DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 2007 pp. 920 926 ON THE UNUSUAL FUČÍK SPECTRUM Ntlij Sergejev Deprtment of Mthemtics nd Nturl Sciences Prdes 1 LV-5400 Dugvpils, Ltvi Abstrct. We construct the spectr for some second order boundry vlue problem of Fučík type. This spectrum differs essentilly from the known Fučík spectr. 1. Introduction. In this pper we study spectr for some second order eqution with piece-wise liner right-hnd sides. Investigtions of Fučík spectr hve strted in sixties of XX century [1]. A number of uthors hve studied the specific cses. Let us mention the cses of the Dirichlet [1] nd the Sturm-Liouville [5] boundry conditions. There re some ppers on higher order equtions. Hbets nd Gudenzi hve studied the third order eqution x = x + + x with the boundry conditions x(0 = x (0 = 0 = x(1 in the work [3], where mny useful references on the subject cn be found. Fučík spectr for the fourth order equtions were considered by Krejčí [2] nd Pope [4]. The pper is orgnized s follows. In Section 2 we formulte the second order boundry vlue problems. In Section 3 we give results on the spectrum for the well-known Fučík problem, but in Section 4 we construct the Fučík spectrum for the second order problem with the boundry conditions y( = 0, y(sds = 0. This spectrum differs essentilly from the known Fučík spectr. To the best of our knowledge Fučík spectr for problems with nonlocl boundry conditions were not considered previously. These re the min results of the work. A connection between the spectr is discussed in Section 5. 2. Formultions of the boundry vlue problems. Consider the second order differentil eqution with the boundry conditions y = y + y,, > 0, (1 y + = mxy, 0, y = mx y, 0, y( = 0, y(b = 0 (2 2000 Mthemtics Subject Clssifiction. Primry: 58F15, 58F17; Secondry: 53C35. Key words nd phrses. Fučík problem, Fučík spectrum, nonlocl integrl condition, bounded Fučík brnches. Supported by ESF project Nr. 2004/0003/VPD1/ESF/PIAA/04/NP/3.2.3.1./0003/0065. 920
921 nd with the boundry conditions y( = 0, y(sds = 0. (3 We cll the problems (1, (2 nd (1, (3 the problem II A nd II B, respectively. We re looking for such points (, so tht the problem II A (II B hs nontrivil solutions (we will cll them the Fučík spectrum. The first result describes decomposition of the spectrum into brnches F + i nd F i (i = 0, 1, 2,... for the problems II A nd II B. Proposition 1. The Fučík spectrum consists of the set of brnches F + i = (, y ( > 0, the nontrivil solution y(t of the problem hs exctly i zeroes in (, b; F i = (, y ( < 0, the nontrivil solution y(t of the problem hs exctly i zeroes in (, b. Proof of Proposition 1 by construction of the spectr in Theorems 1 nd 2. 3. Spectrum of the problem II A. Theorem 1. The Fučík spectrum for the problem II A consists of the brnches given by where i = 1, 2,.... ( ( F 0 + = π 2,, b F 0 = (, ( π 2, b F + 2i 1 = (, + = b, F 2i 1 = (, (, F + 2i 1, F + (i + 1π 2i = (, + = b, F 2i = (, (, F + 2i, Notice tht the problem II A is the clssicl Fučík problem, which ws investigted in the work [1], the proof of this theorem is given in the book. Figure 1 illustrtes the Fučík spectrum structure for the problem II A in the cse of b = 1. 4. Spectrum of the problem II B. Theorem 2. The Fučík spectrum for the problem II B consists of the brnches given by F + 2i 1 (, = 2i (2i 1 cos( (b πi = 0, + (i 1π b, + > b,
922 NATALIJA SERGEJEVA Figure 1. The spectrum for the problems II A. F 2i 1 = (, (, F + 2i 1, F + 2i (, = (2i + 1 2i where i = 1, 2,.... + b, (i+1π cos( (b πi + > b, F 2i = (, (, F + 2i, = 0, Proof. Consider the problem II B. We re looking for the clssicl C 2 -solution y(t, so the derivtive y (t is continuous function nd we must look for continuity of y (t t zero points of y(t. It is cler tht y(t must hve zeroes in (, b. Tht is why F 0 ± =. We will prove the theorem for the cse of F 1 +. Suppose tht (, F 1 + nd let y(t be the corresponding nontrivil solution of the problem II B. The solution hs only one zero in (, b nd y ( > 0. Let this zero be denoted by τ. Consider solution of the problem II B in the intervl (, τ nd in the intervl (τ, b. We obtin tht the problem II B in these intervls reduces to the liner eigenvlue problems. So in the intervl (, τ we hve the problem y = y with boundry conditions y( = y(τ = 0. In the intervl (τ, b we hve the problem y = y with boundry condition y(τ = 0, notice tht y (τ < 0. In view of (3 solution y(t must stisfy the condition τ y(sds = τ y(sds. (4
923 Since y(t = A sin( (t for A > 0 (this reltion holds only for t (, τ nd y(τ = 0 we obtin τ = π +. In view of this equlity it is esy to get tht We hve lso τ y(sds = A ( 1 cos (τ = 2A. y ( π + = A. (5 π Now we consider solution of the problem II B in (τ, b. Since y(t = B sin( (t (B > 0 we obtin We hve lso tht τ y(sds = B ( π 1 cos( (b. It follows from (5 nd (6 tht A = B. In view of the lst equlity nd (4 we obtin y ( π + = B. (6 2 B = B ( 1 cos( (b π. Multiplying both sides by B, we obtin 2 cos( (b π + = 0. (7 Considering the solution of the problem II B it is esy to prove tht π b < π + π. This result nd (7 prove the theorem for the cse of F 1 +. The proof for other brnches is nlogous. Figure 2 illustrtes the Fučík spectrum structure for the problem II B in the cse of b = 1. At the end of this section we would like to give some properties of the spectr of the problems II A nd II B : the brnches of the spectrum for the problem II B re bounded, but the brnches of the spectrum for the problem II A re infinite; ll positive brnches (F + i constitute continuous curve, which is locted below the bisectrix, similrly ll negtive brnches (F i constitute continuous curve, which is locted bove the bisectrix for the problem II B ; the brnches F 2n 1 ± nd F 2n ± hve common point which is the eigenvlue of the corresponding liner problem, the brnches F 2n ± nd F 2n+1 ± hve common point which is not the eigenvlue of the corresponding liner problem for the problem II B ;
924 NATALIJA SERGEJEVA Figure 2. The spectrum for the problem II B. esy computtions shows tht the brnches F 2i nd F 2i+1 intersect t the points ( ( i(i + 1+i 2 π 2 ; ( i(i + 1+i+1 2 π 2. We cn obviously see some of these points in Figure 2. The points of intersections of positive brnches re symmetric to those of the negtive brnches with respect of the bisectrix. 5. Connection between the spectr. Consider the second order eqution (1 together with the boundry conditions (1 αy(b + α We cll this problem the problem II C. y( = 0; F + 2i 1 (, = 2i α (2i 1 α + ( (b sin πi = 0, y(sds = 0, α [0, 1]. Theorem 3. The spectrum for the problem II C consists of the brnches given by (where i = 1, 2,... ( (b α cos πi + (i 1π b, + > b, F 2i 1 = α sin ( (b πi (, (, F + 2i 1 F + 2i (, = (2i + 1 α 2i α + ( (b πi sin α sin = 0, + b, + > b, (i+1π, ( (b πi α cos ( (b πi (8
925 F 2i = (, (, F + 2i. Proof. The proof of Theorem is similr to tht of Theorem 2. Remrk 1. If α = 0 we obtin the problem II A. In cse of α = 1 we hve the problem II B. The brnches F ± 1 to F ± 5 of the spectrum for the problem II C for severl vlues of α re depicted in Figures 3 nd 4 in the cse of b = 1. Figure 3. The spectrum for the problem II C for α = 1 2 nd α = 3 4. Figure 4. The spectrum for the problem II C for α = 8 9 nd α = 19 20.
926 NATALIJA SERGEJEVA Acknowledgements. The uthor would like to thnk F. Sdyrbev for supervising this work nd the referees for their comments. REFERENCES [1] A. Kufner nd S. Fučík, Nonliner Differentil Equtions, Nuk, Moscow, 1988. (in Russin [2] P. Krejčí, On solvbility of equtions of the 4th order with jumping nonlinerities, Čs. pěst. mt., 108 (1983, 29 39. [3] M. Gudenzi nd P. Hbets, Fučík Spectrum for Third Order Eqution, J. Dif. Equtions, 128 (1996, 556 595. [4] P. J. Pope, Solvbility of non self-djoint nd higher order differentil equtions with jumping nonlinerities, PhD Thesis, University of New Englnd, Austrli, 1984. [5] B. P. Rynne, The Fučík Spectrum of Generl Sturm-Liouville Problems, Journl of Differentil Equtions, 161 2000, 87 109. Received September 2006; revised August 2007. E-mil ddress: ntlijsergejev@inbox.lv