Mařík, Robert

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Archivum MathematicumRobert MaříkComparison theorems for half-linear second order difference equationsArchivum Mathematicum, Vol. 36 (2000), No. 5, 513--518Persistent URL: http://dml.cz/dmlcz/107766Terms of use:© Masaryk University, 2000Institute 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.czARCHIVUM MATHEMATICUM (BRNO)Tomus 36 (2000), 513–518, CDDE 2000 issueCOMPARISON THEOREMS FOR HALF–LINEAR SECONDORDER DIFFERENCE EQUATIONSRobert Mař́ık⋆Dept. of Mathematics, Mendel University of Agriculture and Forestry BrnoZemědělská 1, 613 00 Brno, Czech RepublicEmail: marik@mendelu.czAbstract. In the paper new comparison theorems for half–linear differ-ence equation∆RkΦ(∆zk)+ CkΦ(zk+1) = 0are derived. The main tool is variational technique developed for half–lineardifference equations in Řehák [5].AMS Subject Classification. 39A10Keywords. half–linear difference equation, comparison theorem1. IntroductionThe well–known result from the calculus of variations states, that there is anequivalence between disconjugacy of second order ordinary differential equation(r(t)y′)′ + c(t)y = 0on the interval (a, b) and nonnegativity of quadratic functional∫ ba(r(t)η′2(t) − c(t)η2(t))dtdefined on the class of functions η ∈ W 1,20 (a, b). This classical result has beenlater extended in various directions. The generalizations include n−dimensionalproblem with general boundary conditions [1], singular functional [2], p-degree⋆ Author supported by the grant No. 201/98/0677 of Czech Grant Agency.514 ROBERT MAŘÍKfunctionals [3,4] and also discrete p-degree functionals [5]. The relationship betweenthe functional and the corresponding equation is frequently used in the comparisonand oscillation theory of differential equations, see. e.g. [6].In this paper we will study the second order half–linear difference equationL[zk] = ∆(RkΦ(∆zk))+ CkΦ(zk+1) = 0(E)for k ∈ [0, n] and the corresponding discrete scalar p-degree functionalJ(x) = A|x0|p +n∑k=0(Rk|∆xk|p − Ck|xk+1|p).(J)The relationship between Eq. (E) and (J) will be used further in the proof ofcomparison theorems, which compare Eq. (E) with another half–linear discretedifferential equationl[yk] = ∆(rkΦ(∆yk))+ ckΦ(yk+1) = 0.(e)Remark that, unless stated explicitly, under the interval [m,n] we actuallymean the discrete set {m,m + 1, . . . n}. Similarly under the term function weactually mean the sequence.The following lemma presents our main tool – Picone-type identity for half–linear difference equations. It is a simplified version of the Picone identity publishedfor Eq. (E) in Řehák [5].Lemma 1 ([5]). If L[zk] = 0 for k ∈ [0, n] and zk 6= 0 for k ∈ [0, n + 1], then fork ∈ [0, n]∆{−|xk|pRkΦ(∆zk)Φ(zk)}= Ck|xk+1|p − Rk|∆xk|p +Rkzkzk+1Gk(x, z),(P)whereGk(x, z) =zk+1zk|∆xk|p −zk+1Φ(∆zk)zkΦ(zk+1)|xk+1|p +zk+1Φ(∆zk)zkΦ(zk)|xk|pholds. The function Gk(·, ·) satisfiesGk(x, z) ≥ 0(1)with equality if and only if ∆xk = xk∆zkzk, i.e. if and only if xk+1 = xkzk+1zk.Lemma 2. If xk+1 = xkzk+1zkfor k ∈ [0, n], then xk =x0z0zk for k ∈ [0, n + 1].Proof. By induction.HALF-LINEAR SECOND ORDER DIFFERENCE EQUATIONS 5152. Main resultsIn connection with Eq. (E) we will study also the first order Riccati-type differenceequation∆wk + Ck + wk(1 −RkΦ(Φ−1(Rk) + Φ−1(wk)))= 0,(R)where Φ−1(·) denotes the inverse function to the function Φ(·).The relationship between functional (J) and Eqs. (E), (R) has been studiedin [5]. Here it is proved the equivalence between disconjugacy of (E), existence ofsolution of (R) and positive definiteness of (J) on the class of functions satisfyingx0 = 0 = xn+1.The difference between these results and the results from this paper lies inanother type of boundary conditions for the function x. The fact that we useanother types of boundary conditions causes that we obtain information aboutsolution of Eq. (E) given by another initial condition, than in [5].First let us recall the definition of generalized zero, which is known to be theconvenient substitution for zeros of the continuous function.Definition 1. An interval (m,m + 1] is said to contain a generalized zero of asolution zk of Eq. (E) if zm 6= 0 and Rmzmzm+1 ≤ 0.The following theorem establishes the relationship between the half–linearequation, Riccati equation and the p−degree functional. Results of this type aresometimes referred as Reid’s Roundabout–type theorem.Theorem 1. The following statements are equivalent:(i) The solution zk of Eq. (E) given by R0Φ(∆z0z0)= A satisfies Rkzkzk+1 > 0for k ∈ [0, n].(ii) Equation (R) has a solution on [0, n] such that w0 = A and Rk + wk > 0 on[0, n].(iii) Functional (J) is positive definite on the class of functions defined on [0, n+1]satisfying xn+1 = 0.Proof. ”(i)⇐⇒(ii)” If zk is the solution of (E) satisfying Rkzkzk+1 > 0 for k ∈[0, n], then the function wk = RkΦ(∆zkzk)is well-defined on [0, n + 1] and satisfies(R) and Rk + wk > 0 on [0, n], which follows from [5].Conversely, if wk is a solution of Eq. (R) satisfying Rk + wk > 0, then zk+1 =zk(1+Φ−1(wkrk))defines solution of Eq. (E) satisfying Rkzkzk+1 > 0 for k ∈ [0, n].In addition, R0Φ(∆z0z0)= A is equivalent to w0 = A.”(i)=⇒(iii)” Let x be defined on [0, n+1] and xn+1 = 0. Summation of Piconeidentity (P) for k ∈ [0, n] gives516 ROBERT MAŘÍKJ(x) = A|x0|p +n∑k=0[∆(|xk|pRkΦ(∆zkzk))+Rkzkzk+1z2k+1Gk(x, z)]≥A|x0|p + |xn+1|pRn+1Φ(∆zn+1zn+1)− |x0|pR0Φ(∆z0z0)= 0and the functional is positive semidefinite.The equality holds throughout only if Gk(x, z) = 0 for k ∈ [0, n]. From here itfollows xk+1 = xkzk+1zkfor k ∈ [0, n], or equivalently xk = zkx0z0for k = [0, n + 1].In view of the fact xn+1 = 0 6= zn+1, it holds x0 = 0 and x ≡ 0. Hence J(x) = 0only if x ≡ 0 and the functional is positive definite.”(iii)=⇒(i)” Suppose, by contradiction, that the functional is positive definiteand for the solution z of Eq. (E) given (uniquely up to the constant multiple) bythe condition R0Φ(∆z0z0)= A there exists N ∈ [0, n] such thatRkzkzk+1 > 0 for 0 ≤ k < NRNzNzN+1 ≤ 0.Denotexk ={zk k ∈ [0, N ]0 k ∈ [N + 1, n + 1].Since z0 6= 0, clearly x 6≡ 0. Suppose N ≥ 1. From the definition of the function xit follows L[xk] = 0 for k ∈ [0, N − 2]. Summation by parts givesJ(x) = A|x0|p +n∑k=0[Rk|∆xk|p − Ck|xk+1|p]= A|x0|p +[xkRkΦ(∆xk)]n+1k=0−n∑k=0xk+1L[xk]= −N∑k=0xk+1L[xk] = −xNL[xN−1]= −zN[∆(RN−1Φ(∆xN−1)) + CN−1Φ(zN )]= zN[RN−1Φ(∆zN−1) − RNΦ(∆xN ) + ∆(RN−1Φ(∆zN−1))]= zNRNΦ(∆zN ) + zNRNΦ(zN ),since ∆xN = −zN and ∆xN−1 = ∆zN−1. Hence J(x) = RN |zN |p[Φ(∆zNzN)+ 1].Now zN+1 6= 0. Really, if zN+1 = 0 would hold, then J(x) = 0, a contradiction.HenceJ(x) =RNzNzN+1z2N+1|zN |p[zN+1zNΦ(zN+1zN− 1)+zN+1zN].In view of the fact RNzNzN+1 ≤ 0 and with respect to inequality αΦ(α−1)+α ≥ 0we obtain J(x) ≤ 0, a contradiction. This contradiction ends the proof.HALF-LINEAR SECOND ORDER DIFFERENCE EQUATIONS 517Corollary 1 (Leighton type comparison theorem). Let yk be solution of Eq.(e), such that yn+1 = 0 6= y0. Denote a = r0Φ(∆y0y0). Let A be such thatV (y) := (A − a)|y0|p +n∑k=0[(Rk − rk)|∆yk|p − (Ck − ck)|yk+1|p]≤ 0.Then the solution of Eq. (E) given by R0Φ(∆z0z0)= A has a generalized zero on[0, n + 1], i.e., there exists i ∈ [0, n] such that Rizizi+1 ≤ 0 holds.Proof. Define the functional j(x) = α|x0|p +∑nk=0 rk|∆xk|p − ck|xk+1|p. Usingsummation by parts we obtain j(y) = 0 and hence J(y) = J(y)−j(y) = V (y) ≤ 0.Since y 6≡ 0, the statement follows from Theorem 1.An immediate consequence is the followingCorollary 2. Let yk be solution of Eq. (e), such that yn+1 = 0 6= y0. Denotea = r0Φ(∆y0y0). Let A < a, Rk ≤ rk on [0, n] and ck ≤ Ck on [0, n − 1]. Then thesolution of Eq. (E) given by R0Φ(∆z0z0)= A has a generalized zero on [0, n + 1],i.e., there exists i ∈ [0, n] such that Rizizi+1 ≤ 0 holds.Corollary 3. Let yk be solution of Eq. (e), such that yn+1 = 0 6= y0. Denotea = r0Φ(∆y0y0). Let A be such thatV(y) := (A−R0r0a)|y0|p−n∑k=0{∆(Rkrk)rkΦ(∆yk)yk+1+(Ck−Rk+1rk+1ck+1)|yk+1|p}≤ 0.Then the solution of Eq. (E) given by R0Φ(∆z0z0)= A has a generalized zero on[0, n + 1], i.e., there exists i ∈ [0, n] such that Rizizi+1 ≤ 0 holds.Proof. Let yk be solution of (e) on [0, n] satisfying yn+1 = 0 6= y0. ThenL[yk] = ∆(RkΦ(∆yk))+ CkΦ(yk+1)= ∆(RkrkrkΦ(∆yk))+ CkΦ(yk+1)= ∆(Rkrk)rkΦ(∆yk) +Rk+1rk+1∆(rkΦ(∆yk))+ CkΦ(yk+1)= ∆(Rkrk)rkΦ(∆yk) + Φ(yk+1)[Ck −Rk+1rk+1ck].(2)Since the integration by parts shows thatn∑k=0∆(Rkrk)rkΦ(∆yk)yk+1 =[RkΦ(∆yk)yk+1]n+1n=0−n∑k=0Rk+1rk+1∆(rkΦ(∆yk)yk+1)= Rn+1yn+2Φ(∆yn+1) − R0y1Φ(∆y0)−n∑k=0Rk+1rk+1[∆(rkΦ(∆yk))yk+1 + ∆yk+1rk+1Φ(∆yk+1)]518 ROBERT MAŘÍK= Rn+1yn+2Φ(∆yn+1) − R0y1Φ(∆y0)−n∑k=0[−Rk+1rk+1ck|yk+1|p + Rk+1|∆yk+1|p]= Rn+1yn+2Φ(∆yn+1) − R0y1Φ(∆y0)−n∑k=0[−Rk+1rk+1ck|yk+1|p + Rk|∆yk|p]− Rn+1|∆yn+1|p + R0|∆y0|p= Rn+1yn+1Φ(∆yn+1) − R0y0Φ(∆y0)−n∑k=0[Rk|∆yk|p −Rk+1rk+1ck|yk+1|p]the following relation holdsn∑k=0yk+1L[yk] = Rn+1Φ(∆yn+1)yn+1 −R0Φ(∆y0)y0 −m∑k=0[Rk|∆yk|p −Ck|yk+1|p].Then in view of (2) and yn+1 = 0, clearlyJ(y) = |y0|p[A − R0Φ(∆y0y0)]−n∑n=0yk+1L[yk] = V(y)and the statement follows from Theorem 1.3. Open problemsIn the oscillation theory of discrete differential equations the concept of generalizedzeros is used. This is caused by the fact that the sequence Rk is allowed to attainalso negative values. However in the boundary conditions of the functional (J)”exact” zeros are used. It could be interesting to remove this disharmonicity andto find out, whether the concept of generalized zeros in boundary conditions wouldproduce some fruitful extension of discrete variational technique.References1. Z. Došlá, O. Došlý, Quadratic functionals with general boundary conditions, Appl.Math. Opt. 36 (1997), 243–262.2. W. Leighton, M. Morse, Singular quadratic functionals, Trans. Amer. Math. Soc 40(1936), 252-286.3. R. Mař́ık, Nonnegativity of functionals corresponding to the second order half-lineardifferential equation, Arch. Math (Brno) 35 (1999), 155–164.4. R. Mař́ık, On p-degree functionals with one free end point, Proc. Int. Sci. Conf. Math.Žilina 1998, 183–188.5. P. Řehák, Oscillatory properties of second order half–linear difference equations, toappear in Czech. Math. J.6. C. A. Swanson, Comparison and oscillation theory of linear differential equations,Math. in Sci. and Eng. vol 48, Ac. Press (1968).

Comparison theorems for half-linear second order difference equations

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Comparison theorems for half-linear second order difference equations

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Institute of Mathematics AS CR, v. v. i.