Strictly Increasing Solutions of Nonautonomous Difference Equations Arising in Hydrodynamics

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doi:10.1155/2010/714891

Research Article

Strictly Increasing Solutions of Nonautonomous

Difference Equations Arising in Hydrodynamics

Luk ´a ˇs Rach ˚unek and Irena Rach ˚unkov ´a

Department of Mathematics, Faculty of Science, Palack´y University, tˇr. 17. listopadu 12, 77146 Olomouc, Czech Republic

Correspondence should be addressed to Irena Rach ˚unkov´a,[email protected]

Received 19 December 2009; Revised 14 February 2010; Accepted 10 March 2010

Academic Editor: A ˘gacik Zafer

Copyrightq2010 L. Rach ˚unek and I. Rach ˚unkov´a. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The paper provides conditions sufficient for the existence of strictly increasing solutions of the second-order nonautonomous difference equationxn1 xn n/n12_x_n₋_x_n₋ 1 h2_f_x_n_,_n_∈_N_{, where}_{h >} _{0 is a parameter and}_f_{is Lipschitz continuous and has three} real zerosL0 <0< L. In particular we prove that for each sufficiently smallh > 0 there exists a solution{xn}∞_n0such that{xn}∞n1is increasing,x0 x1 ∈L0,0,and limn→ ∞xn> L.

The problem is motivated by some models arising in hydrodynamics.

1. Formulation of Problem

We will investigate the following second-order non-autonomous diﬀerence equation

xn1 xn

_n

n1

2

xn−xn−1 h2fxn, n∈N, 1.1

wherefis supposed to fulfil

L0<0< L, f ∈Lip_locL0,∞, fL0 f0 fL 0, 1.2

xfx<0 forx∈L0, L\ {0}, fx≥0 forx∈L,∞, 1.3

∃B∈L0,0such that

_L

Bfzdz0.

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Let us note thatf ∈ Lip_locL0,∞means that for eachL0, A ⊂ L0,∞there existsKA >0 such that|fx−fy| ≤ K_A|x−y|for allx, y ∈L0, A . A simple example of a functionf

satisfying1.2–1.4isfx cx−L0xx−L, wherecis a positive constant.

A sequence{xn}∞_n₀which satisfies1.1is called a solution of1.1. For each values

B, B1∈L0,∞there exists a unique solution{xn}∞n0of1.1satisfying the initial conditions

x0 B, x1 B1. 1.5

Then{xn}∞_n₀is called a solution of problem1.1,1.5.

In 1 we have shown that 1.1 is a discretization of diﬀerential equations which generalize some models arising in hydrodynamics or in the nonlinear field theory; see2–

6 . Increasing solutions of 1.1,1.5withB B1 ∈L0,0has a fundamental role in these

models. Therefore, in1 , we have described the set of all solutions of problem1.1,1.6, where

x0 B, x1 B, B∈L0,0. 1.6

In this paper, using1 , we will prove that for each suﬃciently smallh >0 there exists at least oneB∈L0,0such that the corresponding solution of problem1.1,1.6fulfils

x0 x1, lim

n→ ∞xn> L, {xn} ∞

n1 is increasing. 1.7

Note that an autonomous case of 1.1 was studied in 7 . We would like to point out that recently there has been a huge interest in studying the existence of monotonous and nontrivial solutions of nonlinear diﬀerence equations. For papers during last three years see, for example,8–22 . A lot of other interesting references can be found therein.

2. Four Types of Solutions

Here we present some results of1 which we need in next sections. In particular, we will use the following definitions and lemmas.

Definition 2.1. Let{xn}∞_n₀be a solution of problem1.1,1.6such that

{xn}∞_n1 is increasing, _nlim_{→ ∞}xn 0. 2.1

Then{xn}∞_n₀is called a damped solution.

Definition 2.2. Let{xn}∞_n₀be a solution of problem1.1,1.6which fulfils

{xn}∞

n1 is increasing, _nlim_{→ ∞}xn L. 2.2

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Definition 2.3. Let {xn}∞_n₀ be a solution of problem 1.1,1.6. Assume that there exists

b∈N, such that{xn}b_n1₁is increasing and

xb≤L < xb1. 2.3

Then{xn}∞_n₀is called an escape solution.

Definition 2.4. Let {xn}∞_n₀ be a solution of problem 1.1,1.6. Assume that there exists

b∈N,b >1, such that{xn}b_n₁is increasing and

0< xb< L, xb1≤xb. 2.4

Then{xn}∞_n₀is called a non-monotonous solution.

Lemma 2.5see1 on four types of solutions. Let{xn}∞_n₀be a solution of problem1.1, 1.6. Then{xn}∞_n₀is just one of the following four types:

I{xn}∞_n₀is an escape solution; II{xn}∞_n₀is a homoclinic solution; III{xn}∞_n₀is a damped solution;

IV{xn}∞_n₀is a non-monotonous solution.

Lemma 2.6see1 estimates of solutions. Let{xn}∞_n₀be a solution of problem1.1,1.6. Then there exists a maximalb∈N∪ {∞}satisfying

xn∈B, L forn1, . . . , b, ifb∈N,

xn∈B, L forn∈N, ifb∞. 2.5

Further, ifb >1, then moreover

{xn}b_n1 is increasing, 2.6

Δxn< hL−2L0M0h2M0 2.7

forn1, . . . , b−1ifb∈N, and forn∈Nifb∞, where

M0max fx :x∈L0, L

. 2.8

In 1 we have proved that the set consisting of damped and non-monotonous solutions of problem 1.1, 1.6 is nonempty for each suﬃciently small h > 0. This is contained in the next lemma.

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InSection 4of this paper we prove that also the set of escape solutions of problem 1.1,1.6is nonempty for each suﬃciently smallh >0. Note that in our next paper23 we prove this assertion for the set of homoclinic solutions.

3. Properties of Solutions

Now, we provide other properties of solutions important in the investigation of escape solutions.

Lemma 3.1. Let {xn}∞_n₀ be an escape solution of problem 1.1, 1.6. Then {xn}∞_n₁ is increasing.

Proof. Due to1.1,{xn}∞_n₀fulfils

Δxn

_n

n1

2

Δxn−1 h2fxn, n∈N. 3.1

According to Definition 2.3 there existsb ∈ N, such that{xn}b_n1₁ is increasing and 2.3

holds. By 1.3 we get fxb 1 ≥ 0. Consequently, by 3.1 and 2.3, Δxb 1 ≥ b12/b22Δxb>0 andfxb2≥0. SimilarlyΔxbj≥bj2/b1j2Δxb

j−1and

Δxbj≥

_b

1

b1j

2

Δxb, j∈N. 3.2

This yields that{xn}∞_n₁is increasing.

Lemma 3.2. Assume thatfx 0forx > L. Choose an arbitrary >0. LetB1, B2∈L0,0and let

{xn}∞

n0and{yn}∞n0be a solution of problem1.1,1.6withBB1andBB2, respectively.

LetK_Lbe the Lipschitz constant forfonL0, L . Then

_x_n₋_y_n _{≤ |}_B₁₋_B₂_|_e2_KL

, 3.3

Δxn−_hΔyn ≤ |B1−B2|KLe2KL, 3.4

wheren∈N,n≤/h.

Proof. By3.1we have

j12Δxj−j2_Δ_x_j₋₁_h2_j2_f_x_j_{, j}_∈_N_. _3.5

Summing it forj1, . . . , k, we get by1.6

Δxk h2 1

k12 k

j1

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Summing it again fork1, . . . , n−1, we get

From this and by using summation by parts we easily obtain

_x_n₋_y_n _{≤ |}_B₁₋_B₂_|_h2n−1

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Moreover, if the sequence {γk}∞k1 is unbounded, then there exists ∈ N such that the solution

{xn}∞n0of problem1.1,1.6withBB∈L0, Bis an escape solution.

Proof. Chooseh0>0 such that

h0

L−2L0M0h2₀M0<|C|. 4.2

Fork∈Ndenote by{x_kn}∞_n₀a solution of problem1.1,1.6withBB_k. The existence of

bkis guaranteed byLemma 2.6. ByLemma 2.5,{xkn}∞n0is just one of the typesI–IV, and

ifh∈0, h0 , then the monotonicity of{xkn}bk_n0yields a uniqueγk ∈N,γk < bk, satisfying

4.1.

Forh∈0, h0, consider the sequence{γk}∞k1 and assume that it is unbounded. Then

we have

lim

k→ ∞γk∞ 4.3

otherwise we take a subsequence.Assume on the contrary that for anyk ∈N,{x_kn}∞_n₀ is not an escape solution. Choosek ∈N. If{xkn}∞n0is damped, then byDefinition 2.1, we

havebk∞and

xkbk: lim

k→ ∞xkn 0, Δxkbk:klim→ ∞Δxkn 0. 4.4

If{xkn}∞n0is homoclinic, then byDefinition 2.2, we havebk∞and

xkbk: lim

k→ ∞xkn L, Δxkbk:klim→ ∞Δxkn 0. 4.5

If{xkn}∞n0is non-monotonous, then byDefinition 2.4, we havebk<∞and

xkbk∈0, L, Δxkbk≤0. 4.6

To summarize if{xkn}∞n0is not an escape solution, then by4.4,4.5, and4.6, we have

xkbk∈0, L, Δxkbk≤0. 4.7

SinceΔx_k0 0, there existsγ_k∈Nsatisfying

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Lettingk → ∞, we obtain, by4.3, that limk→ ∞Δxkγ_k ∞, contrary to4.17. Therefore an escape solution{xn}∞n0of problem1.1,1.6withBB∈L0, Bmust exist.

Now, we are in a position to prove the next main result.

Theorem 4.2On the existence of escape solutions. There existsh∗ > 0such that for anyh ∈ 0, h∗ the initial value problem1.1,1.6has an escape solution for someB∈L0, B.

Proof. We have the following steps.

Step 1. Let us define

fx

⎧ ⎨ ⎩

fx forx≤L,

0 forx > L,

4.23

and consider an auxiliary equation

xn1 xn

_n

n1

2

xn−xn−1 h2fxn, n∈N. 4.24

Leth∗ > 0 be the constant ofLemma 4.1for problem4.24,1.6. Chooseh ∈ 0, h∗ ,C ∈ L0, Band letKLbe the Lipschitz constant forfonL0,∞. Consider a sequence{Bk}∞k1 ⊂

L0, Csuch that limk→ ∞BkL0. Then, for eachm∈Nthere existskm∈Nsuch that

|Bkm −L0|< e−m

2_KL

C−L0. 4.25

Letx00 x0n L0 forn ∈ N. Then the sequence{x0n}∞n0 is the unique solution of

problem4.24,1.6withB L0. Let{xkn}∞n0 be a solution of problem4.24,1.6with

BBk,k∈N, and let{γk}∞k1be the sequence corresponding to{xkn}∞n0byLemma 4.1. We

prove that{γk}∞k1is unbounded. According toLemma 3.2, for eachm∈N,

|xkmn−x0n| ≤ |Bkm−L0|em

2_KL

, n≤ m

h. 4.26

Consequently,4.25and4.26give

|xkmn−x0n| ≤C−L0, n≤ m_h, 4.27

and hence

xkmn≤C, n≤ m_h. 4.28

Therefore

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which yields that{γk}∞k1 is unbounded. By Lemma 4.1, the auxiliary initial value problem

4.24, 1.6 has an escape solution for some B B ∈ L0,B. Denote this solution by

{xn}∞n0.

Step 2. ByDefinition 2.3, there existsb∈Nsuch that

{xn}b1

n1 is increasing, xb≤L <xb1. 4.30

Now, consider the solution{xn}∞n0of our original problem1.1,1.6withB B. Due

to4.23,xn xn forn 0,1, . . . , b1. Using4.30and Definition 2.3, we get that

{xn}∞n0is an escape solution of problem1.1,1.6.

Acknowledgments

The paper was supported by the Council of Czech Government MSM 6198959214. The authors thank the referees for valuable comments.

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