Peiguang, Wang

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Mathematica SlovacaWang PeiguangForced oscillations of a class of nonlinear delay hyperbolic equationsMathematica Slovaca, Vol. 49 (1999), No. 4, 495--501Persistent URL: http://dml.cz/dmlcz/132057Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1999Institute 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 must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMathernatíca Slovaca ©1999 .-.. _.-. /--.-.~\ _. - .-.- --.- Mathematical Institute Math. SlOVaCa, 49 (1999), NO. 4, 495-501 Slovák Academy of Sciences FORCED OSCILLATIONS OF A CLASS OF NONLINEAR DELAY HYPERBOLIC EQUATIONS WANG PEIGUANG (Communicated by Michal Feckan ] ABSTRACT. In this paper, we discuss a class of nonlinear hyperbolic equations with deviating arguments, and obtain sufficient conditions for the oscillation of such equations subject to two kinds of boundary value problems. 1. Introduction Owing to delay partial differential equations arise frequently in many fields of natural science, the oscillation theory of such equations is of growing interest. For the oscillation of hyperbolic equations, we have a few results. For example, we can refer to the contributions by D. G o r g i o u tz K. K r e i t h [1], D. Mis he v [2], Y o s h i d a [3], B.S. L a l l i , Y.H. Yu & B.T. Cu i [4], [5], [6], Y.K. Li [7] and the references cited therein. But the corresponding theory is as yet not well developed. In this paper, we consider the forced oscillation of nonlinear partial differential equations of the form д2 ш u(x, ť) = a(t)Au(x, t) + ^2 a,i(t)Au(x, p{(t)) dt2 2 = 1 -J2pj(x,t)fj(u(x,aj(t)))^F(x,t) ( E ) (x,t) e fix[0,+oo) =G where Q is a bounded domain in W1 with piecewise smooth boundary dQ. R+. = [0, +oo), Ait is the Laplacian in W1. Suppose that the following conditions (H) hold. A M S S u b j e c t C l a s s i f i c a t i o n (1991): Pr imary 34K20, 35K50. K e y w o r d s : hyperbolic partial differential equation, boundary value problem, delay differen­tial inequality, oscillation. 495 WANG PEIGUANG ( H J a , a ť G C ( R + , R + ) , i = l , 2 , . . . , m . (H2) OJ,PÍ € C ( R + , R + ) are nondecreasing in t respectively; a- < t, p{ < t and lim p.(t) = lina a.(í) = +oo; i = 1,2,.. . , m ; j = 1,2,.. . , n . (H3) p- G C ( í í x l + , R + ) ; F G C ( n x R + , M + ) ; j = 1,2,. . . , n . (H4) /• G C(R,R) and / .(u) is convex in R + ; u / ^ u ) > 0, u / 0, z = l , 2 , . . . , r a . We consider two kinds of boundary value conditions: (Bx) u = <p(x,í) on (re,č) G dfž x R+ . (B2) | ^ + / Í ( X , Í ) W = 0 on (x,t) edtt x R + . In which n is the unit exterior normál vector to <9íl, fi is a nonegative continuous function on <9Í7 x R+ . The objective of this páper is to study the oscillatory properties of solutions of equation (E) subject to boundary conditions ( B J and (B2) respectively. The results generalize and improve the some results in [4], [5] and [6]. A solution u{x, ť) of equation (E) satisfying certain boundary conditions is called oscillatory in the domain G if for each positive number r there exists a point (xQ,t0) G G, t0 > r such that u(x0,t0) = 0. 2. Oscillation er i ter ia for p rob lém (E), (Bx) We consider the following problém Au + au = 0 in íí x R+ , u = 0 on <9fí x R+ , where a is a constant. It is well known ([7]) that the smallest eigenvalue a0 is positive and the corresponding eigenfunction <£(#) is also positive. LEMMA 2 . 1 . ([4]) Suppose that y G C2([t0: +oo),R) , and y ( t ) > 0 , y'{ť)>0 and y"{t)<0, t>t0>0. (2.1) Then for any A0 G (0,1), there exists a number tx > tQ such that y(t)>\ty'(t) for t>tx. (2.2) We defme the function p.{ť) by p.(t) = min{p.(x,ť)\ , j = 1,2,. .. , n . 496 FORCED OSCILLATIONS OF A CLASS OF NONLINEAR DELAY HYPERBOLIC EQUATIONS For a solution u(x, t) of equation (E) satisfying boundary condition (B x), we define f u(x,t)$(x) dx U(t) = f Ф(x)dx (2.3) LEMMA 2.2. Suppose that conditions (H) hold, and (Ax) There exists a positive constant e such that fi(u)>eu. (2.4) (A2) There exists an oscillatory function H G C2(M, R) with ^ lim H(t) = 0, and t—> + oo - f \a(ť)<p(x,t) + JTa ť(t)p(.r,p ť(í)) PJO í = 1 ðФ ÔП dct; (2.5) If u(x,t) is a positive solution of problem (E), (Bx) on Q x [t0,+oo), then the delay differential inequality c r Tn " i n ^ p"W + A 0 | a 0 fl(*)v(*) + S a i ( * ) » ( ^ ( ' ) ) + e E P j ( * ) » K " ( * ) ) [ < 0 (2.6) I L i = 1 J = 1 J /iai>e eventually positive solutions y(t) = U(t)-H(t). (2.7) P r o o f . Suppose that u(x, t) is a positive solution of problem (E), (Bx) on fi x [t0, +oo). In view of (H2) there is a number tx > t0 such that u(x,Pi(t)) > 0 , ufaafi)) > 0 . Multiplying both sides of (E) by the $(x) and integrating with respect to x over the domain ft, we have j 2 /» --- y «OM)*(aO dx = a(*) / Au(x,t)$(x) dx + ^ a ^ ) / Au(x,pi(t))^(x) dx - ^ J PjfaQf^ufaafo)))^) dx + j F(x,t)$(x) dx, t>tx. (2.8) 497 WANG PEIGUANG Using Green's theorem, we have / Au(x, t)Ф(x) dx = - ip(x, £) — duj - a0 / u(x, ť)Ф(x) dx, t>tгì (2.9) / Au(x,p{(t))^(x) dx = - </?(x,/>;(*)) ̂ - du-a0 I u(x,p{(t))^(x) dx, n an n t > tt. (2.10) Noticing the definition of pAt), using (H4) and Jensen's inequality, we have ypJ.(a;,t)/ i(U(x, (7 j(t)))$(x)dx n >Pj(t)jfMx>aj(t)))®Wdx (2.11) Ҷu(x,aЛt))Ф(x)dx >pjm ) / Ф(x) dx, t > tг . f $(x) dx n Combing (2.8)-(2.11), yields r m i U"(t) <-a0 a(t)£IW + X;oi(*)^^(*)) i -YtPiWiPW))) - i = l + ( f$(x)dx\ i f F^Mx) - /ja(0?(s,«) + f>.(*M*»P.(*)) i. L i=l :) dx ðФ дn düü>. Let l/(t) = tr(t) ̂  h o -using the condition (Ax) and (A2), we get (2.12) y"rø + «o m -i " aííйíl + S ^ í f t ^ Г & i ^ ^ ^ -°' *- ť l ' (2.13) ѓ = l J = l 498 FORCED OSCILLATIONS OF A CLASS OF NONLINEAR DELAY HYPERBOLIC EQUATIONS from (2.13), we have y"(t) < 0 for t > tx, and we can prove that there exists a t2 > tr, such that y(t) > 0. In fact, if y(t) < 0 then U(t) < H(t), which is impossible in view of the fact that U(t) > 0 and H(t) is oscillatory. From y(t) > 0 and y"(t) < 0, we have y'(t) > 0, t>t2. Furthermore, since y(t) is an increasing function and lim H(t) = 0, it t—>-\-oo follows that there is a number t3 > t2 and A0 G (0,1) such that U(t) > X0y(t), U(Pl(t)) > X0y(Pi(t)) , U(aj(t)) > X0y(aj(t)) , t > t, . (2.14) Then from (2.14) it follows that the function U(t) defined by (2.3) is a positive solution of the delay inequality (2.6). • THEOREM 2.1. Suppose that conditions (H) ; (Ax) and (A2) hold. If the delay inequality (2.6) have no eventually positive solutions, then every solution of the problem (E). (Bx) is oscillatory on fi x 1R+ . P r o o f . Suppose that there is a nonoscillatory solution u(x, t) of the prob-lem, we may assume that u(x,t) > 0, (x,t) G ft x [t0, +co ) , by Lemma 2.2, we get y(t) = U(t) - H(t) is a eventually positive solution, which is a contradiction. If u(x, t) < 0 then set u(x, t) = —u(x, t), using the condition (H4) , it is easy to check that u(x, t) is a positive solution of the problem (E), (Bx) , defining fu(x,t)$(x) dx J $(x) dx n then by Lemma 2.2 y(t) = U(t) - H(t) (2.16) is a eventually positive solution, which is also a contradiction. This completes the proof of the Theorem 2.1. • Remark 1. Theorem 2.1 generalize and improve Theorem 2.2 in [4], Theo-rem 3.1 in [5] and Lemma 2.1 in [6]. 3. Oscillation criteria for problem (E), (B2) With a solution u(x,t) of problem (E), (B2), we define V(t) = ±-ju(x,t)dx, *>0, \Q\ = J dx. 499 WANG PEIGUANG LEMMA 3.1. Suppose that conditions (H), (Ax) hold, and ther and (A3) e exists an oscillatory function H G C2(R, M) with lim H(t)=0, H"(t) = ±f F(x,t)dx (3.1) 7/ u(x,t) is a positive solution of problem (E), (B2) on Q x [t0,+oo). then the delay differential inequality n y;/(*) + AeX;-Pi(*)y(^(*))<0 (3.2) i = i /ms eventually positive solutions y(t) = V(t) - H(t). (3.3) P r o o f . Suppose that u(x, t) is a positive solution of problem (E), (B2) on $1 x [t0, +oo). In view of (H2) there is a number tx > t0 such that u(x,Pi(t)) > 0 , u(x,aj(t)) > 0 . Integrating (E) with respect to x over the domain Q, we have — / u(x, t) dx = a(£) / Atx(z,i) dx + ^ a . ( t ) / Au(x,p-(t)) dx (3.4) n *=1 ^ -^fpj&iVfjHx^ft))) dx + f F(x,t) dx, t>t,. Using the Green's theorem and (B2), we have / Au(x, t) dx = I -^ duo = - I ii(x, t)u(x, t) do; < 0 , (3.5) n an an J Au(x, p.(t)) dx = J ^ / £ i M dw = - y /i(x, Pi(t))u(x, p.(t)) duj < 0 . Q dQ dQ W"^) Noticing the definitions of Pj(t) and (H4), using Jensen's inequality, we have f Pjixrff^ufaafi))) dx > p . ( t ) | n | / Y p f u(x,aj(t)) dx\ . (3.7) 500 d2 dí2 Ln J Let FORCED OSCILLATIONS OF A CLASS OF NONLINEAR DELAY HYPERBOLIC EQUATIONS Combining (3.4)-(3.7), we have < -p.itmfJ^juix^^t)) dzj + j F{x,t) dx. (3.8) y(t) = V(t) - H(t). (3.9) Using conditions (A-_) and (A3) , we have n v"(t) + \eY/Pj(t)V((Tj(t))<0, t>tlt (3.10) 3 = 1 the remainder of the proof is similar to that of Lemma 2.2, we omit it. • Using Lemma 3.1, we have: THEOREM 3 .1 . Suppose that conditions (H) ; (Ax) and (A3) hold. If the delay inequality (3.2) has no eventually positive solutions, then every solution of the problem (E), (B2) is oscillatory on ft x R+ . R e m a r k 2. Theorem 3.1 generalizes and improves Theorem 2.1 in [4], and Theorem 2.1 in [5]. REFERENCES [1] GEORGIOU, D.—KREITH, K. : Function characteristic initial value problem, J. Math . Anal. Appl. 107 (1985), 414-420. [2] MISHEV, D. P.—BAINOV, D. D . : Oscillation properties of a class of hyperbolic equations of neutral type, Funkcial. Ekvac. 29 (1986), 213-218. [3] YOSHIDA, N . : On the zeros of solutions of hyperbolic equations of neutral type, Differen-tial Integral Equations 3 (1990), 155-160. [4] LALLI, B. S.—YU, Y. H.—CUI, B. T . : Oscillations of hyperbolic equations with functional arguments, Appl. Math. Comput . 5 3 (1993), 97-110. [5] LALLI, B. S.—YU, Y. H.—CUI, B. T . : Forced oscillations of hyperbolic differential equa-tions with deviating arguments, Indian J. Pure Appl. Math . 2 5 (1994), 387-397. [6] CUI, B. T.—YU, Y. H.—LIN, S. Z. : Oscillations of solutions of delay hyperbolic differential equations, Acta Math. Appl. Sinica 19 (1996), 80-88. [7] LI, Y. K. : Oscillation of system of hyperbolic differential equations with deviating argu-ments, Acta Math . Sinica. 4 0 (1997), 100-105. Received December 1, 1997 Department of Mathematics Revised April 7, 1998 Hebei University Baoding, Hebei 071002 P. R. CHINA E-mail: pgwang@mail.hbu.edu.cn 501 

Forced oscillations of a class of nonlinear delay hyperbolic equations

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Forced oscillations of a class of nonlinear delay hyperbolic equations

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