LIAPUNOV FUNCTIONALS AND PERIODICITY IN A SYSTEM OF NONLINEAR INTEGRAL EQUATIONS

In this paper, we construct a Liapunov functional for a system of nonlinear integral equations. From that Liapunov functional we are able to establish the existence of periodic solutions to the system by applying some well-known fixed point theorems for the sum of a nonlinear contraction mapping and compact operator.


Introduction
This paper is concerned with the existence of periodic solutions to the system of nonlinear integral equations where x(t) ∈ R n , h : R × R n → R n , D : R × R → R n×n , g : R × R n → R n are continuous, and R = (−∞, ∞).
The existence of periodic solutions of (1.1) or its differential form has been the subject of extensive investigations for many years.Our interest here centers on the use of Liapunov's direct method and fixed point theorems of continuation type, which are nonlinear alternatives of Leray-Schauder degree theory, to derive the existence of periodic solutions.Continuation theorems, such as Schaefer's [26] fixed point theorem without actually calculating degree, require less restrictive growth conditions on the functions involved.For the historical background and discussion of applications, we B. Zhang refer the reader to, for example, the work of Burton [4], Burton, Eloe, and Islam [10], Graef and Kong [12], Miller [24], O'Regan [25], Zeidler [28] and Zhang [29].
It is well-known that Liapunov's direct method has been used very effectively for differential equations.The method has not, however, been used with much success on integral equations until recently.The reason for this lies in the fact that it is very difficult to relate the derivative of a scalar function to the unknown non-differentiable solution of an integral equation.In the present paper, we construct a Liapunov functional for (1.1).From that Liapunov functional we are able to establish an a priori bound for all possible T -periodic solutions of a companion system of (1.1), and then, to prove the existence of a T -periodic solution to (1.1).As in the case for differential equations, once the Liapunov function is constructed, we can take full advantage of its simplicity in qualitative analysis.A good summary for recent development of the subject may be found in Burton [8].
A continuous function x : R → R n is called a solution of (1.1) on R if it satisfies (1.1) on R. If x(t) is specified to be a certain initial function on an initial interval, say we are then looking for a solution of (1.2) Note that the initial function φ is absorbed into the forcing function term h(t, x(t)).
There is substantial literature on equations (1.1) and (1.2).Much of the literature can be traced back to the pioneering work of Levin and Nohel ([18]- [21]) in the study of asymptotic behavior of solutions of the scalar integral equation and the integro-differential equation These equations arise in problems related to evolutionary processes in biology, in nuclear reactors, and in control theory (see Corduneanu [11], Burton [7], Levin and Nohel [20], Kolmanovskii and Myshkis [16]).It is often required that a ∈ C(R + , R) and (1.6) Levin's work was based on (1.6) and the construction of a Liapunov functional for (1.4).
The method was further extended into a long line of investigation drawing together such different notions of positivity as Liapunov functions, completely monotonic functions, and kernels of positive type (see Corduneanu [11], Gripenberg et al [14], Levin and Nohel [21], MacCamy and Wong [23]).In a series of papers ([4]- [6]), Burton obtains results on boundedness and periodicity of solutions for a scalar equation in the form of (1.1) without asking the growth condition on g.Liapunov functionals play an essential role in his proofs.
Many investigators mentioned above frequently use the fact that (1.4) can be put into the form of (1.3) by integration to study the existence and qualitative behavior of solutions by applying fixed point theorems.We now consider the right-hand side of (1.1) as a mapping F x = Bx + Ax on a convex subset M of the Banach space (P T , • ) of continuous T -periodic functions φ : R → R n with the supremum norm, where (Bx)(t) = h(t, x(t)) and Ax represents the integral term, with a view of showing that B is a nonlinear contraction and A is compact.Observe that F in general is a non-self map; that is, F may not necessarily map M into M.This presents a significant challenge to investigators.A modern approach to such a problem is to use topological degree theory or transversality method to prove the existence of fixed points.The later requires the construction of a homotopy U λ and uses conditions on U λ which may be less general, but more easily established in application (see Burton and Kirk [9], Liu and Liu [22], Granas and Dugundji [13], Wu, Xia, and Zhang [27]).The following formulation is from O'Regan [25].
Theorem 1.1 Let U be an open set in a closed, convex set C of a Banach space (E, • ) with 0 ∈ U. Suppose that F : U → C is given by F = F 1 + F 2 and F (U) is a bounded set in C. In addition, assume that F 1 : U → C is continuous and completely continuous and for F 2 : U → C, there exists a continuous, nondecreasing function for all x, y ∈ U .Then either (A1) F has a fixed point in U , or (A2) there is a point u ∈ ∂U and λ ∈ (0, 1) with u = λF (u).
We will apply Theorem 1.1 to show that F = B + A has a fixed point in M which is a T -periodic solution of (1.1).This will be done in Section 2. Our proofs are by Liapunov functionals V .It is to be noted that the technique used allows us to prove that V is bounded without ever asking a growth condition on g that makes the derivative of V negative in any region.In Section 3, we discuss some special cases of ( 1 Similarly, if B is negative semi-definite, then B ≤ 0. Also, if B ≥ 0, we denote its square root by √ B. Concerning the terminology of a completely continuous mapping, we use the usual convention to mean the following: Let E be a Banach space and P : X ⊆ E → E. If P (Y ) is relatively compact for every bounded set Y ⊆ X, we say that P is completely continuous.In particular, X need not be bounded (see Agarwal, Meehan and O'Regan [1, p. 56]).

The Main Result
In this section, we will apply Theorem 1.1 with for any x ∈ P T .
To show that F has a fixed point in M, we must prove that the alternative (A2) does not hold and the homotopy U λ (x) = λF (x) is fixed point free on ∂M for λ = 1.This can be achieved by establishing the existence of an a priori bound for all possible fixed points of λ(B + A) for 0 < λ ≤ 1.To accomplish this, we assume that (H 1 ) there exists a constant T > 0 such that D(t + T, s x) for all t, s ∈ R and all x ∈ R n , (H 2 ) there exist constants K > 0 and η > 0 such that where g T is the transpose of g, , where ψ is continuous, nondecreasing with ψ(r) < r for all r > 0 and We observe that all of these conditions on D(t, s) can be verified if We also see that some of these conditions are interconnected.For example, (H 3 ) nearly implies (H 2 ) if x T g(t, x) ≥ γ|x||g(t, x)| for all |x| ≥ K and a constant γ > 0. It is also to be noted that h(t, x) satisfying (H 3 ) is a nonlinear contraction in the sense of Boyd and Wong [3].
We now prove the following theorem by constructing a Liapunov functional which has its roots in the work of Burton [4], Kemp [15], and Zhang [30].The result here generalizes a theorem of Burton [4] for scalar equations.Proof.Let A and B be defined in (2.1) for each x ∈ P T .A change of variable shows that if φ ∈ P T , then (Aφ)(t+T ) = (Aφ)(t).Thus, A, B : P T → P T are well defined.By (H 3 ), B satisfies the conditions for F 2 in Theorem 1.1.To establish that A : M → P T is continuous and completely continuous, we need several steps which follow.Let us first show that there exists a constant µ > 0 such that x < µ whenever x ∈ P T and x = λ(Bx + Ax) for λ ∈ (0, 1].Suppose now that x ∈ P T satisfying and define

B. Zhang
Then V (t, x(•)) is T -periodic and If we integrate the last term by parts, we have The first term vanishes at both limits by (H 5 ); the first term of V ′ is not positive since D st (t, s) ≤ 0, and if we use (2.4) on the last term, then we obtain By (H 2 ), we see that if |x(t)| ≥ K, then It is clear that V ′ (t, x(•)) is bounded above for 0 ≤ |x(t)| ≤ K since g(t, x) and h(t, x) are bounded for x bounded, and hence, there exists a constant L > 0 depending on K such that for all t ∈ R. By the Schwarz inequality, we have (2.9) and so Thus, This implies that V (t, x(•)) ≤ γL 2 for all t ∈ R, and therefore by (2.9) we obtain a contradiction, and thus, x < µ whenever x is a solution of (2.4) for 0 < λ ≤ 1.We now define M = {x ∈ P T : x < µ}.

It is clear that
M is an open subset of P T .By the argument above, if x = λ(Bx + Ax) for 0 < λ ≤ 1, then x < µ.This implies x ∈ M, and therefore, (A2) of Theorem 1.1 fails to hold.
Next, we show that A : M → P T is continuous and AM is contained in a compact subset of P T .Let φ 1 , φ 2 ∈ M .Then for all t ∈ [0, T ], we have (2.12) Since g is uniformly continuous on {(t, x) : 0 ≤ t ≤ T, |x| ≤ µ}, then for any ε > 0, there exists a δ > 0 such that It then follows from (2.12) that Aφ 1 − Aφ 2 ≤ J * ε, where Thus, A is continuous on M .Now if φ ∈ M and 0 ≤ t 1 ≤ t 2 ≤ T , then where Here we have used (H 6 ) in the last inequality.This implies that AM is equicontinuous.The uniform boundedness of AM follows from the following inequality for all φ ∈ M .So, by the Ascoli-Arzela Theorem, AM lies in a compact subset of P T , and therefore, A is completely continuous.Moreover, for each x ∈ M , we have ) The constant L can be expressed as a function of K and η.In fact, letting

Special Equations and Examples
In this section, we discuss some special cases of (1.1) with examples and remarks concerning conditions (H 1 )-(H 6 ).These special equations not only have deep roots in application, but possess rich properties that provide much needed insight for investigation of highly nonlinear equations.We first consider the system where x(t) ∈ R n , a : R → R n , D : R × R → R n×n , g : R n → R n are continuous and assume there exists a constant K > 0 such that
Proof.Using inequality (3.3) and the convexity of r α+1 for r ≥ 0, we obtain and thus, This yields (3.4).Proof.We first observe that (3.1) is in the form of (1.1) with h(t, x) = a(t).Thus, (H 1 ) and (H 3 ) are satisfied.To verify (H 2 ), we start with the inequality where m is an odd positive integer, or by the bounded function Next, we consider the system with a nonlinear contraction term where x(t) ∈ R n , h : R × R n → R n , D : R × R → R n×n , g : R × R n → R n are continuous, and D(t + T, s + T ) = D(t, s) for all t, s ∈ R.
We now let for all x ∈ R n and t ∈ R, and for  Proof.We first observe that for the function α defined in (3.12), it is a straightforward calculation to obtain for all u, v ∈ R, where It is clear that ψ is continuous, increasing with ψ(r) < r for all r > 0 and lim r→∞ (r − ψ(r)) = ∞.
We now proceed to verify that (H 2 ) and (H 3 ) hold.For any (3.15) Thus, h(t, x) satisfies (H 3 ).To show (H 2 ) holds, we first observe that by (3.10), there exists K 1 > 0 such that |x| * ≥ K 1 implies g(t, x) = b(t) x 3 1 , x Remark 3.2 We again point out that (H 2 ) is a quite mild condition which allows g(t, x) to be highly nonlinear and nearly independent of h(t, x).We also observe that g(t, x) in (3.10) can take the form of (3.6) or (3.7), and Theorem 3.2 is still valid.
Remark 3.3 For h(t, x) defined in (3.9), the mapping (Bx)(t) = h(t, x(t)) for x ∈ P T is almost a contraction, but fails near x = 0. Thus, we don't expect to find a constant 0 < α < 1 satisfying Bx − By ≤ α x − y for all x, y ∈ P T .
Finally, we wish to point out that the sign condition (H 4 ) is essential for the existence of periodic solutions of (1.1).For example, the scalar equation e −(t−s) x(s)ds (3.17) does not have a periodic solution.In fact, all solutions of (3.17) are unbounded.It is clear that (H 4 ) is not satisfied with D(t, s) = −e −(t−s) .
.1) with EJQTDE Spec.Ed.I, 2009 No. 32 B. Zhang illustrative examples to show application of the main result.For x ∈ R n , |x| denotes the Euclidean norm of x.Let C(X, Y ) denote the space of continuous functions φ : X → Y .For an n × n matrix B = (b ij ) n×n , we denote the norm of B by B = sup{|Bx| : |x| ≤ 1}.If B is symmetric, we use the convention for self-adjoint positive operators to write B ≥ 0 whenever B is positive semi-definite.
and m is an odd positive integer.EJQTDE Spec.Ed.I, 2009 No. 32

. 10 )
whenever |x| ≥ K for some positive constant K, where a : R → R n , b : R → R + are continuous and T -periodic, and α 8) Ed. I, 2009 No. 32This implies that BM is bounded, and hence, F = B + A is bounded on M. By Theorem 1.1, F has a fixed point x * ∈ M .In this case, x * ∈ M and x * is a T -periodic solution of (1.1).The proof is complete.