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Existence and uniqueness of positive solution for discrete multipoint boundary value problems.

1. Introduction

Let T > 1 be an integer; [Z.sub.a,b] := {a, a + 1, ..., b}, where a, b are positive integers. Difference equation appears as natural descriptions of observed evolution phenomena because most measurements of time evolving variables are discrete and so it arises in many physical problems, as nonlinear elasticity theory or mechanics, and engineering topics. In recent years, the study of positive solutions for discrete boundary value problems has attracted considerable attention, but most research dealt with two-point boundary value problem; see [1-4] and the references therein. For multipoint boundary value problem, there appeared a small sample of work related to the existence of positive solution; we refer the reader to [57]. However all of them do not address the problem with the uniqueness of positive solution.

In this paper, we consider the existence uniqueness and positive solutions for difference equation

-[[DELTA].sup.2]u(t - 1) = f (t,u(t)) + g (t, u(t)), t [member of [Z.sub.1,T], (1)

subject to boundary conditions

(i) u(0) - [beta][DELTA]u(0) = 0, u(T + 1) = [alpha]u([eta]), (2)

or

(ii) [DELTA]u(0) = 0, u(T + 1) = [alpha]u([eta]), (3)

where 0 < [alpha] < 1, [beta] > 0, and [eta] [member of] [Z.sub.2,T-1].

Let E - C([Z.sub.0,T+1], R) denote the class of real valued functions w on Z0T+1 with norm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Observe that E is a Banach space. Set P = {u [member of] E : u(t) [greater than or equal to] 0,t [member of] [Z.sub.0,T+1]} to be the normal cone in E with the normality constant 1. For u, v [member of] E, the notation u ~ v means that there exist [lambda] > 0 and [mu] > 0 such that [lambda]v [less than or equal to] u [less than or equal to] gv. Clearly, ~ is an equivalence relation. Given h > [theta], we denote by [P.sub.h] the set [P.sub.h] = {x [member of] E | x ~ h}.

Remark 1. As suggested by the notation, by equipping [Z.sub.0,T+1] with the discrete topology, every [omega] [member of] C([Z.sub.0,T+1]) is continuous.

Definition 2 (see [8]). Let D = P or D = [P.sup.0] and let [gamma] be a real number with 0 [less than or equal to] [gamma] < 1. An operator A: P [right arrow] P is said to be [gamma]-concave if it satisfies

A(tx) [greater than or equal to] [t.sup.[gamma]] Ax, t [member of] (0,1), x [member of] D. (4)

Definition 3 (see [8]). An operator A : E [right arrow] E is said to be homogeneous if it satisfies

A (tx) = tAx, t > 0, x [member of] E. (5)

An operator A : P [right arrow] P is said to be subhomogeneous if it satisfies

A (tx) [greater than or equal to] tAx, t > 0, x [member of] P. (6)

The main tool of this paper is the following fixed point theorem.

Theorem 4 (see [9]). Let P be a normal cone in a real Banach space E, A : P [right arrow] P an increasing [gamma]-concave operator, and B: P [right arrow] P an increasing subhomogeneous operator. Assume that

(i) there is h > 0 such that Ah [member of] Ph and [B.sub.h] [member of] [P.sub.h];

(ii) there exists a constant [[delta].sub.0] > 0 such that, for any x [member of] P, Ax [greater than or equal to] [[delta].sub.0][B.sub.x].

Then the operator equation Ax + Bx = x has a unique solution x* [member of] [P.sub.h]. Moreover, constructing successively the sequence [y.sub.n] = [A.sub.yn-1] + [B.sub.yn-1,] n = 1,2, ..., for any initial value [y.sub.0] [member of] [P.sub.h], one has y [right arrow] x* as n [right arrow] [infinity].

Remark 5. When B is a null operator, Theorem 4 also holds.

2. Positive and Uniqueness of Solutions to BVP (1)-(2)

In this section, we will apply Theorem 4 to study the existence and uniqueness of positive solution for (1)-(2).

Lemma 6 (see [6]). If T [member of] {4, 5, ...}, [eta] [member of] [Z.sub.2,T-1], and [alpha], [beta] [member of] R are real numbers with [beta] [not equal to] -1 and (T+1-[alpha][eta])+[beta](1-[alpha]) [not equal to] 0,for any y defined in [Z.sub.0,T+1], the nonlocal boundary value problem

-[[DELTA].sup.2]u (t - 1) = y(t), t [member of] [Z.sub.1,T], u (0) - [beta][DELTA]u (0) = 0, u(T + 1) = [alpha]u ([eta]) (7)

has a unique solution

u(t) = [T.summation over (s=1)] G(t, s)y(s), (8)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Lemma 7. For t [member of] [Z.sub.0,T+1], s [member of] [Z.sub.1,T], the Green function G(t, s) in Lemma 6 has the following properties:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (10)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

Proof. Since (i) and (iii) are obvious, here we just prove (ii). For s < t, s < [eta], notice that t - [eta] < t - s,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

For [eta] [less than or equal to] s < t,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

For t [less than or equal to] s < [eta],

G (t, s) [less than or equal to] (t + [beta]) (T + 1 - s)/(T + 1 - [alpha][eta]) + [beta](1 - [alpha]) [less than or equal to] (t + [beta])[(T + 1 - s) + [alpha]T ([eta] + [beta])]/(T + 1 - [alpha][eta]) + [beta])1 - [alpha]). (14)

For t [less than or equal to] s, [eta] [less than or equal to] s,

G (t, s) [less than or equal to] (t + [beta]) (T + 1 - s)/(T + 1 - [alpha][eta]) + [beta](1 - [alpha]) [less than or equal to] (t + [beta])[(T + 1 - s) + [alpha]T ([eta] + [beta])]/(T + 1 - [alpha][eta]) + [beta])1 - [alpha]). (15)

That is, for any t [member of] [Z.sub.0,T+1] and s [member of] [Z.sub.1,T], G(t,s) [less than or equal to] r(s)h(t).

Theorem 8. Assume that

(A1) f, g: [Z.sub.0,T+1] x [0, [infinity]) [right arrow] [0, [infinity]) are continuous and increasing with respect to the second variable, g(T, 0) [not equal to] 0;

(A2) g(t, [lambda]x) [greater than or equal to] [lambda]g(t, x) for [lambda] [member of] (0,1), t [member of] [Z.sub.0T+1], x [member of] [0, [infinity]) and there exists a constant [gamma] [member of] (0,1) such that f(t, [lambda]x) [greater than or equal to] [[lambda].sup.[gamma]] f(t, x) for X [member of] (0,1), t [member of] [Z.sub.0T+1], x [member of] [0, [infinity]);

(A3) there exists a constant [[delta].sub.0] > 0 such that f(t, x) [greater than or equal to] [[delta].sub.0]g(t,x), t [member of] [Z.sub.0,T+1], x [member of] [0, [infinity]).

The problem (1)-(2) has a unique positive solution u* [member of] [P.sub.h], where h(t) = t + [beta], ([member of] [Z.sub.0,T+1]. Moreover, for any initial value [u.sub.0] [member of] [P.sub.h], constructing successively the sequence

[u.sub.n+1] (t) = [T.summation over (s=1)] (t, s) [f (s, [u.sub.n] (s)) + g (s, [u.sub.n] (s))], n = 0,1,2, ..., (16)

we have [u.sub.n](t) [right arrow] u*(t) as n [right arrow] [infinity], where G(t, s) is given as (9).

Proof. Define two operators A: P [right arrow] E and B : P [right arrow] E by

(Au)(t) = [T.summation over (s=1)) G (t, s) f (s, u(s)), (Bu)(t) = [T.summation over (s=1)) G (t, s) f (s, u(s)). (17)

It is easy to see that u is a solution of (1)-(2) if and only if u = Au + Bu. From (A2) and Lemma 7, we know that A: P [right arrow] P and B : P [right arrow] P. In the sequel we check that A, B satisfy all assumptions of Theorem 4.

Firstly, we prove that A, B are two increasing operators. In fact, by (A1) and Lemma 6, for u, v [member of] P with u [greater than or equal to] v, we know that u(t) [greater than or equal to] v(t), t [member of] [Z.sub.0,T+1], and obtain

(Au)(t) = [T.summation over (s=1)) G (t, s) f (s, u(s)), [greater than or equal to] (Au)(t) = [T.summation over (s=1)) G (t, s) f (s, v(s)), = (Av)(t). (18)

Similarly, Bu [greater than or equal to] Bv.

Next we show that A is a [gamma]-concave operator and B is a subhomogeneous operator. In fact, for any [lambda] [member of] (0,1) and u [member of] P, from (A2), we know that

(A[lambda]u)(t) = [T.summation over (s=1)) G (t, s) f (s, [lambda]u(s)) [greater than or equal to] [[lambda].sup.[gamma]] [T.summation over (s=1)) (t, s) f (s, u(s)), = [[lambda].sup.[gamma]] (Au)(t). (19)

That is, A is a [gamma]-concave operator. At the same time, for any X [member of] (0,1) and u [member of] P, from (A2), we get

(B[lambda]u)(t) = [T.summation over (s=1)) G (t, s) g (s, [lambda]u(s)) [greater than or equal to] [lambda] [T.summation over (s=1)) (t, s) g (s, u(s)), = [lambda] (Bu)(t). (20)

So B is subhomogeneous.

Now we show that Ah [member of] [P.sub.h] and [B.sub.h] [member of] [P.sub.h]. From (A3) and Lemma 7,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (21)

where [l.sub.1] = [[delta].sub.0]p(T)g(T,0) > 0 and [l.sub.2] = [[summation].sup.T.sub.s=1] r(s)f(s,s + [beta]). Hence we have [l.sub.1]h(t) [less than or equal to] (Ah)(t) [less than or equal to] [l.sub.2]h(t), t [member of] [Z.sub.0,T+1]; that is, Ah [member of] [P.sub.h]. We can similarly prove that [B.sub.h] [member of] [P.sub.h]. Thus condition (i) of Theorem 4 is satisfied.

In the following we show that condition (ii) of Theorem 4 holds. From (A3),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

Then we get Au [greater than or equal to] [[delta].sub.0]Bu, u [member of] P. By applying Theorem 4, it can be obtained that the operator equation Au + Bu = u has a unique solution u* [member of] [P.sub.h]. Moreover, constructing successively the sequence [u.sub.n] = A[u.sub.n-1] + B[u.sub.n-1], n= 1,2, ..., for any initial value [u.sub.0] [member of] [P.sub.h], we have [u.sub.n] [right arrow] u* as n [right arrow] [infinity]. That is, problem (1)-(2) has a unique positive solution u* [member of] [P.sub.h]. In addition, for any initial value [u.sub.0] [member of] [P.sub.h], constructing successively the sequence

[U.sub.n+1] (t) = [T.summation over (s=1) G (t, s) [f (S> U (S)) +g(s, u (s))], n = 0,1,2, ..., (23)

we have [u.sub.n](t) [right arrow] u*(t) as n [right arrow] [infinity].

The following result can be obtained by Remark 5 and Theorem 4.

Corollary 9. Assume that

(A1)' f : [Z.sub.0,T+1] x [0, [infinity]) [right arrow] [0, [infinity]) is continuous and increasing with respect to the second variable, f(T, 0) [not equal to] 0;

(A2)' there exists a constant [gamma] [member of] (0,1) such that f(t, [lambda]x) [greater than or equal to] [[lambda].sup.[gamma]]f(t, x) for [lambda] [member of] (0,1), t [member of] [Z.sub.0,T+1], x [member of] [0, [infinity]).

Then problem

-[[DELTA].sup.2]u (t - 1) = f (t, u(t)), t [member of] [Z.sub.1,T], u (0) - [beta][DELTA]u (0) = 0, u(T + 1) = [alpha]u (g) (24)

has a unique positive solution u* [member of] [P.sub.h], where h(t) = t + [beta], t [member of] [Z.sub.0,T+1]. Moreover, for any initial value [u.sub.0] [member of] [P.sub.h], constructing successively the sequence

[u.sub.n+1] (t) = [T.summation over (s=1)] G(t,s)f(s, [u.sub.n] (s)), n = 0,1,2, ..., (25)

we have [u.sub.n](t) [right arrow] u*(t) as n [right arrow] [infinity], where G(t, s) is given as (9).

Remark 10. In a similar way, we can get the corresponding results for the difference equation (1) subject to boundary conditions

u (0) = [alpha]u (g), u(T + 1) + [beta][DELTA]u (T + 1) = 0, (26)

which are symmetric to the boundary condition (2).

Remark 11. The results can also be generalized to discrete m-point boundary value problems:

-[DELTA]u (t - 1) = f (t,u (t)) + g(t,u (t)), t [member of] Z1T,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

Example 12. Consider the following nonlinear discrete problem:

-[[DELTA].sup.2] y (t - 1) = f (t, y(t)) + g(t, y (t)), t [member of] [Z.sub.1,9], y(0) - 2[DELTA]y(0) = 0, y(10) = 1/2 y (4). (28)

If we set

f(t,y) = [y.sup.1/4] + [t.sup.4] + [pi]/2, (t, y) [member of] [Z.sub.0,10] x [0, [infinity]), g (t, y) = arctan y + [t.sup.3], (t, y) [member of] [Z.sub.0,10] x [0, [infinity]), (29) then f, g : [Z.sub.0,10] x [0, [infinity]) [right arrow] [0, [infinity]) are continuous and increasing with respect to the second variable, g(9,0) [not equal to] 0, and for [lambda] [member of] (0,1), t [member of] [Z.sub.0,10], y [member of] [0, [infinity]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (30)

In addition,

f(t,y) = [y.sup.1/4] + [t.sup.4] + [pi]/2 [greater than or equal to] [t.sup.4] + [pi]/2 [t.sup.3] + [pi]/2 [greater than or equal to] 1/2 (arctan y+[t.sup.3]) = 1/2 g (t,y) (31)

Thus, all conditions of Theorem 8 are satisfied and so problem (28) has a unique positive solution in [P.sub.t+2].

3. Positive and Uniqueness of Solutions to BVP (1) and (3)

Lemma 13 (see [6]). If [alpha] [not equal to] 1, the nonlocal boundary value problem

-[[DELTA].sup.2]u (t - 1) = y (t), t [member of] [Z.sub.1,T], [DELTA]u(0) = 0, u(T + 1) = [alpha]u ([eta]) (32)

has a unique solution

u (t) = [T.summation over (s=1)] [??] (t, s) y (s), (33)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (34)

Lemma 14. For t [member of] [Z.sub.0,T+1], s [member of] [Z.sub.1,T], the Green function G(t, s) in Lemma 13 has the following property:

(i) G(t, s) > 0, t [member of] [Z.sub.0,T+1], s [member of] [Z.sub.1,T];

(ii) G(t, s) [less than or equal to] [??](s), t [member of] [Z.sub.0,T+1], s [member of] [Z.sub.1,T], where [??](s) = T + 1 - s + [alpha]T/1-[alpha]. (35)

Proof. We omit it since it is obvious.

Theorem 15. Assume that (A1), (A2), and (A3) are satisfied; then the problem (1)-(3) has a unique positive solution [??]* [member of] [P.sub.h], where h(t) = 1, t [member of] [Z.sub.0,T+1]. Moreover, for any initial value [[??].sub.0] [member of] [P.sub.h], constructing successively the sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (36)

we have [[??].sub.n] (t) [right arrow] [??]*(t) as n [right arrow] m, where [??](t, s) is given as (34).

Proof. It is similar to the proof of Theorem 8.

The following corollary can be obtained by Remark 5 and Theorem 4.

Corollary 16. Assume that (A1)' and (A2)' are satisfied; then the problem

-[[DELTA].sup.2]u (t - 1) = f (t,u(t)), t [member of] [Z.sub.1,T], [DELTA]u(0) = 0, u(T + 1) = [alpha]u ([eta]) (37)

has a unique positive solution [??]* [member of] [P.sub.h], where h(t) = 1, t [member of] [Z.sub.0,T+1]. Moreover, for any initial value [[??].sub.0] [member of] [P.sub.h], constructing successively the sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (38)

we have [[??].sub.n](t) [right arrow] [??]*(t) as n [right arrow] [infinity], where [??](t, s) is given as (34).

Remark 17. In a similar way, we can get the corresponding results for the difference equation (1) subject to boundary conditions

u (0) = [alpha]u([eta]), [DELTA]u(T + 1) = 0, (39)

which are symmetric to the boundary condition (3).

Remark 18. The results can also be generalized to discrete m-point boundary value problems:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (40)

Example 19. Consider the following nonlinear discrete problem:

-[DELTA]y (t - 1) = f(t, y (t)) + g(t, y (t)), t [member of] [Z.sub.1,4], [DELTA]y (0) = 0, y(5)= 3/5 y(3). (41)

If we set

f (t, y) = [y.sup.2/3] + [e.sup.t], (t, y) [member of] [Z.sub.0,5] x [0, [infinity]), g(t, y) = y/1+y [t.sup.2] t + 1, (t, y) [member of] [Z.sub.0,5] x [0, [infinity]] (42)

then f, g : [Z.sub.0,5] x [0, [infinity]) [right arrow] [0, [infinity]) are continuous and increasing with respect to the second variable, g(4,0) [not equal to] 0, and for [lambda] [member of] (0,1), t [member of] [Z.sub.0,5], y [member of] [0, [infinity]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (43).

In addition,

f (t, y) [greater than or equal to] [e.sup.0] = 1/26 * 26 [greater than or equal to] 1/26 * (y/1+y [t.sup.2] +1) = 1/26 g (t, y). (44)

Thus, all conditions of Theorem 15 are satisfied and so problem (41) has a unique positive solution in [P.sub.1].

http://dx.doi.org/10.1155/2014/531978

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (nos. 61363058 and 11061030), the Scientific Research Fund for Colleges and Universities of Gansu Province (2013B-007, 2013A-016), Natural Science Foundation of Gansu Province (145RJZA232, 145RJYA259), and Promotion Funds for Young Teachers in Northwest Normal University (NWNU-LKQN-12-14).

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Huili Ma and Huifand Ma

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Correspondence should be addressed to Huili Ma; mahuili@nwnu.edu.cn

Received 15 May 2014; Revised 9 [DELTA]ugust 2014; Accepted 10 [DELTA]ugust 2014; Published 29 October 2014

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