# First-Order Boundary Value Problem with Nonlinear Boundary Condition on Time Scales

Authors: Ya-Hong Zhao
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

This work is concerned with the following first-order dynamic equation on time scale𝕋,
xΔ(t)+p(t)x(σ(t))=f(t,x(t)),t∈[0,T]𝕋
with the nonlinear boundary condition
x(0)=g(x(σ(T))).
By applying monotone iteration method, we not only obtain the existence of positive solutions, but also establish iterative schemes for approximating the solutions.


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## Body

## 1. Introduction

The theory of time scales was introduced by Hilger in his Ph.D. thesis [1] in 1988 in order to unify continuous and discrete analysis. The study of dynamic equations on time scales is a fairly new subject and research in this area is rapidly growing. For some basic definitions and relevant results on time scales, see [2, 3].Recently, first-order boundary value problems (BVPs for short) on time scales have attracted much attention from many authors. For example, for first-order periodic boundary value problem (PBVP for short) on time scales(1.1)uΔ(t)=f(t,u(t)),t∈[a,b]T,u(a)=u(σ(b)),
Cabada [4] developed the method of lower and upper solutions coupled with the monotone iterative techniques to derive the existence of extremal solutions. In [5], Cabada and Vivero studied the existence of solutions for the first-order dynamic equation with nonlinear functional boundary value conditions(1.2)uΔ(t)=f(t,uσ(t)),forΔ-a.e.t∈[a,b]T,B(u(a),u)=0.
They proved the uniqueness of solutions and developed the monotone iterative technique when B(x,y)=x-g(yσ(b)) and g was a continuous and nonincreasing function. In 2007, Sun and Li [6] considered the first-order PBVP on time scales(1.3)xΔ(t)+p(t)x(σ(t))=g(t,x(σ(t))),t∈[0,T]T,x(0)=x(σ(T)).
Some existence criteria of at least one solution were established by using novel inequalities and the Schaefer fixed point theorem. In [7], by applying several well-known fixed point theorems, Sun and Li obtained some existence and multiplicity criteria of positive solutions for the first-order PBVP on time scales(1.4)xΔ(t)+p(t)x(σ(t))=f(x(t)),t∈[0,T]T,x(0)=x(σ(T)).
In 2010, Zhao and Sun [8] investigated the first-order PBVP on time scales(1.5)xΔ(t)+p(t)x(σ(t))=f(t,x(t)),t∈[0,T]T,x(0)=x(σ(T)).
Some existence criteria of positive solutions were established, and the method used was the monotone iterative technique. For other related results, one can refer to [9–14] and the references therein.Motivated greatly by the above-mentioned works, in this paper, we are interested in the existence and iteration of positive solutions for the following first-order dynamic equation with nonlinear boundary condition on time scales:(1.6)xΔ(t)+p(t)x(σ(t))=f(t,x(t)),t∈[0,T]T,x(0)=g(x(σ(T))),
where 𝕋 is an arbitrary time scale, T>0 is fixed, and 0,T∈𝕋. For each interval I of ℝ, we denote by I𝕋=I∩𝕋. Throughout this paper, we always assume that p:[0,T]𝕋→(0,+∞) is right-dense continuous. Here, a solution x of the BVP (1.6) is said to be positive if x is nonnegative and nontrivial. By applying monotone iteration method, we not only obtain the existence of positive solutions for the BVP (1.6), but also establish iterative schemes for approximating the solutions. It is worth mentioning that the initial terms of our iterative schemes are constant functions, which implies that the iterative schemes are significant and feasible. In our arguments, the following monotone iteration method [15] is very crucial.Theorem 1.1.
LetK be a normal cone of a Banach space E and v0≤w0. Suppose that(a1)
T:[v0,w0]→E is completely continuous,(a2)
T is monotone increasing on [v0,w0],(a3)
v0 is a lower solution of T, that is, v0≤Tv0,(a4)
w0 is an upper solution of T; that is, Tw0≤w0.Then, the iterative sequences
(1.7)vn=Tvn-1,wn=Twn-1(n=1,2,3,…)
satisfy
(1.8)v0≤v1≤⋯≤vn≤⋯≤wn≤⋯≤w1≤w0,
and converge to, respectively, v and w∈[v0,w0], which are fixed points of T.


## 2. Main Results

Theorem 2.1.
Assume thatf:[0,T]𝕋×[0,+∞)→[0,+∞) and g:[0,+∞)→[0,+∞) are continuous and ∫0σ(T)f(s,0)Δs>0. If there exists a constant R>0 such that the following conditions are satisfied: (H1)
f(t,u)≤f(t,v)≤R/2σ(T), t∈[0,T]𝕋, 0≤u≤v≤R,(H2)
g(u)≤g(v)≤R/2, 0≤u≤v≤R,then the BVP (1.6) has positive solutions.Proof.
Let(2.1)E={x∣x:[0,σ(T)]T→Riscontinuous}
be equipped with the norm
(2.2)‖x‖=max⁡t∈[0,σ(T)]T|x(t)|.
Then, E is a Banach space. Denote
(2.3)K={x∈E∣x(t)≥0,t∈[0,σ(T)]T}.
Then, K is a normal cone of E. Now, if we define an operator Φ:K→K by
(2.4)(Φx)(t)=1ep(t,0)[∫0tep(s,0)f(s,x(s))Δs+g(x(σ(T)))],t∈[0,σ(T)]T,
then it is easy to know that fixed points of Φ are nonnegative solutions of the BVP (1.6).
Letv0(t)≡0 and w0(t)≡R, t∈[0,σ(T)]𝕋. Now, we divide our proof into the following steps.
Step 1.
We verify thatΦ:[v0,w0]→K is completely continuous.
First, we will show thatΦ:[v0,w0]→K is continuous.
Letxn(n=1,2,…),x∈[v0,w0] and limn→∞xn=x. Then,(2.5)0≤xn(t)≤R,0≤x(t)≤R,fort∈[0,σ(T)]T.
For any givenϵ>0, since f is uniformly continuous on [0,T]𝕋×[0,R], there exists δ1>0 such that for any u1,u2∈[0,R] with |u1-u2|<δ1,(2.6)|f(s,u1)-f(s,u2)|<ϵ2σ(T)ep(σ(T),0),s∈[0,T]T.
On the other hand, sinceg is continuous at x(σ(T)), there exists δ2>0 such that for any u∈[0,+∞) with |u-x(σ(T))|<δ2,(2.7)|g(u)-g(x(σ(T)))|<ϵ2.
Letδ=min{δ1,δ2}. Then, it follows from limn→∞xn=x that there exists a positive integer N such that for any n>N,(2.8)|xn(s)-x(s)|<δ,s∈[0,σ(T)]T.
In view of (2.6), (2.7), and (2.8), we know that for any n>N,(2.9)|(Φxn)(t)-(Φx)(t)|=1ep(t,0)|∫0tep(s,0)[f(s,xn(s))-f(s,x(s))]Δs+[g(xn(σ(T)))-g(x(σ(T)))]|≤ep(σ(T),0)∫0σ(T)|f(s,xn(s))-f(s,x(s))|Δs+|g(xn(σ(T)))-g(x(σ(T)))|<ϵ,t∈[0,σ(T)]T,
which indicates that limn→∞Φxn=Φx. So, Φ:[v0,w0]→K is continuous.
Next, we will show thatΦ:[v0,w0]→K is compact.
LetX⊂[v0,w0] be a bounded set. Then, there exists a constant L>0 such that for any x∈X and s∈[0,σ(T)]𝕋, 0≤x(s)≤L. Define Q=maxs∈[0,T]𝕋,u∈[0,L]f(s,u) and M=maxu∈[0,L]g(u).
On the one hand, for anyx∈X, we have(2.10)(Φx)(t)=1ep(t,0)[∫0tep(s,0)f(s,x(s))Δs+g(x(σ(T)))]≤ep(σ(T),0)∫0σ(T)f(s,x(s))Δs+g(x(σ(T)))≤Qσ(T)ep(σ(T),0)+M,t∈[0,σ(T)]T,
which shows that Φ(X) is uniformly bounded.
On the other hand, for anyx∈X and t1,t2∈[0,σ(T)]𝕋 with t1≤t2, we have(2.11)|(Φx)(t2)-(Φx)(t1)|=|[1ep(t2,0)-1ep(t1,0)][∫0t1ep(s,0)f(s,x(s))Δs+g(x(σ(T)))]+1ep(t2,0)∫t1t2ep(s,0)f(s,x(s))Δs|≤[1ep(t1,0)-1ep(t2,0)][∫0σ(T)ep(s,0)f(s,x(s))Δs+g(x(σ(T)))]+∫t1t2ep(s,0)f(s,x(s))Δs≤[1ep(t1,0)-1ep(t2,0)][Qσ(T)ep(σ(T),0)+M]+Qep(σ(T),0)(t2-t1),
which implies that Φ(X) is equicontinuous. Consequently, Φ:[v0,w0]→K is compact.Step 2.
We assert thatΦ is monotone increasing on [v0,w0].
Suppose thatu,v∈[v0,w0] and u≤v. Then, 0≤u(t)≤v(t)≤R, t∈[0,σ(T)]𝕋. By (H1) and (H2), we have(2.12)(Φu)(t)=1ep(t,0)[∫0tep(s,0)f(s,u(s))Δs+g(u(σ(T)))]≤1ep(t,0)[∫0tep(s,0)f(s,v(s))Δs+g(v(σ(T)))]=(Φv)(t),t∈[0,σ(T)]T,
which shows that Φu≤Φv.Step 3.
We prove thatv0 is a lower solution of Φ.
For anyt∈[0,σ(T)]𝕋, it is obvious that(2.13)(Φv0)(t)=1ep(t,0)[∫0tep(s,0)f(s,0)Δs+g(0)]≥0=v0(t),
which implies that v0≤Φv0.Step 4.
We show thatw0 is an upper solution of Φ.
In view of (H1) and (H2), we have(2.14)(Φw0)(t)=∫0tep(s,0)ep(t,0)f(s,R)Δs+g(R)ep(t,0)≤∫0tf(s,R)Δs+g(R)≤∫0σ(T)f(s,R)Δs+g(R)≤R=w0(t),t∈[0,σ(T)]T,
which indicates that Φw0≤w0.Step 5.
We claim that the BVP (1.6) has positive solutions.
In fact, if we construct sequences{vn}n=1∞ and {wn}n=1∞ as follows:(2.15)vn=Φvn-1,wn=Φwn-1,n=1,2,3,…,
then it follows from Theorem 1.1 that
(2.16)v0≤v1≤⋯≤vn≤⋯≤wn≤⋯≤w1≤w0,
and {vn}n=0∞ and {wn}n=0∞ converge to, respectively, v and w∈[v0,w0], which are solutions of the BVP (1.6). Moreover, it follows from
(2.17)(Φv0)(σ(T))=1ep(σ(T),0)[∫0σ(T)ep(s,0)f(s,0)Δs+g(0)]≥1ep(σ(T),0)∫0σ(T)f(s,0)Δs>0
that
(2.18)0<(Φv0)(σ(T))≤(Φv)(σ(T))=v(σ(T))≤w(σ(T)),
which shows that v and w are positive solutions of the BVP (1.6).Example 2.2.
Let𝕋=[0,1]∪[2,3]. We consider the following BVP on 𝕋:
(2.19)xΔ(t)+p(t)x(σ(t))=tx(t)+39,t∈[0,3]T,x(0)=x2(3)12.
Sincef(t,x)=tx+3/9 and g(x)=x2/12, if we choose R=6, then all the conditions of Theorem 2.1 are fulfilled. So, it follows from Theorem 2.1 that the BVP (2.19) has positive solutions v and w. Furthermore, if we construct sequences {vn}n=1∞ and {wn}n=1∞ as follows:(2.20)vn=Φvn-1,wn=Φwn-1,n=1,2,3,…,
where v0(t)≡0 and w0(t)≡6, then
(2.21)lim⁡n→∞vn=v,lim⁡n→∞wn=w.



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*Source: 845107-2011-08-01.xml*