ON SOME EQUATIONS y

ON SOME EQUATIONS y

(x) = f (x, y(h(x) + g(y(x)))) Zbigniew Grande

Institute of Mathematics Kazimierz Wielki University

Plac Weyssenhoffa 11, 85–072 Bydgoszcz, Poland e-mail: grande@ukw.edu.pl

Abstract

In [4] W. Li and S.S. Cheng prove a Picard type existence and uniqueness theorem for iterative differential equations of the form y

(x) = f (x, y(h(x) + g(y(x)))). In this article I show some analogue of this result for a larger class of functions f (also discontinuous), in which a unique differentiable solution of considered Cauchy’s problem is obtained.

Keywords: iterative differential equation, existence and uniqueness theorem, Picard approximation, derivative, (S)-continuity, (S)-path continuity.

2010 Mathematics Subject Classification: 34A12, 39B12, 26B05.

1. Introduction

Let IR be the set of all reals. In [4] the following theorem is established.

Theorem 1.1 ([4], Theorem 1). Let Ω = {(x, y) ∈ IR

; |x − x

| ≤ a,

|y − x

| ≤ b}. Suppose

(i) f = f (x, y) is continuous on Ω with a positive real M ≥ sup

|f (x, y)| and such that

|f (x, u) − f (x, v)| ≤ L|u − v|; (x, u), (x, v) ∈ Ω, f or some L > 0, (ii) h = h(x) is continuous on [x

− a, x

+ a] and g = g(x) is continuous

on [x

− M δ, x

+ M δ] where δ = min(a,

), and such that

(a) h(x

) = αx

and g(x

) = βx

, where α, β ≥ 0 and α + β = 1, (b) the functions h and g satisfy

|h(u) − h(v)| ≤ αλ|u − v|, u, v ∈ [x

− a, x

+ a], and

|g(u) − g(v)| ≤ βµ|u − v|, u, v ∈ [x

− M δ, x

+ M δ]

for some λ, µ > 0.

If αλ + βµM ≤ 1, then equation

(1) y

(x) = f (x, y(h(x) + g(y(x)))), with initial condition y(x

) = x

has a unique continuously differentiable solution y = y(x) defined on [x

− δ, x

+ δ] such that |y(x) − x

| ≤ M δ and y(x

) = x

.

Observe that special cases of equation (1) are y

(x) = f (x, y(y(x))), which can be used to model infection disease processes, pattern formation in the plane, and in investigation of dynamical systems. An other special case of (1) is well known classical differential equation y

(x) = f (x, y(x)).

2. The main result

In this article I show some analogy of Theorem 1.1 for a larger class of functions f (also discontinuous), in which a unique differentiable solution of considered Cauchy’s problem is obtained.

For this recall that a locally Lebesgue integrable function φ : [c, d] → IR is a derivative at a point x ∈ [c, d] if

φ(x) = lim

r→0

1 r

Z

x

φ(t)dt.

Observe that for a Lebesgue integrable function φ : [c, d] → IR there is a function F with F

= φ if and only if φ is a derivative at each point x ∈ [c, d]

([1]).

Theorem 2.1. Suppose that the set Ω and the functions g and h are the same as these in Theorem 1.1. Let f : Ω → IR be a bounded function such that

|f (x, u) − f (x, v)| ≤ L|u − v|, (x, u), (x, v) ∈ Ω for some L > 0

and the horizontal sections f

(x) = f (x, u), u ∈ [x

− b, x

+ b], of f are derivatives. Let M ≥ sup

|f (x, u)| be a positive real.

If αλ + βµM ≤ 1, then equation (1) has a unique differentiable solution y = y(x) defined on [x

− δ, x

+ δ] such that y(x

) = x

and |y(x) − y(x

)| ≤ M δ for x ∈ [x

− δ, x

+ δ].

P roof. The proof is a modification of that of Theorem 1 from [4]. First assume that y = y(x) is a differentiable solution of equation (1) defined on [x

− δ, x

+ δ] such that y(x

) = x

and |y(x) − x

| ≤ M δ for x ∈ [x

−δ, x

+δ]. Then for t ∈ [x

−δ, x

+δ] the value y(t) ∈ [x

−M δ, x

+M δ]

and g(y(t)) are well defined. Moreover,

|h(t) + g(y(t)) − x

| ≤ |h(t) − αx

+ g(y(t)) − βx

|

≤ |h(t) − h(x

)| + |g(y(t)) − g(x

)| ≤ αλ|t − x

| + βµ|y(t) − x

|

≤ αλδ + βµM δ ≤ δ, for t ∈ [x

− δ, x

+ δ].

This means that f (t, y(h(t) + g(y(t)))) is well defined on [x

− δ, x

+ δ].

So, by integrating equation (1) we observe that y is a continuous solution of the integral equation

(2) y(x) = x

+ Z

x0

f (t, y(h(t) + g(y(t))))dt, x ∈ [x

− δ, x

+ δ].

Conversely, we assume that y = y(x) is a continuous solution of the above integral equation defined on [x

− δ, x

+ δ]. Evidently the function t → y(h(t) + g(y(t))) is continuous. Since the vertical sections f

(u) = f (x, u), x ∈ [x

− a, x

+ a] are equicontinuous and the horizontal sections f

, u ∈ [x

− b, x

+ b] are derivatives, the Carath´eodory superposition t → f (t, y(h(t) + g(y(t)))) is a derivative on [x

− δ, x

+ δ] ([2]). Consequently, by differentiating the above integral equation we obtain (1). So y is a dif- ferentiable solution of (1) satisfying the initial condition y(x

) = x

.

Now, in the same way as in the proof of Theorem 1 from [4] we define a sequence (y

, y

, . . .) of successive approximations of the desired solution of (2) by

y

(x) = x

for |x − x

| ≤ δ, and for m = 0, 1, . . . ,

y

(x) = x

+ Z

x0

f (t, y

(h(t) + g(y

(t))))dt for |x − x

| ≤ δ

and we verify that y

, i = 1, 2, . . . , are well defined and differentiable on [x

−δ, x

+δ]. The Lebesgue integrability of the functions t → f (t, y

(h(t)+

g(y

(t)))) (and consequently the differentiability of y

), m ≥ 0, follows from the fact that they are bounded derivatives ([2]). For the proof that the graphs of the functions y

are contained in Ω we start from an evident observation that (x, y

(x)) = (x, x

) ∈ Ω for |x − x

| ≤ bδ. Next,

|h(x) − g(y

(x)) − x

| = |h(x) − g(x

) − αx

− βx

|

≤ |h(x) − αx

| + |g(x

) − βx

| = |h(x) − h(x

)| ≤ αλ|x − x

| ≤ αλδ ≤ δ for |x − x

| ≤ δ, as well as

|y

(x) − x

| ≤    

Z

f (t, y

(h(t) + g(y

(t))))dt    

=    

Z

f (t, x

)dt    

≤ M |x − x

|

for |x − x

| ≤ δ, so that (x, y

(x)) ∈ Ω for |x − x

| ≤ δ. Next,

|h(x) + g(y

(x)) − x

| ≤ |h(x) − h(x

)| + |g(y

(x)) − g(x

)|

≤ αλ|x − x

| + βµ|x − x

| ≤ (αλ + βµM )δ ≤ δ for |x − x

| ≤ δ and

|y

(h(x) + g(y

(x))) − x

| =    

Z

f (t, y

(h(t) + g(y

(t))))dt    

≤ M |h(x) + g(y

(x)) − x

| ≤ M δ ≤ b for |x − x

| ≤ δ, so that

|y

(x) − x

| ≤    

Z

f (t, y

(h(t) + g(y

(t))))dt    

≤ M |x − x

| ≤ M δ ≤ b

for |x − x

| ≤ δ. By induction, we deduce that the functions y

, k ≥ 0,

are well defined and differentiable on [x

− δ, x

+ δ] and that the points

(x, y

(x)) and (x, y

(h(x) + g(y

(x)))) belong to Ω for |x − x

| ≤ δ.

Next, similarly as in the proof of Theorem 1 from [4] we show that

|y

(x) − y

(x)| ≤ L

M

k! (1 + βµM )

|x − x

|

, for k = 1, 2, . . . , for x ∈ [x

− δ, x

+ δ]. To this end, observe that for x ∈ [x

− δ, x

+ δ],

|y

(x) − y

(x)| =    

Z

f(t, y

(h(t) + g(y

(t))))dt    

≤ M |x − x

|,

|y

(x) − y

(x)| ≤    

Z

f(t, y

(h(t) + g(y

(t)))) − f (t, y

(h(t) + y

(t)))))dt    

≤ L    

Z

y

(h(t) + g(y

(t))) − y

(h(t) + g(y

(t)))|dt|

= L    

Z

Z

f (t, y

(h(t) + g(y

(t))))|dt|

≤ LM    

Z

h(t) + g(y

(t)) − x

|dt|

≤ LM (αλ + βµM )    

Z

t − x

|dt| ≤ LM 1

2! |x − x

|

, and

|y

(h(t) + g(y

(t))) − y

(h(t) + g(y

(t))|

≤ |y

(h(t) + g(y

(t))) − y

(h(t) + g(y

(t)))|

+ |y

(h(t) + g(y

(t))) − y

(h(t) + g(y

(t)))|

=    

Z

f (s, y

(h(s) + g(y

(s))))ds     + |y

(h(t) + g(y

(t))) − y

(h(t) + g(y

(t)))|

≤ M |g(y

(t)) − g(y

(t))| + LM

2! |h(t) + g(y

(t)) − x

|

≤ M βµ|y

(t) − y

(t)| + LM

2! (αλ + βµM )

|t − x

|

≤

(1 + βµM )|t − x

|

.

Hence,

|y

(x) − y

(x)| ≤ L    

Z

y

(h(t) + g(y

(t))) − y

(h(t) + g(y

(t)))|dt|

≤ L

M

3! (1 + βµM )|x − x

|

.

The same principle can be used to show that

|y

(x) − y

(x)| ≤ L

M

k! (1 + βµM )

|x − x

|

, k = 1, 2. . . . for x ∈ [x

− δ, x

+ δ]. Since the series

∞

X

k=1

L

M

k! (1 + βµM )

δ

converges by means of the ratio test, we obtain that the infinite sum y(x) = y

(x) + (y

(x) − y

(x)) + (y

(x) − y

(x)) + · · · , x ∈ [x

− δ, x

+ δ], is well defined by the Weierstrss M-test and the sequence of functions

y

= y

+ (y

− y

) + · · · + (y

− y

)

uniformly converges on [x

− δ, x

+ δ] to a continuous limit function y. Ob- serve that the composition x → y

(h(x) + g(y

(x))) also tends to y(h(x) + g(y(x))) uniformly as m tends to ∞ because of

|y

(h(x) + g(y

(x))) − y(h(x) + g(y(x)))|

≤ |y

(h(x) + g(y

(x))) − y

(h(x) + g(y(x)))|

+ |y

(h(x) + g(y(x))) − y(h(x) + g(y(x)))|

=    

Z

f (t, y

(h(t) + g(y

(t))))dt     + |y

(h(x) + g(y(x))) − y(h(x) + g(y(x)))|

≤ M |g(y

(x)) − g(y(x))| + |y

(h(x) + g(y(x))) − y(h(x) + g(y(x)))|.

Moreover,

|f (x, y

(h(x) + g(y

(x))))

(x, y(h(x) + g(y(x))))|

≤ L|y

(h(x) + g(y

(x))) − y(h(x) + g(y(x)))|, hence

f (x, y

(h(x) + g(y

(x)))) → f (x, y(h(x) + g(y(x)))) uniformly on [x

− δ, x

+ δ] as m → ∞. Now, we see that

y(x) = lim

y

(x)

= lim

m→∞

 x

+

Z

f (t, y

(h(t) + g(y

(t))))dt



= x

+ Z

x0

f (t, y(h(t) + g(y(t))))dt

for x ∈ [x

− δ, x

+ δ]. This completes the proof of the existence part.

For the proof of the uniqueness let z : [x

− δ, x

+ δ] be another solution of (1) satisfying z(x

) = x

. Then

z(x) = x

+ Z

x0

f (t, z(h(t) + g(z(t))))dt for x ∈ [x

− δ, x

+ δ].

Furthermore,

|y

(x) − z(x)| ≤    

Z

|f (t, z(h(t) + g(z(t))))|dt    

≤ M |x − x

|,

|y

(x) − z(x)| ≤ L    

Z

|y

(h(t) + g(y

(t))) − z(h(t) + g(z(t)))|dt    

= L    

Z

|

Z

f (s, z(h(s) + g(z(s))))ds|dt    

≤ LM    

Z

|h(t) + g(z(t)) − x

|dt    

≤ LM (αλ + βµM )    

Z

|t − x

|dt    

≤ LM

2! |x − x

|

,

|y

(h(t) + g(y

(t)) − z(h(t) + g(z(t))|

≤ |y

(h(t) + g(y

(t)) − y

(h(t) + g(z(t))|

+ |y

(h(t) + g(z(t)) − z(h(t) + g(z(t))|

≤ M |g(y

(t)) − g(z(t))| + LM

2! (αλ + βµM )|h(t) + g(z(t)) − x

|

≤ M βµ|y

(t) − z(t)| +

|x − x

|

, and

|y

(x) − z(x)| ≤ L    

Z

|y

(h(t) + g(y

(t))) − z(h(t) + g(z(t)))|dt    

≤ L

M

2! (1 + βµM )|x − x

|

for x ∈ [x

− δ, x

+ δ]. By induction, we may show that

|y

(x) − z(x)| ≤ L

M

(n + 1)! (1 + βµM )

|x − x

|

, n = 1, 2, . . . . This means that the sequence (y

) converges to z uniformly on [x

−δ, x

+δ].

So, z(x) = y(x) for x ∈ [x

− δ, x

+ δ] and the proof of the uniqueness part is completed.

3. Additional remarks

Observe the assumption of the continuity of f in Theorem 2.1 implies that the solution y is continuously differentiable. We will prove that in Theorem 2.1 some additional conditions concerning the horizontal sections f

of f imply that the derivative y

of the solution y is (S)-continuous with respect to a local system S.

To this end recall the following definitions ([5], p. 3 and p. 70).

Definition 1. By a local system we mean a family S of subsets of IR such that at each point x ∈ IR there is given a nonempty collection of sets S(x) ⊂ S with the following properties:

(i) the singleton {x} is not in S(x),

(ii) if A ∈ S(x) then x ∈ A,

(iii) if A

∈ S(x) and A

⊃ A

then A

∈ S(x),

(iv) if A ∈ S(x) and δ > 0 then A ∩ (x − δ, x + δ) ∈ S(x).

Definition 2. Let S be a local system, let I ⊂ IR be a nonempty open set and let φ : I → IR be a function. We say that φ is (S)-continuous at a point x provided that for every η > 0 the set

{t ∈ I; |φ(t) − φ(x)| < η} ∈ S(x).

Definition 3. Let S be a local system and let I ⊂ IR be a nonempty open set. A function φ : I → IR is (S)-path continuous at a point x if there is a set A ∈ S(x) such that

t∈A, t→x

lim φ(t) = φ(x).

Similarly we can define the (S)-continuity and the (S)-path continuity of functions defined on nondegenerate closed intervals [r, s].

Observe that the (S)-path continuity implies the (S)-continuity ([5], p. 27), but the converse, does not occur (see [[5], p. 27 and p. 29, Theorem (14.3)]).

Remark 3.1. If in Theorem 2.1 we assume additional condition that the horizontal sections f

, u ∈ [x

− b, x

+ b], are (S)-continuous (resp. (S)- path continuous) then the derivative y

of the solution y is the same on [x − δ, x

+ δ].

P roof. It suffices to observe that the vertical sections f

, x ∈ (x

− a, x

+a), are equicontinuous, and apply Theorem 3 from [3]. For the endpoints x

− δ and x

+ δ we can reason similarly.

Remark 3.2. Let L ≥ 0 be a real and let Φ be the family of all bounded functions f : Ω → IR such that the horizontal sections f

, u ∈ [x

−δ, x

+δ], are derivatives and

|f (x, u) − f (x, v)| ≤ L|u − v| for (x, u), (x, v) ∈ Ω.

If Φ is endowed with the metric ρ

(f

, f

) = min(1, sup

|f

(x, u) −

f

(x, u)| of the uniform convergence then the set Φ

= {f ∈ Φ; f is

continuous} is nowhere dense in Φ.

P roof. Since Φ

is closed in Φ, it suffices to prove that for each f ∈ Φ

and for each real η > 0 there is a function g ∈ Φ \ Φ

with ρ

(f, g) < η.

Fix η > 0 and f ∈ Φ

. Let φ : [x

− a, x

+ a] → [0, 1] be a discontinuous derivative ([1]) and let

g(x, u) = f (x, u) + ηφ(x)

2 for (x, u) ∈ Ω.

Evidently,

g ∈ Φ \ Φ

and ρ

(f, g) ≤ η 2 < η.

References

[1] A.M. Bruckner, Differentiation of real functions, Lectures Notes in Math. 659, Springer-Verlag, Berlin, 1978.

[2] Z. Grande, A theorem about Carath´eodory’s superposition, Math. Slovaca 42 (1992), 443–449.

[3] Z. Grande, When derivatives of solutions of Cauchy’s problem are (S)- continuous?, Tatra Mt. Math. Publ. 34 (2006), 173–177.

[4] W. Li and S.S. Cheng, A Picard theorem for iterative differential equations, Demonstratio Math. 42 (2) 2009, 371–380.

[5] B.S. Thomson, Real Functions, Lectures Notes in Math., Vol. 1170, Springer- Verlag, Berlin, 1980.

Received 17 January 2011