Resolution of Semilinear Equations by Fixed Point Methods

P. Amster and M. C. Mariani

Abstract: We give conditions in order to obtain solutions of quasilinear systems with periodic type conditions. Our main tool will be the use of fixed point theorems.

Introduction

In this work we will study some special cases of ordinary semilinear differential equations of the type X′ = F (t, X) , with boundary conditions X(0) = g(X(α)) .

The periodic problem may be regarded as a particular case, when g = I. In this case under some conditions and F Lipschitzian it is possible to obtain solutions by finding a fixed point of the Poincaré operator (see e.g. [8]).

Existence and uniqueness results for second order differential equations and systems with periodic conditions are given in [1], [2], [3], [4], [5], [6], [8], [9].

Existence by Fixed Point methods

We’ll study the system

(2) \{ \begin{matrix}X′ = F (t, X)  in (0, α) \\ X(0) = g(X(α)) \end{matrix}

where F : [0, α] × R^{m} ⟶ R^{m} and g : R^{m} ⟶ R^{m} are continuous.

Let us define

F_{M} = sup_{t∈[0,α],|x|≤M} |F (t, x)|

g_{M} = sup_{|x|≤M} |g(x)|

B_{M} will denote the closed ball of radius M centered in 0 in the space C([0, α], R^{m}) .

Theorem 1

If \frac{g_{M}}{M} + α\frac{F_{M}}{M} ≤ 1 then (2) admits a solution in B_{M} . Furthermore, if F and g are Lipschitz with constants K_{F} and K_{g}, K_{g} +K_{F} α < 1 , (2) has a unique solution.

\underline{Proof} We consider the continuous operator

(3)  T X(t) = g(X(α)) + ∫_{0}^{t} F (s, X(s))ds

If ∥X∥_{∞} ≤ M then

|g(X(α)) + ∫_{0}^{t} F (s, X(s))ds| ≤ g_{M} + αF_{M}

On the other hand

|T X(t_{1}) − T X(t_{2})| ≤ |t_{2} − t_{1}|F_{M}

By Arzela-Ascoli, we conclude that T is compact, and being \frac{g_{M}}{M} + α\frac{F_{M}}{M} ≤ 1 , T (B_{M}) ⊂ (B_{M}) . By Schauder Theorem (see e.g. [7]), we conclude that T has a fixed point in B_{M} .

When F and g are Lipschitz with constants K_{F} , K_{g}, T is a contraction for K_{g} + K_{F} α < 1 .

Remarks i) If g has continuous inverse, we may consider the operator

T X(t) = g^{−1}(X(0)) − ∫_{t}^{α} F (s, X(s))ds

and get solutions of (2) under the same conditions of Theorem 1 for g^{−1} . In particular, if g is a linear isomorphism, ∥g∥ ≠ 1 , the system (2) admits a solution in B_{M} when

∥g∥ + α\frac{F_{M}}{M} ≤ 1 or ∥g^{−1}∥ + α\frac{F_{M}}{M} ≤ 1 .

ii) For constant g, Theorem 1 gives a proof of the well known existence result for ordinary equations with Cauchy data.

For linear g, existence may be obtained from a different operator if I − g is invertible:

Theorem 2

Let g be linear such that I − g is invertible. We consider G = B + φI, where B = (I − g)^{−1}g and φ is defined by

φ(t, s) = \{ \begin{matrix}1  if t ≥ s \\ 0  if t < s \end{matrix}

and assume, for a certain M , that ∫_{0}^{α} |G(t, s)|ds\frac{F_{M}}{M} ≤ 1 for all t. Then (2) admits a solution in B_{M} .

Furthermore, if F is Lipschitz with constant K, ∫_{0}^{α} |G(t, s)|dsK < 1 , (2) has a unique solution. \underline{Proof}

For any X ∈ C([0, α], R^{m}) , we define
X_{0} = ∫_{0}^{α} BF (s, X)ds

and

T X(t) = X_{0} + ∫_{0}^{t} F (s, X)ds = ∫_{0}^{α} G(t, s)F (s, X)ds

As in the previous theorem, T is compact and, for ∥X∥_{∞} ≤ M ,

∥T X∥_{∞} ≤ ∫_{0}^{α} |G(t, s)|dsF_{M} ≤ M

By Schauder Theorem, T has a fixed point in B_{M} .

Moreover, if F is Lipschitz, T is a contraction.

As simple consequence we obtain the following result for g = kI, improving Theorem 1 when k ≤ 0 :

Corollary 3

Let k ≠ 1 , g = kI, and c = inf_{M>0} \frac{F_{M}}{M} . Then the problem (2) admits a solution in C([0, α], R^{m}) in the following cases:

i) |k| ≥ 1 , cα < \frac{k − 1}{k}

ii) |k| < 1 , cα < 1 − k

In particular, if \frac{F_{M}}{M} ⟶ 0 , then for any k ≠ 1 (2) admits a solution in C([0, α], R^{m}) .

\underline{Proof} It is immediate in this case that

G(t, s) = \{ \begin{matrix}\frac{1}{1−k}  if t ≥ s \\ \frac{k}{1−k}  if t < s \end{matrix}

and a simple computation shows that

∫_{0}^{α} |G(t, s)|ds ≤ \frac{k}{k − 1} if |k| ≥ 1

y

∫_{0}^{α} |G(t, s)|ds ≤ \frac{1}{1 − k} if |k| < 1

Remark

For n > 1 assumption \frac{F_{M}}{M} ⟶ 0 is not appliable to the equation u^{(n)} = f(t, u, ..., u^{(n−1)}) .

For the periodic problem, which is not contemplated in the results above, we have the following criteria:

Theorem 4 If X_{n} is a bounded sequence in C([0, α], R^{m}) such that

\{ \begin{matrix}X′_{n} = F_{n}(t, X_{n})  in (0, α) \\ X_{n}(0) = g_{n}(X_{n}(α_{n})) \end{matrix}

with linear g_{n}, and continuous F_{n} such that g_{n} ⟶ I, F_{n} ⟶ F , and α_{n} ⟶ α ( α_{n} ≤ α). Then the periodic problem admits a solution in C([0, α], R^{m}) . \underline{Proof}

We consider the same operator as in (3) for g = I, then

(T X_{n})′ = F (t, X_{n}) = F (t, X_{n}) − F_{n}(t, X_{n}) + X′_{n}

and

(T X_{n} − X_{n})(t) = (T X_{n} − X_{n})(0) + ∫_{0}^{t} F (s, X_{n}) − F_{n}(s, X_{n})

Being T compact we may suppose that T X_{n} ⟶ X. Moreover, taking K compact big enough, we obtain:

| ∫_{0}^{t} F (s, X_{n}) − F_{n}(s, X_{n})| ≤ α∥F − F_{n}∥_{∞,K} ⟶ 0

and

(T X_{n}−X_{n})(0) = X_{n}(α)−g_{n}(X_{n}(α_{n})) = (I−g_{n})(X_{n}(α))+g_{n}(X_{n}(α)−X_{n}(α_{n})) ⟶ 0

since X_{n} is bounded, I − g_{n} ⟶ 0 and X_{n}(α) − X_{n}(α_{n}) = ∫_{α_{n}}^{α} F_{n}(s, X_{n}) ⟶ 0 . Then X_{n} ⟶ X, and X is a fixed point of T .

Theorem 5 Let us assume that the system

( 4_{r}) \{ \begin{matrix}X′ = \frac{1}{r}F (t, X)  in (0, α) \\ X(0) = \frac{1}{r}X(α). \end{matrix}

has no solution in ∂B_{M} for any r ∈ (1, 1 + α\frac{F_{M}}{M}] . Then the periodic problem (r=1) admits a solution in B_{M} .

\underline{Proof} We consider the same compact operator as in (3) for g = I, and define

T^{∗}X = \{ \begin{matrix}T X  if ∥T X∥_{∞} ≤ M \\ \frac{MT X}{∥T X∥_{∞}} if ∥T X∥_{∞} ≥ M. \end{matrix}

T^{∗} : B_{M} ⟶ B_{M} is compact and in consequence it has a fixed point X. If X is not a fixed point of T , then ∥X∥_{∞} = M and T X = rX, with

r = \frac{∥T X∥_{∞}}{M}

Then the system ( 4_{r}) has a solution in ∂B_{M} , and 1 < r ≤ 1 + α\frac{F_{M}}{M} .

Example

Let F be continuous with F (t, x).x < 0 for any (t, x) ∈ [0, α] × R^{m} such that |x| = M . Then any solution of ( 4_{r}) verifies that \frac{1}{2}(X.X)′ = X′.X = \frac{1}{r}F (t, X).X < 0 when |X(t)| is close to M . We conclude that if |X(0)| < M then ∥X∥_{∞} < M . On the other hand, if |X(0)| = M , |X(α)| = r|X(0)| > M . By theorem 5, the periodic problem admits a solution in B_{M} .

Uniqueness for the problem (2)

In theorems 1 and 2 we obtained uniqueness for problem (2) when F and g are Lipschitz with small constants. Now we’ll prove uniqueness under some other assumptions:

Theorem 6

Let g be linear and F continuously differentiable with respect to X and for every (t, x) ∈ (0, α) × R^{m} let A(t, x) denote the matrix D_{x}F (t, x) . Then (2) has at most one solution in any of the following cases:

i) ∥g∥ < 1 y A(t, x) ≤ 0 for any (t, x) ∈ (0, α) × R^{m}.

ii) g invertible, ∥g^{−1}∥ < 1 y A(t, x) ≥ 0 for any (t, x) ∈ (0, α) × R^{m}.

iii) g isometric, A(t, x) > 0 (or A(t, x) < 0 ) for any (t, x) ∈ (0, α) × R^{m}, x ≠ 0 .

\underline{Proof}

Let us suppose that X and Y are solutions of (2) and take Z = Y − X. Then, Z′.Z = (F (t, Y) − F (t, X))Z, and applying for fixed t mean value theorem to φ(u) = F (t, uY + (1 − u)X)Z we see that

(Z.Z)′ = 2A(t, ψ)Z.Z

for a certain ψ(t) .

Assuming i) we obtain that |Z| = (Z.Z)^{1/2} decreases in (0, α) , and the result follows since |Z(0)| = |g(Z(α))| < |Z(α)|. Under condition ii) the proof is analogous, considering Z(α) = g^{−1}(Z(0)) . If we assume iii), we obtain that |Z| is monotone, and if Z(t_{0}) ≠ 0 then |Z| is strictly monotone in a neighborhood of t_{0} , a contradiction.

ACKNOWLEDGEMENT The authors thank specially Prof. J. Mawhin for his careful reading of the man-

uscript and his fruitful suggestions and remarks.

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P.Amster Dpto. de Matemática Fac. de Cs. Exactas y Naturales, UBA Pab. I, Ciudad

Universitaria (1428) Capital, Argentina CONICET M. C. Mariani Dpto. de Matemática Fac. de Cs. Exactas y Naturales, UBA Pab. I, Ciudad

Universitaria (1428) Capital, Argentina CONICET Address for correspondence: Prof. M. C. Mariani, Dpto. de Matemática

Fac. de Cs. Exactas y Naturales, UBA Pab. I, Ciudad Universitaria (1428) Capital, Argentina

E-mail: pamster@dm.uba.ar - mcmarian@dm.uba.ar