Some Properties of a Semi Dynamical System. Generated by von Forester-Losata Type. Partial Equations

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1 Int. Journal of Math. Analysis, Vol. 7, 2013, no. 38, HIKARI Ltd, Some Properties of a Semi Dynamical System Generated by von Forester-Losata Type Partial Equations Eman Samir, Iftichar Muder and Eman Jawad University of Babylon, College of Education for Pure Sciences Mathematics Department, Babylon, Iraq emanbhaya@yahoo.com, iftichar_talb@yahoo.com, im_j2012@yahoo.com Copyright 2013 Eman Samir et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we consider a semi dynamical system, generated by different partial equations of Von Forester-Lasota a type. We investigate some of the properties of dynamical system in the space with 1. Keywords: von Forester-Lasota equation, stability. 1-Introduction First, we consider the partial differential equation,0,01 (1) with the initial condition 0,, 01 (2)

2 1864 Eman Samir, Iftichar Muder and Eman Jawad where belong to some normed vector space of functions defined on 0,1. The function is given by the formula,, 0,1 (3) where is the unique solution of 1 and 2. The second considered partial differential equation is, 0, 01 (4) with the initial condition 0,, 01 (5) where belongs to some normed vector space of functions defined on 0,1 and :0,1 is given continuous function. Let a semi dynamical system : Be given by the formula,. It is clear that the unique of (4), (5) is given by the formula where,, 0,1 (6) with the condition. (7) This can be found in Dawidowicz, Poskrobko (2006). There exists a connection between these two equations. It is easy to check that if and are the solutions of the equation (4) and (1), respectively, we have the equality where,,, (8) and 0. All properties of the system and depend on the value of the constant 0.

3 Properties of a semi dynamical system 1865 Definition1.1 A function is a periodic point of the semi group with a period 0 if and only if. A number 0 is called a principal period of a periodic point if and only if the set of all periods of is equal. Definition1.2 The semi group T is strongly stable in iff for every, lim 0 in. Definition1.3 The semi group is exponentially stable on iff there exists and 0 such that, for 0 where. is the norm of. 2. Properties of dynamical system Theorem 2.1 Assume that C,q0 x 0,1 λ0λx Cx (9) Holds, then we have the equivalence: the function belongs to the space if and only if, 1. Proof: by (9), iff. This can be found in Dawidowic, Poskrobko (2006). Assume that we have, (),,, 0,,,,,,,, So lim, 0 In this same manner we can establish the inverse implication.,,

4 1866 Eman Samir, Iftichar Muder and Eman Jawad,,,,,,,,, lim, 0.,, Theorem 2.2 if 00, then for any there exists such,1 that (10) Moreover, if and only if for some positive integer (11) Proof: Let be an arbitrary function belong to, defined on the interval,1 and satisfying the following conditions : 1, (12) 1 0,. (13) Consider the function on the interval 0,1,. for The function is defining on the whole interval (0,1, and comes into being squeezing the graph of the function into each of the intervals,. By assumption of the continuity of,1,.. 0 Such that for each,1. By the above for, we have the estimation., where.,. from the assumption (7) lim 0 so we deduce that 00. we obtain the continuous function defined on the whole interval0,1. The property (10) follows from (12), while the property (11) from (13). Our next goal is to show that. Under theorem (4.1)6.we know that for

5 Properties of a semi dynamical system , Where is the solution of the equation (1). It clearly the same conclusion for the function by theorem (2.1) Theorem 2.3 if 00 then the set of periodic points of (4 ) is dense in,1. Proof: let 0 and let. let be a periodic solution of (4 )Defined by the formula (6 ) Since and belong to there exist to such that,. and, We know that, where is the periodic solution of (1) The assumption 00, guarantees the density of the set of periodic points of (1), so, and, Thus,,,,,,,,,. Theorem 2.4 if 00 then for every, lim, 0. Moreover, if 00, the semigroup is exponentially stable. Proof: Take any, then we have,,,,, And, so,, Then,, then Since,,,

6 1868 Eman Samir, Iftichar Muder and Eman Jawad Applying theorem (4.3)1 we can assert that, 0, as. This proves the first part of the theorem. The second one follows immediately from the same above inequality and theorem (4.3)6 with and 0,,.. References 1 A.L.Dawidowicz and A. Poskrobko, On asymptotic behavior of the dynamical system generated by Von-Lasota equations. Control and Cybernetics. 35,4 (2006), A. L. Dawidowicz, A. L. On the existence of an invariant measure for a quasi-linear partial differential equation, Zesyty Naukowe UJ, Prace Matematyczene 23, (1982), A. L. Dawidowicz and N. Haribash, On the periodic solution of von Foerster type equation. Universitatis Iagellonice Acts Mathematica 37(1999), A. Lasota and T. Szarek, Dimension of measures invariant wit respect to Wazewska partial differential equation, J. Differential Equations 2 (2004), K. Loskot, Turbulent solution of first order partial differential equation, J. Differential Equations 58 (1985) 1, E. Samir, I. Mudar, and E. Jawad, (2013) Stability and density of period point Of for 01,34,7(2013), Received: April 8, 2013