SINGULAR DISCRETE INITIAL AND BOUNDARY VALUE PROBLEMS VIA INEQUALITY THEORY

(1)

SINGULAR DISCRETE INITIAL AND BOUNDARY VALUE PROBLEMS VIA INEQUALITY THEORY

HAISHEN L ¨U AND DONAL O’REGAN Received 25 May 2004

We present new existence results for singular discrete initial and boundary value problems. In particular our nonlinearity may be singular in its dependent variable and is al- lowed to change sign.

1. Introduction

An upper- and lower-solution theory is presented for the singular discrete boundary value problem

−∆ϕp

∆u(k−1)=q(k)fk,u(k), k∈N= {1,...,T},

u(0)=u(T+ 1)=0, (1.1)

and the singular discrete initial value problem

∆u(k−1)=q(k)fk,u(k), k∈N= {1,...,T},

u(0)=0, (1.2)

whereϕp(s)=|s|^p⁻²s, p >1,∆u(k−1)=u(k)−u(k−1),T∈{1, 2,...},N⁺={0, 1,...,T}, andu:N⁺→R.Throughout this paper, we will assumef :N×(0,∞)→Ris continuous.

As a result, our nonlinearity f(k,u) may be singular atu=0 and may change sign.

Remark 1.1. Recall a map f :N×(0,∞)→Ris continuous if it is continuous as a map of the topological spaceN×(0,∞) into the topological spaceR. Throughout this paper, the topolopy onNwill be the discrete topology.

We will letC(N⁺,R) denote the class of mapucontinuous onN⁺(discrete topology) with normu =max_k_∈_N⁺u(k). By a solution to (1.1) (resp., (1.2)) we mean au∈ C(N⁺,R) such thatusatisfies (1.1) (resp., (1.2)) fori∈N andusatisfies the boundary (resp., initial) condition.

It is interesting to note here that the existence of solutions to singular initial and boundary value problems in the continuous case have been studied in great detail in

(2)

the literature (see [2,4,5,6,7,9,10,11] and the references therein). However, only a few papers have discussed the discrete singular case (see [1,3,8] and the references therein).

In [7], the following result has been proved.

Theorem1.2. Letn0∈ {1, 2,...}be fixed and suppose the following conditions are satisfied:

f :N×(0,∞)−→Ris continuous, (1.3)

q∈CN, (0,∞), (1.4)

there exists a functionα∈CN⁺,R with α(0)=α(T+ 1)=0, α >0onNsuch that q(k)fk,α(k)≥ −∆ϕp

α(k−1) fork∈N,

(1.5)

there exists a functionβ∈C(N⁺,R)with β(k)≥α(k), β(k)≥ 1

n0 fork∈N⁺with q(k)fk,β(k)≤ −∆ϕp

β(k−1) fork∈N.

(1.6)

Then (1.1) has a solutionu∈C(N⁺,R)withu(k)≥α(k)fork∈N⁺. In [1], the following result has been proved.

Theorem1.3. Letn0∈ {1, 2,...}be fixed and suppose the following conditions are satisfied:

f :N×(0,∞)−→Ris continuous, (1.7)

q∈CN, (0,∞), (1.8)

there exists a functionα∈CN⁺,R with α(0)=0, α >0onNsuch that q(k)fk,α(k)≥∆α(k−1) fork∈N,

(1.9)

there exists a functionβ∈CN⁺,R with β(k)≥α(k), β(k)> 1

n0 fork∈N⁺with q(k)fk,β(k)≤∆β(k−1) fork∈N.

(1.10)

Then (1.2) has a solutionu∈C(N⁺,R)withu(k)≥α(k)fork∈N⁺.

Also some results from the literature, which will be needed inSection 2are presented.

Lemma1.4 [8]. Letu∈C(N⁺,R)satisfyu(k)≥0fork∈N⁺. Ifu∈C(N⁺,R)satisfies

−∆²u(k−1)=u(k), k∈N= {1, 2,...,T},

u(0)=u(T+ 1)=0, (1.11)

(3)

then

u(k)≥µ(k)u fork∈N⁺; (1.12)

here

µ(k)=min

T+ 1−k T+ 1 ,k

T

. (1.13)

Lemma1.5 [8]. Let[a,b]= {a,a+ 1,...,b} ⊂N. Ifu∈C(N⁺,R)satisfies

∆ϕp

∆u(k−1)≤0, k∈[a,b],

u(a−1)≥0, u(b+ 1)≥0, (1.14)

thenu(k)≥0fork∈[a−1,b+ 1]= {a−1,a,...,b+ 1} ⊂N⁺.

In Theorems1.2and1.3the construction of a lower solutionαand an upper solution βis critical. We present an easily verifiable condition inSection 2.

2. Main results

We begin with a result for boundary value problems.

Theorem2.1. Let n0∈ {1, 2,...}be fixed and suppose (1.3), (1.4) hold. Also assume the following conditions are satisfied:

there exists a constantc0>0such that

q(k)f(k,u)≥c0 fork∈N, 0< u≤ 1

n0, (2.1)

there existh >0continuous and nondecreasing on[0,∞)such that f(k,u)≤h(u) for(k,u)∈N×

1 n0

,∞

, (2.2)

there existM > 1

n0 such that M− 1

n0> ϕ⁻_p¹h(M)b0;

(2.3)

here

b0=max

k∈N

_k

i=1

ϕ⁻_p¹ _k

j=iq(j)

,

k i=1

ϕ⁻_p¹ _k

j=iq(j)

. (2.4)

Then (1.1) has a solutionu∈C(N⁺,R)withu(k)>0fork∈N.

(4)

Proof. First we construct the lower solutionαin (1.5). Letα(k)=cv(k), k∈N⁺, where v∈C(N⁺, [0,∞)) is the solution of

−∆ϕp

∆v(k−1)=1, k∈N,

v(0)=v(T+ 1)=0, (2.5)

0< c <min

c^1/(p0 ⁻¹⁾, 1 n0v

. (2.6)

Since−∆(ϕp(∆v(k−1)))>0 implies∆²v(k−1)<0 fork∈N, it follows fromLemma 1.4 thatv(k)≥µ(k)vfork∈N⁺. Thus,

0< α(k)≤ 1 n0

fork∈N, (2.7)

−∆ϕp

∆α(k−1)=c^p⁻¹≤c0 fork∈N,

α(0)=α(T+ 1)=0. (2.8)

As a result (1.5) holds, since

q(k)fk,α(k)≥c0≥ −∆ϕp

∆α(k−1) fork∈N. (2.9) Next we discuss the boundary value problem

−∆ϕp

∆u(k−1)=q(k)h(M), k∈N, u(0)=u(T+ 1)= 1

n0. (2.10)

It follows from [8] that (2.10) has a solutionu∈C(N⁺,R). Letv(k)=u(k)−1/n0 for k∈N⁺. Then∆(ϕp(∆u(k−1)))= −∆(ϕp(∆v(k−1)))≤0 fork∈N, andv(0)=v(T+ 1)=0.Lemma 1.5guarantees thatv(k)≥0 and sou(k)≥1/n0fork∈N⁺. Next we prove u(k)≤Mfork∈N⁺. Now since∆(ϕp(∆u(k−1)))≤0 onNimplies∆²u(k−1)≤0 on N, then there existsk0∈Nwith∆u(k)≥0 on [0,k0)= {0, 1,...,k0−1}and∆u(k)≤0 on [k0,T+ 1)= {k0,k0+ 1,...,T}, andu(k0)= u. Supposeu(k0)> M.

Also notice that fork∈N, we have

−∆ϕp

∆u(k−1)=q(k)h(M). (2.11)

We sum (2.11) fromj+ 1 (0≤j < k0) tok0to obtain ϕp

∆u(j)=ϕp

∆uk0

+h(M)

k0

k=j+1q(k). (2.12) Now since∆u(k0)≤0, we have

ϕp

∆u(j)≤h(M) ^k⁰

k=j+1

q(k) for 0≤j < k0, (2.13)

(5)

that is,

∆u(j)≤ϕ⁻_p¹h(M)ϕ⁻_p¹ _k₀

k=j+1

q(k)

for 0≤j < k0. (2.14) Then we sum the above from 0 tok0−1 to obtain

uk0

−u(0)≤ϕ⁻_p¹h(M)

k0−1 j=0

ϕ⁻_p¹ _k₀

k=j+1

q(k)

≤ϕ⁻_p¹h(M)

k0

j=1

ϕ⁻_p¹ _k₀

k=j

q(k)

.

(2.15)

Similarly, we sum (2.11) fromk0toj(k0≤j≤T+ 1) to obtain

−ϕp

∆u(j)= −ϕp

∆uk0−1+h(M)

j k=k0

q(k) for j≥k0. (2.16) Now since∆u(k0−1)≥0, we have

−∆u(j)=ϕ⁻_p¹h(M)ϕ⁻_p¹ _j

k=k0

q(k)

for j≥k0. (2.17) We sum the above fromk0toTto obtain

uk0

−u(T+ 1)≤ϕ⁻_p¹h(M)

T j=k0

ϕ⁻_p¹ _j

k=k0

q(k)

. (2.18)

Now (2.15) and (2.18) imply

M− 1

n0≤b0ϕ⁻p¹

h(M). (2.19)

This contradicts (2.3). Thus 1

n0 ≤u(k)≤M fork∈N⁺. (2.20) Letβ(k)≡u(k) fork∈N⁺. Now (2.7) and (2.20) guarantee

α(k)≤β(k) fork∈N⁺. (2.21)

Now (2.2) and (2.20) implyf(k,β(k))≤h(β(k))≤h(M) so β∈CN⁺,R

with β(k)≥α(k), β(k)≥ 1

n0 fork∈N⁺with q(k)fk,β(k)≤ −∆ϕp

β(k−1) fork∈N.

(2.22)

NowTheorem 1.2guarantees that (1.1) has a solutionu∈C(N⁺,R) withu(k)≥α(k)>0

fork∈N.

(6)

Example 2.2. Consider the boundary value problem

∆²u(k−1)= k

u(k)^α+u(k)^β−A, k∈N, u(0)=u(T+ 1)=0

(2.23) with p=2,α >0, 0≤β <1, andA >0. Then (2.23) has a solutionu∈C(N⁺,R) with u(k)>0 fork∈N.

To see this, we will applyTheorem 2.1with q(k)=1, f(k,u)= k

u^α+u^β−A. (2.24)

Letn0>(2A)^1/αandc0=A. Then fork∈Nand 0< u≤1/n0, q(k)f(k,u)= k

u^α+u^β−A≥ k

u^α⁻A≥ 1

u^α⁻A≥2A−A=A=c0, (2.25) so (2.1) is satisfied. Leth(u)=u^β+n^α0T+A. Then (2.2) is immediate. Also since 0≤β <1, we see that

there existM > 1

n0 such thatM− 1 n0 > b0

M^β+n^α0T+A; (2.26) here

b0=max

k∈N

_k

j=1

(k−j+ 1),

T j=k

(j−k+ 1)

. (2.27)

Thus (2.3) holds.Theorem 2.1guarantees that (2.23) has a solutionu∈C(N⁺,R) with u(k)>0 fork∈N.

Next we present a result for initial value problems.

Theorem2.3. Let n0∈ {1, 2,...}be fixed and suppose (1.2), (1.3) hold. Also assume the following conditions are satisfied:

there exists a constantc0>0such that

q(k)f(k,u)≥c0 fork∈N, 0< u≤ 1

n0, (2.28)

there existh >0continuous and nondecreasing on[0,∞)such that f(k,u)≤h(u) for(k,u)∈N×

1 n0

,∞

, (2.29)

there existM > 1

n0 such that M− 1

n0 > h(M)

T k=1

q(k). (2.30)

Then (1.2) has a solutionu∈C(N⁺,R)withu(k)>0fork∈N.

(7)

Proof. First we construct the lower solutionαin (1.9). Let

α(k)=







 c ^k

i=1

q(i) , k∈N,

0, k=0,

(2.31)

where

0< c < 1 n0_T

i=1q(i), cmax

k∈Nq(k)≤c0. (2.32)

Then (2.7) holds, andα(0)=0,∆α(k−1)=α(k)−α(k−1)=cq(k)≤c0fork∈Nwith (1.9) holding, since

q(k)fk,α(k)≥c0≥∆α(k−1) fork∈N. (2.33) Next we discuss the initial value problem

∆u(k−1)=q(k)f^∗k,u(k), k∈N, u(0)= 1

n0; (2.34)

here

f^∗(k,u)=









 fk, 1

n0

, u≤ 1 n0, f(k,u), 1

n0≤u≤M, f(k,M), u≥M.

(2.35)

Then (2.34) is equivalent to

u(k)=









 1 n0

+

k i=1

q(i)f^∗i,u(i), k∈N, 1

n0

, k=0.

(2.36)

From Brouwer’s fixed point theorem, we know that (2.34) has a solutionu∈C(N⁺,R).

We first show

u(k)≥ 1

n0 fork∈N⁺. (2.37)

Suppose (2.37) is not true. Then there exists aτ∈Nsuch that u(τ)< 1

n0, u(τ−1)≥ 1

n0 (2.38)

(8)

sinceu(0)=1/n0. Thus we have, from (2.28)

∆u(τ−1)=q(τ)f^∗τ,u(τ)=q(τ)fτ, 1 n0

>0, (2.39)

so

u(τ)− 1

n0 > u(τ−1)− 1

n0≥0, (2.40)

a contradiction. Thus (2.37) is satisfied. Next we show

u(k)≤M fork∈N⁺. (2.41)

Suppose (2.41) is false. Then sinceu(0)=1/n0, there existsτ∈Nsuch that

u(τ)> M, u(k)≤M fork∈ {0, 1,...,τ−1}. (2.42) Thus, we have

∆u(τ−1)=u(τ)−u(τ−1)≤q(τ)h(M),

∆u(τ−2)=u(τ−1)−u(τ−2)≤q(τ−1)h(M), ...

∆u(0)=u(1)−u(0)≤q(1)h(M).

(2.43)

Adding both sides of the above formula gives u(τ)−u(0)≤h(M) ^τ

k=1

q(k)≤h(M)

T k=1

q(k), (2.44)

that is,

M− 1

n0 ≤h(M) ^T

k=1

q(k). (2.45)

This contradicts (2.30). Thus, we have (2.20). Letβ(k)≡u(k) fork∈N⁺. By (2.7) and (2.37), we haveα(k)≤β(k) fork∈N⁺. Then

β∈CN⁺,R with β(k)≥α(k), β(k)> 1

n0

fork∈N⁺with q(k)fk,β(k)=∆β(k−1) fork∈N.

(2.46)

(9)

NowTheorem 1.3guarantees that (1.2) has a solutionu∈C(N⁺,R) withu(k)≥α(k)>0

fork∈N.

Example 2.4. Consider the initial value problem

∆u(k−1)=ku(k)⁻^α+u(k)^β−A, k∈N,

u(0)=0 (2.47)

withα >0, 0≤β <1, andA >0. Now (2.47) has a solutionu∈C(N⁺,R) withu(k)>0 fork∈N.

To see this we will applyTheorem 2.3with (2.24). Letn0>(2A)¹^/αandc0=A.Then for k∈Nand 0< u≤1/n0, (2.25) holds and so (2.28) is satisfied. Leth(u)=u^β+n^α0T+A.

Then (2.29) is immediate. Also since 0≤β <1, we see there existM > 1

n0

such thatM− 1

n0> TM^β+n^α0T+A, (2.48) so (2.30) holds. Theorem 2.3guarantees that (2.47) has a solution u∈C(N⁺,R) with u(k)>0 fork∈N.

Acknowledgment

The research is supported by National Natural Science Foundation (NNSF) of China Grant 10301033.

References

[1] R. P. Agarwal, D. Jiang, and D. O’Regan,A generalized upper and lower solution method for singular discrete initial value problems, Demonstratio Math.37(2004), no. 1, 115–122.

[2] R. P. Agarwal, H. L¨u, and D. O’Regan, Existence theorems for the one-dimensional singular p-Laplacian equation with sign changing nonlinearities, Appl. Math. Comput.143(2003), no. 1, 15–38.

[3] R. P. Agarwal and D. O’Regan,Nonpositone discrete boundary value problems, Nonlinear Anal.

39(2000), no. 2, 207–215.

[4] R. P. Agarwal, D. O’Regan, and V. Lakshmikantham,Existence criteria for singular initial value problems with sign changing nonlinearities, Math. Probl. Eng.7(2001), no. 6, 503–524.

[5] R. P. Agarwal, D. O’Regan, V. Lakshmikantham, and S. Leela,Existence of positive solutions for singular initial and boundary value problems via the classical upper and lower solution approach, Nonlinear Anal.50(2002), no. 2, 215–222.

[6] R. P. Agarwal, D. O’Regan, and P. J. Y. Wong,Positive Solutions of Diﬀerential, Diﬀerence and Integral Equations, Kluwer Academic Publishers, Dordrecht, 1999.

[7] P. Habets and F. Zanolin,Upper and lower solutions for a generalized Emden-Fowler equation, J.

Math. Anal. Appl.181(1994), no. 3, 684–700.

[8] D. Jiang, D. O’Regan, and R. P. Agarwal,A generalized upper and lower solution method for singular discrete boundary value problems for the one-dimensionalp-Laplacian, to appear in J. Appl. Anal.

[9] H. L¨u and C. Zhong, A note on singular nonlinear boundary value problems for the one- dimensionalp-Laplacian, Appl. Math. Lett.14(2001), no. 2, 189–194.

(10)

[10] R. Man´asevich and F. Zanolin, Time-mappings and multiplicity of solutions for the one- dimensionalp-Laplacian, Nonlinear Anal.21(1993), no. 4, 269–291.

[11] M. N. Nkashama,A generalized upper and lower solutions method and multiplicity results for nonlinear first-order ordinary diﬀerential equations, J. Math. Anal. Appl.140(1989), no. 2, 381–395.

Haishen L¨u: Department of Applied Mathematics, Hohai University, Nanjing 210098, China E-mail address:[email protected]

Donal O’Regan: Department of Mathematics, National University of Ireland, Galway, Ireland E-mail address:[email protected]

(11)

Special Issue on

Time-Dependent Billiards

Call for Papers

This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.

This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under diﬀerent approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many diﬀerent fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).

We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.

To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.

Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System athttp://

mts.hindawi.com/according to the following timetable:

Manuscript Due March 1, 2009 First Round of Reviews June 1, 2009 Publication Date September 1, 2009

Guest Editors

Edson Denis Leonel,Department of Statistics, Applied Mathematics and Computing, Institute of Geosciences and Exact Sciences, State University of São Paulo at Rio Claro, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil; [email protected]

Alexander Loskutov,Physics Faculty, Moscow State University, Vorob’evy Gory, Moscow 119992, Russia;

[email protected]

Hindawi Publishing Corporation http://www.hindawi.com

10.1155/ADE.2005.205

http://www.hindawi.com/journals/mpe/.

http://mts.hindawi.com/