# Bounded Positive Solutions for a Third Order Discrete Equation

Authors: Zeqing Liu; Ming Jia; Shin Min Kang; Young Chel Kwun
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper studies the following third order neutral delay discrete equationΔ(anΔ2(xn+pnxn-τ))+f(n,xn-d1n,…,xn-dln)=gn,n≥n0, where τ,l∈ℕ, n0∈ℕ∪{0}, {an}n∈ℕn0, {pn}n∈ℕn0, {gn}n∈ℕn0 are real sequences with an≠0 for n≥n0, {din}n∈ℕn0⊆ℤ with limn→∞(n-din)=+∞ for i∈{1,2,…,l} and f∈C(ℕn0×ℝl,ℝ). By using a nonlinear alternative theorem of Leray-Schauder type, we get sufficient conditions which ensure the existence of bounded positive solutions for the equation. Three examples are given to illustrate the results obtained in this paper.


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## Body

## 1. Introduction and Preliminaries

The oscillatory, nonoscillatory and asymptotic behaviors and existence of solutions for various difference equations have received more and more attentions in recent years. For details, we refer the reader to [1–11] and the references therein.In 2005, M. Migda and J. Migda [10] studied the asymptotic behavior of solutions for the second order neutral difference equation
(1.1)Δ2(xn+pxn-k)+f(n,xn)=0,n≥1,
where p∈ℝ, k is a nonnegative integer and f:ℕ×ℝ→ℝ. In 2008, Cheng and Chu [7] established sufficient and necessary conditions of oscillation for the second order difference equation
(1.2)Δ(rn-1Δxn-1)+pnxnγ=0,n≥1,
where γ is the quotient of two odd positive integers and pn,rn∈(0,+∞) for n∈ℕ. In 2000, Li et al. [9] gave several necessary and/or sufficient conditions of the existence of unbounded positive solution for the nonlinear difference equation
(1.3)Δ(rnΔxn)+f(n,xn)=0,n≥n0,
where n0 is a fixed nonnegative integer, r:ℕn0→(0,+∞) and f:ℕn0×ℝ→ℝ. In 2003, using the Leray-Schauder's nonlinear alternative theorem, Agarwal et al. [1] presented the existence of nonoscillatory solutions for the discrete equation
(1.4)Δ(anΔ(xn+pxn-τ))+F(n+1,xn+1-σ)=0,n≥1,
where τ,σ are fixed nonnegative integers, p∈ℝ,a:ℕ→(0,+∞) and F:ℕ×(0,+∞)→[0,+∞) is continuous. In 1995, Yan and Liu [11] proved the existence of a bounded nonoscillatory solution for the third order difference equation
(1.5)Δ3xn+f(n,xn,xn-r)=0,n≥n0
by utilizing the Schauder's fixed point theorem. In 2005, Andruch-Sobiło and Migda [2] studied the third order linear difference equations of neutral type
(1.6)Δ3(xn-pnxσn)±qnxτn=0,n≥n0
and obtained sufficient conditions under which all solutions of (1.6) are oscillatory.The aim of this paper is to study the following third order neutral delay discrete equation(1.7)Δ(anΔ2(xn+pnxn-τ))+f(n,xn-d1n,…,xn-dln)=gn,n≥n0,
where τ,l∈ℕ,n0∈ℕ∪{0},{an}n∈ℕn0,{pn}n∈ℕn0,{gn}n∈ℕn0 are real sequences with an≠0 for n≥n0, {din}n∈ℕn0⊆ℤ with limn→∞(n-din)=+∞ for i∈{1,2,…,l} and f∈C(ℕn0×ℝl,ℝ). By making use of the Leray-Schauder's nonlinear alternative theorem, we establish the existence results of bounded positive solutions for (1.7), which extend substantially Theorem 2 in [11]. Three nontrivial examples are given to illustrate the superiority and applications of the results presented in this paper.Let us recall and introduce the below concepts, signs and lemmas. Letℝ,ℤ and ℕ denote the sets of all real numbers, integers and positive integers, respectively,
(1.8)Nn0={n:n∈Nwithn≥n0},Zβ={n:n∈Zwithn≥β},β=min⁡{n0-τ,inf⁡{n-din:1≤i≤l,n∈Nn0}}
and lβ∞ stand for the Banach space of all bounded sequences on ℤβ with norm
(1.9)‖x‖=sup⁡n∈Zβ|xn|forx={xn}n∈Zβ∈lβ∞.
For any constants M>N>0, put
(1.10)E(N)={x={xn}n∈Zβ∈lβ∞:xn≥Nforn∈Zβ},U(M)={x∈E(N):‖x‖<M}.
It is easy to verify that E(N) is a nonempty closed convex subset of lβ∞ and U(M) is a nonempty open subset of E(N).By a solution of (1.7), we mean a sequence {xn}n∈ℤβ with a positive integer T≥τ+|β| such that (1.7) holds for all n≥T.For any subsetU of a Banach space X, let U¯ and ∂U denote the closure and boundary of U in X, respectively.Lemma 1.1 (see [8]).
A bounded, uniformly Cauchy subsetD of lβ∞ is relatively compact.Lemma 1.2 (Leray-Schauder's Nonlinear Alternative Theorem [1]).
LetE be a nonempty closed convex subset of a Banach space X and U be an open subset of E with p*∈U. Also G:U¯→E is a continuous, condensing mapping with G(U¯) bounded. Then either(A1)
G has a fixed point in U¯; or(A2)
there arex∈∂U and λ∈(0,1) with x=(1-λ)p*+λGx.


## 2. Existence of Bounded Positive Solutions

Now we investigate sufficient conditions of the existence of bounded positive solutions for (1.7) by using the Leray-Schauder's Nonlinear Alternative Theorem.Theorem 2.1.
Assume that there exist constantsk0∈ℕn0 and M,N,p¯ and  p_ satisfying
(2.1)∑s=k0∞‍∑k=s∞‍1|ak|∑j=k∞‍sup⁡{|f(j,w1,…,wl)|:wi∈[N,M],1≤i≤l}<+∞;(2.2)∑s=k0∞‍∑k=s∞‍1|ak|∑j=k∞‍|gj|<+∞;(2.3)0<N<(1-p_-p¯)M,min⁡{p_,p¯}≥0,p_+p¯<1(2.4)-p_≤pn≤p¯,n≥k0.
Then (1.7) possesses a bounded positive solution in U(M)¯.Proof.
LetL∈(p¯M+N,M(1-p_)). It follows from (2.1)–(2.3) that there exists a positive integer T>1+τ+k0+|β| sufficiently large satisfying
(2.5)∑s=T∞‍∑k=s∞‍1|ak|∑j=k∞‍[|gj|+sup⁡{|f(j,w1,…,wl)|:wi∈[N,M],1≤i≤l}]<min⁡{L-p¯M-N,M(1-p_)-L}.
Choose p*=M-ɛ0 with ɛ0∈(0,min{L-p¯M-N,M(1-p_)-L}) and
(2.6)∑s=T∞‍∑k=s∞‍1|ak|∑j=k∞‍[|gj|+sup⁡{|f(j,w1,…,wl)|:wi∈[N,M],1≤i≤l}]≤min⁡{L-p¯M-N,M(1-p_)-L}-ɛ0.
Note that
(2.7)M>M-ɛ0=p*>M-min⁡{L-p¯M-N,M(1-p_)-L}≥N+M(1+p¯)-L>N,
which implies that p*={p*}n∈ℤβ∈U(M). Define two mappings AL,BL:U(M)¯→lβ∞ by
(2.8)ALxn={L-pnxn-τ,n≥T+1L-pTxT,β≤n≤T;(2.9)BLxn={-∑s=Tn-1‍∑k=s∞‍1ak∑j=k∞‍[-gj+f(j,xj-d1j,…,xj-dlj)],n≥T+10,β≤n≤T
for all x={xn}n∈ℤβ∈U(M)¯.
We now show that(2.10)DL=AL+BL:U(M)¯⟶E(N).
For each x={xn}n∈ℤβ∈U(M)¯, by (2.4)–(2.9), we have
(2.11)ALxn+BLxn=L-pnxn-τ-∑s=Tn-1‍∑k=s∞‍1ak∑j=k∞‍[-gj+f(j,xj-d1j,…,xj-dlj)]≥L-p¯M-∑s=Tn-1‍∑k=s∞‍1|ak|∑j=k∞‍[|gj|+|f(j,xj-d1j,…,xj-dlj)|]≥L-p¯M-min⁡{L-p¯M-N,M(1-p_)-L}+ɛ0≥N+ɛ0>N,n≥T+1ALxn+BLxn=L-pTxT≥L-p¯M>N,β≤n≤T,We next assert that(2.12)BL:U(M)¯⟶lβ∞isacontinuous,compactmapping.
Let {xα}α∈ℕ⊆U(M)¯ be an arbitrary sequence and x0∈lβ∞ with
(2.13)‖xα-x0‖⟶0asα⟶∞.
Since U(M)¯ is closed, it follows that x0∈U(M)¯. Given ɛ>0. Using (2.1), (2.13) and the continuity of f, we infer that there exists T**,T*∈ℕ with T*>T+1 satisfying
(2.14)∑s=T*∞‍∑k=s∞‍1|ak|∑j=k∞‍sup⁡{|f(j,w1,…,wl)|:wi∈[N,M],1≤i≤l}<ɛ16;(2.15)∑k=T*∞‍1|ak|∑j=k∞‍sup⁡{|f(j,w1,…,wl)|:wi∈[N,M],1≤i≤l}<ɛ16(T*-T);(2.16)∑s=TT*-1‍∑k=sT*-1‍1|ak|∑j=T*∞‍sup⁡{|f(j,w1,…,wl)|:wi∈[N,M],1≤i≤l}<ɛ16;(2.17)∑s=TT*-1‍∑k=sT*-1‍1|ak|∑j=kT*-1‍|f(j,xj-d1jα,…,xj-dljα)-f(j,xj-d1j0,…,xj-dlj0)|<ɛ2,α≥T**.Combining (2.9) and (2.14)–(2.17), we conclude that
(2.18)‖BLxα-BLx0‖=max⁡⁡{sup⁡β≤n≤T|BLxnα-BLxn0|,sup⁡n≥T+1|BLxnα-BLxn0|}≤max⁡{0,sup⁡n≥T+1∑s=Tn-1‍∑k=s∞‍1|ak|∑j=k∞‍|f(j,xj-d1jα,…,xj-dljα)-f(j,xj-d1j0,…,xj-dlj0)|}≤∑s=TT*-1‍∑k=s∞‍1|ak|∑j=k∞‍|f(j,xj-d1jα,…,xj-dljα)-f(j,xj-d1j0,…,xj-dlj0)|+∑s=T*∞‍∑k=s∞‍1|ak|∑j=k∞‍[|f(j,xj-d1jα,…,xj-dljα)|+|f(j,xj-d1j0,…,xj-dlj0)|]≤∑s=TT*-1‍∑k=sT*-1‍1|ak|∑j=k∞‍|f(j,xj-d1jα,…,xj-dljα)-f(j,xj-d1j0,…,xj-dlj0)|+∑s=TT*-1‍∑k=T*∞‍1|ak|∑j=k∞‍|f(j,xj-d1jα,…,xj-dljα)|+|f(j,xj-d1j0,…,xj-dlj0)|+2∑s=T*∞‍∑k=s∞‍1|ak|∑j=k∞‍sup⁡{|f(j,w1,…,wl)|:wi∈[N,M],1≤i≤l}≤∑s=TT*-1‍∑k=sT*-1‍1|ak|∑j=kT*-1‍|f(j,xj-d1jα,…,xj-dljα)-f(j,xj-d1j0,…,xj-dlj0)|+∑s=TT*-1‍∑k=sT*-1‍1|ak|∑j=T*∞‍[|f(j,xj-d1jα,…,xj-dljα)|+|f(j,xj-d1j0,…,xj-dlj0)|]+2(T*-T)∑k=T*∞‍1|ak|∑j=k∞‍sup⁡{|f(j,w1,…,wl)|:wi∈[N,M],1≤i≤l}+2⋅ɛ16<ɛ2+2⋅ɛ16+2(T*-T)⋅ɛ16(T*-T)+ɛ8<ɛ,α≥T**,
which means that BL is continuous in U(M)¯. On the other hand, in light of (2.6) and (2.9), we get that for each x={xn}n∈ℤβ∈U(M)¯(2.19)‖BLx‖=max⁡{sup⁡β≤n≤T|BLxn|,sup⁡n≥T+1|BLxn|}≤sup⁡n≥T+1∑s=Tn-1‍∑k=s∞‍1|ak|∑j=k∞‍[|gj|+|f(j,xj-d1j,…,xj-dlj)|]≤∑s=T∞‍∑k=s∞‍1|ak|∑j=k∞‍[|gj|+sup⁡{|f(j,w1,…,wl)|:wi∈[N,M],1≤i≤l}]≤min⁡{L-p¯M-N,M(1-p_)-L}-ɛ0≤M,
which yields that BL(U(M)¯) is a bounded subset of lβ∞. By virtue of (2.1) and (2.2), we deduce that for any ɛ>0, there exists T0>T satisfying
(2.20)∑s=T0∞‍∑k=s∞‍1|ak|∑j=k∞‍[|gj|+sup⁡{|f(j,w1,…,wl)|:wi∈[N,M],1≤i≤l}]<ɛ,
which together with (2.9) gives that for any x={xn}n∈ℤβ∈U(M)¯(2.21)|BLxn-BLxm|=|∑s=Tn-1‍∑k=s∞‍1ak∑j=k∞‍[-gj+f(j,xj-d1j,…,xj-dlj)]-∑s=Tm-1‍∑k=s∞‍1ak∑j=k∞‍[-gj+f(j,xj-d1j,…,xj-dlj)]|≤∑s=nm-1‍∑k=s∞‍1|ak|∑j=k∞‍[|gj|+|f(j,xj-d1j,…,xj-dlj)|]<∑s=T0∞‍∑k=s∞‍1|ak|∑j=k∞‍[|gj|+sup⁡{|f(j,w1,…,wl)|:wi∈[N,M],1≤i≤l}]<ɛ,m>n≥T0,
which means that BL(U(M)¯) is uniformly Cauchy. Thus Lemma 1.1 ensures that BL(U(M)¯) is a relatively compact subset of lβ∞.Letx={xn}n∈ℤβ,y={yn}n∈ℤβ∈U(M)¯. In view of (2.3), (2.4) and (2.8), we know that
(2.22)|ALxn-ALyn|=|L-pnxn-τ-L+pnyn-τ|≤|pn|‖x-y‖≤(p_+p¯)‖x-y‖,n≥T+1,|ALxn-ALyn|=|L-pTxT-τ-L+pTyT-τ|≤|pT|‖x-y‖≤(p_+p¯)‖x-y‖,β≤n≤T,
which implies that
(2.23)‖Ax-Ay‖≤(p_+p¯)‖x-y‖,
which together with (2.10) and (2.12) guarantees that DL:U(M)¯→E(N) is a continuous, condensing mapping.In order to show the existence of a fixed point ofDL, we need to prove that (A2) in Lemma 1.2 does not hold. Otherwise there exist x={xn}n∈ℤβ∈∂U(M) and λ∈(0,1) such that x=(1-λ)p*+λDLx. Let
(2.24)S1={x∈lβ∞:N≤xn≤M,∀n≥β,‖x‖=M},S2={x∈lβ∞:N≤xn≤M,∀n≥β,andthereexistsn*≥βsatisfyingxn*=N}.
It is easy to verify that ∂U(M)=S1∪S2. Now we have to discuss two possible cases as follows:Case  1.Let x∈S1. It follows from (2.3), (2.4), (2.8) and (2.9) that
(2.25)xn=(1-λ)p*+λ[ALxn+BLxn]=(1-λ)p*+λ[L-pnxn-τ-∑s=Tn-1‍∑k=s∞‍1ak∑j=k∞‍[-gj+f(j,xj-d1j,…,xj-dlj)]]≤(1-λ)p*+λ[L+p_M+∑s=Tn-1‍∑k=s∞‍1|ak|∑j=k∞‍[|gj|+|f(j,xj-d1j,…,xj-dlj)|]]≤(1-λ)(M-ɛ0)+λ[L+p_M+min⁡{L-p¯M-N,M(1-p_)-L}-ɛ0]≤(1-λ)(M-ɛ0)+λ(M-ɛ0)=M-ɛ0,n≥T+1,(2.26)xn=(1-λ)p*+λ[L-pTxT]≤(1-λ)p*+λ(L+p_M)≤(1-λ)(M-ɛ0)+λ(M-ɛ0)=M-ɛ0,β≤n≤T,
which yield that
(2.27)M=‖x‖≤M-ɛ0<M,
which is a contradiction;Case  2.Let x∈S2. If n*≥T+1, by (2.3), (2.4), (2.8) and (2.9), we deduce that
(2.28)N=xn*=(1-λ)p*+λ[ALxn*+BLxn*]=(1-λ)p*+λ[L-pn*xn*-τ-∑s=Tn*-1‍∑k=s∞‍1ak∑j=k∞‍[-gj+f(j,xj-d1j,…,xj-dlj)]]≥(1-λ)(M-ɛ0)+λ[L-p¯M-∑s=Tn*-1‍∑k=s∞‍1|ak|∑j=k∞‍[|gj|+|f(j,xj-d1j,…,xj-dlj)|]]>(1-λ)N+λ[L-p¯M-min⁡{L-p¯M-N,M(1-p_)-L}+ɛ0]≥(1-λ)N+λ(N+ɛ0)=N+ɛ0>N,
which is impossible; if n*≤T, by (2.3), (2.4), (2.8) and (2.9), we arrive at
(2.29)N=xn*=(1-λ)p*+λ[L-pTxT]≥(1-λ)p*+λ(L-p¯M)≥(1-λ)N+λ(N+ɛ0)=N+ɛ0>N,
which is absurd.Consequently Lemma1.2 ensures that there is x={xn}n∈ℤβ∈U(M)¯ such that DLx=ALx+BLx=x, which is a bounded positive solution of (1.7). This completes the proof.Remark 2.2.
Under the conditions of Theorem2.1 we prove also that (1.7) has uncountably many bounded positive solutions in U(M)¯.
In fact, as in the proof of Theorem2.1, for any different L1,L2∈(p¯M+N,M(1-p_)) we conclude that for each r∈{1,2}, there exist a constant Tr>1+τ+k0+|β| and two mappings Ar,Br:U(M)¯→lβ∞ satisfying (2.6)–(2.9) and
(2.30)∑s=min⁡{T1,T2}∞‍∑k=s∞‍1|ak|∑j=k∞‍[|gj|+sup⁡{|f(j,w1,…,wl)|:wi∈[N,M],1≤i≤l}]≤|L1-L2|4,
where T,L,AL and BL are replaced by Tr,r,Ar and Br, respectively, and Ar+Br has a fixed point zr={znr}n∈ℤβ∈U(M)¯, which is a bounded positive solution of (1.7). In order to prove that (1.7) possesses uncountably many bounded positive solutions in U(M)¯, we need only to prove that z1≠z2. It follows from (2.8), (2.9) and (2.30) that for n≥min{T1,T2}(2.31)|zn1-zn2|=|L1-pnzn-τ1-∑s=T1n-1‍∑k=s∞‍1ak∑j=k∞‍[-gj+f(j,zj-d1j1,…,zj-dlj1)]-L2+pnzn-τ2+∑s=T2n-1‍∑k=s∞‍1ak∑j=k∞‍[-gj+f(j,zj-d1j2,…,zj-dlj2)]|≥|L1-L2|-(p_+p¯)‖z1-z2‖-2∑s=min⁡{T1,T2}∞‍∑k=s∞‍1|ak|∑j=k∞‍[|gj|+sup⁡{|f(j,w1,…,wl)|:wi∈[N,M],1≤i≤l}]≥|L1-L2|-(p_+p¯)‖z1-z2‖-|L1-L2|2,
which implies that
(2.32)‖z1-z2‖≥|L1-L2|2(1+p_+p¯)>0,
which yields that z1≠z2.Remark 2.3.
If eitherp_=0 or p¯=0, then Theorem 2.1 reduces to the below results, respectively.Theorem 2.4.
Assume that there exist constantsk0∈ℕn0 and M,N and p¯ satisfying (2.1), (2.2) and
(2.33)0<N<(1-p¯)M,0≤pn≤p¯<1,n≥k0.
Then (1.7) possesses a bounded positive solution in U(M)¯.Theorem 2.5.
Assume that there exist constantsk0∈ℕn0 and M,N and p_ satisfying (2.1), (2.2) and
(2.34)0<N<(1-p_)M,-1<-p_≤pn≤0,n≥k0.
Then (1.7) possesses a bounded positive solution in U(M)¯.Remark 2.6.
Theorems2.1–2.5 include Theorem 2 in [11] as special cases. Examples 3.1–3.3 in Section 3 explain that Theorems 2.1–2.5 are genuine generalizations of Theorem 2 in [11].


## 3. Examples and Applications

Now we construct three nontrivial examples to explain the superiority and applications of Theorems2.1–2.5, respectively.Example 3.1.
Consider the third order neutral delay discrete equation(3.1)Δ((-1)nn3Δ2(xn+nsin⁡(n2)3n+1xn-τ))+n3xn2-2-nxn3-63n5+5n+nx2n+52=cos⁡(nln⁡(n2+1))n3+1,n≥1,
where τ∈ℕ is fixed. Let l=3,n0=1,k0=2,p_=1/2,p¯=1/3,M=7,N=1,
(3.2)an=(-1)nn3,pn=nsin⁡(n2)3n+1,gn=cos⁡(nln⁡(n2+1))n3+1,d1n=-n2+n+2,d3n=-n-5,f(n,u,v,w)=n3u-nv3n5+5n+nw2,∀(n,u,v,w)∈N×R3.
It is easy to verify that (2.1)–(2.4) hold. It follows from Theorem 2.1 that (3.1) has a bounded positive solution in U(M)¯. However Theorem 2 in [11] is useless for (3.1).Example 3.2.
Consider the third order neutral delay discrete equation(3.3)Δ((n+1)2ln⁡3(n+2)Δ2(xn+3n-44n+2xn-τ))+xn(n+1)/22xn(n-1)/23n3+1=(-1)nn2,n≥1,
where τ∈ℕ is fixed. Let l=2,n0=1,k0=2,p¯=3/4,M=40,N=8,
(3.4)an=(n+1)2ln⁡3(n+2),pn=3n-44n+2,d1n=n(1-n)2,d2n=n(3-n)2,gn=(-1)nn2,f(n,u,v)=u2v3n3+1,∀(n,u,v)∈N×R2.
It is clear that (2.1), (2.2) and (2.33) hold. Consequently Theorem 2.4 guarantees that (3.3) has a bounded positive solution in U(M)¯. But Theorem 2 in [11] is inapplicable for (3.3).Example 3.3.
Consider the third order neutral delay discrete equation(3.5)Δ(n5+1Δ2(xn-n3+13n3+4xn-τ))+nxn2-2n6n4+n+1+x2n+33n2+2=sin⁡(n2-n)n2+1,n≥1,
where τ∈ℕ is fixed. Let l=2,n0=1,k0=3,p_=1/3,M=30,N=19,
(3.6)an=n5+1,pn=-n3+13n3+4,d1n=n(3-n),d2n=-n-3,gn=sin⁡(n2-n)n2+1,f(n,u,v)=u6nn4+n+1+v3n2+2,∀(n,u,v)∈N×R2.
Obviously, (2.1), (2.2) and (2.34) hold. Thus Theorem 2.5 ensures that (3.5) has a bounded positive solution in U(M)¯. While Theorem 2 in [11] is unfit for (3.5)



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