples (a,b)*R2. - 205 -When resonance occurs, we obtain Landesman-Lazer type sufficient con-ditions for the solvability of (1.1) - (1.2) (see (3.6), (3.7) below). Simi-larly as in H],t2] and £33, some upper bounds for s f(t,s) are necessary if s —*• - oo . In case of the Duffing equation we have an explicit dependence of these bounds on the damping. Roughly speaking, the stronger the damping |o( is, the weaker assumptions laid on s~ f(t,s) (as s — • -oo ) are necessa-ry. The paper is organized as follows. In Section 2 we formulate some auxi-liary assertions. The main result of this paper is formulated in Section 3. The proof of the main result is contained in Section 4. Some remarks in Secti-on 5 conclude the paper. 2. Preliminary assertions. In this section we shall summarize some re-sults on BVPs for second order ODE's. Let T>0 be given number, and let *c: = -n7T. Denote u-:=(|u|- u)/2. Lean 2.1. Let (a.bOcR2. The nonlinear BVP (2.1) u'+a u+-b u"=0 on tO,T], (2.2) u(0)=u(T)=0 has a nontrivial solution if and only if (a.W.Cju t kU. C C ^ c p } , where Cj:= \(a,b)*R2;(a-0c2)(b-oC2)=0i, C^:* i(a,b)6R2;a>k2oC2, b>0, b1/2=koca1/2/<(a1/2-k«t)J, C£:= *(a,b)*R2;a>k2oc2, t>>0, b1/2=(k-»-l)oca1/2/(a1/2-koc)J u ui(a,b)*R2;a>(k+l)2oc2, b^O, b1/2=koca1/2/(a1/2-(k+l)oc)}. Renerk 2.1. The proof of this lemma can be found in Fufiik C6, Lemma 42.23. Let CQ:=-i(a,b)€R2; ab=0i. Assuming *,=l/2 and plotting (a1/2,b1/2) for (a,b)eCk or C*, kSO, a£0, b*0, it is possible to get a picture of the sets C. and Cj* which can be found in Drdbek, Invernizzi U , p. 6453. The following assertion can be proved similarly as Lemma 2.1. L S M B 2.2. Let (a,b)eR2. The nonlinear periodic BVP (2.1) u"+a u*-b u"=0 on C0,T1, (2.3) u(T)=u(0), u'(T)=u'(0) - 206 -has a nontrivial solution if and only if (a,b)- kU 0C k. The set of nontrivial solutions of nonlinear periodic BVP is reduced only to constants if we consider a nonzero damping term (c .#0) in the equ-ation (2.1). Lean 2.3. Let (a.b)iR2, ceR, c*Q. The nonlinear periodic BVP (2.4) u"+c u'+a u+-b u"=0 on £0,TJ, (2.3) u(T)=u(Q), u'(T)=u'(0) has a nontrivial solution if and only if (a,b)eCQ. Proof. Multiplying both parts of (2.4) by u' and integrating on 10,T] we get (with respect to (2.3)) that u m const. Let us suppose that (a,b) + 4 C . Then the continuous function R—-*-R defined by s t~#-a s -b s~ vanishes only at s=0. In this case u i O o n tO,TJ. On the other hand, u * 1 and u « -1 is a nontrivial solution of (2.4) - (2.3) if a=0 and b=0, respectively. Substituting a, b by t-dependent mappings, we obtain the following re-sult. Lean 2.4. Let 9* be two mappings in L°*(Q,T;R). Let us assume one of the following hypotheses is valid: (HI) there exist an integer 121, two points (ai>Di)*Ci> ^ai+i»ni+i^€Ci+l such that ai*0+(t)#al+l> b ^ g j t ) * ^ ^ holds a.e. in £Q,TJ, with strict inequality signs on the set of positi-ve measure in £0,TJ; (H2> there is (apb-^c Cj such that g^(t)<lalf g.(t)*bj, holds a.e. in £0,TJ, with strict inequality signs on the set of positi-ve measure in £0,TJ. Then the nonlinear Dirichlet BVP (2.5) u"+g^(t) u*-g_(t) u"*=0 on £0,TJ, (2.2) u(0)=u(T)=0 - 207 -has no nontrivial solutions verifying also "' >'ipn|-H * » (2.6) sign u'(0)=sign u'(T). Reaark 2.2- The proof of Lemma 2.4 can be found in Invernizzi [7, Lem-ma 1.2]. Under more restrictive assumptions than (HI), (H2) laid on g^(t), the assertion of Lemma 2.4 is proved in Dr^bek, Invernizzi 14, Lemma 2.21. 3. Main result. In this section we shall consider the solvability of the nonlinear periodic BVP (3.1) u"+c u'+f(t,u)=e(t) a.e. on 10,T], (3.2) u(T)=u(0), u'(T)=u'(0). We shall suppose that e is a fixed element in the Banach space X:=L (0,T;R), with usual norm 1 • H , cc R, and f is a Caratheodory function (i.e. f(»,s) is measurable for all s, and f(t,») is continuous for a.e. tel0,T3), satisfying the growth restriction (3.3) |f(t,s)|=p1(t)4-p2|s| for a.e. t€l0,T3 and for all s€R, with p ^ X , p2cR, p 2&0. Suppose that there are numbers r+ £ s+ such that (3.4) r«. i lim inf s""1f(t,s), (3.5) lim sup s*"1f(t,s)^sH. S—»t o° ~* a.e. in [0,TJ. The solution u of (3.1) - (3.2) is a continuously differentiable map-ping u:[0,T]—• R, such that u' is absolutely continuous and (3.1) - (3.2) held. Let us denete by A^ the d-th closed quadrant of R . Theoren 3.1 (Nonuniform nonresonance case). Let tr+,s+1x lr_,s} c C A J U A J and let either (r+-c2/4,r_-c2/4)6Ci, (s+-c2/4,s_-c2/4)«Ci+1 for some fixed ifcl, or (s+-c /4,s_-c2/4)*Cp or s+-c /4<0,s_-c /4#0. Moreover, let us suppose that (3.4) and (3.5) hold with strict inequality signs for t in a subset of positive measure in £0,T3. Then the BVP (3.1) - (3.2) has a solution for arbitrary e«X. In the resonance case we shall suppose that the following hypothesis is fulfilled: - 208 -(f) f*a>(t)= lim sup f( t ,s) and f _ ( t ) = lim inf f ( t ,s) S--.--00 + 0° £-++00 are bounded from above and from below, respectively, for a.e. tc£0,Tl. Reaark 3.1. Note that hypothesis ( f ) implies that r+ from (3.4) satis-fy r* S O . 2 2 Theorem 3.2 (Resonance case). Let us suppose (f), (s+-c /4,s_ -c /4) ft «.C, and (3.5) hold with strict inequality signs for t in a subset of positi-ve measure in CO,VI. Then the BVP (3.D - (3.2) has a solution provided that (3.6) / V w(t)dt < Ĵ Te(t)dt < ̂ Tf+a>(t)dt. Consider, instead of (f), the following hypothesis: (f*) f + a > ( t ) = lim sup f ( t , s ) and *f _ ( t ) = lim inf f(t,s) S-H-OO -0° 6-»-00 are bounded from above and from below, respectively, for a.e. t€ .L0 ,T3 . Remark 3.2. The hypothesis ( f ) implies that S+ from (3.5) satisfy Theorem 3.3 (Resonance case). Let us suppose (f') is valid. Then the BVP (3.D - (3.2) has a solution provided that (3.7) j;V"(t)dt< j[Te(t)dt < jkTf_„(t)dt. 4. Proof of the wain result. The proofs of Theorems 3.1 - 3.3 will be performed in several steps. The main tool we shall use is the homotopy in-variance property of the Leray-Schauder degree, o Fix A €30,(2ar/T) £, Consider the linear operator K:X—* X defined by Kw:= the unique solution u of the linear periodic BVP: u"+c u'+Au=w, u(T)= =u(0), u'(T)=u'(0). A straightforward argument shows that K is completely continuous. Moreover, the standard regularity argument for ODE's proves that K maps bounded sets in L (0,T;R) into relatively compact sets in C(£0,TJ). The Nemytskii operator induced by f and the mappings U H * U - are all conti-nuous X—*• X and map bounded sets into bounded sets. Step 1. Let e e K be given and c4»0. Let us consider the continuous path (4.1) ** C0,U*-*(^,>V ) € ( Äi uAз> ч c, o - 209 -with endpoints (|40, v0> aQd (<*p »{)* with ^ = y^ Then there is R>0 sufficiently large such that the mapping X—i*X defined by (4.2) ut—•u-K(-jM^u+ + % u ~ + Au+(l-«r)e) has the Leray-Schauder degree at 0 relative to the ball f xc X; |xH<R$ equal to a fixed odd number. Proof. The mapping (4.2) is odd for «f=l (we have (U.,= *>,). Hence by the Leray-Schauder continuation theorem it is sufficient to show the existen-ce of such R>0 that H([er,u).= u-K(-(i^u+ + ^ u " +au+(l-tf)e)=0 , for («f ,u)« [0,1 .3xX, implies 8u|«R. Let us suppose the contrary. Then the-re are «nCtO,lU and un« X, Iunft—*co and (4.3) Tf(«n,un)=0, for all ne H. The equation (4.3) is equivalent to the periodic BVP (4.4) u n' +c un * f ^ - % u n =(1- ern)e, (4.5) un(T)=un(0), un(T)=un(0). Put v =u /Ilu il. Then passing to a subsequence, if necessary, we obtain from (4.4) - (4.5) v n — * v in X, ^ - ^ ^ € [ 0 , 1 1 and v"+c v'+<Ufc, v* - % v"* =0, *o o. v(T)=v(0), v'(T)= v'(0). This, with respect to the assumption (4.1), contradicts the assertion of Lem-ma 2.3. Step 2. Let eftX be given and c=0. Let us consider the continuous path (4.6) €Te tO,ll^^( (« f e,y 0)c(A 1uA3)\ kU oC k with endpoints ((JLQ$ V Q ) and (fc^, V , ) , with ft,= V^. Then the assertion from Step 1 remains true. Proof. By the same way as in the proof of Step 1 we get v#X, I v t =1, 0 o 4tO, l ] and v " + f i ~ v* -»£. v** =0, o o v(T)=v(0), v'(T)=v'(0). This, with respect to the assumption (4.6), contradicts the assertion of - 210 -Lemma 2.2. Step 3. Let us suppose that the hypotheses of Theorems 3.1 - 3.3 are fulfilled. Let either ( c V V 0)e Cr+,s+]x[r ,sJ\C Q, for c*0, or ( (a 0,p 0)etr +,s +3Ktr_,sJ\ lU oC | <, for c=0. Consider the homotopy H:[0,l}xX—* X defined by H(t,u):= u-K(-^f(.,u)-(l-r)C(Uou+ -VQu""J + Au+e). Then there exists R>Q such that H(r,u)=0, for ( T , U ) C [ 0 , 1 ] * X implies ftuft<R. Proof. Let us suppose the contrary. Then there is a sequence ( ^ n , u n ) £ e [0,13 xX such that (4.7) un=K(- *„«••"-,>-<-- V ^ o V V n ^ V5 ' and 11 u„ S--*• oo . The normalized sequence v := u„/llu„ll verifies n ^ n n n (4.8) vn=K(-1Tn ftunr1f(.,un)-(l- r n) ltfn- * 0v n] +Av n + Hu^" 1 e). According to (3.3) the sequence f := tuj f(*,u ) is bounded in X. Therefo-re, passing if necessary to subsequence, we can assume that v •—#• v uniform-ly on [0,T]. In this case, (3.3) implies |fn(t)|.ip1(t)llu r |r1+p2|vn(t)|.ip(t), for all nfeN, with some peX. Hence _,t„ ft |fn(t)|dt-*0, for | t г t 2 | - ^ 0 , uniformly with respect to n*N. Therefore the sequence i On =i C X is weakly sequentially compact (see Dunford, Schwartz 15, Corollary 8.11.1), i.e. there is g e X such that some subsequence of tt-^^i converges weakly to g in X. We can suppose t — • X e [0,1], too. Since any bounded linear mapping X — * X is both continuous and weakly continuous, we can pass to the weak limit in (4.8) and we get (4.9) v=K(-rg+(l- X) L ^ v + - >>0v"J + Av). It is a direct consequence of Lebesgue's theorem, Fatou's lemma, (3.3), (3.4) and (3.5) that - 211 -(4.10) g(t)=h+(t)v+(t)-h_(t)v~(t) a.e. on [0,T3, (4.11) r«.4 h (t)*s+ a.e. on E0,TJ The mappings ht :L0,T3—•* R are defined as follows: ht(t):= g(t)/v(t), for teV ± , h+(t):= <-i0,hJt): = »Q, for t6VQ, where VQ:= itc L0,T1; v(t)=0|, V± := 4teLO,T3; v(t) * 0*. The re-ader is referred to Dra"bek, Invernizzi L4, p. 6483 for details. Note that (4.9) - (4.11) imply that v, Iv| =1, is a solution of the pe-riodic BVP (4A2 )« v"+c v'+y+(t)v+ - y_(t)v~ =0 a.e. on L0,T3, (4.13) v(T)=v(0),V(T)=v'(0), where the coefficients Y+(t):=Zh+(t)+(l-Z) ftQ and ^_(t): = %hJt)+(1- v) i> verify (4.14) r t * jt(t)ii3t a.e. on L0,TJ. Our aim is to show that (4.12) - (4.14) yield to the contradiction under the assumptions of Theorems 3.1 - 3.3. Let us suppose that the assumptions of Theorem 3 A are satisfied. Then either T+(*-)»0 or Y+(t).£0 with strict inequality signs on the set of po-sitive measure in E0,T3. Then integrating (4.12) on L0,T3, we obtain (with respect to (4.13)) that there is neither v > 0 nor v<0 on L0,TJ. Hence v has at least two zero points in L0,T1, i.e. we can find t*e L0,T1 such that v(t*)= =0 and C:= l/v'(t*) >0. Let us extend v and f± °y T-periodicity on the whole real line and define v, y..:L0,T3—»- R by the relations V(t):= Cv(t+t*), Tp.»(t):= y ±(t+t*). We obtain v"+c v ' + ^ W v * -flf_(tjv" =0 a.e. on L0,Tl, v(T)=v(0)=0, v'(T)=v'(0)=l. Introducing z:L0,T3—* R, z(t):= exp((c/2)t)v(t), a simple computation shows that z is a solution of the nonlinear Oirichlet BVP (4.15) z"+(f+(t)-c2/4)z+ -(yjt)-c2/4)z~ =0, (4.16) 2(0)=z(T)=0, which verifies (4.17) sign z'(T)=sign z'(0). The inequalities (4.14) and the hypotheses of Theorem 3.1 imply that the - 212 -functions g+(t):= yv(t)-c /4 fulfil the assumptions of Lemma 2.4. Then (4.15) - (4.17) yield to z(t) 21 0 which contradicts IvH =1. Let the assumptions of Theorem 3.2 be satisfied. Then TV(t)=0 a#e# ±n [0,T3 . If these inequalities hold with the strict inequality signs on the subset of positive measure in [0,T3 (the resonance does not occur), we can reach the contradiction by the same way as in the previous case. With resp-ect to the assumptions of Theorem 3.2, we have that either v>0, or v < 0 on [0,T3 (see Lemma 2.4, (H2 ) ) . If <y+(t)=0 a-e- on £°>T3 and r J t ) * 0 in a subset of positive measure in [0,TJ then integrating (4.12) on [0,TJ, we ob-tain v > 0 . Then it follows directly from (4.12) - (4.13) that v S l / T on [0,T3. Similarly, we have v m -1/T on [0,T1 if f_(t)=0 a.e. on [0,TJ and ^ ( t ) ^ 0 in a subset of positive measure in [0,T3. Finally, it is either v m 1/T, or v m -1/T on [0,T3 if ?+ (t)=0 a.e. on [0,T3. Let us suppose that v m 1/T on [0,T) (the other case v » -1/T on [0,T1 can be treated similarly). Then with respect to the uniform convergence v —* 1/T on [0,T3, we have u ( x ) — * + 00 uniformly on [0,T3. Hence the operator equation (4.7) is, for n sufficiently large, equivalent to the periodic BVP (4.18) un'+c v rnf(t,unWl- *n) fvfce, (4.19) un(T)=-Jn(0), up(T)=un(0). Integrating (4.18) on [0,T3,'we obtain by using Fatou's lemma and (4.19): (4-20) £}&,ln' L*n f ( t '% )+ (1-*n )^J- t* . i la, i n ' ti 'V^'V* + < 1 - e n ) ^ o u n J d t = / T e ( t ) d t -On the other hand, the assumption (3.6) yields jj e(t)dt < JT lim inf [f.f(t,un)+(l- tn) (MLu*Mt, V0 JQ (i-+co n n n ' 0 n which contradicts (4.20). Let us suppose that the assumptions of Theorem 3.3 are fulfilled. Then #t.(t)40 a.e. in [Q,T3. If these inequalities hold with the strict inequa-lity signs on the subset of positive measure in [0,T3 (resonance does not occur), we can proceed again as in the case of Theorem 3.1. In the opposite case we can derive v m 1/T, or v s -1/T on [0,T3, and the proof can be per-formed in a similar way as in the case of Theorem 3.2 but by using the as-sumption (3.7) instead of (3.6) (note that it is (*o<0> ^o*** under the - 213 -assumptions of Theorem 3.3). Step 4. let us suppose that the hypotheses of Theorems 3.1 - 3.3 are fulfilled. Then the nonlinear periodic BVP (3.1) - (3.2) has a solution. Proof. The nonlinear periodic BVP (3.1) - (3.2) is equivalent to the operator equation (4.21) u-K(-f(.,u)+Au+e)=0. By Step 3 the Leray-Schauder degree of the operator standing on the left hand side of (4.21), relative to {xcX; i.xH<Ri, is equal to the degree of the operator (4.22) u H~* u-K(- .440u+ + *>Qu~ + Au+e), provided that R > 0 is large enough. But according to Steps 1,2 the degree of (4.22) is different from zero. Hence the operator equation (4.21) has at le-ast one solution which is simultaneously the solution of the nonlinear peri-odic BVP (3.1) - (3.2): 5. Concluding remarks Reaark 5.1. The assertion of Theorem 3.1 is proved under more restric-tive assumptions in Ora*bek, Invernizzi £43. More precisely, in £4, Theorem 3.11, we impose uniform nonresonance conditions on nonlinearity f, while the nonresonance conditions in Theorem 3.1 are nonuniform ones. Reaark 5.2. The proof of Theorem 3.1 can be found in Invernizzi £7] if c=0. Reaark 5.3. To prove our result, we construct a slightly different ho-motopy than in £4, 73. The horootopy of compact perturbations of the identity used here allows us to treat also the resonance case. This is the topic of Theorems 3.2, 3.3. The relation of our result to the paper of Ahmad £21 was already pointed out in the introduction. The author of £2] studies nonself-adjoint resonance problems for POE's with unbounded perturbations. Note that in case of Duffing equation we received an explicit bound for the ratio s" f(t,s) (if |s| is large enough) by means of the curve C, and the damping c. Reaark 5.4. Our result also generalizes the result contained in Ward £9, Theorem 11 for the case of the Duffing equation. - 214 -R e f e r e n c e s til AHMAD S.: A resonance problem in which the nonlinearity may grow line-arly, Proc. Amer. Math. Soc. 93(1984), 381-384. C21 AHMAO S.: Nonselfadjoint resonance problems with unbounded perturbati-ons, Nonlinear Analysis T.M.A. 10(1986), 147-156. D l DRfeEK P.: On the resonance problem with nonlinearity which has arbitra-ry linear growth, J.Math.Analysis and Appl. 127(1987), 435-442. 14] DR/teEK P. and INVERNIZZI S.: On the periodic BVP for the forced Duffing equation with jumping nonlinearity, Nonlinear Analysis T.M.A. 10 (1986), 643-650. t53 DUNFORD N. and SCHWARTZ 3.1.: Linear operators. Part I, Interscience Publ., New York, 1958. t63 FllCfK S.: Solvability of Nonlinear Equations and Boundary Value Problems, D. Reidel, Dordrecht, 1980. t7l INVERNIZZI S.: A note on nonuniform nonresonance for jumping nonlineari-ties, Comment. Math. Univ. Carolinae 27(1986), 285-291. 181 LANDESMAN E.M. and LAZER A.C.: Nonlinear perturbations of linear ellip- > tic boundary value problems at resonance, 3. Math. Mech. 19(1970), 609-623. 19} WARD J.R. Jr.: Asymptotic conditions for periodic solutions of ordinary differential equations, Proc. Amer. Math. Soc. 81(1981), 415-420. Department of Mathematics, Technical University of Plzen, NejedleTio sady 14, 306 14, Plzen, Czechoslovakia (Oblatum 18.11. 1987) - 215 -

A resonance problem for nonlinear Duffing equation

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A resonance problem for nonlinear Duffing equation

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