Existence of solutions for a second order discrete boundary value problem with mixed periodic boundary conditions

Journal Pre-proof Existence of solutions for a second order discrete boundary value problem with mixed periodic boundary conditions

Lingju Kong, Min Wang

PII: DOI: Reference:

S0893-9659(19)30462-8 https://doi.org/10.1016/j.aml.2019.106138 AML 106138

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Applied Mathematics Letters

Received date : 12 September 2019 Revised date : 28 October 2019 Accepted date : 28 October 2019 Please cite this article as: L. Kong and M. Wang, Existence of solutions for a second order discrete boundary value problem with mixed periodic boundary conditions, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106138. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.

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EXISTENCE OF SOLUTIONS FOR A SECOND ORDER DISCRETE BOUNDARY VALUE PROBLEM WITH MIXED PERIODIC BOUNDARY CONDITIONS LINGJU KONG AND MIN WANG1

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Abstract. In this paper, a second order discrete boundary value problem with a pair of mixed periodic boundary conditions is considered. Sufficient conditions on the existence and multiplicity of solutions are obtained by using variational methods. A particular Banach space and an associated functional are presented to overcome the asymmetry of the mixed periodic boundary conditions. Examples are also given to illustrate the applications of the main result.

1. Introduction

In this paper, we consider a boundary value problem (BVP) consisting of a second order difference equation − ∆2 u(t − 1) = f (u(t)),

t ∈ [1, N ]Z ,

(1.1)

and a pair of mixed periodic boundary conditions (BCs) u(0) = −u(N ),

∆u(0) = ∆u(N ),

(1.2)

ur

where f ∈ C(R), N ≥ 1 is an integer, [a, b]Z denotes the discrete interval {a, . . . , b} for any integers a and b with a ≤ b, ∆ is the forward difference operator defined by ∆u(t) = u(t + 1) − u(t) and ∆2 u(t) = ∆(∆u(t)). By a solution of BVP (1.1), (1.2), we mean a function u : [0, N + 1]Z → R that satisfies (1.1) and (1.2). BVPs have been a focus in research for decades. The existence, multiplicity, and uniqueness of solutions to BVPs for both differential equations and difference equations subject to both local and nonlocal BCs have been extensively studied, see for example [1–9, 11–13] and the references therein. Among the existing works, the problems involving periodic or anti-periodic BCs have been investigated by many authors. To name a few, Graef et al. [6] investigated a second order periodic BVP ( y 00 − ρ2 y + λg(t)f (y) = 0, 0 < t < 2π, y(0) = y(2π), y 0 (0) = y 0 (2π).

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Liang and Weng [11] considered a second order discrete periodic BVP ( ∆(p(t − 1)∆x(t − 1)) + q(t)x(t) + f (t, x(t)) = 0, t ∈ [1, N ]Z , x(0) = x(N ), p(0)∆x(0) = p(N )∆x(N ). Lyons and Neugebauer [12] considered a second order BVP with anti-periodic BCs ( x00 + f (x) = 0, t ∈ (0, T ), x(0) + x(T ) = 0, x0 (0) + x0 (T ) = 0.

(1.3)

2010 Mathematics Subject Classification. 39A10; 34B15. Key words and phrases. Discrete boundary value problem; mixed periodic boundary conditions; variational methods; mountain pass lemma. 1 Corresponding author. 1

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L. KONG AND M. WANG

In [13], they further studied a discrete version of (1.3) ( ∆2 u(t) + f (u(t + 1)) = 0, t ∈ [0, N ]Z , u(0) + u(N + 2) = 0, ∆u(0) + ∆u(N + 1) = 0. In [12], the authors commented that it may seem natural to consider the problems involving the mixed periodic BCs x(0) + x(T ) = 0, x0 (0) − x0 (T ) = 0 instead of the anti-periodic BCs. However, this modification on BCs will lead to BVPs at resoance that are difficult to handle. Motivated by this comment, we decide to consider the problem with mixed periodic BCs, i.e., BVP (1.1), (1.2). It is notable that the linear BVP consisting of the equation − ∆2 u(t − 1) = 0,

t ∈ [1, N ]Z ,

(1.4)

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and BCs (1.2) has an infinite number of solutions uτ defined by uτ (t) = tτ −

Nτ , 2

t ∈ [0, N + 1]Z , τ 6= 0.

Therefore, there is no Green’s function for BVP (1.4), (1.2). Hence the fixed point theory cannot be directly used to investigate the existence of solutions of BVP (1.1), (1.2). In this paper, we will apply the variational method to study the existence and multiplicity of solutions to BVP (1.1), (1.2). To the best of our knowledge, the variational methods have not been applied to study the discrete BVPs with mixed periodic BCs. Our work will fill this void. One obstacle in the construction of the functional is the functions at boundary points cannot be cancelled due to the asymmetry of BCs. We will select a particular Banach space and establish the functional by manipulating the terms in the functional to overcome this challenge. The Banach space and functional presented in this paper will benefit other scholars interested in this area. This will be a major contribution of our work. This paper is organized as follows: after this introduction, the Banach space, the functional, and the needed lemmas are given in Section 2. Section 3 contains the main results. Two examples are given in Section 3 as well. 2. Preliminary

We first introduce a few definition and lemmas needed in the proofs.

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Definition 2.1. Assume X is a real Banach space. We say that a functional J ∈ C 1 (X; R) satisfies the Palais-Smale (PS) condition if every sequence {un } ⊂ X, such that J(un ) is bounded and J 0 (un ) → 0 as n → ∞, has a convergent subsequence. The sequence {un } is called a PS sequence.

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Lemma 2.1. ( [14]) Let X be a real reflexive Banach space, and let J be a weakly upper semicontinuous functional such that limkuk→∞ J(u) = −∞. Then, there exists u0 ∈ X such that J(u0 ) = supu∈X J(u). Furthermore, if J ∈ C 1 (X, R), then J 0 (u0 ) = 0. We now recall the mountain pass lemma of Ambrosetti and Rabinowitz (see, for example, [10, Theorem 7.1]). Below, we denote by Bρ (u) the open ball centered at u ∈ X with radius ρ > 0, ¯ρ (u) its closure, and ∂Bρ (u) its boundary. B Lemma 2.2. Let (X, k · k) be a real Banach space and J ∈ C 1 (X, R). Assume that J satisfies the PS condition and there exist e0 , e1 ∈ X and ρ > 0 such that (A1) e1 6∈ Bρ (e0 ); (A2) max{J(e0 ), J(e1 )} < inf u∈∂Bρ (e0 ) J(u).

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BVP WITH MIXED PERIODIC BCS

3

Then, J possesses a critical value which can be characterized as d = inf max J(γ(s)) ≥ γ∈Γ s∈[0,1]

inf

u∈∂Bρ (e0 )

J(u),

where Γ = {γ ∈ C([0, 1], X) : γ(0) = e0 , γ(1) = e1 }.

(2.1)

In the sequel, we let X be defined by X = {u : [0, N + 1]Z → R | u(0) = −u(N ), ∆u(0) = ∆u(N ), u(1) = 0} and for any u ∈ X, define kuk =

P

1

N 2 t=1 u (t)

2

(2.2)

.

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Remark 2.1. (a) It is clear that any u ∈ X satisfies BC (1.2). By (1.2), for any u ∈ X, we have u(N + 1) = 2u(N ) = −2u(0). Therefore, it is easy to see that X is a (N − 1)-dimensional reflexive Banach space. (b) The condition u(1) = 0 will be needed in Lemma 2.3 below to overcome the asymmetry of BC (1.2). Let f˜ : [1, N ]Z × R → R and F˜ : [1, N ]Z × R → R be defined by ( f (x), f˜(t, x) = f (x) + 2x,

F˜ (t, x) =

Z

x

t 6= N, t = N,

(2.3)

f˜(t, s)ds.

(2.4)

0

It is clear that f˜(t, x) and F˜ (t, x) are continuous in x. Define J : X → R by J(u) =

N

N

t=1

t=1

X 1X (∆u(t − 1))2 − F˜ (t, u(t)). 2

(2.5)

Lemma 2.3. u ∈ X is a critical point of J if and only if u is a solution of BVP (1.1), (1.2).

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Proof. By (2.5), (2.4), (2.3), and (2.2), for any u ∈ X, N

N

Jo

X 1X J(u) = (∆u(t − 1))2 − 2 1 = 2

t=1 N X t=1

2

(∆u(t − 1)) −

Z

u(t)

t=1 0 N Z u(t) X t=1

0

f (s)ds − 2 f (s)ds − 2

Z

u(N )

sds

0

Z

u(0)

sds.

0

Then J is continuously differentiable and its derivative J 0 (u) at u ∈ X is given by 0

hJ (u), vi =

N X t=1

∆u(t − 1)∆v(t − 1) −

N X t=1

f (u(t))v(t) − 2u(0)v(0)

for any v ∈ X.

(2.6)

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L. KONG AND M. WANG

Note that by summation by parts and (2.2), N X t=1

∆u(t − 1)∆v(t − 1) =∆u(N )v(N ) − ∆u(0)v(0) − = − 2v(0)∆u(0) −

N X t=1

=2u(0)v(0) −

Next, let us consider an equivalent form  [4],    # "    2 0    ,   0 2     A =  2 −1 0   −1 2 −1      0 −1 2        ...      1 0 0

N X t=1

∆2 u(t − 1)v(t)

∆2 u(t − 1)v(t).

PN

t=1 ∆

2 u(t − 1)v(t) −

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Then by (2.6) and (2.7), we have hJ 0 (u), vi = − completes the proof of the lemma.

t=1

t=1

∆2 u(t − 1)v(t)

∆2 u(t − 1)v(t)

= − 2u(1)v(0) + 2u(0)v(0) − N X

N X

(2.7) PN

t=1 f (u(t))v(t).

This 

of J. Let u = (u(1), u(2), . . . , u(N ))0 and

... ... ... ... ...

 1  0  0    −1 2 0 0 0

N = 1, N = 2, (2.8) ,

N ≥ 3.

N ×N

Then it can be verified by direct computation that for any u ∈ X, N

X 1 F˜ (t, u(t)). J(u) = u0 Au − 2

(2.9)

t=1

Jo

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Remark 2.2. (a) It is clear that A is positive definite when N = 1, 2. When N ≥ 3, A = A˜0 A˜ with   1 0 0 ... 0 1 −1 1 0 . . . 0 0    . A˜ =   0 −1 1 . . . 0 0   ... ... 0 0 0 . . . −1 1 N ×N Therefore, for any u 6= (0, 0, . . . , 0)0 ,

˜ 0 (Au) ˜ = (u(1) + u(N ))2 + u0 Au =(Au)

N X t=2

(u(t) − u(t − 1))2 > 0.

Hence A is positive definite as well. (b) Let 0 < λ1 ≤ λ2 ≤ . . . λN be the eigenvalues of A. Then we have λ1 kuk2 ≤ u0 Au ≤ λN kuk2 ,

u ∈ RN .

(2.10)

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BVP WITH MIXED PERIODIC BCS

5

3. Main results In this section, we will apply the lemmas presented in Section 2 to study the existence and multiplicity of solutions to BVP (1.1), (1.2). Define   f (x) f (x) , +2 , (3.1) f˜0 = lim sup max x x |x|→0   f (x) f (x) ˜ f∞ = lim inf min , +2 . (3.2) x x |x|→∞

Then we obtain the following result.

Theorem 3.1. Let λ1 and λN be the eigenvalues of A defined in Remark 2.2. Assume f˜∞ > λN . Then BVP (1.1), (1.2) has at least one nontrivial solution. If in addition, f˜0 < λ1 , then BVP (1.1), (1.2) has at least two nontrivial solutions.

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Proof. Let f˜(t, x) and F˜ (t, x) be defined by (2.3) and (2.4). Since f˜∞ > λN , there exists ε1 ∈ (0, f˜∞ − λN ) and C > 0 such that for any t ∈ [1, N ]Z and x ∈ (−∞, −C) ∪ (C, ∞), f˜(t, x) ≥ λN + ε1 . x Therefore, there exists c ∈ R such that for any (t, |x|) ∈ [1, N ]Z × (C, ∞), λ N + ε1 2 F˜ (t, x) ≥ x + c. (3.3) 2 Then by (2.9), (2.10), and (3.3), λ N + ε1 ε1 J(u) ≤ λN kuk2 − kuk2 − cN = − kuk2 − cN. (3.4) 2 2 Hence J(u) → −∞ as kuk → ∞. By Lemma 2.1, there exists u0 ∈ X such that J(u0 ) = supu∈X J(u) and J 0 (u0 ) = 0. By Lemma 2.3, u0 is a solution of BVP (1.1), (1.2). When f˜0 < λ1 , there exists ε2 > 0 and D > 0 such that for any t ∈ [1, N ]Z and x ∈ [−D, D], f˜(t, x) ≤ λ1 − ε2 . x Therefore, for any (t, x) ∈ [1, N ]Z × [−D, D], λ 1 − ε2 2 F˜ (t, x) ≤ x . (3.5) 2 Let e0 (t) ≡ 0 on [0, N + 1]Z . It is easy to see that J(e0 ) = 0. For any u ∈ ∂BD (e0 ), by (2.9), (2.10), and (3.5), λ 1 − ε2 ε2 J(u) ≥ λ1 kuk2 − kuk2 = D2 > 0. (3.6) 2 2 ¯D (e0 ) such that J(e1 ) < 0. Hence (A1) and (A2) of Lemma By (3.4), there exists e1 6∈ B 2.2 are satisfied. In view of (3.4), any PS sequence is bounded and so has a convergent subsequence. Then, the PS condition holds as well. By Lemma 2.2, J has a critical value d = inf γ∈Γ maxs∈[0,1] J(γ(s)) ≥ ε22 D2 > 0 with Γ defined by (2.1). Let u1 be a critical point of J associated with d. If u0 6= u1 , then by Lemma 2.3, BVP (1.1), (1.2) has two nontrivial solutions u0 and u1 . If u0 = u1 , then supu∈X J(u) = inf γ∈Γ sups∈[0,1] J(γ(s)). Choose distinct γ1 , γ2 ∈ Γ such that {γ1 (s) : s ∈ (0, 1)} ∩ {γ2 (s) : s ∈ (0, 1)} = ∅. Then, maxs∈[0,1] J(γ1 (s)) = maxs∈[0,1] J(γ2 (s)) = supu∈X J(u). Thus, there exist s1 , s2 ∈ (0, 1) such that γ1 (s1 ) and γ2 (s2 ) are two different critical points of J in X that are solutions of BVP (1.1), (1.2). This completes the proof of the theorem.  The following corollary is a direct consequence of Theorem 3.1.

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Corollary 3.1. Assume that f˜0 = 0 and f˜∞ = ∞. Then BVP (1.1), (1.2) has at least two nontrival solutions. We conclude this paper with two examples to demonstrate the application of our results. Example 1. Consider BVP (1.1), (1.2) with N = 100 and 2

f (x) = e3x .

(3.7)

We claim BVP (1.1), (1.2) has at least one nontrivial solution. In fact, by (3.7) and (3.2), we have f˜∞ = ∞ > λ100 . Then by Theorem 3.1, BVP (1.1), (1.2) has at least one nontrivial solution. Example 2. Consider BVP (1.1), (1.2) with N = 10 and −6x + 4x5 . (3.8) 3 + x4 We claim BVP (1.1), (1.2) has at least two nontrivial solutions. In fact, let A be defined by (2.8). Then the eigenvalues λ1 and λ10 of A may be estimated by MATLAB as λ1 ' 0.0979 and λ10 ' 3.9021. By (3.8), (3.1), and (3.2), we have f˜0 = 0 < λ1 and f˜∞ = 4 > λ10 . By Theorem 3.1, BVP (1.1), (1.2) has at least two nontrivial solutions.

na lP repr oo f

f (x) =

References

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[1] J. M. Davis, P. W. Eloe, J. R. Graef, and J. Henderson, Positive solutions for a singular fourth order nonlocal boundary value problem, International Journal of Pure and Applied Mathematics 109 (2016), 67 – 84. [2] Y. Feng, J. R. Graef, L. Kong, and M. Wang, The forward and inverse problems for a fractional boundary value problem, Applicable Analysis 97 (2018), 2474 – 2484. [3] A. E. Garcia and J. T. Neugebauer, Solutions of boundary value problems at resonance with periodic and antiperiodic boundary conditions, Involve 12 (2019), 171 – 180. [4] J. R. Graef, S. Heidarkhani, L. Kong and M. Wang, Existence of solutions to a discrete fourth order boundary value problem, Journal of Difference Equations and Applications 24 (2018), 849 – 858. [5] J. R. Graef, L. Kong, Q. Kong, and M. Wang, On a fractional boundary value problem with a perturbation term, Journal of Applied Analysis and Computation 7 (2017), 57 – 66. [6] J. R. Graef, L. Kong, and H. Wang, Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem, Journal of Differential Equations 245 (2008), 1185 – 1197. [7] J. R. Graef, L. Kong, and M. Wang, Multiple solutions to a periodic boundary value problem for a nonlinear discrete fourth order equation, Advances in Dynamical Systems and Applications 8 (2013), 203 – 215. [8] J. Henderson and R. Luca, Positive solutions for a system of coupled fractional boundary value problems, Lithuanian Mathematical Journal 58 (2018), 15 – 32. [9] J. Henderson and R. Luca, Existence of positive solutions for a system of semipositone coupled discrete boundary value problems, Journal of Difference Equations and Applications 25 (2019), 516 – 541. [10] Y. Jabri, The Mountain Pass Theorem, Variants, Generalizations and some Applications, Encyclopedia of Mathematics and its Applications. Vol. 95, Cambridge University Press, Cambridge, New York, 2003. [11] H. Liang and P. Weng, Existence and multiple solutions for a second-order difference boundary value problem via critical point theory, Journal of Mathematical Analysis and Applications 326 (2007) 511 – 520. [12] J. W. Lyons and J. T. Neugebauer, Existence of an antisymmetric solution of a boundary value problem with antiperiodic boundary conditions, Electronic Journal of Qualitative Theory of Differential Equations 2015, No. 72, 1 – 11. [13] J. W. Lyons and J. T. Neugebauer, A difference equation with anti-periodic boundary conditions, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 22 (2015) 47 – 60. [14] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, Vol. 74, Springer, NewYork, 1989. Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA. Email address: [email protected] Department of Mathematics, Kennesaw State University, Marietta, GA 30060, USA. Email address: [email protected]