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let us remind the differential equations on time scales , proposed by hilger @xcite . the main element of the equations is the time scale , which is understood as a nonempty closed subset , @xmath0 of the real numbers . on a time scale @xmath1 the functions @xmath2 and @xmath3 are called the forward and backward jump operators , respectively . the point @xmath4 is called right - scattered if @xmath5 and right - dense if @xmath6 similarly , it is called left - scattered if @xmath7 and left - dense if @xmath8    the @xmath9derivative of a continuous function @xmath10 at a right - scattered point is defined as @xmath11 and at a right - dense point it is defined as @xmath12 if the limit exists . a differential equation @xmath13 is said to be a differential equation on time scale , where function @xmath14 in ( [ tsde ] ) is assumed to be rd - continuous on @xmath15 @xcite . if a point @xmath16 is right - dense , then the @xmath17-derivative is the ordinary derivative . otherwise , @xmath4 is a right - scattered point , and the @xmath17-derivative is defined by means of the quotient . let us discuss the second case more attentively . by the definition of the @xmath9derivative , assuming that @xmath16 is not an isolated point , one has that @xmath18 now , compare this expression with the numerical approximation of the function on the interval @xmath19.$ ] setting @xmath20 and @xmath21 we have that @xmath22 thus , one can see that the idea of the @xmath9derivative , as well as of the @xmath23derivative is close to the basic one for the numerical analysis @xcite . see figures [ fig1 ] and [ fig2 ] for illustration . that is , in its heart the concept of the differential equations on time scales is closely adjoint with numerical methods . however , for numerical methods , of course as far as the definition in @xmath24 is considered , smaller @xmath25 gives a better approximation . in the former theory of differential equations on time scales , the request was not considered , and the elements of linear change over the intervals near right - scattered ( left - scattered ) points are strong there . to return the non - linearity sense in the discussion , one needs to modify the equations . in our papers @xcite , the concept of transition conditions were proposed , when we consider the simple version of the time scale as a union of disjoint sections . in @xcite , these systems emerge as an auxiliary instrument of a discontinuous dynamical system investigation . in section [ sec3 ] , we generalize this idea for all possible time scales . in this part we shall answer two questions : why the differential equations on time scales are needed by real world problems ? how the hilger s equations can be extended for these ? consider a real world process , which can be symbolically given as a function @xmath26 where @xmath16 is from a time scale . suppose for this moment that the time scale admits only non - isolated moments . if the time is increasing , we have to prescribe what the value at @xmath27 is equal to for the right - scattered point @xmath28 in hilger s theory , the value is evaluated by using the derivative of @xmath29 at the moment @xmath30 and this is very restrictive , since the difference @xmath25 is not supposed to be small ! it is reasonable to assume that there exists more general law of transition for @xmath29 from the moment @xmath16 to @xmath31 in general , it can be given with a function @xmath32 we call this expression as the transition condition . to illustrate , consider a population developing in time . we may assume that the size satisfies a local law in some borders , say in summer time , and it is not under the law in winter time . moreover , the local law in winter time is not known . nonetheless , we can evaluate the change in the time , that is the size at the beginning of the season and that at the end of the season . assume that the change satisfies a new formula , which is naturally a discrete one . in that case , we come to the idea of the equations on time scales , but more general than those investigated before . consider the time scale @xmath33 given above , and the same @xmath9derivative of a continuous function @xmath10 defined by ( [ dd1 ] ) and ( [ dd2 ] ) . denote by @xmath34 the set of all right - scattered points of @xmath35 consider a function @xmath36 which is assumed to be rd - continuous on @xmath37 define a function @xmath38 now , on the time scale , we introduce a piecewise function @xmath39 which is equal to the function @xmath40 if @xmath16 is a right - dense point , and it is equal to @xmath41 if @xmath16 is a right - scattered point . we suggest to consider the following differential equation on the time scale @xmath42    condition that all solutions are continuous on the time scale should be assumed .    to demonstrate the opportunities , which one can have with the newly introduced equations , consider the following theorem of existence and uniqueness . let @xmath43 be a time scale , @xmath44 @xmath45 @xmath46 with @xmath47 and @xmath48 and put @xmath49 suppose that @xmath50 is rd - continuous , bounded with bound @xmath51 lipschitz continuous with lipschitz constant @xmath52 and @xmath53 for all @xmath54 and @xmath55 then the ivp @xmath56 has exactly one solution on @xmath57,$ ] where @xmath58 if @xmath59 and @xmath60 then the unique solution exists on the interval @xmath61.$ ] [ thmeu ]    one could compare the last assertion with the following , from @xcite . the condition on the transition function @xmath62 is added , in this particular case . in general , properties of the function will generate new theoretical , and what is most important , application prospects for these equations . one can see that many results in the theory , which have been obtained earlier can , now , be generalized , for newly proposed systems with additional interesting properties that transition conditions can admit . another opportunity is to consider state - dependent time scales @xcite . let a set @xmath63 in @xmath64 be given , such that for each @xmath65 the projection of the intersection , @xmath66 of the line @xmath67 with @xmath68 on the time axis is a time scale in the sense of hilger . thus , one can define the function @xmath69 for each point @xmath70 we assume that the function @xmath71 is continuous in @xmath72 if the moment @xmath16 is such that @xmath73 is right - dense in the correspond time scale , @xmath74 then the @xmath9derivative is defined by ( [ dd2 ] ) . otherwise it is equal to @xmath75 now , define a function @xmath76 which is equal to a continuous function @xmath77 at a right - dense point @xmath78 and to function @xmath79 if the point is a right - scattered one . then , one can discuss the following equation @xmath80 it is obvious that the equations with state - dependent time scales can also play a certain role in this theory , and they will be used in the modeling of real world problems . the analysis may be of great interest , since new opportunities related to the non - linear properties of the time scales will emerge . finally , we want to mention that very profitable ways for differential equations on time scales is to investigate them together with different types of discontinuities . these days there are several well extended theories such that systems with discontinuities in the right - hand - side , arguments , impulsive differential equations . they are realized in very interesting application projects as well @xcite-@xcite . we are sure that integration of the methods and results of these theories with above mentioned extensions of differential equations on time scales can provide really interesting horizons , which can give a light on new ways for modeling processes in mechanics , electronics , medicine , biology . one of the particular results in this sense is our recent paper @xcite , where certain class of differential equations on time scales is embedded in differential equations with fixed moments of impulses . a. katok , j .- strelcyn , f. ledrappier and f. przytycki , _ invariant manifolds , entropy and billiards ; smooth maps with singularities _ , lecture notes in mathematics , 1222 , springer - verlag , berlin , 1986 .
we give a proposal to generalize the concept of the differential equations on time scales @xcite , such that they can be more appropriate for the analysis of real world problems , and give more opportunities to increase the theoretical depth of investigation . 
 differential equations on time scales ; transition condition ; modeling