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Dynamic Systems and Applications 13 (2004) 351-379
PARTIAL DIFFERENTIATION ON TIME SCALES
MARTIN BOHNER AND GUSEIN SH. GUSEINOV
University of Missouri–Rolla, Department of Mathematics and Statistics, Rolla,
Missouri 65401, USA. E-mail: bohner@umr.edu
Atilim University, Department of Mathematics, 06836 Incek, Ankara, Turkey.
E-mail: guseinov@atilim.edu.tr
ABSTRACT. In this paper a differential calculus for multivariable functions on time scales is
presented. Such a calculus can be used to develop a theory of partial dynamic equations on time
scales in order to unify and extend the usual partial differential and partial difference equations.
AMS(MOS)Subject Classification. 26B05, 39A10, 26B10.
1. INTRODUCTION
A time scale is an arbitrary nonempty closed subset of the real numbers. The
calculus of time scales was initiated by B. Aulbach and S. Hilger [4,9] in order to
create a theory that can unify discrete and continuous analysis. For a treatment of
the single variable calculus of time scales see [5,6,11] and the references given therein.
The present paper deals with the differential calculus for multivariable functions
on time scales and intends to prepare an instrument for introducing and investigating
partial dynamic equations on time scales. Note that already two papers related to
this subject appeared [2,10]. An integral calculus of multivariable functions on time
scales will be given in forthcoming papers of the authors.
Thereareanumberofdifferencesbetweenthecalculusofoneandoftwovariables.
The calculus of functions of three or more variables differs only slightly from that of
two variables. The study in this paper will be therefore limited largely to functions
of two variables. Also we mainly consider partial delta derivatives. Partial nabla
derivatives and combinations of partial delta and nabla derivatives can be investigated
in a similar manner.
The paper is organized as follows. In Section 2 we introduce partial delta and
nabla derivatives for multivariable functions on time scales and offer several new
concepts related to differentiability. Section 3 deals with a geometric interpretation
of delta differentiability. Section 4 contains several useful mean value theorems for
derivatives. In Section 5 sufficient conditions to ensure differentiability of functions
c
Received August 15, 2003 1056-2176 $15.00
Dynamic Publishers, Inc.
352 MARTIN BOHNER AND GUSEIN SH. GUSEINOV
are provided. In Section 6 we present sufficient conditions for equality of mixed partial
delta derivatives. Section 7 is devoted to the chain rule for multivariable functions on
time scales, while Section 8 treats the concept of the directional derivative. Finally,
in Section 9, we study implicit functions on time scales.
2. PARTIAL DERIVATIVES AND DIFFERENTIABILITY
Let n ∈ N be fixed. Further, for each i ∈ {1,...,n} let T denote a time scale,
i
that is, T is a nonempty closed subset of the real numbers R. Let us set
i
n
Λ =T ×...×T ={t=(t ,...,t ): t ∈T for all i ∈ {1,...,n}}.
1 n 1 n i i
n n
Wecall Λ an n-dimensional time scale. The set Λ is a complete metric space with
the metric d defined by
!
n 1/2
X 2 n
d(t,s) = |t −s | for t, s ∈ Λ .
i i
i=1
Consequently, according to the well-known theory of general metric spaces, we have
n
for Λ the fundamental concepts such as open balls, neighbourhoods of points, open
sets, closed sets, compact sets, and so on. In particular, for a given number δ > 0,
0 0 n n
the δ-neigbourhood U (t ) of a given point t ∈ Λ is the set of all points t ∈ Λ such
δ
0 0 n n
that d(t ,t) < δ. By a neigbourhood of a point t ∈ Λ is meant an arbitrary set in Λ
0 n
containing a δ-neighbourhood of the point t . Also we have for functions f : Λ → R
the concepts of the limit, continuity, and properties of continuous functions on general
complete metric spaces.
Our main task in this section is to introduce and investigate partial derivatives
n
for functions f : Λ → R. This proves to be possible due to the special structure of
n
the metric space Λ .
Let σ and ρ denote, respectively, the forward and backward jump operators in
i i
T. Remember that for u ∈ T the forward jump operator σ : T → T is defined by
i i i i i
σ (u) = inf {v ∈ T : v > u}
i i
and the backward jump operator ρ : T → T is defined by
i i i
ρ (u) = sup{v ∈ T : v < u}.
i i
In this definition we put σ (maxT ) = maxT if T has a finite maximum, and
i i i i
ρ (minT ) = minT if T has a finite minimum. If σ (u) > u, then we say that
i i i i i
u is right-scattered (in T ), while any u with ρ (u) < u is called left-scattered (in
i i
T). Also, if u < maxT and σ (u) = u, then u is called right-dense (in T ), and
i i i i
if u > minT and ρ (u) = u, then u is called left-dense (in T ). If T has a left-
i i i i
κ κ
scattered maximum M, then we define T = T \{M}, otherwise T = T . If T has
i i i i i
a right-scattered minimum m, then we define (T ) = T \{m}, otherwise (T ) = T .
i κ i i κ i
PARTIAL DIFFERENTIATION ON TIME SCALES 353
n
Let f : Λ → R be a function. The partial delta derivative of f with respect to
κ
t ∈ T is defined as the limit
i i
f(t ,...,t , σ (t ), t , . . . , t ) −f(t ,...,t , s , t , . . . , t )
lim 1 i−1 i i i+1 n 1 i−1 i i+1 n
s →t σ (t ) − s
i i i i i
s 6=σ (t )
i i i
provided that this limit exists as a finite number, and is denoted by any of the
following symbols:
∂f(t ,...,t ) ∂f(t) ∂f
1 n , , (t), f∆i(t).
t
∆t ∆t ∆t i
i i i i i i
If f has partial derivatives ∂f(t),..., ∂f(t), then we can also consider their partial delta
∆ t ∆ t
1 1 n n
derivatives. These are called second order partial delta derivatives. We write
∂2f(t) ∂2f(t) h ∆i∆i ∆i∆j i
and or f (t) and f (t)
t t t t
2 i i i j
∆t ∆t∆t
i i j j i i
for the partial delta derivatives of ∂f(t) with respect to t and with respect to t ,
∆t i j
i i
respectively. Thus
∂2f(t) = ∂ ∂f(t) and ∂2f(t) = ∂ ∂f(t).
2
∆t ∆t ∆t ∆t∆t ∆t ∆t
i i i i i i j j i i j j i i
Higher order partial delta derivatives are similarly defined. The partial nabla deriva-
tive of f with respect to t ∈ (T ) is defined as the limit
i i κ
f(t ,...,t , ρ (t ), t , . . . , t ) −f(t ,...,t , s , t , . . . , t )
lim 1 i−1 i i i+1 n 1 i−1 i i+1 n
si→ti ρ (t ) − s
s 6=ρ (t ) i i i
i i i
and denoted by ∂f(t), provided that this limit exists as a finite number. In an obvious
∇t
i i
way we can define higher order partial nabla derivatives and also mixed derivatives
obtained by combining both delta and nabla differentiations such as, for instance,
∂2f(t) ∂3f(t)
or 2 .
∆t∇t ∆t ∇ t
i i j j i i j j
n
Definition 2.1. We say that a function f : Λ → R is completely delta differentiable
0 0 0 κ κ
at a point t = (t ,...,t ) ∈ T × ... × T if there exist numbers A1,...,An inde-
1 n 1 n
n 0
pendent of t = (t ,...,t ) ∈ Λ (but, in general, dependent on t ) such that for all
1 n
0
t ∈ U (t ),
δ
n n
0 0 X 0 X 0
(2.1) f(t ,...,t ) − f(t ,...,t ) = A(t −t)+ α(t −t)
1 n 1 n i i i i i i
i=1 i=1
0
and, for each j ∈ {1,...,n} and all t ∈ U (t ),
δ
0 0 0 0 0
(2.2) f(t ,...,t , σ (t ),t , . . . , t ) −f(t ,...,t , t , t , . . . , t )
1 j−1 j j j+1 n 1 j−1 j j+1 n
n n
0 X 0 0 X 0
=A σ(t )−t + A(t −t)+β σ (t )−t + β (t −t ),
j j j j i i i jj j j j ij i i
i=1 i=1
i6=j i6=j
354 MARTIN BOHNER AND GUSEIN SH. GUSEINOV
0 0
where δ is a sufficiently small positive number, Uδ(t ) is the δ-neighbourhood of t in
n 0 0 0
Λ , α = α (t ,t) and β =β (t ,t) are defined on U (t ) such that they are equal
i i ij ij δ
0
to zero at t = t and such that
0 0
lim α (t ,t) = 0 and lim β (t ,t) = 0 for all i, j ∈ {1,...,n}.
0 i 0 ij
t→t t→t
In the case T = ... = T = R, this definition coincides with the classical (total)
1 n
differentiability of functions of n real variables (see, for example, [3,12]).
In the one-variable case, Definition 2.1 becomes the following: A function f :
0 κ
T → R is called completely delta differentiable at a point t ∈ T if there exists a
number A such that
0 0 0 0
(2.3) f(t ) − f(t) = A(t −t)+α(t −t) for all t ∈ U (t )
δ
and
0 0 0 0
(2.4) f(σ(t )) −f(t) = A σ(t )−t +β σ(t )−t for all t ∈ Uδ(t ),
0 0 0
where α = α(t ,t) and β = β(t ,t) are equal to zero at t = t and
0 0
lim α(t ,t) = 0 and lim β(t ,t) = 0.
0 0
t→t t→t
0
If t is right-dense, then the conditions (2.3) and (2.4) coincide and are equivalent to
0 0
the existence of a usual derivative of f at t , being equal to A. If t is right-scattered
0
and left-dense, then (2.3) and (2.4) mean, respectively, that the function f has at t
a usual left-sided derivative and a delta derivative and that these derivatives coincide
and are equal to A. In this place we see a difference between the completely delta dif-
ferentiability and delta differentiability, where the latter means, simply, the existence
of a delta derivative. This is why we use the term “completely delta differentiable”
0
rather than “delta differentiable”. Finally, if t is right-scattered and left-scattered
at the same time (i.e., an “isolated” point), then the condition (2.3) disappears be-
cause both of its sides are zero independent of A and α (for sufficiently small δ, the
0 0
neighbourhood Uδ(t ) consists of the single point t ), and (2.4) means the existence
0
(which holds, in this case) of a delta derivative of f at t , being equal to A.
In the two-variable case, Definition 2.1 becomes the following: A function f :
0 0 κ κ
T ×T →Riscompletely delta differentiable at a point (t ,s ) ∈ T ×T if there
1 2 1 2
exist numbers A and A such that
1 2
0 0 0 0 0 0
(2.5) f(t ,s ) − f(t,s) = A (t −t)+A (s −s)+α (t −t)+α (s −s)
1 2 1 2
and
0 0 0 0
(2.6) f(σ (t ),s ) − f(t,s) = A σ (t ) − t +A (s −s)
1 1 1 2
+β 0 0
σ (t ) − t +β (s −s),
11 1 12
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