1Introduction Existenceofperiodicsolutionsforaclassoffunctionalintegralequations

Electronic Journal of Qualitative Theory of Differential Equations 2012, No. 57, 1-11;http://www.math.u-szeged.hu/ejqtde/

Existence of periodic solutions for a class of functional integral equations

Wei Long^†, Xiong-Jun Zheng and Lu Li

College of Mathematics and Information Science, Jiangxi Normal University Nanchang, Jiangxi 330022, People’s Republic of China

Abstract

In this paper, we investigate the existence of periodic solution for a class of nonlin- ear functional integral equation. We prove a fixed point theorem in a Banach algebra.

As an application, an existence theorem about periodic solutions to the addressed functional integral equation is presented. In addition, an example is given to illustrate our result.

Keywords: functional integral equation; periodic solution.

2000 Mathematics Subject Classification: 45G10, 34K13.

1 Introduction

This paper has four main motivations. The first motivation is that recently, the study on the existence of solutions to various kinds of functional integral equations has became one of the most attractive topics in the theory of integral equations. Many authors have made a lot of interesting contributions on this topic. For example, we refer the readers to [1–7, 9–

15, 17, 18, 20] and references therein. The second motivation is that in recent years, some authors have focused on the resolution of the operator equationx=AxBx+Cxin Banach algebras, and obtained many valuable results (see, e.g., [2–5, 7, 9–13, 18] and references

∗The work was supported by the NSF grant of China (11101192), the Key Project of Chinese Ministry of Education (211090), the NSF grant of Jiangxi Province, and the Foundation of Jiangxi Provincial Education Department (GJJ12205).

†Corresponding author. E-mail address: [email protected]

therein). Moreover, in these papers, the authors applied successfully their abstract results to the study on the existence of solutions to functional integral equations. The third motivation is that the authors of [19] studied the existence of periodic solutions for the following Fredholm integral equation:

y(t) =h(t) + Z

R

k(t, s)f(s, y(s))ds, t∈R,

by using nonlinear alternative of Leray-Schauder type. The fourth motivation is that in [16], the authors investigated the existence of almost periodic type solutions to the following functional integral equation:

y(t) =e(t, y(α(t))) +g(t, y(β(t)))

h(t) + Z

R

k(t, s)f(s, y(γ(s)))ds

, t∈R. Motivated by all the above works, in this paper, we first establish a fixed point theorem in a Banach algebra, and then, with its help, we discuss the existence of periodic solution for the following general functional integral equation:

x(t) = Xn i=1

f_i(t, x(a_i(t)))· Z

R

k_i(t, s)g_i(s, x(b_i(s)))ds, t∈R, (1.1) where n is a fixed positive integer, and f_i, a_i, k_i, g_i and b_i (i = 1, . . . , n) satisfy some conditions recalled in Section 2.

Throughout the rest of this paper, we denote byRthe set of real numbers,R⁺ the set of nonnegative real numbers, by Nthe set of positive integers, byC(R⁺,R⁺) the set of all continuous and nondecreasing functions φ:R⁺ → R⁺ withφ(0) = 0, and by PT(R) the Banach algebra of all T-periodic continuous functions from Rto Rwith the usual norm

kxk= sup

t∈R|x(t)|= max

t∈[0,T]|x(t)|, x∈ PT(R) and the multiplication defined by

(x·y)(t) =x(t)·y(t), x, y∈ PT(R), t∈R.

Definition 1.1. LetXbe a Banach space. A mappingA:X→X is calledD-Lipschitzian if there exists a function φ∈C(R⁺,R⁺) such that

kAx−Ayk ≤φ(kx−yk)

for all x, y∈X. In addition, the function φis called a D-function of A.

2 Main results

Theorem 2.1. Let n be a positive integer, and C be a nonempty, closed, convex and bounded subset of a Banach algebra X. Assume that the operators A_i : X → X and B_i :C→X, i= 1,2, . . . , n, satisfy

(a) for each i∈ {1,2, . . . , n}, A_i is D-Lipschitzian with aD-function φ_i; (b) for each i∈ {1,2, . . . , n}, B_i is continuous and B_i(C) is precompact;

(c) for each y∈C, x=Pⁿ

i=1

A_ix·B_iy implies that x∈C;

Then, the operator equation x= Pⁿ

i=1

A_ix·B_ix has a solution provided that Xn

i=1

M_iφ_i(r)< r, ∀r >0, where M_i= sup

x∈CkB_ixk, i= 1,2, . . . , n.

Proof. For each y∈C, define an operator onX by Syx=

Xn i=1

A_ix·B_iy, x∈X.

Denote

ψ(r) :=

Xn i=1

M_iφ_i(r), r >0.

Then ψ is continuous and nondecreasing. Moreover, ψ(r) < r for all r > 0. For all x₁, x₂∈X, we have

kSyx₁− Syx₂k

=

Xn i=1

A_ix₁·B_iy− Xn

i=1

A_ix₂·B_iy

≤ Xn

i=1

kA_ix₁−A_ix₂k · kB_iyk

≤ Xn

i=1

M_iφ_i(kx₁−x₂k)

= ψ(kx₁−x₂k).

Then, by using the well-known results in [8], we know that Sy has a unique fixed pointx_y in X.

Now, define an operator S on C by

Sy=x_y, y∈C, where x_y is the unique fixed point of S^y inX. Then,

Sy=x_y =Syx_y = Xn i=1

A_ix_y·B_iy, y∈C.

By the assumption (c), we know that Sy = x_y ∈ C for all y ∈ C. In addition, for all y, z ∈C, we have

kSy− Szk

=

Xn i=1

A_ix_y·B_iy− Xn

i=1

A_ix_z·B_iz

≤ Xn i=1

kA_ix_y·B_iy−A_ix_z·B_iy+A_ix_z·B_iy−A_ix_z·B_izk

≤ Xn i=1

M_iφ_i(kx_y−x_zk) + Xn

i=1

kA_ix_zk · kB_iy−B_izk

= ψ(kSy− Szk) + Xn

i=1

kA_ix_zk · kB_iy−B_izk

≤ ψ(kSy− Szk) +M · Xn i=1

kB_iy−B_izk, (2.1)

where

kA_ix_zk ≤ kA_iek+kA_ix_z−A_iek

≤ kA_iek+φ_i(kx_z−ek)

≤ max

1≤i≤nkA_iek+φ_i kek+ sup

y∈Ckyk

!

≤ max

1≤i≤nkA_iek+ max

1≤i≤n

"

φ_i kek+ sup

y∈Ckyk

!#

:=M for a fixed element e∈C.

Next, let us show thatS(C) is precompact andS :C→C is continuous. Let{y_m}be a sequence inC. Noting that every B_i(C) is precompact, there exists a subsequence{y_k} of {y_m} such that every {B_iy_k} is convergent for each i= 1,2. . . , n. For all k₁, k₂ ∈ N, by (2.1), we have

kSy_k1 − Sy_k2k ≤ψ(kSy_k1 − Sy_k2k) +M · Xn

i=1

kB_iy_k1−B_iy_k2k. (2.2)

Since ψ is continuous and nondecreasing, we have lim sup

k1,k2→∞

ψ(kSy_k1 − Sy_k2k) := inf

k∈N sup

k¹,k²≥k

ψ(kSy_k1 − Sy_k2k)

= ψ inf

k∈N sup

k¹,k²≥kkSy_k1 − Sy_k2k

!

:= ψ lim sup

k¹,k²→∞kSy_k1 − Sy_k2k

! , which together with (2.2) yield that

lim sup

k1,k2→∞kSy_k1− Sy_k2k ≤ψ lim sup

k1,k2→∞kSy_k1− Sy_k2k

!

since every {B_iy_k} is convergent. Noting thatψ(r)< r for all r >0, we conclude that lim sup

k¹,k²→∞kSy_k1− Sy_k2k= 0,

which means that {Sy_k}is a Cauchy sequence, and thus{Sy_k}is convergent. SoS(C) is precompact. In addition, letting y_k→y inC, it follows from (2.1) that

kSy_k− Syk ≤ψ(kSy_k− Syk) +M · Xn

i=1

kB_iy_k−B_iyk. Noting that B_iy_k→B_iy,i= 1,2, . . . , n, we conclude

lim sup

k→∞ kSy_k− Syk ≤ψ

lim sup

k→∞ kSy_k− Syk

, which yields that

k→∞lim kSy_k− Syk= 0, i.e., Sy_k→ Sy. Thus,S :C →C is continuous.

Now, by using Schauder’s fixed point theorem, we know that S has a fixed point y₀∈C. Then, we have

y₀=Sy₀=x_y0 = Xn i=1

A_ix_y0 ·B_iy₀= Xn

i=1

A_iy₀·B_iy₀,

i.e., y₀ is a solution of the operator equation x= Pⁿ

i=1

A_ix·B_ix.

Remark 2.2. In the case of n = 1, Theorem 2.1 is due to [12, Theorem 2.1]. However, due to some misprints, [12, Theorem 2.1] is essentially proved in the case of n = 1 and φ₁(r) =αr for some constant α >0.

Next, we consider the existence of periodic solution for Eq. (1.1).

Theorem 2.3. Let p≥1 and ¹_p +¹_q = 1. Assume that the following assumptions hold:

(H1) For each i ∈ {1,2, . . . , n}, a_i, b_i : R → R are continuous functions such that x(a_i(·))∈ PT(R) for all x∈ PT(R).

(H2) For each i ∈ {1,2, . . . , n}, f_i(·, x) ∈ PT(R) for any fixed x ∈ R and there exists a function φ_i∈C(R⁺,R⁺) such that

|f_i(t, x)−f_i(t, y)| ≤φ_i(|x−y|), ∀t∈R, ∀x, y∈R.

(H3) For each i∈ {1,2, . . . , n}, g_i(·, x) is measurable for all x ∈R, g_i(t,·) is continuous for almost all t ∈ R, and for each r > 0, there exists a function µ^r_i ∈ L^p(R) such that |g_i(t, x)| ≤µ^r_i(t) for all |x| ≤r and almost allt∈R.

(H4) For each i ∈ {1,2, . . . , n}, k_i : R×R → R satisfies that the map t → ek_i(t) is a continuousT-periodic function fromRtoL^q(R), where[ek_i(t)](s) =k_i(t, s),∀t, s∈R. (H5) There exists a constant M >0 such that

Xn i=1

K_ikµ^M_i kp·φ_i(r)< r, ∀r >0, where K_i = max

t∈[0,T]kek_i(t)kq; and Xn

i=1

"

sup

t∈R,|x|≤λ|f_i(t, x)| ·K_i· kµ^M_i kp

#

< λ, ∀λ > M.

Then Eq. (1.1) has a continuous T-periodic solution.

Proof. Let

(A_ix)(t) =f_i(t, x(a_i(t))), x∈ PT(R), t∈R, and

(B_ix)(t) = Z

R

k_i(t, s)g_i(s, x(b_i(s)))ds, x∈ PT(R), t∈R.

For each x ∈ PT(R), it follows from (H1) and the periodicity of f_i and k_i that A_ix and B_ixare bothT-periodic; in addition, it is not difficult to verify thatA_ixandB_ixare both continuous. Thus, bothA_i and B_i mapPT(R) intoPT(R).

We will use Theorem 2.1 to prove that Eq. (1.1) has a T-periodic solution. Next, let us verify all the assumptions of Theorem 2.1. Denote

C={x∈ PT(R) :kxk ≤M}. First, by (H2), for allx, y∈ PT(R), we have

kA_ix−A_iyk = max

t∈R |f_i(t, x(a_i(t)))−f_i(t, y(a_i(t)))|

≤ max

t∈R φ_i

|x(a_i(t))−y(a_i(t))|

≤ φ_i(kx−yk),

which means that A_i is D-Lipschitzian with a D-function φ_i, i.e., the assumption (a) of Theorem 2.1 holds.

Next, let us show that for each i ∈ {1,2, . . . , n}, B_i is continuous. Let x_k → x in PT(R). We have

|(B_ix_k)(t)−(B_ix)(t)| ≤ Z

R|k_i(t, s)| · |g_i(s, x_k(b_i(s)))−g_i(s, x(b_i(s)))|ds

≤ Z

R|k_i(t, s)|^qds 1/q

· Z

R|g_i(s, x_k(b_i(s)))−g_i(s, x(b_i(s)))|^pds 1/p

≤ sup

t∈Rkek_i(t)kq· Z

R|g_i(s, x_k(b_i(s)))−g_i(s, x(b_i(s)))|^pds 1/p

≤ K_i· Z

R|g_i(s, x_k(b_i(s)))−g_i(s, x(b_i(s)))|^pds 1/p

. (2.3)

On the other hand, Let r^′ = sup

k kx_kk+ 1. Thenr^′ <+∞.By (H3), for almost all t∈R, we have

|g_i(t, x_k(b_i(t)))−g_i(t, x(b_i(t)))| ≤2µ^r_i^′(t) and

k→∞lim g_i(t, x_k(b_i(t))) = g_i(t, x(b_i(t))).

Thus, by using the Lebesgue’s dominated convergence theorem, we get

k→∞lim Z

R|g_i(s, x_k(b_i(s)))−g_i(s, x(b_i(s)))|^pds= 0, which and (2.3) yield that B_ix_k→B_ixin PT(R).

Now, let us prove that everyB_i(C) is precompact. Since for allt∈Rand x∈C,

|(B_ix)(t)| ≤ Z

R|k_i(t, s)| · |g_i(s, x(b_i(s)))|ds

≤ Z

R|k_i(t, s)| · |µ^M_i (s)|ds

≤ Z

R|k_i(t, s)|^qds 1/q

· Z

R|µ^M_i (s)|^pds 1/p

≤ K_i· kµ^M_i kp <+∞,

B_i(C) is uniformly bounded. In addition, for all t₁, t₂∈Rand x∈C, we have

|(B_ix)(t₁)−(B_ix)(t₂)| ≤ Z

R|k_i(t₁, s)−k_i(t₂, s)| · |g_i(s, x(b_i(s)))|ds

≤ Z

R|k_i(t₁, s)−k_i(t₂, s)|^qds 1/q

· Z

R|µ^M_i (s)|^pds 1/p

= kek_i(t₁)−ek_i(t₂)kq· kµ^M_i kp. (2.4) Sincet→ek_i(t) is a continuousT-periodic function fromRtoL^q(R),t→ek_i(t) is uniformly continuous on R. Combining this with (2.4), we know that B_i(C) is equicontinuous.

Then, by using the well-known Arz´ela-Ascoli Theorem, B_i(C) is precompact. Thus, the assumption (b) of Theorem 2.1 holds.

Next, we show that the assumption (c) of Theorem 2.1 holds. Let y ∈ C and x = Pn

i=1

A_ix·B_iy. Denotekxk=λ. We claim thatλ≤M. In fact, ifλ > M, by (H5), we have

λ=kxk =

Xn i=1

A_ix·B_iy

≤ sup

t∈R

Xn i=1

|f_i(t, x(a_i(t)))| · Z

R

k_i(t, s)g_i(s, y(b_i(s))) ds

≤ Xn

i=1

"

sup

t∈R,|x|≤λ|f_i(t, x)| ·K_i· kµ^M_i kp

#

< λ,

which is a contradiction. So λ≤M, and thus x∈C.

At last, it follows from Xn i=1

K_ikµ^M_i kp·φ_i(r)< r, ∀r >0 and

sup

x∈CkB_ixk ≤K_ikµ^M_i kp

that _n X

i=1

sup

x∈CkB_ixk ·φ_i(r)

< r, ∀r >0.

Now, by Theorem 2.1, there existsx₀ ∈C such that x₀ =

Xn i=1

A_ix₀·B_ix₀,

which means that x₀(t) is a continuousT-periodic solution of Eq. (1.1).

To complete this paper, we give an example to illustrate how Theorem 2.3 can be used.

Example 2.4. Letn= 2,p= 1,q =∞,

a₁(t) =t−1, b₁(t) =t², a₂(t) = 2t, b₂(t) =|t|, f₁(t, x) = x

10sint, g₁(t, x) = sin(xe^t2)

2(1 +t²), k₁(t, s) = cost 1 +s², and

f₂(t, x) = costsinx

20 , g₂(t, x) = arctan(tx)

1 +t² , k₂(t, s) =e^−s2sint.

It is easy to see that (H1) and (H2) hold withT = 2π,φ₁(r) = ₁₀^r and φ₂(r) = ₂₀^r. In addition, we have

|g₁(t, x)| ≤ 1

2(1 +t²), |g₂(t, x)| ≤ π 2 · 1

1 +t².

Thus (H3) holds with µ^r₁(t)≡ _2(1+t^{1 2}₎ andµ^r₂(t)≡ ^π₂·_1+t¹². By a direct calculation, we can get (H4) holds and

K₁=π, K₂ =√ π.

Letting M = 1, we have X2

i=1

K_ikµ^M_i k1·φ_i(r)≤ π²r

20 +π²√ π·r

40 < r, ∀r >0,

and X2

i=1

"

sup

t∈R,|x|≤λ|f_i(t, x)| ·K_i· kµ^M_i k1

#

≤ π²λ

20 +π²√ π

40 < λ, ∀λ >1.

Thus, (H5) holds.

By using Theorem 2.3, we know that the following functional integral equation x(t) = sintcost·x(t−1)

20 ·

Z

R

sin[x(s²)e^s2]

(1 +s²)² ds+sintcostsin[x(2t)]

20 ·

Z

R

arctan[sx(|s|)]

1 +s² e^−s2ds has a continuous 2π-periodic solution.

3 Acknowledgements

The authors would like to thank the referee for his/her careful reading of this paper and valuable comments.

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(Received January 9, 2012)