A MIN-MAX THEOREM AND ITS APPLICATIONS TO NONCONSERVATIVE SYSTEMS

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1 IJMMS 3:7, PII. S Hindawi Publishing Corp. A MIN-MAX THEOREM AND ITS APPLICATIONS TO NONCONSERVATIVE SYSTEMS LI WEIGUO and LI HONGJIE Received August and in revised form 4 February A nonvariational generation of a min-max principle by A. Lazer is made. And it is used to prove a new existence results for a nonconservative systems of ordinary differential equations with resonance. Mathematics Subject Classification: 34C5, 49J35.. Introduction and lemmas. Let X and Y be two subspaces of a real Hilbert space H such that H X Y.Letf : H R be of class C and denote by f and f the gradient and the Hessian of f, respectively. In 975, Lazer et al. 3] under the following conditions: fu)h,h m h, m >, h X, u H; fu)k,k m k, m >, k Y, u H.) proved that f has a unique critical point, that is, there exists a unique v H such that fv ). Moreover, this critical point is characterized by the equality f v ) max In 5], with the following conditions: fx+y)h,h β x ) h, fx+y)k,k α y ) k, min x X y Y fx+y)..) βs)ds, x,h X, y Y ; αs)ds, x X, y,k Y,.3) where αs) and βs) are two continuous nonincreasing functions from, ) to, ), it is proved that f has a unique critical point v such that fv ) max x X min y Y fx+y). These results were generalized in 6] and especially for a nonselfadjoint extension of the results of Lazer. This extension was applied in 6] to prove that if the following conditions hold: N <γ γ <N+), γ I Gu) γ I,.4)

2 L. WEIGUO AND L. HONGJIE where N is a nonnegative integer and I is an n n matrix, then the following differential equations system has a unique π-periodic solution: u t)+au t)+ Gu) et),.5) where A is a constant symmetric matrix. System.5) is included in the following nonconservative system.6), and assume the following: u t)+au t)+ Gu,t) et)..6) With the use of a nonvariational version of a max-min principle inspired by 5, 6], in Section we generalize these unique existence results of system.6) to a more general case. To be more precise, we apply a min-max lemma to the periodic boundary value problem of the nonconservative system.6) and assume that the following conditions hold: B +α u ) I Gu,t) B β u ) I, min { αs),βs) } ds +,.7) where u R n, B and B are two real symmetric matrices, and the eigenvalues of B and B are N i and N i +), i,...,n, respectively; here, N i, i,...,n are nonnegative integers and αs) and βs) are two positive nonincreasing functions for s, ). In Section 3, we show with some examples that our main results extend the results known so far. We firstly employ the following lemma from 9]. Lemma. see 9]). Assume that H is a Hilbert space. Let T C H,H), T u) IsomH;H),forallu H. Then, T is a global diffeomorphism onto H if there exists a continuous map ω : R + R + \{} such that ds ωs) +, T u) ω u )..8) With this lemma, we can prove the following lemma. Lemma. see 4]). Let X and Y be two closed subspaces of a real Hilbert space H, and H X Y. Suppose that T : H H is a C -mapping. If there exist two continuous functions α :, ), ) and β :, ), ) such that T u)x,x α u ) x, T u)y,y β u ) y,.9) T u)x,y x,t u)y.)

3 A MIN-MAX THEOREM AND ITS APPLICATIONS... 3 for arbitrary u H, x X, y Y, and min { αs),βs) } ds +,.) then T is a diffeomorphism from H onto H. The following lemma is required in the proof of Theorem.. Lemma.3 see ]). Let H be a vector space such that for subspaces Y and Z, H Z Y.IfZ is finite dimensional and X is a subspace of H such that X Y {} and dimensionx dimensionz, then H X Y.. Unique existence. Assume that Gu,t) is continuous for u,t) R n, π] and twice continuously differiable about u. Denote by Gu, t) and Gu,t) the gradient and the Hessian of Gu,t), respectively. We will investigate the unique existence of periodic solutions for system.6). Firstly, we introduce the following definition. Definition.. The real symmetric matrix A is called admissible with two real symmetric matrices B and B if there exist orthogonal matrices P and P such that P T B P, P T B P, and P T AP are simultaneously diagonal matrices. Theorem.. If conditions.7) hold for all t,π],allu R n, and for A is admissible with the matrices B and B, then there exists a unique π-periodic solution to system.6). Proof. Because A is admissible with the matrices B and B, and conditions.7) hold for all t,π] and all u R n, we can get orthogonal matrices P a,a,...,a n ), P b,b,...,b n ), P T B P diagn,...,n n), P T B P diagn +),...,N n +) ), and P T AP diagγ,γ,...,γ n ). Clearly, a i and b i are the eigenvectors of B and B, respectively, corresponding to the eigenvalues N i and N i +), which satisfy a T i a j b T i b j δ ij, i,j,,...,n,.) where δ ij, i j; δ ij, i j. Define V { vt) v t),...,v n t) ) T vi ) v i π), i,,...,n; vt) absolutely continuous and v t) L,π] },.) and it is easy to see that V is a Hilbert space with the following inner product: u,v u T t)v t)+u T t)vt) ] dt..3)

4 4 L. WEIGUO AND L. HONGJIE Denote by V the norm induced by this inner product, and define subspaces of V as follows: n N X xt) i f i t)a i f i t) c i + cim cosmt +d im sinmt ) ; m n Y yt) g i t)b i g i t) pim cosmt +q im sinmt ) ;.4) mn i + n Z zt) h i t)b i h i t) p i + N i m pim cosmt +q im sinmt ), where N i, i,...,n are as in.7) and c im, d im, p im, and q im are constants. Obviously, V Z Y. Using the Riesz representation theorem, define a mapping T : V V by π Tu),v u T t)v t) v T t)au t) v T t) G ut),t )] dt.5) for arbitrary v V. We observe that T is defined implicitly. From.5) and the fact that G is C, it can be proved that T is a C -mapping and that T u)w,v w T v t) v T t)aw T t) v T t) Gu,t)wt) ] dt.6) for all vt),ut),wt) V. Again, from the Riesz representation theorem, there exists an element d V satisfying d,v v T t)et)dt..7) It can be proved that u is a π-periodic solution to.6) if and only if u satisfies the operator equation Tu) d..8) We will next show that T satisfies the conditions of Lemma.. This will, in turn, imply that.6) has a unique π-periodic solution. For any x X and u V, we have that T u)x,x x T t)x t) x T t)ax t) x T t) Gu,t)xt) ] dt, where x T t)x t)dt f i t)dt x T t)ax t)dt xt t)axt) π. N i f i t)dt;.9).)

5 By.7), we have A MIN-MAX THEOREM AND ITS APPLICATIONS... 5 x T t) Gu,t)xt)dt x T t)b xt)dt+α ) u V x T t)xt)dt x V N i j f i t)f j t)a T i B a j dt +α ) u V x T t)xt)dt f i t)dt +α ) u V x T t)xt)dt; x T t)xt)dt + M + ) x T t)xt)dt, x T t)x t)dt.) where M max i n {N i }, therefore T u)x,x α ) u V M + x V..) Similarly, from y T t)y t)dt y T t) Gu,t)yt)dt Ni + ) g i t)dt, y T t)b yt)dt +β ) u V y T t)yt)dt,.3) we can get that for all y Y and all u V, { +M +) ] y T t)y t) y T t) Gu,t)yt) ] β u V ) y T y t)+y T t)yt) ]} dt +M +) β )] u V y T t)y t)dt +M +) ] y T t) Gu,t)yt)dt β ) u V y T t)yt)dt +M +) β )] n u V Ni + ) g i t)dt +M +) ] y T t)b yt)dt+m +) β ) u V y T t)yt)dt

6 6 L. WEIGUO AND L. HONGJIE +M +) β )] n u V Ni + ) g i t)dt +M +) ] n Ni + ) g i t)dt +M +) β ) n u V β ) n u V M +) N i + ) ] and from g i t)dt, g i t)dt.4) y T t)ay t)dt y T t)ayt) π,.5) we can prove that for all y Y and all u V, T u)y,y y T t)y t) y T t)ay t) y T t) Gu,t)yt) ] dt β u V ) M +) + y V..6) Obviously, for all x X and all y Y, we have the following: T u)x,y x,t u)y x T t)ay t) y T t)ax t) ] dt f t),...,f n t) ) P T AP g t),...,g n t) ) T dt γ i f i t)g i t)dt..7) Let α αs)/m +) and β s) βs)/m +) +), then cs) min { α s),β s) } min { αs),βs) } / M +) + )..8) Based on conditions.7), cs)ds +.SinceT u) is positive definite on Y and negative definite on X, we see that X Y {}. Moreover, it is readily seen that dimensionx dimensionz Ni + )..9) Thus, since it was shown above that V Z Y, it follows, by application of Lemma.3, thatv X Y. We may, therefore, apply Lemma. to get the conclusion of the theorem.

7 A MIN-MAX THEOREM AND ITS APPLICATIONS... 7 If we set V {vt) v t),...,v n t)) T v i ) v i π), i,...,n; vt) to be absolutely continuous and v t) L,π]}, it is easy to know that V is a Hilbert space about the following inner product: π u,v u T t)v t)+u T t)vt) ] dt..) Again, define the norm induced by this inner product and subspaces X, Y, and Z, correspondingly; we can prove the following theorem similarly. Theorem.3. Assume that Gu,t) is continuous and C -mapping with respect to u and that conditions.7) holdforallt,π],allu R n, and for A is admissible with matrices B and B.Letet) be a continuous function. Then, there exists a unique solution to.6), which satisfies boundary value condition u) uπ). Especially, when B N I and B N + ) I, where N is natrual and I is n n identity matrix, A is admissible with B and B as long as A is real symmetric. So, we have the following corollary. Corollary.4. Assume that A is real symmetric and there exist two positive continuous functions δ and δ : R n R such that for all u R n and all t,π], N I<δ u)i Gu,t) δ u)i < N +)..) Let ρr) min{ max n ξ i r δ ξ)/n + ) ),max n ξ i r δ ξ)/n ) }, and if ρr)dr +, then the system.6) has a unique π-periodic solution. If we set A, system.6) becomes a conservative system and admissibility is trivial. So, the main conclusion in 7] the method there is different from ours) is a corollary of Theorem.. Corollary.5. Assume that there exist integers N i such that for all u R n and all t,π], N i <λ iu,t) < N i + ), δ u,t ) { max v u min i n i,...,n; { λ i u,t) N i, N i + ) λi u,t)} },.) where λ i u,t), i,,...,n denote the eigenvalues of Gu). If δs, t)ds + for all t,π], then there exists a π-periodic solution to.5). Let Gu,t) Gu) and cs) min { αs),βs) } c > ;.3) we can get the following unique existence corollary.

8 8 L. WEIGUO AND L. HONGJIE Corollary.6. If the real symmetric matrix A is admissible with real symmetric matrices B and B, then assume that B Gu) B, N i <λ i µ i < N i + ),.4) where λ i and µ i are eigenvalues of B and B, respectively, and N i, i,...,n are nonnegative integers; there exists a unique π-periodic solution to system.5). 3. Examples. It should be pointed out that conditions.9) and.) are not completely the same as.3). In fact, from.3), we know that α x ) depends on subspace X and β y ) depends upon subspace Y. So, condition.3) is more strict than conditions.9). But, from.), we can deduce the following conditions: αs)ds +, βs)ds +. 3.) Conversely, note that 3.) does not imply.). Now, we give an example to illustrate it. Example 3.. First of all, we define two nondecreasing functions as follows: αx) i+), x i x<x i+ ; βx) i+), x i x<x i+, 3.) where x i i k k, i,,..., x, and βx), when x<. It is easy to see that αx) and βs) are two nondecreasing positive functions for all x,+ ); and the number of noncontinuous points is countable infinite. We also have + αx)dx βx)dx min { αx),βx) } + dx i + i x i+ x i i+) i+) + i) i+) +, x i x + i i) + i ) +, i) i) x i+ x + i x i x i + i+) i i+) i+) i+) + + i i) < +. i+) 3.3) Secondly, from the definition of αx) and βx), it is easy to make them continuous and even continuously differentiable, and then they are still positive nondecreasing and satisfy 3.), but they do not satisfy conditions.).

9 A MIN-MAX THEOREM AND ITS APPLICATIONS... 9 We can state, from the following example, that Theorem.3 is more general than the results of,, 3, 4, 5, 6, 7, 8]. Example 3.. Assume that ft) is continuous and π-periodic in.6). Let Gu,t) ) ) 4 5 sin t u +u ) 3 sin t u u ) ) +u ln u + +u +u ln u + +u +u +u +C u +C u, 3.4) then 5 Gu,t) u + 4 ) 5 sin t ) +u 3 sin t) ) ), 5 sin t + +u Gu,t) +u u. +u 3.5) sin t) It is easy to see that Gu,t) satisfies.7). Therefore, there exists a unique π-periodic solution to.6) by Theorem., but we cannot make this conclusion from,, 3, 4, 5, 6, 7, 8]. References ] K.J.BrownandS.S.Lin,Periodically perturbed conservative systems and a global inverse function theorem, Nonlinear Anal. 4 98), no., 93. ] A. C. Lazer, Application of a lemma on bilinear forms to a problem in nonlinear oscillations, Proc. Amer. Math. Soc ), ] A. C. Lazer, E. M. Landesman, and D. R. Meyers, On saddle point problems in the calculus of variations, the Ritz algorithm, and monotone convergence, J.Math. Anal. Appl ), no. 3, ] W. Li, Periodic solutions for kth order ordinary differential equations with resonance, J. Math. Anal. Appl. 59 ), no., ] R. F. Manasevich, A min max theorem, J. Math. Anal. Appl. 9 98), no., ], A nonvariational version of a max-min principle, Nonlinear Anal ), no. 6, ] Z.H.Shen,On the periodic solution to the Newtonian equation of motion, Nonlinear Anal ), no.,

10 L. WEIGUO AND L. HONGJIE 8] S.A.Tersian,A minimax theorem and applications to nonresonance problems for semilinear equations, Nonlinear Anal. 986), no. 7, ] G. Zampieri, Diffeomorphisms with Banach space domains, Nonlinear Anal. 9 99), no., Li Weiguo: Department of Applied Mathematics, University of Petroleum, Dongying 576, Shandong Province, China address: liwg@mail.hdpu.edu.cn Li Hongjie: Department of Mathematics, Linyi Teacher s College, Linyi 765, Shandong Province, China. address: lyjk@63.com

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