Nonlinear Oscillations, Vol. 11, No. 2, 2008

SYSTEMS OF NONLINEAR DIFFERENTIAL EQUATIONS WITH BOUNDED SOLUTIONS I. M. Hrod

UDC 517.9

We establish sufficient conditions for the existence of solutions bounded on R for the equation dx = F (t, x)x + f (t), dt

t ∈ R,

in a finite-dimensional Banach space E.

1. Statement of the Problem Let E be a finite-dimensional Banach space with norm  · E , let C0 be the Banach space of functions x = x(t) bounded and continuous on R = (−∞, ∞) and taking values in the space E with the norm xC0 = sup x(t)E , t∈R

and let C1 be the Banach space of all functions x ∈ C0 for which dx ∈ C0 dt with the norm xC1

 = max xC0 ,

    dx     dt  0 . C

Let F (t, x) denote a function defined and continuous on R × E and taking values in L(E, E) (L(E, E) is the Banach space of all linear continuous operators acting in the space E). Consider the nonlinear differential equation dx = F (t, x)x + f (t), dt

t ∈ R,

f (t) ∈ C0 .

(1)

Of special importance is the problem of the existence of solutions of Eq. (1) in the space C1 for every function f ∈ C0 . We give sufficient conditions for the existence of at least one solution x ∈ C1 of Eq. (1) if f ∈ C0 . Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine; e-mail: [email protected]. Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 160–167, April–June, 2008. Original article submitted August 23, 2006; revision submitted April 20, 2007. 168

1536–0059/08/01102–0168

c 2008 

Springer Science+Business Media, Inc.

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2. Main Result Assume that the following conditions are satisfied for Eq. (1): (i) F (t, x) depends continuously on (t, x) ∈ R × E, and sup

lim

u−v→0 t∈R, u ≤r, v ≤r E E

  F (t, u) − F (t, v) =0 L(E,E)

for every r > 0; (ii) for every r > 0, one has sup

(t,x)∈R×B[0,r]

F (t, x)L(E,E) < ∞,

  where B[0, r] = x ∈ E : xE ≤ r is a closed ball of radius r. We introduce operators L : C1 → C0 and Ly : C1 → C0 , y ∈ C0 , by the equalities (Lx)(t) =

dx(t) − F (t, x(t)) x(t), dt

t ∈ R,

(Ly x)(t) =

dx(t) − F (t, y(t)) x(t), dt

t ∈ R,

where t ∈ R and x ∈ C1 . Note that, since F (t, x) is continuous on R × E and the space E is finite-dimensional, the operators L and Ly defined above are continuous and bounded on C1 . Moreover, Ly is a linear operator for every fixed y ∈ C0 . Let R(L) and R(Ly ) denote the ranges of values of the operators L and Ly , respectively. It is clear that Eq. (1) has at least one solution for every f ∈ C0 if and only if R(L) = C0 .

(2)

Assume that the following conditions are satisfied for the operator (Ly x): (iii) for every y ∈ C0 , the operator Ly : C1 → C0 has the inverse (Ly )−1 : C0 → C1 ;   (iv) sup (Ly )−1 L(C0 ,C1 ) < +∞. y∈C0

We now formulate the main theorem. Theorem 1. Suppose that the function F (t, x) is such that conditions (i)–(iv) are satisfied. Then the differential equation (1) has at least one solution x ∈ C1 for every function f ∈ C0 .

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3. Auxiliary Statements Consider the equation dx = F (t, y(t))x + f (t), dt

t ∈ R,

(3)

where y = y(t) is an arbitrary element of the space C0 . Since this equation is linear, we can apply the theory presented, e.g., in [1–3] to this equation. By virtue of condition (iii), the unique solution x ∈ C1 of Eq. (3) that corresponds to a function f ∈ C0 can be represented by using the operator (Ly )−1 in the form x = (Ly )−1 f.

(4)

Further, assuming that a function f ∈ C0 is fixed, we consider the mapping Uf : C0 → C0 that associates every element y ∈ C0 with the element (Ly )−1 f of the same space. It is obvious that this mapping is defined by the equality Uf = (Ly )−1 f,

(5)

where y ∈ C0 . We now give several properties of the mapping Uf , f ∈ C0 . Lemma 1. The operator Uf : C0 → C0 is continuous for every f ∈ C0 . Proof. We fix arbitrary elements y = y(t) and z = z(t) of the space C0 and consider the differential equations   dx = F t, y(t) x + f (t), dt

t ∈ R,

  dx = F t, z(t) x + f (t), dt

t ∈ R.

Let xy = xy (t) and xz = xz (t) be the corresponding solutions of these equations, i.e.,   dxy (t) ≡ F t, y(t) xy (t) + f (t), dt

t ∈ R,

  dxz (t) ≡ F t, z(t) xz (t) + f (t), dt

t ∈ R.

(6)

Then xy = (Ly )−1 f = Uf y

(7)

xz = (Lz )−1 f = Uf z.

(8)

and

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We represent (6) in the form

  dxz (t) − F t, y(t) xz (t) ≡ F (t, z(t)) − F (t, y(t)) xz (t) + f (t), dt

t ∈ R.

This yields xz = (Ly )−1 (Bz,y xz + f ) = (Ly )−1 f + (Ly )−1 Bz,y xz = (Ly )−1 f + (Ly )−1 Bz,y (Lz )−1 f, where Bz,y xz is defined by the equality (Bz,y w) (t) = [F (t, z(t)) − F (t, y(t))] w(t). By virtue of (7) and (8), we obtain Uf z − Uf y = (Ly )−1 Bz,y Uf z.

(9)

Further, we consider an arbitrary sequence (yn )n≥1 of elements yn , n ≥ 1, of the space C0 for which lim yn − yC0 = 0.

(10)

Uf yn − Uf y = (Ly )−1 Byn ,y Uf yn

(11)

n→∞

Using (9), we get

for all n ≥ 1. According to (10), with regard for condition (iv) and equality (5), we can conclude that there exists a number a > 0 such that sup Uf yn C0 ≤ a. n≥0

Taking into account that [F (t, yn ) − F (t, y)] Uf yn C0 = sup [F (t, yn (t)) − F (t, y(t))] (Uf yn )(t)E t∈R

≤ sup F (t, yn (t)) − F (t, y(t))L(E,E) Uf yn C0 t∈R

≤ a sup F (t, yn (t)) − F (t, y(t))L(E,E) , t∈R

n ≥ 1,

the operator function F (t, x) is continuous on R × E, and the Banach space E is finite-dimensional, by virtue of (10) we obtain lim sup F (t, yn (t)) − F (t, y(t))L(E,E) = 0.

n→∞ t∈R

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Thus, according to condition (iv), we get lim Byn ,y Uf yn C0 = 0

n→∞

and   lim (Ly )−1 Byn ,y Uf yn C0 = 0.

n→∞

Using these results and (11), we obtain lim Uf yn − Uf yC0 = 0.

n→∞

(12)

Thus, if relation (10) is true, then equality (12) is also true. This means that the mapping Uf : C0 → C0 is continuous at the point y ∈ C0 . Since the point y ∈ C0 has been chosen arbitrarily, Uf is continuous on C0 for every f ∈ C0 . Lemma 1 is proved. Further, we note that, according to condition (iv), there exists a certain finite number d such that   d = sup (Ly )−1 L(C0 ,C1 ) . y∈C0

Using this result, we write the relation     Uf yC0 = (Ly)−1 f C0 ≤ (Ly)−1 L(C0 ,C1 ) f C0 ≤ d f C0 , which is true for all y ∈ C0 and f ∈ C0 by virtue of (5). The following statement is true:    Lemma 2. The closed ball B 0, df C0 df C0 is the radius of this ball is invariant under the action of the operator Uf , i.e., Uf yC0 ≤ df C0 for y ≤ df C0 . Remark 1. According  to the examples given in [4], the operator Uf may not be completely continuous on the closed ball B 0, df C0 , and, hence, the Schauder fixed-point theorem [5] cannot be applied to it. We now introduce several necessary definitions. Definition 1. A sequence xk ∈ C1 , k ∈ N, is called locally convergent to an element x ∈ C1 as k → ∞ if this sequence is bounded and

lim max xk (t) − x(t) = 0 k→∞ |t|≤p

for each p ∈ N.

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Definition 2. An operator G : C1 → C0 is called c-continuous if, for arbitrary functions y ∈ C1 and yk ∈ C1 , k ∈ N, such that loc., C0

yk −−−−→ y

as

k → ∞,

one has loc., C0

Gyk −−−−→ Gy as k → ∞. Note that the notion of the c-continuity of an operator in the space C0 was introduced in terms of “ ε, δ ” by Mukhamadiev in [6], and the definition presented above was given in [7]. Lemma 3. The mapping Uf : C0 → C1 , f ∈ C0 , is c-continuous. Proof. Note that, by virtue of the c-continuity of the operator Ly and the fact that the space E is finite-dimensional, the operator (Ly )−1 : C0 → C1 is c-continuous [6–9]. Now consider an arbitrary y ∈ C0 and an arbitrary sequence yk ∈ C0 , k ∈ N, for which loc., C0

yk −−−−→ y

as k → ∞.

(13)

Let xy ∈ C1 and xyk ∈ C1 , k ∈ N, be functions such that   dxy ≡ F t, y(t) xy (t) + f (t) dt

(14)

and   dxyk ≡ F t, yk (t) xyk (t) + f (t). dt We represent the second relation in the form    dxyk ≡ F t, y(t) xyk (t) + F (t, yk (t)) − F (t, y(t)) xyk (t) + f (t). dt

(15)

According to conditions (iii) and (iv) and relations (18) and (19), we get xy = (Ly )−1 f and xyk = (Ly )−1 (f +Byk ,y xyk ),

k ≥ 1.

By virtue of relation (5) and Lemma 2, the sequence (xyk )k≥1 is bounded. Therefore, according to relation (15), the condition of the continuity of F (t, x) on R × E, and the fact that the space E is finite-dimensional, we get loc., C0

Byk ,y xyk −−−−→ 0 as k → ∞.

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Therefore, loc., C0

f + Byk ,y xyk −−−−→ f

as k → ∞.

With regard for the c-continuity of the operator (Ly )−1 : C0 → C1 , we obtain loc., C0

xyk −−−−→ xy

as k → ∞.

Note that, by virtue of (5), we have xy = Uf y = (Ly )−1 f and xyk = Uf yk = (Lyk )−1 f,

r ≥ 1.

Hence, loc., C0

Uf yk −−−−→ Uf y

as k → ∞,

which proves Lemma 3. Lemma 4. The mapping Uf : C0 → C0 , f ∈ C0 , is c-completely continuous. Indeed, by virtue of condition (iii) and equality (5), not only the equality Uf yC0 ≤ df C0 holds for all y ∈ C0 and f ∈ C0 , but also the inequality Uf yC1 ≤ df C0 ,

y, f ∈ C0 ,

is true. Therefore, we have    d(Uf y)(t)   ≤ d f  0 sup  C   dt t∈R for an arbitrary function y ∈ C0 . The analysis of the last inequalities shows that the set   (Uf y)|[a,b] : y ∈ C0 is bounded and jointly continuous in C0 [a, b] for arbitrary a, b ∈ R. This means that this set is precompact in C0 [a, b] (Arzel`a theorem [5]). Here, z|[a,b] is the restriction of a function z = z(t) to a segment [a, b], a < b. With regard for the c-continuity of the mapping Uf : C0 → C0 , f ∈ C0 , this implies that these mappings are c-completely continuous. Lemma 4 is proved.

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4. Proof of Main Theorem By virtue of (4) and (5), it is clear that each fixed point of the mapping U is a bounded solution of Eq. (1). On the other hand, taking into account the lemmas proved, one can verify that the fact that the ball B[0, r] is invariant under the action of the c-completely continuous operator Uf : C0 → C0 , i.e., Uf B[0, r] ⊂ B[0, r], implies that there exists a nonempty set of fixed points of the mapping Uf in B[0, r], which means that Theorem 1 is true. 5. Stability of Solutions of Eq. (1) in the Space C 0 under Small Perturbations Consider the perturbed differential equation  dx  = F (t, x) + B(t, x) x + f (t), dt

(16)

where B(t, x) is a certain function sufficiently small in norm, defined and continuous on R × E, and taking values in L(R × E). The following statement is true: Theorem 2. Suppose that the function F (t, x) in Eq. (16) is such that the conditions of Theorem 1 are satisfied. Then there exists a sufficiently small ε0 > 0 such that the differential equation (16) has at least one solution x ∈ C1 for every function f ∈ C0 if B(t, x)L(E,E) ≤ ε0 . Proof. We introduce the auxiliary operator (Ly,B )x =

    dx − F t, y(t) x − B t, y(t) x. dt

  For every y ∈ C0 , denoting (Bx)(t) = B t, x(t) x(t), we obtain (Ly )−1 Ly,B = (Ly )−1 (Ly − B) = I − (Ly )−1 B.

(17)

It is clear that if ε0 is a sufficiently small number, then we can assume that there exists a number α < 1 for which   (Ly )−1 B  0 1 ≤ α. L(C ,C ) Therefore, the operator I − (Ly )−1 B has the continuous inverse. In view of (17), this enables us to write  −1 , (Ly,B )−1 Ly = I − (Ly )−1 B or  −1 (Ly,B )−1 = I − (Ly )−1 B (Ly )−1 .

(18)

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Further, taking inequality (18) into account, we obtain ∞    (Ly,B )−1  0 1 ≤ αi = L(C ,C ) i=0

1 . 1−α

Thus, the following estimate is true:   (Ly,B )−1  0 1 ≤ L(C ,C )

 1  (Ly )−1  0 1 ≤ b, L(C ,C ) 1−α

where b is a certain finite number. Using this inequality and the proof of the main theorem, we establish that Theorem 2 is true. REFERENCES 1. Yu. A. Mitropol’skii, A. M. Samoilenko, and V. L. Kulik, Investigation of Dichotomy of Linear Systems of Differential Equations Using Lyapunov Functions [in Russian], Naukova Dumka, Kiev (1990). 2. Yu. M. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1970). 3. J. L. Massera and J. J. Schaffer, Linear Differential Equations and Function Spaces, Academic Press, New York (1966). 4. I. M. Hrod, “Conditions for the existence of bounded solutions of one class of nonlinear differential equations,” Ukr. Mat. Zh., 58, No. 3, 317–325 (2006). 5. L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences, New York University, New York (1974). ´ Mukhamadiev, “On the invertibility of functional operators in the space of functions bounded on the axis,” Mat. Zametki, 11, No. 3, 6. E. 269–274 (1972). 7. V. Yu. Slyusarchuk, Invertibility of Nonlinear Operators [in Ukrainian], Rivne Technical University, Rivne (2002). 8. V. E. Slyusarchuk, “Necessary and sufficient conditions for the invertibility of uniformly c-continuous functional differential operators,” Ukr. Mat. Zh., 41, No. 2, 201–205 (1989). 9. V. E. Slyusarchuk, “P-continuous operators and their application to the solution of problems of mathematical physics,” in: Integral Transformations and Their Application to Boundary-Value Problems [in Russian], Issue 15, Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, (1997), pp. 188–226.