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Rev. Anal. Num´er. Th´eor. Approx., vol. 38 (2009) no. 1, pp. 45–53 ictp.acad.ro/jnaat

SOME FUNCTIONAL DIFFERENTIAL EQUATIONS WITH BOTH RETARDED AND ADVANCED ARGUMENTS

VERONICA ANA ILEA

Abstract. In this paper we shall study a functional differential equations of mixed type. This equation is a generalization of some equations from medicine.

Related to this equation we study the existence of the solution by contraction’s principle and Schauder’s fixed point theorem.

MSC 2000. 47H10, 34H10.

Keywords. Functional-differential equations, boundary value problems, con- traction’s principle, Schauder’s fixed point principle.

1. INTRODUCTION

Functional differential equations with advanced and retarded arguments (called here as functional differential equations of mixed type – MFDE) have had a slower contributions in mathematical researches, that is if we compare them with functional differential equations with delay. This is due the fact that in our type of equations we have in the same time both advanced and retarded argument, each of them having different behavior. These equations have the important feature that both the history and the future status of the system affect their change rate at the present time.

Here we have a general form of MFDE problem

(1) x0(t) =f(t, x(t), x(t−h), x(t+h)) +λ, t∈(a, b),

(2) x(t) =ϕ(t), t∈[a−h, a],

(3) x(t) =ζ(t), t∈[b, b+h],

where f ∈ C((a, b) ×R3;R), ϕ ∈ C([a−h, a];R), ζ ∈ C([b, b+h];R) and a, b∈R,a < b,h >0.

The first part of the results from this paper appear in I.A. Rus-V. Dˆarzu- Ilea [6] and contains the existence and the uniqueness of the solution of the problem (1)–(2)–(3) studied in different spaces.

This type of equations came from different fields of applications. For exam- ple, A. Rustichini [7], [8] investigated a specific mixed functional differential

Department of Applied Mathematics, “Babe¸s-Bolyai” University Cluj-Napoca, Kog˘alniceanu 1, 400084, Cluj-Napoca, Romania, e-mail: [email protected].

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equation arising in a special way from a competitive economy; Schulman [9]

gave a physical justification to MFDE. Some other fields of interest for this type of equations would be: population genetic (D.G. Aronson and H.F. Wein- berger [1]), population growth, mathematical biology. Some other names in the study of MFDE are: J. Mallet-Paret [2], [3], [4], J. Wu and X. Zou [10], R. Precup [5].

The biologic signification of a MFDE can be given as follows.

The equation (1) is a model for a certain disease. This depends on the physical state of the subject – the delay argument; the treatment that should be given to the patient – the advanced argument; the parameter is an out- side factor that can influence the physical state of the subject; condition (2) – represents the statistics observations obtained before from other subjects; con- dition (3) – represents also, from a statistical point of view, the expectations of the evolution of the disease.

From the study of population growth we can explain the above model as follows:

ϕ-the state of some population in an chosen environment,ζ-the state that should have the population,λ-a control parameter,x0(t) is the speed of grow of the population with the lowx0=f+λ.

If some part of the population is sacrificed thenλ <0, and if the population is extending numerically thenλ >0.

2. HOW TO OBTAIN A FREDHOLM-VOLTERRA INTEGRAL EQUATION

Let (x, λ) a solution for (1)–(2)–(3). It follows that:

(4) x(t) =





ϕ(t), t∈[a−h, a],

ϕ(a) +Rt

af(s, x(s), x(s−h), x(s+h))ds+

+λ(t−a), t∈[a, b],

ζ(t) t∈[b, b+h].

From the continuity in t=b we have

(5) λ= ζ(b)−ϕ(a)b−ab−a1 Z b

a

f(s, x(s), x(s−h), x(s+h))ds.

Thus the problem (1)–(2)–(3) is equivalent with

x=A(x) andλ = the right hand side of (5),

(3)

whereA:C[a−h, b+h]→C[a−h, b+h] and (6)

A(x)(t) :=









ϕ(t), t∈[a−h, a]

ϕ(a) +b−at−a(ζ(b)−ϕ(a))−

b−at−aRb

af(s, x(s), x(s−h), x(s+h))ds+

+Rt

af(s, x(s), x(s−h), x(s+h))ds, t∈[a, b]

ζ(t), t∈[b, b+h].

By the contraction principle we obtain the following existence theorem.

Theorem 1. (I.A. Rus,V. Dˆarzu-Ilea [6]) If we have the following conditions

(i) there existLf >0 such as:

kf(t, u1, v1, w1)−f(t, u2, v2, w2)k≤Lf(ku1−u2 k+kv1−v2 k+kw1−w2 k), for allt∈[a, b], ui, vi, wi ∈R, i= 1,2;

(ii) 6Lf(b−a)<1.

Then the problem(1)–(2)–(3)has a unique solution. More than that, if(x, λ) is the unique solution for (1)–(2)–(3), then

x = lim

n→∞An(x), f or all x∈C[a−h, b+h], and

(7) λ= ζ(b)−ϕ(a)b−ab−a1 Z b

a

f(s, x(s), x(s−h), x(s+h))ds.

3. MAIN RESULT

We generalize the above problem considering the next Fredholm-Volterra integral equation

x(t) =g(t) + Z t

a

K(t, s, x(s), x(s−h), x(s+h))ds−

(8)

t−ab−a Z b

a

K(t, s, x(s), x(s−h), x(s+h))ds, t∈[a, b]

with (9)

x(t) =ϕ(t), t∈[a−h, a], x(t) =ζ(t), t∈[b, b+h],

whereK∈C([a, b]×[a, b]×R3) andg∈C([a, b];R) given by the formula (F1) g(t) = ζ(b)−ϕ(a)b−a t+bϕ(a)−aζ(b)

b−a .

Our purpose here is to study the existence of the solution of the equation (8) with contraction’s principle and Schauder’s types theorems.

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The problem (8)–(9) is equivalent withx=T(x), whereT :C[a−h, b+h]→ C[a−h, b+h] and

(10)

T(x)(t) :=













ϕ(t), t∈[a−h, a]

ζ(b)−ϕ(a)

b−a t+bϕ(a)−aζ(b)

b−a

t−ab−aRb

aK(s, t, x(s), x(s−h), x(s+h))ds+

+Rt

aK(s, t, x(s), x(s−h), x(s+h))ds, t∈[a, b]

ζ(t), t∈[b, b+h].

By the contraction principle we obtain the following existence theorem.

Theorem 2. If we have the following conditions:

(i) there existLf >0 such as

kK(s, t, u1, v1, w1)−K(s, t, u2, v2, w2)k≤Lf(ku1−u2k+kv1−v2 k+kw1−w2k), for allt∈[a, b], ui, vi, wi ∈R, i= 1,2;

(ii) 6Lf(b−a)<1

Then the problem (8)–(9) has a unique solution, moreover the solution x can be obtain by the method of successive approximation beginning from any element from the space C[a−h, b+h].

Proof. Let the operator (11)

T(x)(t) :=





ϕ(t), t∈[a−h, a]

g(t) +Rt

aK(t, s, x(s), x(s−h), x(s+h))ds−

t−ab−aRb

aK(t, s, x(s), x(s−h), x(s+h))ds, t∈[a, b]

ζ(t), t∈[b, b+h].

|T x(t)−T y(t)|=

=| Z t

a

(K(t, s, x(s), x(s−h), x(s+h))−K(t, s, y(s), y(s−h), y(s+h)))ds−

t−ab−a Z b

a

(K(t, s, x(s), x(s−h), x(s+h))−K(t, s, y(s), y(s−h), y(s+h)))ds|

≤3Lf(b−a)kx−yk+3Lf(b−a)kx−yk

= 6Lf(b−a)kx−yk

But 6Lf(b−a) < 1, follows T is a contraction. We can apply now the principle of contraction and follows the conclusion from the theorem.

This is a classical problem of Krasnoselskii type, but by applying this type of theorem in spaceC[a−h, b+h] we obtain to many conditions on the date K, thus we apply Schauder type theorem in order to obtain optimality of the conditions on equation’s data.

Let the Banach space C[a−h, b+h] with the Chebyshev norm,k · k.

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Theorem 3. If we have the following conditions

(i) K∈C([a, b]×[a, b]×J3),J - is a compact interval,g∈C([a, b])given by (F1);

(ii) there existsM ∈R+ such that

kK(t, s, u, v, w)k≤M, t, s∈[a, b], u, v, w∈J,

then the equation(8)has at least one solutionx∈C([a, b]) with the propriety that kxk≤R, where R is a number greater than2M(b−a).

Proof. Let the operatorT :C[a−h, b+h]→C[a−h, b+h] be defined by

(12) x(t) =





ϕ(t), t∈[a−h, a]

g(t) +Rt

aK(t, s, x(s), x(s−h), x(s+h))ds−

t−ab−aRb

aK(t, s, x(s), x(s−h), x(s+h))ds, t∈[a, b]

ζ(t). t∈[b, b+h]

From (i) the operatorT is well defined and complete continuous.

In what follows we use the conditions (ii) in order to prove the invariance on sphere

k T(x)(t)−g(t) k≤ R, with k x k≤k g k +R and R > 0 (x ∈ B(g;R) ⇒ x(t)∈J, whereJ = [−j, j], with j =kgk+R),

kT(x)(t)−g(t)k ≤ Z t

a

kK(t, s, x(s), x(s−h), x(s+h))kds+

+t−ab−a Z b

a

kK(t, s, x(s), x(s−h), x(s+h))kds≤

≤ Z t

a

kK(t, s, x(s), x(s−h), x(s+h))kds +

Z b a

kK(t, s, x(s), x(s−h), x(s+h))kds

≤2M(b−a)≤R.

Therefore

(13) kT(x)(t)−g(t)k≤2M(b−a).

Now we can say that for R greater than 2M(b−a) the operator T satisfy the invariance condition. Thus by applying Schauder’s theorem it follows that there exist at least one solution x and for this solution we have established that

kx k≤ϕ(b) +R.

(6)

Now we consider the following equation x(t) =g(t) +b−ab−t

Z t a

K1(t, s, x(s), x(s−h), x(s+h))ds+

(14)

+(b−t)(t−a)b−a Z b

a

K2(t, s, x(s), x(s−h), x(s+h))ds, t∈[a, b]

with the initial conditions (15)

x(t) =ϕ(t), t∈[a−h, a]

x(t) =ζ(t), t∈[b, b+h]

where K1, K2 ∈ C([a, b]×[a, b]×R3;R) and g ∈ C([a, b];R) given by the formula

(F1) g(t) = ζ(b)−ϕ(a)b−a t+bϕ(a)−aζ(b)

b−a .

The problem (14)–(15) is equivalent withx=T(x), whereT :C[a−h, b+h]→ C[a−h, b+h] and

(16)

T(x)(t) :=













ϕ(t), t∈[a−h, a]

ζ(b)−ϕ(a)

b−a t+bϕ(a)−aζ(b)

b−a +

+(b−t)(tb−a−a)Rb

aK2(s, x(s), x(s−h), x(s+h))ds+

+b−ab−tRt

aK1(s, x(s), x(s−h), x(s+h))ds, t∈[a, b]

ζ(t), t∈[b, b+h].

The existence of the solution of the equation (14) with contraction’s prin- ciple is trivial.

Theorem 4. If we have the following conditions:

(i) there existLi>0 such as

kKi(t, u1, v1, w1)−Ki(t, u2, v2, w2)k≤Li(ku1−u2 k+kv1−v2k+kw1−w2k), for allt∈[a, b], uj, vj, wj ∈R, i, j= 1,2;

(ii) 3(L1+ (b−a)L2)(b−a)<1

Then the problem(14)–(15)has a unique solution, more the solutionx can be obtain by the method of successive approximation beginning from any element from the space C[a−h, b+h].

Now we consider the Banach spaceC[a−h, b+h] with the Chebyshev norm, k · k.

Theorem 5. If we have the following conditions

(i) K1, K2∈C([a, b]×[a, b]×J3),J - is a compact interval,g∈C([a, b];R) given by (F1);

(ii) there exist real numbers α, β, γ, δ such as

k K1(t, s, u, v, w) k≤ α k u k +β k v k +γ k w k +δ, t, s ∈ [a, b], u, v, w∈J;

(7)

(iii) there existsM ∈Rsuch that

kK2(t, s, u, v, w)k≤M, t, s∈[a, b], u, v, w∈J;

then the equation (14) has at least one solution x ∈ C([a, b];R) with the propriety that kx k≤R, where R is a number greater than

(b−a)[ϕ(b)(α+β+γ)+M(b−a)+δ]

1−(α+β+γ)(b−a)

with (α+β+γ)(b−a)<1.

Proof. Let the operatorT :C[a−h, b+h]→C[a−h, b+h] be defined by (17)

x(t) =





ϕ(t), t∈[a−h, a]

g(t) + b−ab−tRt

aK1(t, s, x(s), x(s−h), x(s+h))ds+

+(b−t)(t−a)b−a Rb

aK2(t, s, x(s), x(s−h), x(s+h))ds, t∈[a, b]

ζ(t). t∈[b, b+h]

From (i) the operatorT is well defined and complete continuous.

In what follows we use the conditions (ii), (iii) in order to prove the invari- ance on sphere

k T(x)(t)−g(t) k≤ R, with k x k≤k g k +R and R > 0 (x ∈ B(g;R) ⇒ x(t)∈J, whereJ = [−j, j], with j =kgk+R),

kT(x)(t)−g(t)k≤

b−ab−t Z t

a

kK1(t, s, x(s), x(s−h), x(s+h))kds+

+(b−t)(t−a)b−a Z b

a

kK2(t, s, x(s), x(s−h), x(s+h))kds

≤ Z t

a

[αkx(s)k+β kx(s−h)k+γ kx(s+h)k+δ]ds+ (b−a) Z b

a

Mds≤

≤(α+β+γ)kxk(b−a) +δ(b−a) +M(b−a)2

≤(α+β+γ)(kgk+R)(b−a) + (M(b−a) +δ)(b−a)

≤(α+β+γ)(ϕ(b) +R)(b−a) + (M(b−a) +δ)(b−a).

Therefore

(18) kT(x)(t)−g(t)k≤(α+β+γ)(ϕ(b) +R)(b−a) + (M(b−a) +δ)(b−a).

Now we can say that if (α+β+γ)(b−a)<1, forR greater than

(b−a)[ϕ(b)(α+β+γ)+M(b−a)+δ]

1−(α+β+γ)(b−a)

the operatorTsatisfy the invariance condition. Thus by applying the Schauder theorem it follows that there exist at least one solutionx and for this solution

we have established thatkx k≤ϕ(b) +R.

(8)

Example 6. Let the equation (19) x(t) =t+b−ab−t

Z t a

[x(s) +x(s−1)]ds+(b−t)(t−a)b−a Z b

a

x(s+ 1)ds, t∈[a, b]

with the initial conditions (20)

x(t) =t, t∈[a−1, a], x(t) =t, t∈[b, b+ 1].

The problem (19)–(20) is equivalent with x=T(x), where T :C[a−h, b+ h]→C[a−h, b+h] and

(21) T(x)(t) :=





t, t∈[a−1, a]

(b−t)(t−a) b−a

Rb

a[x(s) +x(s−1)]ds+

+b−ab−tRt

ax(s+ 1)ds, t∈[a, b]

ζ(t), t∈[b, b+ 1].

The existence of the solution of the equation (19) with contraction’s prin-

ciple is trivial.

Theorem 7. If we have the condition:

(ii) (b−a)(2 +b−a)<1 (i.e. a= 0, b= 1/3)

Then the problem(19)–(20)has a unique solution, more the solutionx can be obtain by the method of successive approximation beginning from any element from the space C[a−1, b+ 1].

Proof. We have to estimate if the operatorT is Lipschitz

|T x−T y| ≤ |b−ab−t Z t

a

[x(s)−y(s) +x(s−1)−y(s−1)]ds|+

+|(b−t)(t−a)b−a Z b

a

[x(s+ 1)−y(s+ 1)]ds|

≤(b−a)(2 +b−a).

Let now the Banach space C[a−1, b+ 1] with the Chebyshev norm.

Theorem 8. The equation (19) with the condition (20) has at least one solution x ∈ C([a, b];R) with the propriety that k x k≤ R, where R is a number greater than b(b−a)(1+b−a)

1−(b−a)(1+b−a) with (b−a)(1 +b−a)<1 (i.e. a= 0, b=

1 3).

Proof. Let the operatorT :C[a−1, b+ 1]→C[a−1, b+ 1] defined by

(9)

(22) x(t) =





t, t∈[a−1, a]

t+b−ab−t Rt

a[x(s) +x(s−h)]ds+

+(b−t)(t−a)b−a Rb

ax(s+h)ds, t∈[a, b]

t. t∈[b, b+ 1]

T is well defined and complete continuous.

In what follows we prove the invariance on sphere

kT(x)(t)−tk≤R, withkxk≤kgk+R andR >0 (x∈B(g;R)⇒x(t)∈ J, whereJ = [−j, j], with j =kgk+R),

kT(x)(t)−tk≤

b−ab−t Z t

a

[|x(s)|+|x(s−h)|]ds+(b−t)(t−a)b−a Z b

a

|x(s+h)|ds

≤(b−a)(1 +b−a)kxk≤(b−a)(1 +b−a)(b+R)< R.

Now we can say that if (b−a)(1+b−a)<1, forRgreater than b(b−a)(1+b−a) 1−(b−a)(1+b−a)

the operatorTsatisfy the invariance condition. Thus by applying the Schauder theorem it follows that there exist at least one solutionx and for this solution

we have established thatkx k≤ϕ(b) +R.

REFERENCES

[1] Aronson, D.G. and Weinberger, H.F., Nonlinear diffusion in population genetics, combustion and nerve propagation, “Lectures Notes in Mathematics”, 446, Springer- Verlag, Berlin, pp. 5–49, 1975.

[2] Mallet-Paret, J.,The global structure of traveling waves in spatially discrete dynam- ical systems, J. Dyn. Diff Eq.,11, no. 1, pp. 49–127, 1999.

[3] Mallet-Paret, J., The Freedholm alternative for functional differential equations of mixed type, J. Dyn. Diff Eq.,11, no. 1, pp. 1–46, 1999.

[4] Mallet-Paret, J. and Lunel, S.V., Exponential dichotomies and Wiener- Hopf factorizations for mixed-type functional differential equations, 2001, www.dam.brown.edu/lcds/publications.textonly.html.

[5] Precup, R., Some existence results for differential equations with both retarded and advanced arguments, Mathematica,44(67), no. 1, pp. 31–38, 2002.

[6] Rus, I.A.andarzu-Ilea, V.A.,First order functional-differential equations with both advanced and retarded arguments, Fixed Point Theory,5, no. 1, pp. 103–115, 2004.

[7] Rustichini, A.,Functional differential equation of mixed type: The linear autonomous case, J. Dyn. Diff. Eq.,1, pp. 121–143, 1989.

[8] Rustichini, A., Hopf bifurcation for functional differential equation of mixed type, J.

Dyn. Diff. Eq.,1, pp. 145–177, 1989.

[9] Schulman, L.S., Some differential difference equations containing both advance and retardation, J. Math. Phys.,15, pp. 195–198, 1974.

[10] Wu, J.andZou, X.,Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations, J. Diff. Eq.,135, pp. 315–357, 1997.

Received by the editors: March 26, 2009.

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