Existence of periodic solutions for nonlinear Lienard systems

265 Internat. J. Math. & Math. Sci. VOL. 18 NO. 2 (1995) 265-272 EXISTENCE OF PERIODIC SOLUTIONS FOR NONLINEAR LIENARD SYSTEMS WAN SE KIM Department of Mathematics Dong-A University Pusan 604 714 Republic of Korea (Received January 26, 1993 and in revised form March 29, 1993) ABSTRACT. We prove the existence and multiplicity of periodic solutions for nonlinear Lienard System of the type d x"(t) + -[VF(x(t))] / g(x(t)) + h(t,x(t)) e(t) under various conditions upon the functions g, h and e. KEY WORDS AND PHRASES: Nonlinear Lienard system, multiplicity of periodic solution. 1991 AMS SUBJECT CLASSIFICATION CODES: 34B15, 34C25 1. INTRODUCTION LetR" be n-dimensional Euclidean space. We define xll [. 1 x,I ] forx (xl, x2,...,x,) E R By L 2([0, 2 :t],R")we denote the space of all measurable functions x: [0, 2hi . R" for which integrable. The norm is given by 1/2 By C*([0.2n],R") we denote the Banach space of 2g-periodic continuous functions x" [0,2g] whose derivatives up to order k are continuous. The norm is given by where Ilyll(R) sup,..lly(t)ll which is a norm in C([0,2],R"). We use the symbol (o,o) for the Euclidean inner product in the space R ". For x, y in C([0,2],R ) we define the L2-inner product as follows 2 (x,y)- fo (x(t),y(t))dt. The mean value x of x and the function of mean value zero are defined by , , - fx(t)dt and respectively. f(t) x(t) We define inequalities in R" componentwise, i.e. x,y _R x y if and only if xi s yl for 1,2,...,n, andx < y if and only ifxi < yi for 1,2 ,n. In this work, we will study the existence of periodic solutions and multiple periodic solutions for the problem x"(t) + (B) -[VF(x(t))] g(x) h(t,x) e(t) + + x(0) x(2 n) x’(0) x’(2 n).= 0 W.E. KIM 266 where F :R" R is a C2-function, g :R" R" is continuous, h [0,2] xR" R is continuous in both variables and 2n-periodic in t, and e :[0,2n]---,R is in L ’([0, 2 n], R" ). We assume that g(x) (gl(xl),g2(x2), ...,gn(x)) for all x (x,x2,...,x) R and h(t,x) (h(t,x),h2(t,x),...,hn(t,x)) for all (t,x)[O,2n]xR Moreover, we assume the following: (HI) h is bounded; i.e., for each 1,2,3 n, there exists Ki > 0 such that . h,(t,x)] g for all (t,x)[O,2n]R n. (Hz) for each 1, 2 n, a OF(x).o(X)x. at ox? Ox, and there exists Ci > 0 such that OF(x) for all x (xl,x2 xn) The purpose of this work is to give existence and multiplicity results for periodic solutions of coupled Lienard system in R". This paper was motivated by the results in [1 and so our results in this work extend some results in [1]. To prove our results we adapt Mawhin’s continuation theorem in [2], and we give appropriate region for the system’s multiplicity by finding an a’priori bound. A’priori Bound To prove our assertion, we consider the following homotopy: x"(t) + dt [VF(x(t))] + .g(x) + Xh(t,x) Xe(t) Let X (0,1) and let x(t) be a possible solution of the problem (E)(B). Taking L 2-inner product by ., x’(t) on both sides of (E), we have 2t 2x x,’. foOF.(-x(t"[x,’(t,dt+Xoxi fo g’(x’(t),x’’(t)dt 2t + X,.. hi(t,x(t))xi’(t)dt ei(t)xi’(t)dt. ". dF0’) By the continuity of --?, (H2) and the periodicity of x(t) in t, we have , 2f , i-1 c, tx,’(t)at 2"a F(x) i-1 dt 1/2 Hence 1/2 By the Sobolev inequality, we have 6M0 1/’2 EXISTENCE OF PERIODIC SOLUTIONS FOR NONLINEAR LIENARD SYSTEMS Suppose there exist a --(al, a2 a,),b (bl, b2 (E) (B) such that a .g[[ b and b,,) in R such that a < b; if x(t) is a solution of M1, then 1/2 ]lx]l(R)[,.l[max(lai],]b,])]2 +Mx. . Taking L Z-inner product by x"(t) on both sides of (E0, we have 2n fo 2n Io [x’"(t)]2dt + ’,’l +’i O2F(x) ’( x, ,t)xi"(t)dt Ox, 2 2 fo g,(x,(t))x,"(t)dt+?i.1 fo h,(t,x(t))x,"(t)dt 2 ?,Y:I ei(t)xi"(t)dt Since F is a C-function, for each 1,2 n, there exists > 0 such that o2F(x) x O, and also since g is continuous, for each 1,2 n, there exists Li > 0 such that g,(x,)l L,. Hence i-1 fo[Xi"(t)]2dt(maxD,) \1 li.n ix,’(t)iat i-l i..1 1/2 + + i,,,1 2n 1/2 fo f01 x,’’(t)l x’’(t)l 2dt 1/2 2n and thus we have gz ( max O liin ),,o ’: + 1/2 + By the Sobolev inequality for every solution of the problem (E0 (B) where M2 depends on a, b, M0 and 3. OPERATOR FORMULATION Define L’D(L)C_ C([0, 2 x],g ") L :([0, 2 x],R ") by (xx(t),xz(t), .,x,(t))-where D(L) 267 C2([0, 2t],R"). Then KerL R and t),x2 t), .,x,, ’(t)) 268 W.E. KIM f te 2([O, 2n]’R’)I fo ImL EL e(t)dt 0 k Consider two continuous projections P: C*([0, 2n],R ") C’([0, 2n],R’) such that ImP KerL and Q" L 2([0, 2 n],R’) L 2([0, 2 n],R’) defined by (Qe)(t)-- -n e(t)dt Then KerQ lmL, C([0, 2 n],R’) KerL O KerP and L :’([0, 2:x],R n) ImL O) ImQ as a topological sum. Since dim [L 2([0, 2 n],R")/ImL dim Jim Q dim[KerL n, L is a Fredholm mapping of index zero and hence there exists an isomorphism J" lm Q KerL. The operator L is not bijective but the restriction of L on DomL NKerP is one-to-one and onto lmL, so it has its algebraic right inverse Ks and, as well known, it is compact. Define L 2([0, 2 n],Rn) N: C 1([0, 2 n],R ") by -t x(t) where x(t) (x(t),x(t) [VF(x(t))] g(x(t)) h(t,x(t)) + e(t) x,(t)). Then N is continuous and maps bounded sets into bounded sets. Let G be any open bounded subset of CI([0,2n],R"), then QN:G----L2([0,2n],R n) is bounded and L :’([0, 2n],R") is compact and continuous. Hence N is L-compact on G. Now we see KR(I Q): x D(L) is a solution to the problem (Ex)(B) if and only if Lx Nx " . MAIN RESULTS THEOREM 4.1. Besides conditions on F, g, e, and (H1), (H2), we assume (Ha) there exists r (r,r2, ...,r,),s (s,s, such that r < s andA B sn),A (A,A An) andB (B1,B, ...,Bn) inR" 2 2 2-" g(r /.(t))dt + 1 2n g(s +X(t))dt h(t, /.(t))dt A and for every " R" such that +- h(t,+X(t))dtaB EXISTENCE OF PERIODIC SOLUTIONS FOR NONLINEAR LIENARD SYSTEMS 269 and for every .f (E CI([0,2t],R ") having mean value zero, satisfying the boundary condition (B) and such that Then (E)(B) has at least one solution if 2 in C(([0,2]),R ") to apply Mawhin’s continuation PROOF. We construct a bounded open set theorem in [2]. Using a’priori estimate, we have (0,11. Hence I111- for any solution x(t) of (EO(B 1, M0- M. Define a bounded set n by and . where L, depends on r, s and M. Thus x’ll V/’-M:’’ Define a bounded open set {x C ([ 0, 2 hi, R")I r < f2 :e < s, < Mt }. Then, for any solution x(t) of (E) (B) lying in fo, we have 1/2 [" I111. [max( ir, l,ls, i)] +M, by -{xeC’([O,2],R")lr < ,llll 2M,,IIx’II VM=} < < < Let (x, :k) [D(L)NO] (0,1) and if (x,k) is any solution to Lx Nx, then (x,k) is a solution to the problem (EO(B ), l[2[l and there exists some I[2[I "M n} such that $-r or s. Take L-inner product with 0,1, 0,..., 0) on both sides of (EO, we have (0, 0 ei {1,2 [il[max(lri[’[si[ ) 2 2x 2x fog,(x,(t))dt+foh,(t,x(t)t-foe,(tt, or 2 2x 2 fogi(xi(t))dt+ fo hi(t’x(t))dt- fo ei(t)dt-0 if x -ri, then, by assumption fo If x-i (...truncated)