On the existence of bounded solutions for nonlinear second order neutral difference equations

On the existence of bounded solutions for nonlinear second order neutral difference equations

Marek Galewski

, Robert Jankowski

,

Magdalena Nockowska–Rosiak

and Ewa Schmeidel

1Lodz University of Technology, Lodz, Poland

2University of Bialystok, Bialystok, Poland

Received 18 December 2013, appeared 6 January 2015 Communicated by Ondˇrej Došlý

Abstract. Using the techniques connected with the measure of noncompactness we investigate the neutral difference equation of the following form

∆ rn(∆(xn+pnxn−k))^γ+qnx^α_n+anf(x_n+1) =_0,

where x:Nk →_R,a,p,q:N0 →_R,r: N0 →_R\ {0}, f:R →_Ris continuous andk is a given positive integer, α≥ 1 is a ratio of positive integers with odd denominator, and γ≤ 1 is ratio of odd positive integers;N_k := {k,k+1, . . .}. Sufficient conditions for the existence of a bounded solution are obtained. Also a special type of stability is studied.

Keywords:difference equation, Emden–Fowler equation, measures of noncompactness, Darbo’s fixed point theorem, boundedness, stability.

2010 Mathematics Subject Classification: 39A10, 39A22, 39A30.

1 Introduction

As it is well known, difference equations serve as mathematical models in diverse areas, such as economy, biology, physics, mechanics, computer science, finance, see for example [1,7].

One of such models is the Emden–Fowler equation which originated in the gaseous dynamics in astrophysics and further was used in the study of fluid mechanics, relativistic mechanics, nuclear physics and in the study of chemically reacting systems, see [28]. For the reader’s convenience, we note that the background for difference equations theory can be found in numerous well-known monographs: Agarwal [1], Agarwal, Bohner, Grace and O’Regan [2], Agarwal and Wong [3], Elaydi [7], Kelley and Peterson [12], and Koci´c and Ladas [13].

In the present paper we study using techniques connected with the measure of noncom- pactness the existence of a bounded solution and some type of its asymptotic behavior to a nonlinear second order difference equation, which can be viewed as a generalization of a dis- crete Emden–Fowler equation or else it can be viewed as a second order difference equation

BCorresponding author. Email:marek.galewski@p.lodz.pl

with memory. This makes a problem which we consider different from those already inves- tigated via techniques of measure of noncompactness, see for example [25] since we do not expect a direct connection between a fixed point of a suitably defined operator investigated on a non-reflexive space l^∞ a solution to the problem under consideration. Indeed, this is the case. What we obtain is that starting from some index which we define the solution is taken from the fixed point while the previous terms have to be iterated. This also makes the definition of the operator different from this which would be used for the problem without dependence on previous terms. It seems that the method which we sketch here would prove applicable for several other problems. We also note that due to the type of space which we use, namelyl^∞ we cannot apply standard fixed point techniques such as Banach theorem or Schauder theorem and related results. We expect that our method would apply for systems of difference equations. However, what we cannot obtain here is the asymptotic stability of the solution. We will use axiomatically defined measures of noncompactness as presented in the paper [5] by Bana´s and Rzepka.

The problem we consider is as follows

∆ r_n(_∆(x_n+p_nx_n−k))^γ+q_nx^α_n+a_nf(x_n+1) =0, (1.1) whereα≥1 is a ratio of positive integers with odd denominator,γ≤1 is ratio of odd positive integers,x: Nk →_R a,p,q: N0 → _R, r: N0 →_R\ {₀}_{, and} f: R →_R is a locally Lipschitz function with no further growth assumptions. HereN0 := {0, 1, 2, . . .}, Nk := {k,k+1, . . .} where k is a given positive integer, and R is a set of all real numbers. By a solution of equation (1.1) we mean a sequencex: Nk →_Rwhich satisfies (1.1) for everyn∈_Nk.

There has been an interest of many authors to study properties of solutions of the second- order neutral difference equations; see the papers [6,8–10,15–18,21–23,26,27] and the refer- ences therein. The interesting oscillatory results for first order and even order neutral differ- ence equations can be found in [14], [19] and [20].

2 Preliminaries

Let (E,k·k)be an infinite dimensional Banach space. If X is a subset of E, then ¯X, ConvX denote the closure and the convex closure ofX, respectively. Moreover, we denote byM_E the family of all nonempty and bounded subsets ofE and byN_E the subfamily consisting of all relatively compact sets.

Definition 2.1. A mappingµ: M_E →[0,∞)is called a measure of noncompactness in Eif it satisfies the following conditions:

1⁰ kerµ= {X∈ M_E: µ(X) =0} 6=_∅and kerµ⊂ N_E, 2⁰ X⊂Y⇒µ(X)≤µ(Y),

3⁰ µ(X^¯) =µ(X) =µ(ConvX),

4⁰ µ(cX+ (1−c)Y)≤cµ(X) + (1−c)µ(Y)for 0≤c≤1, 5⁰ IfX_n∈ M_E, X_n+1⊂ X_n, X_n=X^¯_n forn=1, 2, 3, . . .

and lim_n→_∞µ(X_n) =0 then ∩^∞_n=1X_n6=_∅.

The following Darbo’s fixed point theorem given in [5] is used in the proof of the main result.

Theorem 2.2. Let M be a nonempty, bounded, convex and closed subset of the space E and let T: M→ M be a continuous operator such that µ(T(X)) ≤ kµ(X) for all nonempty subset X of M, where k∈[0, 1)is a constant. Then T has a fixed point in the subset M.

We consider the Banach spacel^∞ of all real bounded sequencesx:Nk →_Requipped with the standard supremum norm, i.e.

kxk= sup

n∈_Nk

|xn| forx∈ l^∞.

Let X be a nonempty, bounded subset of l^∞, Xn = {xn :x∈ X} (it meansXn is a set of n-th terms of any sequence belonging to X), and

diamXn=sup{|xn−yn|: x,y∈ X}. We use a following measure of noncompactness in the spacel^∞ (see [4])

µ(X) =lim sup

n→_∞

diamXn.

3 Main result

In this section, sufficient conditions for the existence of a bounded solution of equation (1.1) are derived.

Theorem 3.1. Assume that a,p,q:N0 →_{R, r}: N0 →_R\ {0}, and f: R→_{R. Let}

α≥1 be a ratio of positive integers with odd denominator, (3.1) γ∈(0, 1] be a ratio of odd positive integers, (3.2) and let k be a fixed positive integer. Assume that

f:R→_Ris a locally Lipschitz function, (3.3) and that the sequences r: N0 →_R\ {0}, a,q: N0 →_Rsatisfy

∑

1 r_n

1

γ ∞

i

∑

|a_i|<+_∞ and

∑

1 r_n

1

γ ∞

i

∑

|q_i|< +_∞. (3.4) Let the sequence p: N0→_Rsatisfy the following condition

−₁<_{lim inf}

n→_∞ p_n≤_{lim sup}

n→_∞

p_n<_{1. (3.5)}

Then there exists a bounded solution x:Nk → _Rof equation(1.1).

Proof. Condition (3.5) implies that there existn₀ ∈_N0 and a constantP∈ [0, 1)such that

|p_n| ≤P<_{1 for}n≥n₀. (3.6)

Condition (3.4) implies that

∑

|a_i|<+_∞,

∑

|q_i|<+_∞.

(3.7)

Then there existsn₁ ∈_{N0 such that ∑}^∞

i=n1

|a_i|<1. Hence, by (3.2),

∑

1 r_n

∑

|a_i|

!_γ¹

≤

∑

n=n1

1 r_n

1

γ ∞

i

∑

|a_i|.

The above and condition (3.4) imply that

∑

1 r_n

∑

|a_i|

!¹

γ

<+_∞. (3.8)

Analogously, we get

∑

1 rn

∑

|q_i|

!¹_γ

≤

∑

1 rn

1

γ ∞

i

∑

|q_i|. (3.9)

Recalling that remainder of a series is the difference between then-th partial sum and the sum of a series, we denote byαnand by βnthe following remainders

α_n =

∑

1 r_j

∑

|a_i|

!¹_γ

and β_n=

∑

1 r_j

∑

|q_i|

!_γ¹

(3.10)

We see, by (3.8) and (3.9) that

nlim→_∞αn=0 and lim

n→_∞βn =0. (3.11)

Fix any numberd >0. From (3.3), there exists a constant M_d >0 such that

|f(x)| ≤ M_d for allx ∈[−d,d]. (3.12) Choose a constantCsuch that

0< C≤ ^d−Pd

2^1γ−1(M_d)^1γ +2^γ1−1(d^α)^1γ

. (3.13)

By condition (3.9) there exists a positive integer n₂such that

α_n≤C and β_n≤C forn∈_Nn2. (3.14) Define setBas follows

B:={(x_n)^∞_n=0:|x_n| ≤d, for n∈_N0}.

Define a mapping T: B→l^∞ as follows

(Tx)_n =











−pnx_n−k− _∑^∞

j=n 1 r_j

∑∞ i=j

a_if(x_i+1) +q_ix^α_i

!¹_γ

, for anyn≥ n3,

x_n, for any 0≤n<n₃,

(3.15)

where n₃ = max{n₁,n₂}+k. Observe that B is a nonempty, bounded, convex and closed subset ofl^∞.

We will prove that the mappingThas a fixed point in B. This proof will follow in several subsequent steps.

Step 1. Firstly, we show thatT(B)⊂B.

We will use classical inequality

(a+b)^s≤2^s−1(a^s+b^s), a,b>0, s ≥1 (3.16) and the fact t → t^1/γ is nondecreasing. If x ∈ B, then for n < n₃ |(Tx)_n| = |x_n| ≤ d and by (3.15), we get for anyn≥n₃

|(Tx)_n| ≤ |pn| |x_n−k|+

∑

1 r_j

∑

(a_if(x_i+1) +q_ix_i^α)

!_γ¹

≤ |p_n| |x_n−k|+

∑

1 r_j

∑

(a_if(x_i+1) +q_ix_i^α)

!¹

γ

≤ |p_n| |x_n−k|+

∑

1 r_j

∑

|a_i| |f(x_i+₁)|+|q_i| |x_i|^α

!_γ¹ .

From (3.12), taking into account thatx_n−k ∈ B, and because ofx_i ∈Bwe have|x_i|^α ≤ d^α. Thus

|(Tx)_n| ≤ |pn|d+

∑

1 r_j

∑

(|a_i|M_d+|q_i|d^α)

!_γ¹

≤ |p_n|d+

∑

1 r_j

∑

|a_i|M_d+

∑

|q_i|d^α

!!¹

γ

. By inequality (3.16), we have

|(Tx)_n| ≤ |p_n|d+₂^γ1−1

∑



 1 r_j

∑

|a_i|M_d

!_γ¹ +

1 r_j

∑

|q_i|d^α

!_γ¹



≤ |pn|d+2^γ1−1(M_d)^1γ

∑

1 r_j

∑

|a_i|

!_γ¹

+2^1γ−1(d^α)^γ1

∑

1 r_j

∑

|q_i|

!_γ¹ . By using (3.6), (3.14) and (3.13), we estimate

|(Tx)_n| ≤ Pd+2^γ1−1(M_d)^1γ C+2^1γ−1(d^α)^1γ C

≤ Pd+2^1γ−1(M_d)^1γ +2^1γ−1(d^α)^{1γ d}−Pd

2^1γ−1 M_d¹_γ

+2^1γ−1(d^α)^1γ

=d. (3.17)

From the above, we have estimation

|(Tx)_n| ≤d, forn∈_Nn3. (3.18) Step 2. T is continuous

By assumption (3.3), (3.7), and by definition ofB, there exists a constantc^∗ >0 such that

∑

|a_if(x_i+1) +q_ix^α_i| ≤c^∗

for all x ∈ B. From (3.2), t → t^1/γ is locally Lipschitz then it is Lipschitz on closed and bounded intervals. Hence, there exists a constantL_γ such that

t^1/γ−s^1/γ

≤L_γ|t−s| for allt,s∈[−c^∗,c^∗]. (3.19) From (3.3), function f is Lipschitz on[−d,d]. So, there is a constantL_d>0 such that

|f(x)−f(y)| ≤L_d|x−y| (3.20) for allx,y ∈ [−d,d]. From (3.1),x → x^α is also Lipschitz on [−d,d]. Then there is a constant Lα such that

|x^α−y^α| ≤L_α|x−y| for all x,y ∈[−d,d]. (3.21) Let(y^(p))be a sequence in Bsuch that

y^(p)−x

→0 as p→_{∞. Since} Bis closed, x∈ B.

By (3.15) and (3.17), we get for alln ≥n₃

(Tx)_n−(Ty^(p))_n≤ |p_n|x_n−k−y⁽_n^p₋⁾_k

+

∑

1 r_j

1 γ

∑

a_if(x_i+1) +q_i(x_i)^α

!¹

γ

−

∑

i=j

a_if(y⁽_i+^p)₁) +q_i

y⁽_i^p)α

!¹

γ

. From (3.19), we have for all n≥n₃

(Tx)_n−(Ty^(p))_n

≤ |pn|x_n−k−y⁽_n^p₋⁾_k +

∑

1 r_j

1 γ

L_γ

∑

a_if(x_i+1) +

∑

q_i(x_i)^α−

∑

i=j

a_if(y⁽_i+^p)₁)−

∑

i=j

q_i y⁽_i^p)α

≤ |p_n|x_n−k−y⁽_n^p₋⁾_k +Lγ

∑

1 r_j

1 γ ∞

∑

|a_i|f(x_i+1)− f(y⁽_i+^p)₁)

+L_γ

∑

1 r_j

1

γ ∞

∑

|q_i|(x_i)^α−y⁽_i^p)α . Hence, by (3.20) and (3.21), we obtain for alln ≥n3

(Tx)_n−(Ty^(p))_n

≤ |p_n|x_n−k−y⁽_n^p₋⁾_k

+LγL_d

∑

1 r_j

1

γ ∞

∑

|a_i|x_i+1−y⁽_i+^p)₁

+LγLα

∑

1 r_j

1

γ ∞

∑

|q_i|x_i−y⁽_i^p)

≤ sup

i∈_N0

y⁽_i^p)−x_i

|p_n|+L_γL_d

∑

1 r_j

1 γ ∞

∑

|a_i|+L_γL_α

∑

1 r_j

1

γ ∞

∑

|q_i|

! .

Moreover,

∀0≤ n<n₃

(Tx)_n−(Ty^(p))_n ≤ y^(p)−x Thus, by (3.4) and (3.5), we have

plim→_∞

Ty^(p)−Tx

=_{0 as lim}

p→_∞

y^(p)−x =_0.

This means that Tis continuous.

Step 3. Comparison of the measure of noncompactness

Now, we need to compare a measure of noncompactness of any subsetX of B andT(X). Let us fix any nonempty set X ⊂ B. Take any sequences x,y ∈ X. Following the same calculations which led to the continuity of the operatorTwe see that

∀n ≥n₃ |(Tx)_n−(Ty)_n| ≤ |p_n| |x_n−k−y_n−k|+L_γL_αβ_n|x_n−y_n|+L_γL_dα_n|x_n+1−y_n+1|. Taking sufficiently large n, by (3.10) and (3.11), we get

LγL_dαn ≤c₁< ¹−P

4 and LγLαβn≤c₂ < ¹−P 4 Herec₁, c₂ are some real constants. From (3.6), we have

P+c₁+c₂ < ¹+P 2 . We see that existsn₅such that

∀n≥n₅ diam(T(X))_n≤ PdiamX_n−k+c₁diamX_n+c₂diamX_n+1. This yields by the properties of the upper limit that

lim sup

n→_∞

diam(T(X))_n≤ P lim sup

n→_∞

diamX_n−k+c₁ lim sup

n→_∞

diamX_n+c₂lim sup

n→_∞

diamX_n+1. From the above, for any X⊂ B, we haveµ(T(X))≤(c₁ +c₂+P)µ(X).

Step 4. Relation between fixed points and solutions

By Theorem2.2we conclude thatThas a fixed point in the setB. It means that there exists x∈ Bsuch that

xn= (Tx)_n. Thus

x_n=−p_nx_n−k−

∑

j=n

1 r_j

∑

(a_if(x_i+1) +q_ix^α_i)

!¹

γ

, forn∈_Nn3 (3.22) To show that there exists a correspondence between fixed points ofTand solutions to (1.1) we apply operator∆to both sides of the following equation

xn+pnx_n−k =−

∑

1 r_j

∑

(a_if(x_i+1) +q_ix^α_i)

!¹_γ , which is obtained from (3.22). We find that

∆(x_n+p_nx_n−k) = ¹ r_n

∑

(a_if(x_i+1) +q_ix^α_i)

!¹

γ

, n∈_Nn3.

and next

(_∆(x_n+p_nx_n−k))^γ = ¹ r_n

∑

(a_if(x_i+1) +q_ix^α_i), n∈_Nn3. Taking operator∆again to both sides of the above equation we obtain

∆ rn(_∆(xn+_pnxn−k))^γ=−anf(_xn+1)−qnx^α_n, n∈_Nn3.

So, we get equation (1.1) forn ∈ _Nn3. Sequencex, which is a fixed point of mapping T, is a bounded sequence which fulfills equation (1.1) forn ≥ n3. If n3 ≥ k the proof is ended. We find previousn₃−k+1 terms of sequence xby formula

x_n−k+l = ¹ p_n+l



−x_n+l+

∑

1 r_j

∑

(a_if(x_i+1) +q_ix^α_i)

!¹_γ

,

where l ∈ {0, 1, 2, . . . ,k−1}, which results leads directly from (1.1). It means that equa- tion (1.1) has at least one bounded solutionx: Nk →_R.

This completes the proof.

Now, we give an example of equation which can be considered by our method.

Example 3.2. Take k = 3, an arbitrary C¹ function f: R→_R and consider the following problem

∆ (−1)ⁿ_∆

xn+¹ 2xn−3

1/3! + ¹

2ⁿ (xn)⁵+ f(x_n+1) =0. (3.23) Takingγ= ¹₃,α=5,r_n= (−1)ⁿ,p_n = ¹₂,a_n=q_n= ₂¹_n with f(x) =x⁵³ we see thatx_n= (−1)ⁿ is a bounded solution to (3.23).

Remark 3.3. We note that the previous terms of the solution sequence are not obtained through a fixed point method, but through backward iteration. It is common that one has a 1–1 correspondence between fixed points to a suitably chosen operator and solutions to the problem under consideration. Here we get as a fixed point solution some sequence which, starting from some index, is a solution to the given problem and in which the first terms must be iterated. This procedure must be applied since we see that in equation (1.1) we have to know also earlier terms in order to start iteration; this is the so called iteration with memory.

We recall that in recent works concerning application of the measure of noncompactness to discrete equations, only problems without memory have been considered. That is why we had to alter to established procedure to overcome the difficulty arising in this problem. We believe our method would be applicable for several other problems.

4 A special type stability

The type of stability investigated in this paper is contained in the following theorem.

Theorem 4.1. Assume that

q_n≡0, (4.1)

and conditions(3.2),(3.4) and(3.5)hold. Assume further that there exists a positive constant D such that

|f(u)− f(v)| ≤D|u−v|

for any u,v∈R. Then equation(1.1)has at least one solution x: Nk →_Rwith the following stability property: given any other solution y: Nk → _Rand ε > 0there exists n₄ > n3 such that for every n≥n₄the following inequality holds

|x_n−y_n| ≤ε.

Proof. From Theorem3.1, equation (1.1) has at least one bounded solution x: N0 →_Rwhich can be rewritten in the form

x_n= (Tx)_n,

where mapping Tis defined by (3.15) forn≥ n₃. From the above and condition (4.1), we see that

|x_n−y_n|=|(Tx)_n−(Ty)_n|

≤ |p_n| |x_n−k−y_n−k|+LγD

∑

1 r_j

1

γ ∞

∑

|a_i| |x_i+1−y_i+1|.

Note that for nlarge enough, sayn≥n₄ ≥n₃, we have

ϑ:=|p_n|+L_γD

∑

1 r_j

1

γ ∞

∑

|a_i|<1.

Let us denote

lim sup

n→_∞

|x_n−y_n|=l, and observe that

lim sup

n→_∞

|x_n−y_n|=lim sup

n→_∞

|x_n−k−y_n−k|=lim sup

n→_∞

|x_n+1−y_n+1|. Thus, from the above, we have

l≤ ϑ·l.

This means that lim sup_n→∞|xn−yn|=0. This completes the proof since forε>0 there exists n₄∈_N0 such that for everyn≥n₄ ≥n₃ the following inequality holds

|x_n−y_n| ≤ε.

Now, we give another example.

Example 4.2. Take k = 3, an arbitrary C¹ function f: R→_R and consider the following problem

∆ (−1)ⁿ_∆

x_n+¹ 2x_n−3

1/3! + ¹

2ⁿf(x_n+1) =0. (4.2) Taking γ = ¹₃, r_n = (−1)ⁿ, p_n = ¹₂, a_n = ₂¹_n, q_n = 0 with f(x) = −x+sin(^π₂x) we see that x_n = (−1)ⁿ is a bounded solution to (4.2). By Theorem 4.1, this solution has the stability property.

5 Comments

In [25], the authors consider a special type of problem (1.1), namely they investigate the existence of a solution and Lyapunov type stability to the following equation

∆(r_n∆xn) =a_nf(x_n+1). (5.1) Their main assumption is the linear growth assumption on nonlinear term f. More precisely, they assume that there exists a positive constant Msuch that|f(xn)| ≤ M |xn|for allx ∈ N0. Using ideas developed in this paper we get the following result.

Corollary 5.1. Assume that f: R → _R satisfies the condition (3.3) and the sequences r: N0 → R\ {0}, a: N0→_Rare such that

∑

1 r_j

∑

|a_i|<+_∞.

Then, there exists a bounded solution x: N0→_Rof equation(5.1).

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