EXISTENCE OF POSITIVE PERIODIC SOLUTIONS OF DISCRETE MODEL FOR THE INTERACTION OF DEMAND AND SUPPLY. S. H. Saker

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1 Nonlinear Funct. Anal. & Appl. Vol. 10 No. 005 pp EXISTENCE OF POSITIVE PERIODIC SOLUTIONS OF DISCRETE MODEL FOR THE INTERACTION OF DEMAND AND SUPPLY S. H. Saker Abstract. In this paper we derive the nonlinear discrete economic model for the interaction of demand and supply with periodic coefficients with a common period. We prove the global existence of positive periodic solutions with strictly positive components by using the method of continuation theorem in coincidence degree theory. 1. Introduction In recent decades the ideas from the theory of complex dynamics and Biological systems has been used to describe the economics and finance problems. In the early 1950s Richard Goodwin used the nonlinear techniques in the study of the dynamic economic processes [5]. With the progress of the research on nonlinear complex Biological systems some concepts and methods in nonlinear dynamics such as stability oscillation periodicity bifurcation catastrophe chaos synchronization control etc. have been applied to economic problems and some results have been established see [1] and the references cited therein. In [1] the authors suggests a two-dimensional logistic model to describe the interactions of potential demand and supply. The logistic equations that determined the evolution of the potential demand and the potential supply Received March Revised May Mathematics Subject Classification: 34C5 91B6. Key words and phrases: Positive periodic solutions economic model demand and supply.

2 31 S. H. Saker are given by dy 1 dt = ay 1 dy dt = by 1 cy 1 y 1 y 1 y y cy where c = M d /M s M d is the sub-capacity for the potential demand and M s is the sub-capacity for the potential supply and the parameters a and b are greater than or equal zero. In [1] the authors showed that beside the trivial fixed point 0 0 there is a stationary point y1 y and proved that with any nonzero initial condition the solution tends to it. However most of the above results are obtained when the coefficients are constants. The variation of the environment plays an important role in many biological ecological and economical dynamical systems. In particular the effects of a periodically varying environment are important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Thus the assumption of periodicity of the parameters in the system in a way incorporates the periodicity of the environment. For qualitative behavior of some periodic models of one dimension we refer the reader to the papers [8-11] and the references cited therein. In view of this it is realistic to assume that the parameters in the models are periodic functions of period. Thus the modification of 1.1 according to the environmental variation is the nonautonomous differential equations dy 1 t dt dy t dt = aty 1 t = bty t 1 cty 1t y t y 1t 1 y t cty 1 t y t. 1. Analytically nonlinear differential equations are difficult to manage and therefore many articles have examined the models as the difference equations [6] and the references cited therein. In practice one can formulate a discrete model directly from experiments and observations. Some times for numerical purpose one wants to purpose a finite-difference scheme to numerically solve a given differential model especially when the differential equation cannot be solved explicitly. For a given differential equation a difference equation approximation would be most acceptable if the solution of the difference equation is the same as

3 Existence of positive periodic solutions of discrete model 313 the differential equation at the discrete points. But unless we can explicitly solve both equations it is impossible to satisfy this requirement. Most of the time it is desirable that a difference equation when derived from a differential equation preserves the dynamical features of the corresponding continuous time model such as equilibria oscillation their local and global stability characteristics and bifurcation behaviors. If such discrete models can be derived from continuous models then the discrete time models can be used without loss of any functional similarity to the continuous-time models and it will preserve the considered realities; such discrete time models can be called Dynamically consistent with the continuous time models. There is no unique way of deriving discrete time version of dynamical systems corresponding to continuous time formulations. In this approach differential equations with piecewise constant arguments have been useful see for example [7]. Before stating our main results about global existence of positive periodic solutions we will derive the discrete analogy of Eq.1.. Thinks to differential equations with piecewise constant arguments we can go on with the discrete analogy of Eq.1.. Let us assume that the average growth rate in 1. changes at regular intervals of time then we can incorporate this aspect in 1. and obtain the following modified equation 1 dy 1 t y 1 t dt = a[t] 1 dy t = b[t] y t dt 1 1 c[t]y 1[t] y t y 1[t] y [t] c[t]y 1 [t] y [t] 1.3 where [t] denotes the integer part of t t 0. Equation of type 1.3 is known as differential equation with piecewise with constant argument and this equation occupy a position midway between differential and difference equation. By a solution of 1.3 we mean a function Y t = y 1 t y t T which is defined for t [0 and satisfy the properties: a Y is continuous on [0. b The derivative dy t dt = dy1 t dt T dy t dt exists at each point t [0 with the possible exception of the points t {0 1...} where left side derivative exists. c The equation 1.3 is satisfied on each interval [n n + 1 with n = By integrating 1.3 on any interval of the form [n n + 1 n = we

4 314 S. H. Saker obtain y 1 t = y 1 n exp an y t = y n exp bn Letting t n + 1 we obtain that y 1 n + 1 = y 1 n exp an y n + 1 = y n exp bn 1 cny 1n y n y 1n 1 y n cny 1 n y n 1 cny 1n y n t n t n y 1n 1 y n cny 1 n y n. 1.4 which is a discrete time analogy of 1.. We note that the equilibrium point of 1.4 is the same as the equilibrium point of the system of differential equations 1.. So the derived discrete analogy preserves on the equilibria. By a solution of Eq.1.4 we mean a sequences {y 1 n} {y n} which are defined for n 0 and which satisfies 1.4 for n 0. Considering the economical significance of 1.4 we specify y 1 0 y 0 > The exponential forms of 1.4 assure that the solution y 1 n y n T with respect to any initial condition 1.5 remains positive. In recent years the investigation of the theory of difference equations has assumed a greater importance as well deserved discipline. Many results in the theory of difference equations have been obtained as more or less natural discrete analogous of corresponding results of differential equations [1 ]. Nevertheless the theory of difference equations is a richer than the corresponding theory of differential equations. For example a simple logistic difference equation resulting from a first order logistic differential equation exhibits the chaotic behavior which can only happen in higher order differential equations. The purpose of this paper is to prove the global existence of periodic solutions with strictly positive components for the discrete two-dimensional logistic model 1.4. Such an existence problem is highly nontrivial and to the best of our knowledge no work has been done for the discrete model 1.4. The method used here will be the continuation theorem in coincidence degree theory proposed by Gaines and Mawhin [4] which has been widely used in the study of ordinary differential equations and recently some authors applied to the study of the global existence of periodic solutions for some mathematical models we refer to [3] and the references cited therein.

5 Existence of positive periodic solutions of discrete model 315. Existence of Positive Periodic Solutions In this section we will assume that the parameters in 1.4 are periodic sequences of period. i.e. an + = an bn + = bn and cn + = cn..1 A very basic and important problem in the study of dynamics of the dynamical system in a periodic environment is the global existence of a positive periodic solution which plays a similar role played by the equilibrium of the autonomous models. Thus it is reasonable to ask for a condition under which the resulting periodic nonautonomous equation have a positive periodic solution. For the reader s convenience we now recall some basic tools in the frame of Mawhin s continuation theorem in coincidence degree theory that will be used to prove the existence of periodic solution of 1.4 borrowing notations and terminology from [4]. Let X and Y be two Banach spaces let L : DomL X Y be a linear mapping and let N : X Y be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if the following three conditions hold: i KerL has a finite dimension; ii ImL is closed and has a finite codimension. iii dim KerL = codimiml <. If L is a Fredholm mapping of index zero and there exist continuous projectors P : X X and Q : Y Y such that ImP = KerL ImL = KerQ = ImI Q it follows that L DomL KerP : I P X ImL is invertible. We denote the inverse of that map by K p. If Ω is an open bounded subset of X the mapping N will be called L- compact on Ω if the mapping QN : Ω Y is continuous QNΩ is bounded and K p I QN : Ω X is compact i.e. it is continuous and K p I QNΩ is relatively compact where K p : ImL DomL KerP is the inverse of the restriction L p of L to DomL KerP so that LK p = I and K p L = I P. Since Q is isomorphic to KerL there exists an isomorphic J : ImQ KerL. Now we are ready to cite the continuation theorem Gaines and Mawhin [10 p.40].

6 316 S. H. Saker Lemma.1. Continuation Theorem Let X and Y be two Banach spaces and L a Fredholm mapping of index zero. Assume that N : Ω Y is L-compact on Ω with Ω is open and bounded in X. Furthermore assume: a for each λ 0 1 every solution of Lx = λnx is such that x / Ω b QNx 0 for each x Ω KerL and deg{qnx Ω KerL 0} = 0. Then the operator equation Lx = Nx has at least one solution in DomL Ω. Let Z Z + N R R + denote the sets of all integers nonnegative integers natural numbers real numbers and nonnegative real numbers respectively. For convenience in what follows we shall let I = { } f = 1 fn F = 1 fn where fn is an -periodic sequence of real numbers defined for all n Z. We need the following Lemma in the proof of our main results. Lemma.. [13] Let f : Z R be periodic i.e. fn + = fn. Then for any fixed n 1 n I and for any n Z one has fn fn 1 + fs + 1 fs s=0 fn fn fs + 1 fs. s=0 Our main result on the global existence of a positive -periodic solution of 1. is the following theorem. Theorem.1. Assume that.1 holds. Then 1.4 has at least one positive -periodic solution yn = y 1 n y n T with strictly positive components and there exist positive constants z i w i such that z i y i n w i for i = 1. Proof. Let y i n = exp{x i n} i = 1.

7 Existence of positive periodic solutions of discrete model 317 Then the system 1.4 takes the form x 1 n + 1 x 1 n = an x n + 1 x n = bn 1 cn exp{x 1n} exp{x n} exp{x 1n} 1 exp{x n} cn exp{yn} exp{x n}.. In order to apply Lemma.1 to. we first define and X = Z = {xn = x 1 n x n T R : xn + = xn} x = x 1 n x n T = max n I x 1 n + max n I x n x X or Z. Then X and Z are Banach spaces when they are endowed with the norm. Let an 1 cn exp{x 1n} exp{x n} exp{x 1n} and let Nx = bn 1 exp{x n} cn exp{yn} exp{x n} x X. Lx = xn = xn + 1 xn P x = 1 xn x X Qz = 1 zn z Z. where xn = x 1 n x n T. Then from Lemma.1 in [13] we have KerL = R ImL = {z Z : zn = 0} is closed in Z dimkerl = = codimiml. and P Q are continuous projectors such that ImP = KerL KerQ = ImL = ImI Q.

8 318 S. H. Saker Therefore L is a Fredholm mapping of index zero. Furthermore the generalized inverse of L K P : ImL KerP DomL exists which is given by n 1 K P z = zs 1 n 1 zs. s=0 s=0 Then QN : X Z and K P I Q : X X read 1 an 1 cn exp{x 1n} exp{x n} exp{x 1n} QNx = 1 bn 1 exp{x n} cn exp{yn} exp{x n} n 1 s=0 K P I QNx = n 1 s=0 an 1 cn exp{x 1n} exp{x n} exp{x 1n} bn 1 exp{x n} cn exp{yn} exp{x n} 1 n 1 an s=0 1 n 1 s=0 n 1 n 1 bn 1 cn exp{x 1n} exp{x n} exp{x 1n} 1 exp{x n} cn exp{yn} exp{x n} an 1 cn exp{x 1n} exp{x n} exp{x 1n} bn 1 exp{x n} cn exp{yn} exp{x. n} From the definitions of QN and K P I QN we have that they are continuously and also they are map bounded functions to bounded functions. By the Ascoli-Arzela Theorem we see that QNΩ and K P I QNΩ are relatively compact for any open bounded set Ω X. Thus N is L compact on Ω for any open bounded set Ω X. Now we reach the position to search for an appropriate open bounded subset Ω for the application of Lemma.1. Corresponding to the operator

9 Existence of positive periodic solutions of discrete model 319 equation Lx = λnx λ 0 1 we have { x 1 n+1 x 1 n = λ an 1 cn exp{x 1n} exp{x n} exp{x } 1n} { x n+1 x n = λ bn 1 exp{x n} cn exp{yn} exp{x } n}..3 Suppose that x = xn X is a solution of.3 for a certain λ 0 1. Summing both sides of.3 from 0 to 1 and using the fact that xn+ 1 xn = 0 we have { an 1 cn exp{x 1n} exp{x n} exp{x } 1n} = 0 { bn 1 exp{x n} cn exp{yn} exp{x } n} = 0 that is { ancn exp{x1 n} exp{x n} From.3-.5 it follows that x 1 n + 1 x 1 n + an exp{x } 1n} = } { bn exp{x n} cn exp{yn} + cn exp{x n} an = a.4 = b.5 = λ an 1 cn exp{x 1n} exp{x n} exp{x 1n} ancn exp{x1 n} < an + + an exp{x 1n} exp{x n} = A + a.6 { x n + 1 x n = λ bn 1 exp{x n} cn exp{yn} exp{x } n}

10 30 S. H. Saker that is { bn exp{x n} < bn + cn exp{yn} + bn exp{x } n} = B + b.7 x 1 n + 1 x 1 n A + a.8 x n + 1 x n B + b.9 Since xn = x 1 n x n T X there exist ζ i η i I such that x i ζ i = min n I x i n x i η i = max n I x i n i = From.4 and.10 we get { an a = exp{x 1n} + ancn exp{x } 1n} exp{x n} an exp{x 1ζ 1 dt = exp{x 1 ζ 1 a/ that is x 1 ζ 1 ln..11 Then from Lemma.1.8 and.11 we have x 1 n x 1 ζ 1 + x 1 n + 1 x 1 n < ln + A + a := M 1..1 Also from.5 and.10 we find that Then from Lemma.1.9 and.13 we have x ζ ln..13 x n x ζ + x n + 1 x n < ln + B + b := M..14

11 Existence of positive periodic solutions of discrete model 31 From.4 and.10 we have a + ac { } an expx 1 η 1 exp{x 1n} + ancn exp{x 1 n} { an exp{x 1n} + ancn exp{x } 1n} exp{x n} = an = a and then we have Thus by Lemma.1 x 1 η 1 ln := M c x 1 n x 1 η 1 x 1 n + 1 x 1 n M 3 A + a := M 4.15 From.5.10 we have 1 c + b bn expx η b cn + bn expx n { bn cn exp{x n} + bn exp{x } n} dt { bn exp{x n} cn exp{x 1 n} + bn exp{x } n} dt = b that is Thus by Lemma.1 x η ln c b b := M 5. + b x n x η x n + 1 x n M 5 B + b := M 6.16

12 3 S. H. Saker Equations.1 and.15 imply that max x 1 n max{ ln + A + a M 3 A + a } := M 1. n I Equations.14 and.16 imply that max x n max{ ln + B + b M 6 } := M. n I ClearlyM i M j i = and j = 1 are independent of λ. Under the assumption in theorem.1 the system of algebraic equations a1 cu 1 u u 1 = 0 b1 u cu 1 u = 0 has a unique solution u 1 u R see [1]. Denote M = M 1 + M + M 3 where M 3 is taken sufficiently large such that ln{u 1} ln{u } = ln{u 1} + ln{u } M 3 and take Ω := {xn = x 1 n x n T X : x < M}. It is clear that Ω verifies requirement a of Lemma.1. When x Ω KerL = Ω R x is a constant with x = M. Then we have QNx = Furthermore it is easy to see that deg{jqnx Ω KerL 0} a a exp{x 1} ca exp{x 1} exp{x } b b exp{x } b exp{x } c exp{x 1 } 0. = deg{jqnx Ω R 0} a = sign exp{x 1} ca exp{x 1} ca exp{x 1 +x } exp{x } exp{x } b exp{x +x 1 } c exp{x 1 b } exp{x } b exp{x } c exp{x 1 } ab expx1 + x = sign + ab expx + ac expx

13 Existence of positive periodic solutions of discrete model 33 where deg the Brouwer degree and J can be the identity mapping since ImQ = KerL. By now we have proved that Ω verifies all the requirements of Lemma.1. Then it follows that Lx = Nx has at least one solution in DomL Ω say xn = x 1 n x n T. Set y i n = expx i n then yn = y 1 n y n T is an -periodic solution of 1.4 with strictly positive components. The boundedness of xn implies that the existence of positive constants z i w i such that z i y i n w i for i = 1. The proof is complete. Remark.1. Theorem.1 tells us that the system 1.4 has one positive periodic solution provided the periodicity of the coefficients this means that the periodicity of the sub-capacity of the demand and supply lead to the periodicity of the total potential demand and total potential supply. Remark.. The change of the system is usually determined by the developing capacity. But in a real socio-economic system the evolution of the market is always influenced by the policies of the government. In [1] the authors assumed that the demand policies changed the developing capacity of potential demand and introduced a parameter µ to describe its effect and assumed that the model with demand polices is given by dy 1 dt = aty 1 1 cty 1 µ y 1 µ y dy dt = bty 1 y y cty 1.17 We note that by the same approach above we can derive the discrete analogy of.17 and proved the existence of positive periodic solutions. Remark.3. The results of Theorem.1 remains valid if some or all terms are replaced with time delays. But it stills to explain the physical meaning of these delays. Acknowledgment. The author thanks Prof. M. A. Noor for his interesting to read the first version of the manuscript. References 1. R. P. Agarwal Difference Equations and Inequalities Theory Methods and Applications Second Edition Revised and Expanded Marcel Dekker New York M. Fan and S. Agarwal Periodic solutions of nonautonomous of discrete predator-prey system of voltera type Appl. Anal

14 34 S. H. Saker 3. M. Fan and K. Wang Periodic solutions of a discrete time nonautonomous ratiodependent predator-prey system Mathl. Comp. Modelling R. E. Gaines and J. L. Mawhin Coincidence degree and nonlinear differential equations 1977 Springer Berlin. 5. R. M. Goodwin The nonlinear accelerator and the persistence of business cycle Econometria XIX I. Kubiaczyk and S. H. Saker Oscillation and global attractivity of discrete survival red blood cells model Applicationes Mathematicae P. Liu and K. Gopalsamy Global stability and chaos in a population model with piecewise constant arguments Appl. Math. Comp S. H. Saker and S. Agarwal Oscillation and global attractivity in a nonlinear delay periodic model of population dynamics Appl. Anal S. H. Saker and S. Agarwal Oscillation and global attractivity in a nonlinear delay periodic model of respiratory dynamics Computers Math. Appl S. H. Saker and S. Agarwal Oscillation and global attractivity in a periodic Nicholson s blowflies model Mathl. Comp. Modelling S. H. Saker and S. Agarwal Oscillation and global attractivity of a periodic survival red blood cells model Journal Dynamics of Continuous Discrete and Impulsive Systems Series B: Applications & Algorithms to appear. 1. D. Zengru and M. Sanglier A two-dimensional logistic model for the interaction of demand and supply and its bifurcations Chaos Solitons and Fractals R. Y. Zhang Z. C. Wang Y. Chen and J. Wu Periodic solutions of a single species discrete population model with periodic harvest/stock Comp. Math. Appl S. H. Saker Mathematics Department Faculty of Science Mansoura University Mansoura Egypt address: shsaker@mans.edu.eg