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Basic existence, uniqueness and approximation results for positive
solutions to nonlinear dynamic equations on time scales
L. Erbe a , A. Peterson a , Christopher C. Tisdell b,∗
a Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA
b School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW 2052, Australia
Received 23 June 2007; accepted 3 August 2007
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Abstract
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This paper focuses on the qualitative and quantitative properties of solutions to certain nonlinear dynamic equations on time
scales. We present some new sufficient conditions under which these general equations admit a unique, positive solution. These
positive (and hence non-oscillatory) solutions: extend across unbounded intervals; and tend to a finite limit as the independent
variable becomes large and positive. Our methods include: Banach’s fixed-point theorem, including the method of Picard iterations;
and weighted norms and metrics in the time scale setting. Due to the wide-ranging nature of dynamic equations on time scales our
results are novel: for ordinary differential equations; for difference equations; for combinations of the two areas; and for general
time scales — this is demonstrated via some examples. Furthermore, we state an open problem of interest.
c 2007 Published by Elsevier Ltd
MSC: 39A10
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Keywords: Positive solution; Second-order dynamic equations; Terminal condition; Time scale
1. Introduction
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“Oscillation theory” forms an important area in the qualitative analysis of differential equations. This includes the
existence of oscillatory or non-oscillatory solutions to second-order, ordinary differential equations. A solution to a
(real) second-order ordinary differential equation is termed to be “oscillatory on [a, ∞)” provided that it exists on
[a, ∞) and has an arbitrarily large number of zeros on [a, ∞). A nontrivial solution that has a finite number of zeros
is called non-oscillatory.
The theory of oscillations has a long and rich history dating back to the work of Sturm in the 1830s [21,15] and
has since enjoyed numerous applications to a wide range of areas in science, engineering and technology [19, pp.
xiii–xiv]. Moreover, interest in the area continues to grow as oscillation theory naturally stimulates the emergence of
new mathematical theories and methods [19, p. xiv], [1], including the improvement of modern numerical methods
and their application to computing.
∗ Corresponding author.
E-mail addresses: [email protected] (L. Erbe), [email protected] (A. Peterson), [email protected] (C.C. Tisdell).
c 2007 Published by Elsevier Ltd
0362-546X/$ - see front matter doi:10.1016/j.na.2007.08.010
Please cite this article in press as: L. Erbe, et al., Basic existence, uniqueness and approximation results for positive solutions to nonlinear
dynamic equations on time scales, Nonlinear Analysis (2007), doi:10.1016/j.na.2007.08.010
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[r (t)x 1 (t)]1 + F(t, x σ (t), x 1σ (t)) = 0,
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t ∈ [a, ∞)T := [a, ∞) ∩ T;
(1.1)
where F : [a, ∞)T × R2 → R may be a nonlinear function; t is from a so-called “time scale” T (which is a nonempty
closed subset of R); x 1 is the generalised “delta” derivative of x; a ∈ T; r : [a, ∞)T → (0, ∞); and x σ := x ◦ σ
with σ a function to be defined a little later.
We will also consider the following special case of (1.1)
[r (t)x 1 (t)]1 + F(t, x σ (t)) = 0,
t ∈ [a, ∞)T .
(1.2)
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If T = R then x 1 = x 0 with σ (t) = t and (1.1) becomes the familiar ordinary differential equation
[r (t)x 0 (t)]0 + F(t, x(t), x 0 (t)) = 0,
t ∈ [a, ∞);
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while if T = Z then x 1 = x(t + 1) − x(t) := 1x(t) with σ (t) = t + 1 and (1.1) becomes the well-known difference
equation
1[r (t)1x(t)] + F(t, x(t + 1), 1x(t + 1)) = 0,
t ∈ {a, a + 1, . . .}.
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There are many more time scales than just T = R and T = Z and hence there are many more dynamic equations.
We shall be primarily concerned with proving the existence of positive solutions which are asymptotic to a positive
limit at ∞.
In the field of differential equations, some of the earliest work concerned with proving the existence of positive
solutions
which are asymptotic to a positive limit at ∞ dates back to the result of Atkinson [2] who showed that if
R∞
t
f
(t)dt
< ∞, then the superlinear equation
0
x 00 + f (t)g(x) = 0;
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One exciting aspect of these new movements includes linking the field of oscillation theory with solutions to
“dynamic equations on time scales”. The area of dynamic equations on time scales is a new, modern and progressive
component of applied analysis that acts as the framework to effectively describe processes that feature both continuous
and discrete elements. Created by Hilger in 1990 [14] and developed by others (see [3,6,7,17,22] and references
therein), this novel and fascinating type of mathematics is more general and versatile than the traditional theories
of differential and difference equations as it can, under one framework, mathematically describe continuous–discrete
hybrid processes and hence is the optimal way forward for accurate and malleable mathematical modelling. In fact, the
field of dynamic equations on time scales contains, links and extends the classical theory of differential and difference
equations.
In this paper, we are concerned with proving the existence of positive (and so non-oscillatory) solutions to the
nonlinear dynamic equation
(1.3)
has a bounded, non-oscillatory solution. This idea was extended in various ways by many authors over the last
50 years. Recently, Dubé and Mingarelli [10] presented a general non-oscillation result, which unifies a number
of cases for the equation
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x 00 + f (t, x) = 0;
where
f (t, x) ≥ 0,
∀ (t, x) ∈ [0, ∞) × [0, ∞).
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L. Erbe et al. / Nonlinear Analysis xx (xxxx) xxx–xxx
This result was further extended by Wahlén in [23] and by Ehrnström in [13], using a renormalisation technique.
Here we present three sorts of results. In the first instance we show existence of a positive solution on a given fixed
interval and with a given asymptote at ∞. In the second case, the limit is prescribed and the solution is shown to be
eventually positive, and in the special case of Eq. (1.2), we fix an interval of positivity but not the limit.
Our methods include: the Banach fixed-point theorem, including the method of Picard iterations; and weighted
norms and metrics in the time scale setting. Due to the wide-ranging nature of dynamic equations on time scales our
results are novel: for ordinary differential equations when r 6≡ 1; for difference equations; for combinations of the two
areas; and for general time scales. We present some examples to illustrate how our results advance existing knowledge
and state an open problem of interest.
Please cite this article in press as: L. Erbe, et al., Basic existence, uniqueness and approximation results for positive solutions to nonlinear
dynamic equations on time scales, Nonlinear Analysis (2007), doi:10.1016/j.na.2007.08.010
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2. The time scales
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For more recent results on oscillation and non-oscillation for solutions to dynamic equations on time scales we
refer the reader to [8,12] and the references therein.
For results on oscillation and non-oscillation to linear ordinary differential equations and difference equations via
Volterra–Stieltjes integral equations, we refer the reader to the monograph of Mingarelli [18].
Finally, we mention the blanket assumption, initial value problems for (1.1) and (1.2) have unique solutions and
our time scale T is unbounded above.
To understand the notation used above, some preliminary definitions are needed, which are now presented. For
more detail see [6, Chapter 1].
Definition 2.1. A time scale T is a nonempty closed subset of the real numbers R.
Since a time scale may or may not be connected, the concept of the jump operator is useful to define the generalised
derivative x 1 of a function x.
Definition 2.2. The forward (backward) jump operator σ (t) at t for t < sup T (respectively ρ(t) at t for t > inf T)
is given by
σ (t) := inf{τ > t : τ ∈ T},
(ρ(t) := sup{τ < t : τ ∈ T}),
for all t ∈ T.
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Define the graininess function µ : T → [0, ∞) as µ(t) := σ (t) − t.
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Definition 2.3. The jump operators σ and ρ allow the classification of points in a time scale in the following way: If
σ (t) > t, then the point t is called right-scattered; while if ρ(t) < t, then t is termed left-scattered. If t < sup T and
σ (t) = t, then the point t is called right-dense; while if t > inf T and ρ(t) = t, then we say t is left-dense.
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If T has a left-scattered maximum value m, then we define
:= T − {m}. Otherwise
The following gives a formal ε − δ definition of the generalised delta derivative.
Tκ
:= T.
Definition 2.4. Fix t ∈ Tκ and let x : T → R. Define x 1 (t) to be the number (if it exists) with the property that given
ε > 0 there is a neighbourhood U of t with
We call
x 1 (t)
RR
|[x(σ (t)) − x(s)] − x 1 (t)[σ (t) − s]| ≤ ε|σ (t) − s|,
for all s ∈ U.
the delta derivative of x(t) and say that x is delta-differentiable.
Converse to the delta derivative, we now state the definition of the delta integral.
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Definition 2.5. If K 1 (t) = k(t) then define the (Cauchy) delta integral by
Z t
k(s) 1s = K (t) − K (a).
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Throughout this work the assumption is made that T has the topology that it inherits from the standard topology
on the real numbers R.
Tκ
1
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Rt
Rt
Rt
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If T = R then a k(s) 1s = a k(s) ds, while if T = Z then a k(s) 1s = at−1 k(s). Once again, there are many
more time scales than just R and Z and hence there are many more delta integrals. For a more general definition of
the delta integral see [6].
The following theorem will be fundamental.
Theorem 2.6 ([14]). Assume that k : T → R and let t ∈ Tκ .
(i) If k is delta-differentiable at t then k is continuous at t.
(ii) If k is continuous at t and t is right-scattered then k is delta-differentiable at t with
k(σ (t)) − k(t)
k 1 (t) =
.
σ (t) − t
Please cite this article in press as: L. Erbe, et al., Basic existence, uniqueness and approximation results for positive solutions to nonlinear
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The following gives a generalised idea of continuity on time scales.
Definition 2.7. Assume that k : T → R. Define and denote k ∈ Cr d (T) as right-dense continuous (rd-continuous) if:
k is continuous at every right-dense point t ∈ T; and lims→t − k(s) exists and is finite at every left-dense point t ∈ T.
For functions F of two variables, t ∈ T and u ∈ R, we say F is rd-continuous if it is rd-continuous in t and continuous
in u.
Of particular importance is the fact that every Cr d function is delta-integrable [6, Theorem 1.73].
3. Banach space construction
In this section we construct a suitable Banach space to accommodate solutions to (1.1) by introducing a novel
weighted norm in the time scale environment.
Since we will assume that r and F are rd-continuous functions in Eqs. (1.1) and (1.2), we are searching for
the existence and uniqueness of solutions x : [a, ∞)T → R such that x 1 : [a, ∞)T → R is continuous and
x 11 : [a, ∞)T → R is rd-continuous.
We denote, by CT ([a, ∞)T ), the space of continuous functions x : [a, ∞)T → R that satisfy
|x(t)| < ∞;
sup
t∈[a,∞)T
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kxkT
0 :=
sup
|x(t)|;
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and couple this linear space with the norm
t∈[a,∞)T
so that (CT ([a, ∞)T ), k · k0 ) forms a Banach space. In particular, the following closed subspace of CT ([a, ∞)T )
H := {x ∈ CT ([a, ∞)T ) : x(t) ≥ 0, t ∈ [a, ∞)T };
coupled with the above sup-norm furnishes a Banach space.
Similarly, we denote, by CT1 ([a, ∞)T ), the space of continuously delta-differentiable functions x : [a, ∞)T → R
that satisfy
(
)
max
sup
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|x 1 (t)| < ∞;
t∈[a,∞)T
and couple this linear space with the norm
(
kxkT
1
:= max
sup
t∈[a,∞)T
|x(t)|,
sup
1
)
|x (t)| ;
(3.1)
t∈[a,∞)T
to obtain a Banach space. For a non-negative constant M, we define the closed subspace of CT1 ([a, ∞)T ) by
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sup
|x(t)|,
t∈[a,∞)T
RR
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(iv) If k is delta-differentiable at t then k(σ (t)) = k(t) + µ(t)k 1 (t).
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(iii) If k is delta-differentiable and t is right-dense then
k(t) − k(s)
.
k 1 (t) = lim
s→t
t −s
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X M := {x ∈ CT1 ([a, ∞)T ) : 0 ≤ x(t) ≤ M; x 1 (t) ≥ 0; t ∈ [a, ∞)T };
(3.2)
and form a Banach space when coupling this with the norm in (3.1).
Finally, for any bounded function w : [a, ∞)T → [c, d] with 0 < c ≤ d < ∞, we introduce the weighted norm
k · kw on CT1 ([a, ∞)T ) by
(
)
|x(t)|
|x 1 (t)|
kxkw := max
sup
, sup
.
t∈[a,∞)T w(t) t∈[a,∞)T w(t)
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We note that the existence of positive numbers c and d above ensure that k · kw and k · kT
1 are equivalent norms, as
kxkT
kxkT
1
1
≤ kxkw ≤
,
d
c
∀x ∈ CT1 ([a, ∞)T ).
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Thus we observe that (CT1 ([a, ∞)T ), k · kw ) is also a Banach space.
Weighted norms date back to the pioneering arguments of Bielecki [9, pp. 25–26], [11, pp. 153–155], [20, p. 44]
and have been recently introduced into the time scale setting by Tisdell and Zaidi [22].
Our particular weighted norm of interest will involve the exponential function on a time scale and we now provide
a short discussion of this special function.
Define the so-called set of regressive functions, R, by
R := { p ∈ Cr d (T) and 1 + p(t)µ(t) 6= 0, ∀t ∈ T}
and the set of positively regressive functions, R+ , by
R := { p ∈ Cr d (T) and 1 + p(t)µ(t) > 0, ∀t ∈ T}.
+
x
x(a) = 1.
= p(t)x,
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We now provide our first Banach space related result.
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Lemma 3.1. Let p : [a, ∞)T → (0, ∞) be a rd-continuous function such that
Z ∞
p(t)1t < ∞.
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t ∈ [a, ∞)T .
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p(t)
,
1 + µ(t) p(t)
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If p ∈ R+ then e p (t, a) > 0 for all t ∈ T [6, Theorem 2.48]. Finally, for p ∈ R we define p by
( p)(t) := −
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For p ∈ R we define [6, Theorem 2.35] the exponential function e p (·, a) on the time scale T as the unique solution
to the scalar initial value problem
1
1
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(3.3)
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Proof. Let
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w(t) := e p (t, a),
RR
Then for w(t) := e p (t, a) the pair (CT1 ([a, ∞)T ), k · kw ) forms a Banach space.
t ∈ [a, ∞)T .
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Since p ∈ R, w is well defined and since p > 0 on [a, ∞)T we have p ∈
Thus, by [6, Theorem 2.44] w > 0 on
[a, ∞)T . By [6, Theorem 2.71], w is the unique solution of the initial value problem
w(a) = 1.
(3.4)
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w 1 (t) = − p(t)wσ (t),
R+ .
We show that 0 < L ≤ w(t) ≤ 1 for t ∈ [a, ∞)T where L := limt→∞ w(t). Thus, the equivalence of the norms k · kw
1
and k · kT
1 will be established and by the earlier discussion in this section the pair (C T ([a, ∞)T ), k · kw ) will form a
Banach space.
From (3.4) we see w 1 (t) < 0 for t ∈ [a, ∞)T and so w is decreasing on [a, ∞)T . Thus w ≤ 1 on [a, ∞)T .
We claim that (3.3) implies
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L := lim w(t) > 0.
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t→∞
To see this note that
w(t) := e p (t, a) = exp
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Z
a
t
ξµ(s) (( p)(s))1s ,
t ∈ [a, ∞)T ,
where ξh is the cylinder transformation [6, Definition 2.21] defined by
(1
Log[1 + hx], h > 0;
ξh (x) := h
x, h = 0;
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ξµ(s) (( p)(s)) =
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=
6
=
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=
8
≥
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Therefore, in either case we have
ξµ(s) (( p)(s)) ≥ − p(s)
for all s ∈ [a, ∞)T .
It follows from (3.3) that
L := lim w(t) > 0.
t→∞
In particular, we have
0 < L ≤ w(t) ≤ 1,
so that k · kw and
k · kT
1
t ∈ [a, ∞)T ;
are equivalent norms.
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for all y ≥ 0.
In the following definition, we introduce a particular form of the function p that will be of use in our investigation.
Definition 3.2. Let k, r : [a, ∞)T → (0, ∞) be rd-continuous. For t ∈ [a, ∞)T we define
Z σ (t)
1
P(t, s) :=
1u, a ≤ s ≤ σ (t);
r (u)
s
RR
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log(1 + y) ≤ y,
(3.5)
and for any constant K > 1, define
p(t) := K [P(t, a) + 1]k(t),
t ∈ [a, ∞)T .
(3.6)
As a special case, consider r ≡ 1 in (3.5). Then (3.5) becomes
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where we have used the inequality
P(t, s) = σ (t) − s,
and (3.6) becomes
a ≤ s ≤ σ (t);
p(t) = K [σ (t) − s + 1]k(t),
t ∈ [a, ∞)T .
For our investigation of (1.1) and (1.2) the function r in the preceding context, of course, is determined from (1.1)
or (1.2), while k will appear from the following generalised Lipschitz condition on F
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1
Log[1 + µ(s)( p)(s)]
µ(s)
1
Log[1 + µ(s)( p)(s)]
µ(s)
1
1
log
µ(s)
1 + µ(s) p(s)
1
−
log[1 + µ(s) p(s)]
µ(s)
− p(s),
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and Log is the principal logarithmic function.
There are two cases to discuss: µ(s) = 0; and µ(s) > 0.
If µ(s) = 0, then ξ(( p)(s)) = − p(s); while if µ(s) > 0 then we have
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|F(t, u, z) − F(t, v, w)| ≤ k(t)[|u − v| + |z − w|],
t ∈ [a, ∞)T , (u, v, w, z) ∈ R4 ;
or the special case
|F(t, u) − F(t, v)| ≤ k(t)|u − v|,
t ∈ [a, ∞)T , (u, v) ∈ R2 .
Our analysis of (1.1) (and (1.2)) will involve the existence and uniqueness of fixed points to certain delta integral
operators. We now state an equivalence result between solutions to the dynamic equation (1.1) and solutions to a
particular delta integral equation.
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Lemma 3.3. Consider (1.1). Let r : [a, ∞)T → (0, ∞) be rd-continuous and let F : [a, ∞)T × R2 → R be
rd-continuous. Let P be defined in (3.5). If there is a constant M > 0 such that for all x ∈ X M
Z ∞
P(t, a)F(t, x σ (t), x 1σ (s))1t ≤ M, t ∈ [a, ∞)T ;
1
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3
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then the dynamic equation (1.1) has a solution with limt→∞ x(t) = M and limt→∞ r (t)x 1 (t) = 0 if and only if the
delta integral equation
Z ∞
x(t) = M −
P(s, t)F(s, x σ (s), x 1σ (s))1s, t ∈ [a, ∞)T ;
(3.7)
t
has a solution on [a, ∞)T .
t ∈ [a, ∞)T .
t
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t ∈ [a, ∞)T .
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Conversely, assume that x is a solution of (3.7) on [a, ∞)T , that is
Z ∞
x(t) = M −
P(s, t)F(s, x σ (s), x 1σ (s))1s, t ∈ [a, ∞)T .
t
Therefore
∞
Z
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(3.8)
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t ∈ [a, ∞)T ;
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F(s, x σ (s), x 1σ (s))1s,
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r (t)x 1 (t) =
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Note that limt→∞ x(t) = M. Taking the delta derivative in (3.8) we obtain
Z ∞
1
1
σ
1σ
x (t) = P(t, σ (t))F(t, x (t), x (t)) +
F(s, x σ (s), x 1σ (s))1s,
r (t) t
Z ∞
1
=
F(s, x σ (s), x 1σ (s))1s.
r (t) t
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Integrating by parts and using the definition of P(t, s) in (3.5) we obtain
Z ∞
P(s, t)F(s, x σ (s), x 1σ (s))1s, t ∈ [a, ∞)T .
x(t) = M −
t
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Dividing by r (t) and integrating again we have
Z ∞
Z ∞
1
F(u, x σ (u), x 1σ (u)) 1u1s,
M − x(t) =
r
(s)
s
t
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Taking the delta integral on both sides from t to ∞ we obtain
Z ∞
1
r (t)x (t) =
F(s, x σ (s), x 1σ (s))1s, t ∈ [a, ∞)T .
5
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Proof. Assume that the dynamic equation (1.1) has a solution x with limt→∞ x(t) = M and limt→∞ r (t)x 1 (t) = 0.
Then
[r (t)x 1 (t)]1 = −F(t, x σ (t), x 1σ (t)),
4
t ∈ [a, ∞)T ;
(3.9)
23
t
which implies
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lim r (t)x 1 (t) = 0.
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t→∞
Differentiating both sides of (3.9) we obtain the desired result
[r (t)x 1 (t)]1 = −F(t, x σ (t), x 1σ (t)),
t ∈ [a, ∞)T .
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15
16
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21
kT (x) − T (y)kY ≤ αkx − ykY ,
Theorem 4.1 (Banach). Let (Y, k · kY ) be a Banach space and let T : Y → Y be contractive. Then T has a unique
fixed point u and T i (y) → u for each y ∈ Y . If we start at an arbitrary y ∈ Y then the following estimate on the
“error” between the ith iteration T i y and the fixed point u, holds
kF i y − ukY ≤
αi
ky − F ykY .
1−α
Proof. See [9, p. 10] or [16, Theorem 7.5].
The following two results furnish the existence of a unique, positive solution to (1.2) and (1.1) with solutions
possessing a horizontal asymptote.
Theorem 4.2. Consider (1.2). Let r, k : [a, ∞)T → (0, ∞) be rd-continuous and let F : [a, ∞)T × R → R be
rd-continuous. Let P and p be defined as in Definition 3.2 with k and F satisfying
F(t, u) ≥ 0,
∀(t, u) ∈ [a, ∞)T × [0, ∞);
|F(t, u) − F(t, v)| ≤ k(t)|u − v|,
Z ∞
p(t)1t < ∞.
26
27
(4.1)
(4.2)
Then there is a unique positive solution x ∈ X M of (1.2) satisfying limt→∞ x(t) = M. Furthermore
Z t
1
1
x(t) ≥ r (t)x (t)
1u, t ∈ [a, ∞)T .
r
(u)
a
(4.3)
CO
25
t ∈ [a, ∞)T , (u, v) ∈ R ;
2
Let M > 0 be a constant such that for all x ∈ X M
Z ∞
P(t, a)F(t, x σ (t)) 1t ≤ M, t ∈ [a, ∞)T .
a
24
RR
23
∀x, y ∈ Y.
The constant α is called the contraction constant of T .
For any given y ∈ Y we define the sequence {T i (y)} recursively by: T 0 (y) := y; and T i+1 (y) := T (T i (y)).
a
22
F
5
PR
OO
4
In this section we state and prove our main results which will be obtained by use of Banach’s fixed-point theorem.
Banach’s fixed-point theorem is one of the most basic, yet most compelling, ideas from fixed-point theory — mainly
because the theorem produces a wide range of qualitative and quantitative information about solutions.
Let (Y, k · kY ) be a Banach space and T : Y → Y . The map T is said to be contractive if there exists a positive
constant α < 1 such that
D
3
TE
2
4. Main results
EC
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L. Erbe et al. / Nonlinear Analysis xx (xxxx) xxx–xxx
In addition, if x0 ∈ X M and we define the “Picard iterates” xi recursively by
Z ∞
P(s, t)F(s, xiσ (s))1s, t ∈ [a, ∞)T ;
xi+1 (t) := M −
t
29
30
31
32
then limi→∞ xi (t) = x(t) uniformly on [a, ∞)T .
UN
28
Proof. Consider X M defined in (3.2). Let w(t) := e p (t, a) for all t ∈ [a, ∞)T where p(t) := K [P(t, a) + 1]k(t)
and K > 1 is an arbitrary constant. Consider the pair (X M , k·kw ), which forms a Banach space by Lemma 3.1. Define
an operator T on X M by
Z ∞
[T x](t) := M −
P(s, t)F(s, x σ (s))1s, t ∈ [a, ∞)T .
t
33
34
35
We will now use Banach’s fixed-point theorem to show that T has a unique fixed point in X M . If x ∈ X M , then by
(4.1), F(t, x σ (t)) ≥ 0 and hence
0 ≤ [T x](t) ≤ M,
t ∈ [a, ∞)T .
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Also, for all t ∈ [a, ∞)T ,
1
F(s, x σ (s)) 1s
2
t
3
F
1
[T x]1 (t) = P(t, σ (t))F(t, x σ (t)) +
r (t)
Z ∞
1
F(s, x σ (s)) 1s
=
r (t) t
≥ 0.
∞
Z
(T x)1
5
6
7
8
9
(4.4)
D
k(t)wσ (t) = −
PR
OO
Since T x : [a, ∞)T → R and
: [a, ∞)T → R are continuous we have that T : X M → X M .
We now show that T is a contraction mapping on X M . For x, y ∈ X M , consider
Z ∞
|[T x](t) − [T y](t)|
1
P(s, t)|F(s, x σ (s)) − F(s, y σ (s))| 1s
≤
w(t)
w(t) t
Z ∞
1
≤
P(s, t)k(s)|x σ (s) − y σ (s)| 1s
w(t) t
Z ∞
|x σ (s) − y σ (s)|
1
1s
P(s, t)k(s)w σ (s)
=
w(t) t
w σ (s)
Z ∞
1
≤ kx − ykw
P(s, t)k(s)w σ (s) 1s.
w(t) t
Using (3.4) we obtain
4
w 1 (t)
RR
EC
TE
, t ∈ [a, ∞)T .
K [P(t, a) + 1]
Therefore, from (4.4), we obtain, for t ∈ [a, ∞)T ,
Z
|[T x](t) − [T y](t)|
kx − ykw ∞ P(s, t)w 1 (s)
≤−
1s
w(t)
K w(t) t
P(s, a) + 1
Z ∞
kx − ykw
w 1 (s) 1s
≤−
K w(t) t
w(t) − L
=
kx − ykw
K w(t)
1
≤ kx − ykw .
K
Taking the sup over t we obtain
1
kx − ykw = αkx − ykw ;
K
where α := 1/K < 1, and hence T is a contraction mapping on X M . Therefore, from Banach’s fixed-point theorem,
T has a unique fixed point x ∈ X M and so by Lemma 3.3 there is a unique positive solution of (1.2) in X M satisfying
limt→∞ x(t) = M.
To see that (4.3) holds note that the solution x satisfies, for each t ∈ [a, ∞)T ,
Z ∞
x(t) = M −
P(s, t)F(s, x σ (s))1s
t
Z ∞
Z t
1
= M−
P(s, a) −
1u F(s, x σ (s))1s
t
a r (u)
Z ∞
Z t
Z ∞
1
σ
= M−
P(s, a)F(s, x (s))1s +
1u
F(s, x σ (s))1s
t
a r (u)
t
Z ∞
Z t
Z ∞
1
σ
≥ M−
P(s, a)F(s, x (s)) 1s +
1u
F(s, x σ (s))1s
a
a r (u)
t
Z t
Z ∞
1
≥
1u
F(s, x σ (s)) 1s.
a r (u)
t
UN
CO
kT x − T ykw ≤
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Since
1
Z ∞
1
F(s, x σ (s))1s,
r (t) t
we see that (4.3) holds. x 1 (t) =
2
3
t ∈ [a, ∞)T ;
6
7
F(t, u, w) ≥ 0,
8
∀t ∈ [a, ∞)T , (u, w) ∈ R2 ;
|F(t, u, w) − F(t, v, z)| ≤ k(t) (|u − v| + |w − z|) ,
Z ∞
p(t) 1t < ∞.
9
10
t ∈ [a, ∞)T ; (u, v) ∈ R ;
2
a
12
Q1
Let M > 0 be a constant such that for all x ∈ X M
Z ∞
P(t, a)F(t, x σ (t), x 1σ (t))1t ≤ M, t ∈ [a, ∞)T .
a
15
Then there is a unique positive solution x ∈ X M of (1.1) satisfying limt→∞ x(t) = M. Furthermore (4.3) holds. In
addition, if x0 ∈ X M and we define the “Picard iterates” xi recursively by
Z ∞
xi+1 (t) := M −
P(s, t)F(s, xiσ (s), xi1σ (s))1s, t ∈ [a, ∞)T ;
t
17
18
then limi→∞ xi (t) = x(t) uniformly on [a, ∞)T .
Proof. Much of this proof is similar to the proof of Theorem 4.2 and so we only sketch the details. Define T on X M by
Z ∞
[T x](t) := M −
P(s, t)F(s, x σ (s), x 1σ (s))1s, t ∈ [a, ∞)T .
EC
16
t
22
23
24
25
26
27
28
29
30
RR
21
It follows that T : X M → X M . We want to show that T is a contraction mapping on X M . As in the proof of Theorem 4.2 we get for x, y ∈ X M
|[T x](t) − [T y](t)|
1
≤ kx − ykw .
w(t)
K
Now for x, y ∈ X M , consider, for t ∈ [a, ∞)T ,
Z ∞
|[T x]1 (t) − [T y]1 (t)|
1
≤
|F(s, x σ (s), x 1σ (s)) − F(s, y σ (s), y 1σ (s))|1s
w(t)
r (t)w(t) t
Z ∞
1
≤
k(s) |x σ (s) − y σ (s)| + |x 1σ (s) − y 1σ (s)| 1s
r (t)w(t) t
σ
Z ∞
1
|x (s) − y σ (s)| |x 1σ (s) − y 1σ (s)|
≤
k(s)w σ (s)
+
1s
r (t)w(t) t
w σ (s)
w σ (s)
Z ∞
1
≤ kx − ykw
k(s)w σ (s) 1s
r (t)w(t) t
Z ∞
−w 1 (s)
1
= kx − ykw
1s
r (t)w(t) t K [P(s, a) + 1]
Z ∞
kx − ykw
≤−
w 1 (s) 1s
K r (t)w(t) t
1 w(t) − L
=
kx − ykw
K R w(t)
1
≤
kx − ykw .
KR
CO
20
UN
19
(4.6)
D
14
(4.5)
TE
13
PR
OO
Theorem 4.3. Consider (1.1). Let r, k : [a, ∞)T → (0, ∞) be rd-continuous and let F : [a, ∞)T × R → R be
rd-continuous. Let K > 1 and R be positive constants with: r > R on [a, ∞)T ; K R > 1. For P and p defined as in
Definition 3.2, let k and F satisfy
5
11
F
We now generalise the previous theorem to include the case (1.1).
4
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It follows that
1
kT x − T ykw ≤ αkx − ykw ;
2
where
1 1
,
K KR
< 1.
PR
OO
α := max
F
3
4
The following result bounds the delta derivative of solutions furnished by the previous two theorems.
5
Corollary 4.4. In Theorems 4.2 and 4.3 if we replace X M by
6
Y M := {x ∈ X M : x 1 (t) ≤ M, t ∈ [a, ∞)T }
and assume that the inequalities
Z ∞
1
F(t, x σ (t), x 1σ (t))1t ≤ M,
r (t) a
Z ∞
1
F(t, x σ (t), x 1σ (t))1t ≤ M,
r (t) a
t ∈ [a, ∞)T ;
t ∈ [a, ∞)T ;
TE
D
hold for all x ∈ Y M , then (1.1) (resp. (1.2)) has a unique positive solution in Y M . In particular, 0 ≤ x 1 ≤ M on
[a, ∞)T .
In the following result, we obtain the existence and uniqueness of “eventually positive” solutions.
EC
Theorem 4.5. Consider (1.1). Let r, k : [a, ∞)T → (0,R ∞) be rd-continuous and let F : [a, ∞)T × R → R be
∞
rd-continuous. Let P be defined as in Definition 3.2 with a P(s, a)k(s)1s < ∞. Assume that F satisfies (4.6) with
Z ∞
1
k(s)1s < 1, for sufficiently large t.
r (t) t
RR
Let the constant M > 0 be given such that there is a v ∈ CT1 ([a, ∞)T ) so that the two integrals
Z ∞
Z ∞
P(s, a)|F(s, v σ (s), v 1σ (s))| 1s,
|F(s, v σ (s), v 1σ (s))| 1s;
a
7
8
9
10
11
12
13
14
15
16
17
18
a
are finite. Then there is a unique solution x ∈ CT1 ([a, ∞)T ) such that for some t0 ∈ [a, ∞)T , x > 0 on
[a, ∞)T ∩ [t0 , ∞) and limt→∞ x(t) = M. Furthermore, (4.3) holds on [a, ∞)T ∩ [t0 , ∞).
CO
Proof. Let M > 0 be given and define TM on CT1 ([a, ∞)T ) by
Z ∞
[TM x](t) := M −
P(s, t)F(s, x σ (s), x 1σ (s)) 1s, t ∈ [a, ∞)T .
19
20
21
22
t
UN
To see that TM is well defined, let x ∈ CT1 ([a, ∞)T ) and consider
Z ∞
0≤
P(s, t)|F(s, x σ (s), x 1σ (s))|1s
a
Z ∞
Z ∞
σ
1σ
σ
1σ
≤
P(s, t)|F(s, x (s), x (s)) − F(s, v (s), v (s))|1s +
P(s, t)|F(s, v σ (s), v 1σ (s))|1s
a
a
Z ∞
Z ∞
T
≤ kx − vk1
P(s, t)k(s)1s +
P(s, t)|F(s, v σ (s), v 1σ (s))|1s
a
a
Z ∞
Z ∞
≤ kx − vkT
P(s,
a)k(s)1s
+
P(s, a)|F(s, x σ (s), x 1σ (s))|1s
1
< ∞.
a
23
24
25
26
27
a
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Therefore,
Z ∞
2
P(s, t)|F(s, x σ (s), x 1σ (s))|1s < ∞,
a
5
6
F
4
and TM is well defined on CT1 ([a, ∞)T ). Clearly, TM (x) is continuous on [a, ∞)T . Also for x ∈ CT1 ([a, ∞)T ),
Z ∞
|F(s, x σ (s), x 1σ (s))|1s
a
Z ∞
Z ∞
|F(s, v σ (s), v 1σ (s))|1s
|F(s, x σ (s), x 1σ (s)) − F(s, v σ (s), v 1σ (s))|1s +
≤
a
a
Z ∞
Z ∞
|F(s, v σ (s), v 1σ (s))|1s
k(s)1s +
≤ kx − vk1
9
10
11
12
13
14
Hence
[TM x]1 (t) =
1
r (t)
Z
∞
F(s, x σ (s), x 1σ (s))1s;
a
exists and is continuous on [a, ∞)T . Thus TM : CT1 ([a, ∞)T ) → CT1 ([a, ∞)T ).
Next we show that TM is a contraction mapping on a subset of CT1 ([a, ∞)T ), where our norm k · k1 is given by
(3.1). To this end let x, y ∈ CT1 ([a, ∞)T ) and consider
Z ∞
P(s, t)|F(s, x σ (s), x 1σ (s)) − F(s, y σ (s), y 1σ (s))|1s
|[TM x](t) − [TM y](t)| ≤
Zt ∞
≤
P(s, t)k(s)[|x σ (s) − y σ (s)| + |x 1σ (s) − y 1σ (s)|]1s
D
8
a
a
< ∞.
7
PR
OO
3
for all x ∈ CT1 ([a, ∞)T ),
t
∞
Z
P(s, t)k(s)1s.
EC
≤ kx − yk1
TE
1
L. Erbe et al. / Nonlinear Analysis xx (xxxx) xxx–xxx
15
(4.7)
t
19
20
21
22
23
24
25
26
27
28
29
RR
18
CO
17
Also, for x, y ∈ CT1 ([a, ∞)T ), consider
Z ∞
1
1
1
|F(s, x σ (s), x 1σ (s)) − F(s, y σ (s), y 1σ (s))| 1s
|[TM x] (t) − [TM y] (t)| ≤
r (t) t
Z ∞
1
k(s)[|x σ (s) − y σ (s)| + |x 1σ − y 1σ |]1s
≤
r (t) t
Z ∞
1
≤ kx − yk1
k(s)1s.
r (t) t
Now we choose t1 ∈ [a, ∞)T , sufficiently large so that:
Z ∞
Z ∞
1
k(s)1s ≤ α < 1,
P(s, t) 1s ≤ α < 1;
r (t) t
t
(4.8)
for t ∈ [a, ∞)T ∩ [t1 , ∞).
Then using (4.7) and (4.8) we see that TM is a contraction mapping on CT1 ([a, ∞)T ∩ [t1 , ∞)). By Banach’s fixedpoint theorem we conclude that TM has a unique fixed point in CT1 ([a, ∞)T ∩ [t1 , ∞)) and this leads to a solution on
all of [a, ∞)T by the assumption that initial value problems for (1.2) have unique solutions. UN
16
We present our final result, which concerns non-negative solutions.
Theorem 4.6. Consider (1.2). Let r, k : [a, ∞)T → (0, ∞) be rd-continuous and let F : [a, ∞)T × R → R be
rd-continuous. Let P be defined as in Definition 3.2. Assume that F satisfies (4.1) and the Lipschitz condition (4.2)
with
Z ∞
P(s, a)k(s)1s < 1.
(4.9)
a
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If there exists a v ∈ CT1 ([a, ∞)T ) such that
Z ∞
P(s, t)F(s, v σ (s)) 1s < ∞;
1
(4.10)
2
a
then there is a unique solution x of (1.2) satisfying
x(t) ≥ 0,
F
x(a) = 0;
3
for t ∈ [a, ∞)T ;
Z
lim x(t) =
t→∞
∞
PR
OO
with
P(s, a)F(s, x σ (s))1s.
a
Furthermore, x satisfies (4.3).
Proof. Consider the Banach space (H, k · k0 ) defined in Section 3. We then define T̂ on H by
Z ∞
Z ∞
[T̂ x](t) :=
P(s, a)F(s, x σ (s))1s −
P(s, t)F(s, x σ (s))1s, t ∈ [a, ∞)T .
a
t
t→∞
TE
so T̂ x is increasing with
Z ∞
lim [T̂ x](t) =
P(s, a)F(s, x σ (s))1s < ∞
D
It follows similar to the proof of Theorem 4.5 that the assumption (4.10) ensures T̂ is well defined on H and T̂ x is
continuous on [a, ∞)T . Furthermore, [T̂ x](a) = 0.
Next, note that
Z ∞
1
1
F(s, x σ (s))1s > 0, t ∈ [a, ∞)T ;
[T̂ x] (t) =
r (t) t
a
RR
EC
and hence [T̂ x](t) ≥ 0 and is bounded for all t ∈ [a, ∞)T . Therefore T̂ : H → H.
We now find a different expression for [T̂ x](t), namely
Z ∞
Z ∞
[T̂ x](t) =
P(s, a)F(s, x σ (s))1s −
P(s, t)F(s, x σ (s))1s
a
t
Z t
Z ∞
=
P(s, a)F(s, x σ (s))1s +
[P(s, a) − P(s, t)]F(s, x σ (s))1s
a
t
Z t
Z t
Z ∞
1
=
P(s, a)F(s, x σ (s))1s +
1u
F(s, x σ (s))1s.
a
a r (u)
t
To show that T̂ is a contraction mapping on H, let x, y ∈ H and consider
UN
CO
|[T̂ x](t) − [T̂ y](t)|
Z t
Z t
Z ∞
1
σ
σ
≤
P(s, a)|F(s, x (s)) − F(s, y (s))| 1s +
1u
|F(s, x σ (s)) − F(s, y σ (s))|1s
a
a r (u)
t
Z t
Z t
Z ∞
1
σ
σ
≤
P(s, a)k(s)|x (s) − y (s)| 1s +
1u
k(s)|x σ (s) − y σ (s)|1s
a
a r (u)
t
Z t
Z ∞ Z t
1
≤ kx − yk0
P(s, a)k(s)1s + kx − yk0
1 uk(s)1s
a
t
a r (u)
Z ∞
≤ kx − yk0
P(s, a)k(s)1s = αkx − yk0 ,
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
a
where
α :=
Z
27
∞
P(s, a)k(s)1s < 1
28
a
by (4.9). Hence T̂ is a contraction mapping on H.
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5. Examples and open problems
1
3
5
PR
OO
Example 5.1. Consider the time scale T = R and the ordinary differential equation
1
1 0 0
x
= 0, t ∈ [1, ∞);
x + γ
δ
t
t 1 + x2
4
F
We now present some simple examples to illustrate the application of our new theorems and how they advance
existing knowledge.
2
6
where γ > δ + 2 > 1 are constants. We claim that this example satisfies the conditions of Theorem 4.2.
7
Proof. For this problem we can choose, for all t ∈ [a, ∞)T :
1
;
tγ
t δ+1 − 1
;
P(t, 1) =
δ+1
p(t) = 3[P(t, 1) + 1]k(t);
k(t)
8
9
10
and show that the hypotheses of Theorem 4.2 hold.
We note that the theorems in [13,23] do not directly apply to the previous example.
12
15
where
M
∂F
∂u (t, u) ≤ t β ,
16
17
where β >
18
21
22
25
31
32
33
34
35
Q2
is still open when f (t) is a bounded function that can be of both signs. This problem is still open even for the special
case of the differential equation (1.3). Results on the above open problems would be very well-received by the time
scales and dynamical systems community (see [4]).
Uncited references
[5].
UN
24
t ∈ [a, ∞)T ;
CO
[r (t)x 1 (t)]1 + f (t)g(x σ (t)) = 0,
23
30
and (5.1) satisfies the hypotheses of Theorem 4.2.
RR
20
29
t ∈ [a, ∞)T , u ∈ R,
No pre-existing results apply to the previous example in the general time scale setting.
Finally, we state the following open problem: It would be interesting to advance the results in this paper by
investigating the more general case of (1.2) where F can assume both positive and negative values. In particular,
the problem
19
28
3
2
EC
14
TE
Example 5.2. For an arbitrary, unbounded time scale interval of the form [a, ∞)T , a > 0, we can obtain a result for
dynamic equations of the form
√
1
p
( t + σ (t))x 1 + F(t, x σ ) = 0,
(5.1)
13
27
D
11
26
=
References
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Please cite this article in press as: L. Erbe, et al., Basic existence, uniqueness and approximation results for positive solutions to nonlinear
dynamic equations on time scales, Nonlinear Analysis (2007), doi:10.1016/j.na.2007.08.010
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