ON A DIFFERENCE EQUATION WITH MIN-MAX RESPONSE

Transcription

1 IJMMS 2004:55, PII. S Hindawi Publishing Corp. ON A DIFFERENCE EQUATION WITH MIN-MAX RESPONSE GEORGE L. KARAKOSTAS and STEVO STEVIĆ Received 25 February 2004 We investigate the global behavior of the positive solutions of the difference equation x n+ = α n + Fx n,...,x n k, n = 0,,..., where α n is a sequence of positive reals and F is a min-max function in the sense introduced here. Our results extend several results obtained in the literature Mathematics Subject Classification: 39A0.. Introduction. Let k be a positive integer and let R + be the set of all positive reals. We give the following definition. Definition.. A function F : R k+ + R + is called a min-max function if it satisfies the inequality k+ j= u j k+ j= u j j= u j F u,u 2,...,u k+ k+ k+ j= u j,. for all u j > 0, j =,...,k+, where, as usual, the symbol n j= u j stands for the maximum of the variables u j, j =,...,n, and n j= u j stands for their minimum. In Section 2, we give exact information on the form which a min-max function may have. Simple examples of min-max functions are F u,u 2 := u 2 u, F 2 u,u 2 := u u 2.2 which appear as the response functions, respectively, in the difference equation studied in [] and in the difference equation y n+ = α+ y n y n.3 y n+ = α+ y n y n.4 studied in [2]. These two equations have completely different behavior; see Remark 3.6. Also in [3, 4], the second author considered the closely related equation x n+ = α n + x n x n,.5

2 296 G. L. KARAKOSTAS AND S. STEVIĆ where α n is either a periodic sequence with period two or a convergent sequence of nonnegative real numbers. Motivated by the above-mentioned works, in this paper, we study the behavior of the difference equation x n+ = α n +F x n,...,x n k, n= 0,,...,.6 where the initial conditions x k,...,x 0 are positive real numbers, α n is a sequence of positive real numbers, and F is a min-max function. Since a min-max function takes the value at the diagonal of the space R k+ +, it follows that in case the sequence α n converges to a certain α, the positive real number K := α+.7 is the unique asymptotic equilibrium of.6. Our purpose here is to discuss the boundedness and persistence of.6, as well as the attractivity of the asymptotic equilibrium α +, where α is the limit of α n whenever this exists. This follows immediately by Theorem 3.2, where we show that, if < liminf α n limsupα n < +, then any solution x n satisfies the relation limsupx n liminf x n limsupα n liminf α n..8 Thus, if the sequence α n converges to some α>, then any solution with positive initial values converges to the asymptotic equilibrium K = α +. This generalizes [, Theorem 5.2] and part of [2, Theorem ]. For the case α n =, for all n in Theorem 3.3, we show that any nonoscillatory solution converges to 2, while if F satisfies the additional sufficient conditions u i < j i u j u i > j i u j F u,u 2,...,u k+ < j i u j u i,.9 F u,u 2,...,u k+ > j i u j u i,.0 then it is shown in Theorem 3.4 that all solutions converge to 2. Comparing this fact with the results in [], we see that the pair of conditions.9.0 seems also to be necessary. Indeed, these conditions are not satisfied in case of.3 and, as it is shown in [, Theorem 4.], it has nontrivial solutions which are periodic with period 2. In Theorem 3.5, we show that if α n = α<, for all n, then there is a large class of equations of the form.6 which have unbounded positive solutions. This result extends [, Theorem 3.]. In the Section 4, we give two examples of difference equations with min-max response to illustrate our results. Also the so-called 2, 2-type equation defined in [6] where about 50 types of difference equations are presented includes the equation x n+ = A x n +B x n A 2 x n +B 2 x n..

3 ON A DIFFERENCE EQUATION WITH MIN-MAX RESPONSE 297 Under appropriate choice of the parameters,. can be written as x n+ = α+ β+γx n βx n +γx n,.2 which is of the type.6. Thus in this paper, we push further the investigation originated in [6] for such a form of 2, 2-type difference equations. For other closely related results, which mostly deal with difference equations and inequalities whose response is or it can be transformed into a min-max function, see, for instance, [7, 8, 9, 0,, 2, 3, 4] and the references cited therein. 2. On the min-max functions. In this section, we give a characterization of min-max functions. The result is incorporated in the following theorem. Theorem 2.. A function F : R k+ + R + is a min-max function if and only if there are nonnegative real-valued functions a j u,u 2,...,u k+, b j u,u 2,...,u k+, j =,2,..., k+, such that k+ k+ a j u,u 2,...,u k+ = b j u,u 2,...,u k+ =, j= j= F j= a j u,u 2,...,u k+ uj u,u 2,...,u k+ =, j= b j u,u 2,...,u k+ uj 2. for all u,u 2,...,u k+ R k+ +. Proof. The if part is easily proved by using the form of F and the conditions on the coefficients a j, b j. To show the inverse, assume that Fu,u 2,...,u k+ is a min-max function and fix any element u,u 2,...,u k+ R k+ +.Welet v := k+ j= u j, w := k+ j= u j, 2.2 thus v = u j and w = u j2, for two indices j,j 2,2,...,k+. From the definition of the min-max functions, we know that the value Fu, u 2,...,u k+ lies in the interval [v/w,w/v], thus there is a number a [0,] such that F w u,u 2,...,u k+ = a v + a v w. 2.3 Let b := av 2 aw 2 + av It is clear that b belongs to the interval [0,], and it depends on v, w thus on u,u 2,...,u k+. By some simple calculations, we obtain bw + bv a w v + a v = aw + av 2.5 w

4 298 G. L. KARAKOSTAS AND S. STEVIĆ and consequently we get F w u,u 2,...,u k+ = a v + a v aw + av = w bw + bv. 2.6 This proves the theorem since we can set a j u,u 2,...,u k+ := 0, if j j,j 2, while a j u,u 2,...,u k+ = a and a j2 u,u 2,...,u k+ = a. Similar substitutions are used for the denominator. The proof is complete. Remark 2.2. The quotient of any two elements of the class of all f : R k+ + R + which satisfy an inequality of the form k+ j= u j f u,u 2,...,u k+ k+ j= u j 2.7 produces a min-max function. 3. The main results. Our first result refers to the boundedness of the solutions. Theorem 3.. Consider.6, where F is a min-max function and the sequence α n satisfies <C:= inf α n supα n =: B<+. 3. Then any solution x n with positive initial values satisfies the condition for all n =,2,..., where min k+ j= x j, LC L L := max k+ j= x j, Also, if α n = α =, foralln, then it holds that for all n, where M := min M x n k+ j= x j, x n L, 3.2 BC C. 3.3 M M, 3.4 k+ j= x j k+ j= x j. 3.5 Proof. Let n>k+. From.6, for all j, we have C<x j n i= x i. 3.6

5 ON A DIFFERENCE EQUATION WITH MIN-MAX RESPONSE 299 Also, for all j = k+2,k+3,...,n,weget These facts imply that x j B + j i=j k x i C B + n i= x i. 3.7 C n j= x j max k+ i= x i,b+ n i= x i, 3.8 C from which we get x n n i= x i max k+ i= x i, BC C, 3.9 and therefore, C<x m L, 3.0 for all m =,2,... Next let n>k+. From 3.0 and.6, it follows that for all j = k+2,k+3,...,n,it holds that Therefore, we have x j C + j i=j k x i L C + n i= x i. 3. L x j min k+ i= x i,c+ n i= x i, 3.2 L for all j =,2,... This implies that n i= x i min k+ i= x i,c+ n i= x i L 3.3 and so n i= x i min k+ i= x i, LC L. 3.4 This gives x n n i= x i min k+ i= x i, LC L, 3.5 which, together with 3.0, proves the first result of the theorem.

6 2920 G. L. KARAKOSTAS AND S. STEVIĆ Next assume that α n =, n = 0,,... To show inequality 3.4, we observe that M x n for all n =,2,...,k+. Also from.6, we get M M, 3.6 x k+2 + k+ j= x j k+ j= x j M + M/M = M, x k+2 + k+ j= x j k+ j= x + M/M = M j M M. 3.7 These arguments and the induction complete the proof. Theorem 3.2. Consider.6, where F is a continuous min-max function and the sequence α n satisfies the condition < liminf α n limsupα n < Then any positive solution x n satisfies relation.8. Hence, if the sequence α n converges to some α>, then x n converges to a constant, which, therefore, is equal to α+ =: K. Proof. Let x n be a solution. From Theorem 3., the solution is bounded, thus there are two-sided sequences, y m upper full limiting sequence and z m lower full limiting sequence of x n see, e.g., [3, 4, 5], satisfying.6, for all integers m, and such that liminf x n = z 0 z m, y m y 0 = limsupx n, 3.9 for all m. Leta 0 := liminf α n and a 0 := limsupα n. Then from.6, we have y 0 a 0 + y 0 z 0, z 0 a 0 + z 0 y Combining these two relations, we obtain.8. Theorem 3.3. Consider.6, where α n =, n = 0,,..., and F is a min-max function. Then every nonoscillatory positive solution converges to the equilibrium K = 2. Proof. Assume first that x n 2, for all n k.setu n := x n 2. From Theorem 2., we know that F may take the form 2., where the nonnegative functions a j, b j satisfy k+ k+ a j xn,...,x n k = b j xn,...,x n k =. 3.2 j= j=

7 ON A DIFFERENCE EQUATION WITH MIN-MAX RESPONSE 292 Then we obtain u n+ = j= a j xn,...,x n k un+ j j= b j xn,...,x n k xn+ j j= a j xn,...,x n k un+ j j= b j xn,...,x n k xn+ j 2 n n k u j. j= b j xn,...,x n k un+ j j= b j xn,...,x n k xn+ j 3.22 Our intention is to show that limu n = 0. To this end, we can either use [7, Lemma ] or proceed as follows. Let Y m be an upper full limiting sequence of u n with Y m Y 0 = limsupu n, for all integersm. Then, from the previous arguments, it follows that it satisfies the inequality Y 0 2 Y 0, 3.23 thus we have Y 0 = 0. This and the fact that u n 0 imply that limx n = 2. Next, assume that x n 2, for all n k. Setv n := 2 x n. From.3 and by using the form of the function F, weobtain v n+ = j= a j xn,...,x n k vn+ j j= b j xn,...,x n k xn+ j j= a j xn,...,x n k vn+ j j= b j xn,...,x n k xn+ j M n j=n k v j, j= b j xn,...,x n k vn+ j j= b j xn,...,x n k xn+ j, 3.24 where M> is the number defined in Theorem 3.. By using this fact and following the same procedure as in the first case, we derive that lim n v n = 0, which implies that limx n = 2, as desired. Theorem 3.4. Consider.6, where α n =, n = 0,,..., and F is a continuous minmax function satisfying the properties.9 and.0. Then every positive solution converges to the equilibrium K = 2. Proof. Let x n be a solution. Then by Theorem 3., x n is bounded. Consider an upper full limiting sequence y m and a lower full limiting sequence z m of x n,as above. From.6, we have and therefore, we get This gives y 0 + y 0 z 0, z 0 + z 0 y y 0 z 0 = y 0 +z y 0 + z 0 =. 3.27

8 2922 G. L. KARAKOSTAS AND S. STEVIĆ If it happens that y 0,z 0 > 2, or y 0,z 0 < 2, then we should have /y 0,/z 0 < /2 and /y 0,/z 0 > /2, respectively. Both these arguments contradict Therefore, we must have z 0 2 y Assume that there is some j k,..., such that y j <y 0 and let j 0 be an index such that y j0 = j= k y j Then from.9, we get y j0 < j j0 y j y and so from.6 and condition.9, we have y 0 = +F y,...,y k < + y 0 y j0 + y 0 z This gives y 0 z 0 <y 0 + z 0, contradicting Thus we have y j = y 0, for all j = k,...,, and therefore, y 0 = +F y,...,y k = +F y0,...,y 0 = Similarly, we can use condition.0 to obtain z 0 = 2. The proof is complete. Our final result refers to the case α [0,. We show that in this case, there are equations of the form.3 which admit unbounded solutions. Theorem 3.5. Consider the equation x n+ = α+ i=0 a i x n 2i i=0 b i x n 2i, 3.33 where m N, α [0,, and where the coefficients a j and b j, j = 0,...,m, are nonnegative constants which satisfy the conditions m a i = b i i=0 Then there exist unbounded solutions of Proof. Obviously, without loss of the generality, we can assume that i=0 a i = i=0 b i =. i=0

9 ON A DIFFERENCE EQUATION WITH MIN-MAX RESPONSE 2923 Assume that α 0,. We choose the initial conditions such that x 2m+,...,x > α > +α, α<x 2m,...,x 0 < We set D := m i=0 x 2i and observe that D> α From 3.33, we have x = α+ x 2 = α+ i=0 a i x 2i+ i=0 b i x 2i i=0 a i x 2i+ i=0 b ix 2i m >α+ a i x 2i+ >α+d, i=0 <α+ i=0 b ix 2i = α+ α+ b 0 x +b x + +b m x 2m+ b 0 α+d+ b 0 / α α+ = α+ b 0 α+/ α + b0 / α b 0 α+/ α, i=0 a i x 2 2i+ m x 3 = α+ >α+ a i x 2 2i+ i=0 b i x 2 2i i=0 α+min x,x,...,x 2m+ α+min x,x,...,x 2m 3.38 = α+min x,min x,...,x 2m α+d. Following the same procedure, we get for all j = 0,,...,m. By induction, we obtain for all j N and s = 0,,...,m,aswellas x 2j+ >α+d, x 2j+2 <, 3.39 x 2m+2j 2s+ >αj+d, 3.40 α<x 2n <, n= m, m,...,, Inequality 3.40 implies the desired result in case α>0. Assume that α = 0. Choose ε 0, and the initial conditions such that x 2m+,...,x > ε, 0 <x 2m,...,x 0 < ε. 3.42

10 2924 G. L. KARAKOSTAS AND S. STEVIĆ From 3.33, we have x = i=0 a i x 2i+ i=0 b i x 2i > / ε ε = ε 2 > ε, x 2 = i=0 a i x 2i+ ε m < i=0 b i x 2i b 0 x + b 0 / ε ε b 0 / ε 2 + b 0 / ε < ε Following the same procedure, we get x 2j+ > ε > 2 ε, x 2j+2 < ε, 3.44 for all j = 0,,...,m. By induction, we obtain x 2m+2j 2s+ > for all j N and s = 0,,...,m,aswellas, 3.45 ε j+ 0 <x 2n < ε, n =,2, From 3.45, the result follows. Remark 3.6. Equation 3.33 includes the special case.3. Thus for α 0,, Theorem 3.5 applies and therefore,.3 has unbounded solutions with positive initial values. On the other hand, 3.33 does not include the case.4 and as proved in [2], for the same values of α,.4 has a global attractor. Remark 3.7. By some modifications of the proof of Theorem 3.5, wecanprovethe following result. Theorem 3.8. Consider the equation x n+ = α n + i=0 a i x n 2i i=0 b i x n 2i, 3.47 where m N, α n is a sequence of positive real numbers such that lim n α n =: A [0,, and where the coefficients a j and b j, j = 0,...,m, are nonnegative constants which satisfy the conditions m a i = b i i=0 i=0 Then there exist unbounded solutions of 3.47.

11 ON A DIFFERENCE EQUATION WITH MIN-MAX RESPONSE Some illustrative examples Example 4.. Consider the difference equation x n+ = α+ βx n +γxn 2 +δxn 2 βx n +γx n x n +δxn 2, 4. where all the coefficients are positive real numbers. The rational function on the righthand side is a min-max function, since it can be written in the form β+γxn / β+γxn +δx n xn + δx n / β+γx n +δx n xn β/ β+γxn +δx n xn γx n +δx n / β+γxn +δx n xn Thus, from Theorems 3.2 and 3.4, we conclude that, for every fixed α, any solution of 4. converges to the equilibrium α +. Notice that conditions.9 and.0 are also satisfied. Example 4.2. Consider the difference equation j x n+ = α+ i n,n,n 2 x j x j2 x j3 xn 3 +xn 3 +x3 n 2 +6x, 4.3 nx n x n 2 where α 0. This is a third-order difference equation whose response on the right-hand side is a min-max function. Indeed, this can be written in the form x 2 2 j i n,n,n 2, j j 2 j 3 j j +x j x j2 +x j x j3 / xn +x n +x n 2 x j x j i n,n,n 2, j j 2 j 3 j j +2x j2 x j3 / xn +x n +x n 2 x j Here, again, Theorems 3.2 and 3.4 apply and we conclude that in case α, any solution of 4.3 converges to α+. References [] A.M.Amleh,E.A.Grove,G.Ladas,andD.A.Georgiou,On the recursive sequence x n+ = α+x n /x n, J. Math. Anal. Appl , no. 2, [2] R. DeVault, G. Ladas, and S. W. Schultz, On the recursive sequence x n+ = A/x n +/x n 2, Proc. Amer. Math. Soc , no., [3] G. L. Karakostas, A discrete semi-flow in the space of sequences and study of convergence of sequences defined by difference equations, M. E. GreekMath. Soc , [4] G. L. Karakostas, Ch. G. Philos, and Y. G. Sficas, The dynamics of some discrete population models, Nonlinear Anal. 7 99, no., [5] G. L. Karakostas and S. Stević, Slowly varying solutions of the difference equation x n+ = fx n,...,x n k +gn,x n,x n,...,x n k, J. Difference Equ. Appl , no. 3, [6] M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations. With Open Problems and Conjectures, Chapman & Hall/CRC, Florida, [7] S. Stević, Behavior of the positive solutions of the generalized Beddington-Holt equation, Panamer. Math. J , no. 4, [8], A generalization of the Copson s theorem concerning sequences which satisfy a linear inequality, Indian J. Math , no. 3, [9], A note on bounded sequences satisfying linear inequalities, Indian J. Math , no. 2,

12 2926 G. L. KARAKOSTAS AND S. STEVIĆ [0], On the recursive sequence x n+ = /x n + A/x n, Int. J. Math. Math. Sci , no., 6. [], A global convergence result, Indian J. Math , no. 3, [2], A global convergence results with applications to periodic solutions, Indian J. Pure Appl. Math , no., [3], On the recursive sequence x n+ = α n + x n /x n, Int. J. Math. Sci , no. 2, [4], On the recursive sequence x n+ = α n +x n /x n.ii, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal , no. 6, George L. Karakostas: Department of Mathematics, University of Ioannina, 450 Ioannina, Greece address: gkarako@cc.uoi.gr Stevo Stević: Mathematical Institute of the Serbian Academy of Sciences and Arts, Knez Mihailova 35, 000 Beograd, Serbia address: sstevic@ptt.yu; sstevo@matf.bg.ac.yu

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