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Boundary value problems for first-order dynamic equations.

1 Introduction

Throughout this paper, we denote by T any time scale (nonempty closed subset of the real numbers R). We assume that 0, T [member of] T and denote by J = [0, T] a subset of T such that [0,T] = {t [member of] T : 0 [less than or equal to] t [less than or equal to] T}. By a we denote the forward jump operator [sigma](t) = inf{s [member of] T : s > t}. The graininess function [mu] : T [right arrow] [R.sub.+] is defined by [mu]t(t) = [sigma](t) - t with [R.sub.+] = [0, [infinity]). Let C (J, R) denote the set of continuous functions u : J [right arrow] R.

In this paper, we investigate the following first-order dynamic equation on time scales of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where f [member of] C (J x R x R, R), [alpha] [member of] C (J, J), g [member of] C (R x R, R).

We apply a monotone iterative method together with a Banach's fixed point theorem to prove that nonlinear dynamic problems of type (1.1) have extremal solutions. Using a corresponding result for a dynamic inequality we are able to show that our problem has a unique solution. The monotone iterative method was earlier applied also to dynamic problems on time scales, see for example [1,2,4-8]. According to our knowledge it is a first paper when this method is applied to dynamic equations with deviating arguments. The results are new. This paper extends the application of this method to such problems. We give also some remarks and examples showing the applications to differential equations.

2 Dynamic Inequalities

In this section we present some linear dynamic inequalities which are needed in Section 4.

Lemma 2.1. Assume that

([H.sub.1]) there exist functions n [member of] C (J, [R.sub.+]), [alpha] [member of] C (J, J), [alpha](t) [less than or equal to] t and [alpha](J) [not equal to] J.

In addition, we assume that

[p.sub.i] = [[integral].sup.T.sub.0] n(t)[DELTA]t [less than or equal to] 1.

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2 1)

Then x(t) [less than or equal to] 0, t [member of] J.

Proof. We need to prove that x(t) [less than or equal to] 0, t [member of] J Suppose that the inequality x(t) [less than or equal to] 0, t [member of] J is not true. Then, we can find [t.sub.0] [member of] (0, T] such that x([t.sub.0]) > 0. Put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Integrating the dynamic inequality in (2.1) from [t.sub.1] to [t.sub.0], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It contradicts the assumption that x(to) > 0. This proves that x(t) < 0 on J and the proof is complete.

Remark 2.2. Let n(t) = 0, t [member of] J. Then [p.sub.1] = 0 < 1. In this case, x is nonincreasing, so x(t) [less than or equal to] x(0) [less than or equal to] 0, t [member of] J.

Example 2.3. Let T = N, J = {0 [less than or equal to] j [less than or equal to] L, j [member of] N}. Assume that n(i) [member of] [R.sub.+] for

i [member of] P = {0, 1, ***,L - 1} and [(L-1.summation over (i=0)] n(i) [less than or equal to] 1. Let 0 [less than or equal to] [alpha](i) [less than or equal to] i, i [member of] P and [alpha](P) [not equal to] P

(for example [alpha](i) = [k.sub.i] [member of] N, [k.sub.i] [less than or equal to] i, i [member of] P and [k.sub.0] = [k.sub.1] = 0). Assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [DELTA]x(i) = x(i + 1) - x(i). Then, x(i) [less than or equal to] 0, i [member of] J, by Lemma 2.1.

Lemma 2.4. Assume ([H.sub.1]) and

([H.sub.2]) there exist a continuous function m : J [right arrow] R such that - m [member of] [R.sub.+], i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

In addition, we assume that

[p.sub.2] [equivalent to] [[integral].sup.T.sub.0] N(t)[DELTA]t [less than or equal to] 1 with N(t) = n(t) [e.sub.(-m)] ([alpha](t), [sigma](t)). (2.3)

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

Then x(t) [less than or equal to] 0, t [member of] J.

Proof. Let p(t) = [e.sub.[??](-m)](t, 0)x(t) = x(t)/[e.sub.(-m)](t,0). Note that, [e.sub.(-m)](t, 0) > 0, by assumption ([H.sub.2]), see [2, Theorem 2.48(i)]. Indeed, [??](m)(t) = - m(t)/1 + [mu](t)m(t). Computing [p.sup.[DELTA]], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

because

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by [2, Theorem 2.36]. Then problem (2.4) takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It yields p(t) [less than or equal to] 0 on J, by Lemma 2.1. This shows that x(t) [less than or equal to] 0 on J. The proof is complete.

Remark 2.5. If m(t) [equivalent to] 0, then [e.sub.m](s, t) = 1, by [2, Theorem 2.36(i)]. In this case Lemma 2.4 reduces to Lemma 2.1.

Remark 2.6. Let T = R. Then [mu](t) = 0, [sigma](t) = t, condition (2.2) holds and

[e.sub.(-m)]([alpha](t), [sigma](t)) = exp ([[integral].sup.t.sub.[alpha](t)] m(s)ds).

In this case, [p.sub.2] from condition (2.3) has the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and we have [3, Lemma 2.2].

Remark 2.7. Let T = R and m(t) [greater than or equal to] 0 on J. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If we assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then condition (2.3) holds. Note that [p.sub.3] does not depend on [alpha]. Moreover, if m(t) = m > 0, n(t) = n > 0, then [p.sub.3] takes the form

[p.sub.3] = n/m ([e.sup.mT] - 1).

3 Dynamic Equations

Now we consider the linear dynamic equation of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

Theorem 3.1. Let assumptions ([H.sub.1]), ([H.sub.2]) hold and let h [member of] C (J, R). Then problem (3.1) has a unique solution.

Proof. We first show that solving (3.1) is equivalent to solving a fixed point problem. Let x be any solution of problem (3.1). We use the product rule and [2, Theorem 2.36(ii)], so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

see [2, Theorem 2.36]. Similarly it is easy to see that if x [member of] [C.sub.rd] (J, R) is any solution of x = [A.sub.h]x, then x is a solution of problem (3.1).

Now we use Banach's fixed point theorem. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with a positive constant [[lambda].sub.0] such that [[lambda].sub.0] sup [mu](t) < 1, [[lambda].sub.0] [greater than or equal to] [n.sub.0] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note X is a Banach space. For x, y [member of] X we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in view of assumption ([H.sub.2]). It shows that [??](-[[lambda].sub.0]) [member of] [[Real part].sub.+], so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by [2, Theorem 2.48(i)]. We see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It shows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is nondecreasing, so by [2, Theorem 1.76], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

because 0 < 1 - [[lambda].sub.0][mu]t(s) [less than or equal to] 1 and

[??](-[[lambda].sub.0])(s) = [[[lambda].sub.0]/1-[[lambda].sub.0][mu](s)] [greater than or equal to] [[lambda].sub.0].

It yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As a result

[parallel][A.sub.h] x - [A.sub.h]y[parallel] [less than or equal to] [parallel]x - y[parallel][[x.sub.1] ].

In view of Banach's fixed point theorem, problem (3.1) has a unique solution. This ends the proof.

Remark 3.2. Let T = R. Then [mu](t) = 0, t [member of] J, so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

4 Existence of Solutions for Problem (1.1)

First we formulate the following result.

Theorem 4.1 (See [1, 8]). Let X be a partially ordered Banach space. Assume that there exist [x.sub.0], [y.sub.0] [member of] X, [x.sub.0] [less than or equal to] [y.sub.0], D = [[x.sub.0], [y.sub.0]], A : D [right arrow] X such that

(i) A is a continuous and increasing operator,

(ii) [x.sub.0] and [y.sub.0] are the lower and upper solutions of x = Ax, respectively,

(iii) A(D) [subset] X is relatively compact.

Then A has a minimalfixed point x* and a maximal fixed point [y.sup.*] in [[x.sub.0], [y.sub.0]]. Moreover [x.sub.n] [right arrow] [x.sup.*] and [y.sub.n] [right arrow] [y.sup.*], where [x.sub.n] = [Ax.sub.n-1], yn = [Ay.sub.n-1], n =1, 2, * * *, and

[x.sub.0] [less than or equal to] [x.sub.1] [less than or equal to] * * * [less than or equal to] [x.sub.n] [less than or equal to] * * * [less than or equal to] [y.sub.n] [less than or equal to] * * * [less than or equal to] [y.sub.1] [less than or equal to] [y.sub.0].

In the next theorem we give sufficient conditions so that problem (1.1) has extremal solutions.

Theorem 4.2. Let f [member of] C(J x R x R, R), k [member of] C(J x J, R). Assume that there exist differentiable functions [x.sub.0], [y.sub.0] : J [right arrow] R, [x.sub.0](t) [less than or equal to] [y.sub.0](t), t [member of] J, and such that they are lower and upper solutions of problem (1.1), respectively i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Moreover, we assume that all assumptions of Lemma 2.4 hold with functions m, n such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.1)

for t [member of] J, [x.sub.0](t) [less than or equal to] [[bar.x].sub.1] [less than or equal to] [x.sub.1] [less than or equal to] [y.sub.0](t), [x.sub.0]([alpha](t)) [less than or equal to] [[bar.x].sub.2] [less than or equal to] [x.sub.2] [less than or equal to] [y.sub.0]([alpha](t)). In addition, we assume that g is nonincreasing in the second variable and there exist a constant a > 0 such that

g(u, v) - g([bar.u], v) [greater than or equal to] - a([bar.u] - u) if [x.sub.0](0) [less than or equal to] u [less than or equal to] [bar.u] [less than or equal to] [y.sub.0](0). (4.2)

Then problem (1.1) has minimal and maximal solutions in the region D = {u [member of] C(J, R) : [x.sub.0](t) [less than or equal to] u(t) [less than or equal to] [y.sub.0](t), t [member of] J}.

Proof. Let G(h) be nondecreasing with respect to h. Choose [h.sub.1], [h.sub.2] [member of] C(J, R) such that [h.sub.1](t) [less than or equal to] [h.sub.2](t) on J. Let [x.sub.1], [x.sub.2] denote the solutions of problem (3.1) with [h.sub.1], [h.sub.2] instead of h, and with G([h.sub.1]), G([h.sub.2]) instead of xo, respectively. Since problem (3.1) has a unique solution for each h [member of] C( J, R), then [x.sub.1], [x.sub.2] are well defined, so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In view of Lemma 2.4, [x.sub.1] (t) [less than or equal to] [x.sub.2] (t) on J, so the operator [A.sub.h] is increasing. It is also continuous. Take u [member of] D. Put

Fu = Fu + Lu, G(u) = - [1/a] g(u(0),u(T )) + u(0),

where operator F is defined as in problem (1.1). Take [u.sub.1], [u.sub.2] [member of] D and let [u.sub.1](t) [less than or equal to] [u.sub.2](t), t [member of] J. Then, in view of conditions (4.1) and (4.2), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It proves that F and G are increasing in D. We define the operator A = [A.sub.F]. Let [x.sub.1] = [Ax.sub.0], [x.sub.2] = [Ay.sub.0] so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Put x(t) = [x.sub.0](t) - [x.sub.1] (t). Using the definition of the lower solution [x.sub.0], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It yields, [x.sub.0](t) [less than or equal to] [x.sub.1] (t) = ([Ax.sub.0])(t), by Lemma 2.4. Similarly, we can show ([Ay.sub.0])(t) = [x.sub.2] (t) [less than or equal to] [y.sub.0](t) on J. Now, using again Lemma 2.4 with x(t) = [x.sub.1] (t) - [x.sub.2] (t) and properties of F and G, we see that [x.sub.1] (t) [less than or equal to] [x.sub.2] (t) on J, so the operator A is increasing. The same argument as in [8, Theorem 3.2] guarantees that A(D) is relatively compact. The result now follows from Theorem 4.1. This ends the proof.

Remark 4.3. Let T = R. In this case Theorem 4.2 reduces to one which is a special case of paper [3].

5 Unique Solution for Problem (1.1)

To show the next result we need the following lemma.

Lemma 5.1. Assume that b [member of] C( J, R), b [member of] [R.sub.+], i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.1)

In addition, we assume that

[e.sub.b](T, 0) < c. (5.2)

Then x(t) [greater than or equal to] 0, t [member of] J.

Proof. First, note that [e.sub.b](t, 0) > 0, t [member of] J, by [2, Theorem 2.48(i)]. Now, we replace the inequality in (5.1) by the following relation:

[x.sup.[DELTA]](t) = b(t)x(t) + B(t), t [member of] J, B [member of] C(J, [R.sub.+]).

Hence,

x(t) = [e.sub.b](t, 0) [x(0)+ [[integral].sup.t.sub.0] [e.sub.b] (0, [sigma](s))B(s)[DELTA]s],

by [2, Theorem 2.77]. It yields

x(t) [greater than or equal to] [e.sub.b](t, 0)x(0), t [member of] J. (5.3)

Using to this the boundary condition x(T) [less than or equal to] cx(0), we see that

x(0) [c - [e.sub.b](T, 0)] [greater than or equal to] 0.

In view of condition (5.2), we obtain x(0) [greater than or equal to] 0. It shows that x(t) [greater than or equal to] 0, t [member of] J, by (5.3).

The proof is complete.

Theorem 5.2. Let all assumptions of Theorem 4.2 be satisfied. In addition, we assume ([H.sub.3]) f is nonincreasing in the last argument, there exist a function b [member of] C ( J, R) such that m(t) + b(t) [greater than or equal to] 0, t [member of] J,

f (t,u, v) - f (t, [bar.u], v) [greater than or equal to] -b(t)[[bar.u] - u] if [x.sub.0](t) [less than or equal to] u [less than or equal to] [bar.u] [less than or equal to] [y.sub.0] (t),

([H.sub.4]) there exist constants [M.sub.1], [M.sub.2] such that a [greater than or equal to] [M.sub.1] > 0, [M.sub.2] [greater than or equal to] 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

([H.sub.5]) [M.sub.2][e.sub.b](T, 0) < [M.sub.1].

Then problem (1.1) has, in the region D, a unique solution.

Proof. Theorem 4.2 guarantees that problem (1.1) has extremal solutions in D. By x we denote the minimal solution of (1.1), and by y the maximal solution of (1.1). Indeed, x [less than or equal to] y. To show that x = y, we put p = x - y, so p(t) [less than or equal to] 0, t [member of] J. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It means that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In view of Lemma 5.1, x(t) [greater than or equal to] y(t), t [member of] J. It proves that x = y, so problem (1.1) has a unique solution. This ends the proof.

Example 5.3 (See [5]). Let T = R. Consider the problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.4)

where J = [0,[pi]], 0 [less than or equal to] [beta] [less than or equal to] 1/4, [alpha] [member of] C(J, J), [alpha](t) [less than or equal to] t on J. Put [x.sub.0](t) = -1, [y.sub.0](t) = 0, t [member of] J. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It proves that xo, yo are lower and upper solutions of problem (5.4), respectively. Note that m(t) = 0, n(t) = 2[beta] sin t, a =1 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so assumption ([H.sub.2]) holds. Assumption ([H.sub.4]) holds with [M.sub.1] = [1.sub.M2] = [e.sup.-1]. Moreover b(t) = [beta] sin t, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By Theorem 5.2, problem (5.4) has, in the region D, a unique solution.

References

[1] Ravi P. Agarwal, Tadeusz Jankowski, and Donal O'Regan. Dynamic inequalities and equations of Volterra type on time scales. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 16(4):471-480, 2009.

[2] Martin Bohner and Allan Peterson. Dynamic equations on time scales. Birkhauser Boston Inc., Boston, MA, 2001. An introduction with applications.

[3] Tadeusz Jankowski. On delay differential equations with nonlinear boundary conditions. Bound. Value Probl., (2):201-214, 2005.

[4] Tadeusz Jankowski. Boundary value problems for dynamic equations of Volterra type on time scales. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 16(5):649-659, 2009.

[5] Tadeusz Jankowski. On dynamic equations with deviating arguments. Appl. Math. Comput., 208(2):423-426, 2009.

[6] Ruyun Ma and Hua Luo. Existence of solutions for a two-point boundary value problem on time scales. Appl. Math. Comput., 150(1):139-147, 2004.

[7] Yepeng Xing, Wei Ding, and Maoan Han. Periodic boundary value problems of integro-differential equation of Volterra type on time scales. Nonlinear Anal., 68(1):127-138, 2008.

[8] Yepeng Xing, Maoan Han, and Guang Zheng. Initial value problem for first-order integro-differential equation of Volterra type on time scales. Nonlinear Anal., 60(3):429-442, 2005.

Tadeusz Jankowski

Gdansk University of Technology Department of Differential Equations 11/12 Narutowicz Str., 80-233 Gdansk, Poland tjank@mifgate.mif.pg.gda.pl

Received November 29, 2009; Accepted April 9, 2010 Communicated by Martin Bohner
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Author:Jankowski, Tadeusz
Publication:International Journal of Difference Equations
Article Type:Report
Geographic Code:4EXPO
Date:Jun 1, 2010
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