(0) < fl (27r), c (0) > c (27r), u = f(t, u), u(0) = u(27r), -u"= f(t, u), u(0) = u(27r), u (0) = u (27r).

Transcription

1 Applied Mathematics and Simulation Volume 2, Number 3, PERIODIC BOUNDARY VALUE PROBLEMS OF FIRST AND SECOND ORDER DIFFERENTIAL EQUATIONS V. Lakshmikantham Department of Applied Mathematics Florida Institute of Technology Melbourne, FL AMS Subject Classification: 34B15, 34C25. Key Words: periodic boundary value problems. 1. INTRODUCTION. Recently the method of upper and lower solutions coupled with Lypunov Schmidt method and monotone iterative technique has been employed fruitfully to prove theoretical as well as constructive existence results relative to periodic boundary value problems, among others, of first and second order differential equations [2-9], namely, u = f(t, u), u(0) = u(27r), -u"= f(t, u), u(0) = u(27r), u (0) = u (27r). These considerations crucially depend upon lower and upper solutions c, fl sagisfying ghe relagions c _< fl, c(0) _< c(27r), fl(0) _> fl(27r), and c (0) > c (27r), (0) < fl (27r), in addition to other assumptions. The problem of proving the existence results when some or all of the foregoing relations are violated is an interesting and important question. In this paper, we shall discuss some known results and raise some open questions. Received November 1988, Revised- August 1989

2 132 Applied Mathematics and Simulation Volume 2, Number 3, AN ABSTRACT EXISTENCE RESULT AT RESONANCE. Let E be a reed Hilbert space. Consider the nordinear operator equation (1) Lu = Nu where L" D(L) C E E is a linear operator and N" D(N) C E E is a nonlinear operator with D(L)D(N) Suppose that P" E E0 is the projection operator and H: E E the partial inverse of L on E. Then it is well known that the problem (1) is equivalent to the coupled system of operator equations (2) ux (3) o = H(1 P)N(uo + ux), = PN(uo + u). Concerning the problem (1), the following result is known [1]. Theorem 1. Suppose that. i, Ii J0, e D(N), ii. there exist r0, R0 > 0 such that [g(u0 + u), u0] _< 0 or _< 0 whenever I1 Uo I1= Ro and II u x 11< ro where Uo e Eo and u Ex. Then the problem (1) admits at least one solution. We shall employ this result in our discussion. 3. FIRST ORDER PERIODIC BOUNDARY VALUE PROBLEMS. Let us consider the PBVP = f(t, = where f e C[[0,. 27r] x R, R]. As n application of Theorem 1, we can prove the following result. Theorem Assume that a, fl C[[0,27r],R], a < f(t,a), a(o) < a(2r), fl >_ f(t, fl), fl(0) >_ fl(2r). Then either a _< fl or a >_ fl on [0, 2r] implies that there exists a solution u of PBVP 4 which lies between a, ft.

3 Periodic Boun"dary Value Problems: Lakshmikantham 133 Proof. The case a < fl is discussed in [4]. We shall indicate the proof of the other case, namely fl _< c on [0, 27r]. We consider the modified problem F(t, -- if where F(t,u) = f(t,p(t,u))+ r(t,u), p(t,u) = max{fl, rain{u, c}}, ad u>c, if / _< u_< c, i/;, if u_<fl. Clearly F(t, u)i_< J on [0,27r] x R, where J depends only on/, a and F is continuous. We let E = L[0,27r], Lu = u, D(L) = {u e E" u is absolutely continuous, u e E, and u(0) u(27r)} and g be the operator generated by F. Then the PBVP (5) is equivalent to the operator equation (1). We note E0 = N(L) consists of constant functions and hence E, where E = E is the class of all functions whose average is zero. The projection operator P can be defined by Pu = fou(s) ds. It is easily seen that the partial inverse operator H also exists and the assumptions required by Theorem 1 are satisfied. Since F is bounded, we can find a constant A which depends on c, fl such that any solution u of (2) satisfies the estimate lux(t)i<_ A on [0,27r]. Hence by Theorem 1, it is enough to find an Ro > 0 such that < N(uo + u), u0 >> 0 or _< 0, for all u0 and u such that u0 l= r0 and u(t)i_< A on [0,2r]. uo Eo and Eo-< 1 >, the relation (6) with _> 0 is equivalent to 2r jr0 2r (7) F(s, Ro + ux(s)) ds >_ 0 and F(s,-ao + ux(s))ds <_ O. Since (We would like to point out that (6) with _< 0 is utilized in proving the theorem when c < ft.) Let us choose R0 > 0 large enough so that we hve Ro+u(t) > max{c(t)} and -Ro+u(t) < min{fl(t)}. This choice of R0, in view of the definition of F, reduces the inequalities (7) to f f(s, a(s)) ds >_ 0 and f0 f(s, fl(s)) ds <_ 0. Since the boundary conditions c(0) _< c(27r) mad fl(0) >_ (27r)imply the foregoing relations, the PBVP (5) admits at least one solution u by Theorem 1.

4 134 Applied Mathctnatics and Simulation Volume 2, Number 3, 1989 We shall now show that fl _< u _< a on [0, 27r], in which case, u will actually be a solution of the PBVP (4). We shall only prove fl _< u since proving u _< a is similar. Suppose that fl _< u on [0, 2r] is not true. hen there exists a to E [0,2r] and an e > 0 such that /3(to) = u(to)+e and (t) <_ u(t)+ e, t e [0,2r]. If to e [0,2r), we have/3 (to) <_ u (t0)and (to) > u(to) which implies that p(to, u(to)) = (t0) and r(to, U(to)) = [u(to)- ;(to)]/[ + u(t0)]. Hence f(to, Z(to))_< Z (to)_< (to)- f(to, fl(to))+ i -I- uz(to) < f(to, fl(to)), which is a contradiction. If to 27r, fl(0) >_ fl(27r) = u(27r) + e = u(0) + e. Consequently, we obtain (0) < u (0) which again leads to a contradiction. Thus we have fl _< u _< a on [0, 27r] and this proves the theorem. Note that when a, violate the boundary conditions a(0) < a(27r), #(0) >_ (2r), proving Theorem 2 is an open question. Employing the ideas of Theorem 2 repeatedly, one can prove the following result which gives a criterion for the existence of k solutions where k is any given number. Theorem ȯ Assume that a e C[[0,27r],R], and c < c _<... _< a+x. Suppose further that for i = 1,2,..., k + 1, either (-1)i[a- f(t, ai)] _< 0 nd (-) [a,(0)-,(2r)] _< o or (--1)i[a- f(t, i)1 >_ 0 nd (-1)i[i(0)- a(27r)] _> 0 holds. Then there exist at least k solutions u for the PBVP (4) such that ai < ui < ai+x on [0, 27r] for i- 1, 2,..., k. Since constructive methods of proving existence resttlts, which can also provide numerical procedures for the computation of solutions, are of greater value than theoretical results, we shall next consider monotone iterative technique. The results we discuss depend on the following lemmas. I,emma 1. Let m 6 CX[[0, 27r], R] and m <_ -Mm- 7, where 0 if m(0) < m(27r), =. M[m(0)- m(27r)] e,,,_l d if m(0) > m(27r). Lemma Let m e C x [[0, 27r], R] and m > Mm +-), where 0 if M(O) >_ m(2r), e2m M[m(27r)- m(0)];,._i if m(0)< m(27r). Then re(t) <_ 0 on [0, 27r1.

5 Periodic Boundary Value Problems: Lakshmikantham 135 Then re(t) < 0 on [0, 2r]. Employing monotone iteraive technique, result. we can prove the following Theorem 4. Let c, fl C[[0,27r],R] and a _< f(t,c)- 7, fl >- y(t, Z)+ 7 wae, fo M > 0, Suppose further that i. if a _< fl, then f(t, u)- f(t, u) >_ -M(ux- u) whenever (t) _< _< _< (t); ii. if fl < c, then y(t, uxo- f(t,u) < M(ux- u)whenever (t) _< _< _< (t). Then there exist monotone sequences {ten ), {fin ) such that lim,..,oo an(t) = p(t), lim,..,oo fl(t) = r(t) uniformly on [0,27r] and that p, r axe minimal and maximal solutions of PBVP (4) respectively. Proof. The case relative o (i) is proved in [15]. Here we shall indicate the proof of (ii). For any u e [fl, c]- {u C[[0,27r],R]I fl _< u _< c, t e [0,2r]}, we consider the linear PBVP (8) u Mu = f (t, (t)) M(t), u(o) u(27r). Setting 5 = f(t, u)- My, we see that u(t) u(o)emt + 5(s)e M(t-s) ds and (0) = (2)= :M i 1 fo2 5(s)e -Ms ds, satisfy the PBVP (8). By Lemma 2, uniqueness of solutions of (8) follows. Hence for any u [fl, o], we define a mapping A by Au u, where u is the unique solugion of (8). Then we can show by a repeated application of Lemma 2 glaag _< Aft, Ao <_ a and A is monotone nondecreasing on [fl, a]. Ig then follows that we can define the sequences {o,}, {fl,} such ghag fin = Afln-1, an Aan_l

6 136 Applied Mathematics and Simulation Volume 2, Number 3, 1989 and conclude _< / _< _<, _< cn _< _< c _< c on [0,27r]. The rest of the arguments are similar to the case (i) discussed in [5]. Hence the proof is complete. A repeated application of the ideas of Theorem 4 yields the following result. Theorem 5. Assume that ai CX[[0, 27r], R], i = 1, 2,..., k + 1 such that cx _< c <... < Ck+X. Suppose further that, for i = 1, 2,..., k + 1, either (-)[ f(t,),] >_ 0 (9) and (-)[f(t, ux)- f(t, u)] _< M(ux u) whenever ci < u2 < ux < ai+, i 7, = 1,2,..., k with 0 if (-1)i[c,(0) ci(27r)] >_ 0 e2mr M[ci(27r) ci(0)., i i (-) [(0) (e)] < 0 or the same relations in (9) hold with reversed inequalities. Then there exist 2k monotone sequences {c}, i = 1,2,...,2k, which converge uniformly to 2k extremal solutions (pi, ri), i = 1, 2,..., k satisfying c < pi < ri _< c+x, i = 1,2,...,k. Extension of even the simplest result to the PBVP u = f(t, u, ut), u(0) = u(27r), uo 0, with f(t, ux, Cx) f(t, u,) > -M(ux u)- g fo,( ) ds whenever (t) <_ u2 <_ u <_ fl(t), cet <_ 2 <_ Cx <_ fl,, t e [0, 27r1, where c, fl are lower and upper solugions such ghag c _< fl seems to be difflculg and is open. 4. SECOND ORDER PERIODIC BOUNDARY VALUE PROBLEMS. Consider the PBVP (10) u"= f(t, u), u(0) = u(27r), u (0) = u (27r), where y 6 C[[0, 27r] x R, R]. As an application of Theorem 1, it is easy to prove the following result. See [2,3].

7 Periodic Boundary Value Problems: Lakshmikantham 137. Tho As tnt., 3 e c[[0,e],n],-."< I(t,-),.(0)= c(27r), cd(0) > cd(27r),-"> f(t,), /(0) = fl(27r), / (0) _< fl (27r) and c _< on [0, 2r]. Then there exists on [0, 27r]. We do not know whether Theorem 6 is true when fl < a. Using monotone iterative technique we can prove the following result. See [5]. Theorem 7. Assume that a(0) = a(27r), -fl" _> f(t, fl)+ 7, fl(0) = fl(2r) and a <_ fl on [0, 27r], with 0 M[cg (2r)-c (0)] (e 2Mr + 1 2(2Mr--l) if if c (0) > a (27r), cd(0) < c (27r), uppose further that f(t,u)- f(t, 2) -M(ux- u), whenever c(t) < u _< ux < fl(t) with M > 0. Then there exist monotone sequences {fin ) such that limn_ = p, lim fl = r, uformly on [0, 2], where p, r are nimal and mimal solutions of PBVP (10) respectively. The proof of Theorem 7 crucially depends on the following Lena. emma 3. 7 where Let m e C[[0,2r],R], m(0)= m(2) d -m" -Mm- 0 M[m, (27r)_m, (0)] (e2m=+l) 2(e2Mr--1) f if m (0) _> m (0) < m (2r). Then re(t)<_ 0 on [0, 27r]. We do not know whether a result similar to case (ii) of Theorem 4 is true Ṅote: This paper was presented t the conference on "Boundary Value Problems" in Trieste, Italy, in May The proceedings of this conference have never been published.

8 138 Applied Mathematics and Simulation Volume 2, Number 3, 1989 REFERENCES. 1. Cesari, L., and Kannan, R. (1977) An abstract theorem at resonance, Proc. Amer. Math. Soc., 63: Kannan, R., and Lakshmikantham, V. (1982) Periodic solutions of nonlinear boundary value problems, Nonlinear Analysis, 6" Kannan, R., and Lakshmikantham, V. (1984) Existence of periodic solutions of nonlinear boundary value problems and the method of upper and lower solutions, Appl. Anal., 17" Lakshmikantham, V., and Leela, S. (1983) Existence and monotone method for periodic solutions of first order differential equations, J.M.A.A., 91" Lakshmikantham, V., and Leela, S. (1984) Remarks on first and second order periodic boundary value problems, Nonlinear Analysis, 8: Leela, S. (1983) Monotone method for second order periodic boundary value problems, Nonlinear Analysis, 7: Vatsala, A.S. (1983) On the existence of periodic quasi-solutions for first-order systems, Nonlinear Analysis, 7: Bebernes, J.W., and Schmitt, K. (1973) Periodic boundary value problems for systems of second order differential equations, Jour. Diff. Equations, 13: Lakshmikantham, V., Ladde, G.S., and Vatsala, A.S. Monotone Iteratire Technology for Nonlinear Differential Equations, Pitman, Boston, 1985.

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