On Bounded Solutions in a Given Set of Systems of Differential Equations

Entirely bounded solutions in given sets of differential systems are proved sequentially. This enables us to Drove several bounded solutions separated in different domains.

(1.2) tE]R Although many papers deal with the existence problem for entirely bounded solutions of differential systems (see e.g. [A1,AGG] and the references therein), only few results can be used for proving the existence of two or more bounded solutions (see e.g. [A2,AK,K]).
In [AK] the techniques which consist mainly in invariantness of prescribed sets and transversality arguments on their boundaries were applied to derive multiplicity results.In this paper we develop mentioned method for another class of differential systems.For a comparison with analogous results related to the periodic boundary value problem we refer the reader e.g. to the paper [FZ].
2. NOTATIONS AND PRELIMINARIES 2.1.In the whole paper we assume that a,/3, % 6, l, m, n, o are natural numbers and e, w are positive real numbers. 2.2.By {F} we denote a set whose elements satisfy condition F. So, we can write x E {(1.2)} for an entirely bounded solution x_ of the system (1.1).
The general quantifier will be denoted by V and for the existence quantifier we reserve the symbol B. 2.3.If Ad is a subset of some topological space then by cl fr 3//we mean the closure, interior, boundary of M, respectively.
By Vr we denote the gradient of a continuously differentiable real function r.
2.5.For a 4-tuple (a, 3,% 6) and n=a + fl+'7+ 6, define in the sequel, 2.6.Let A4 C ]1m, 6(./, ]1; n) be a set ofall continuous vector functions defined in A//and with values in n.If A//is compact we add the norm Ilqll max Iq()l tO obtain the standard Banach space.If A/[ is not compact we endow the set C(A4, I n) with a topology of the uniform convergence on compact subsets of A4 to make the Fr6chet space.
The following lemmas give appropriate estimates for solutions of scalar equations and differential systems.
Proof For abbreviation, let {_x(t)}t+= stand for the sequence of appropriate extensions of solutions x_t) in .Duet o the well-known Arzelfi-Ascoli theorem and the diagonalization arguments, we are able to choose a subsequence {x_(ti))i+= which converges to a solution_x cl Q of the system (1.1) (for more details, see e.g. [K, pp. 178-180]).
To ensure the solvability of problems (2.17.1), we apply the special form of the Leray-Schauder continuation principle (see e.g. [LS]).
The last statement of this section follows immediately from the result developed in [CFM].
Then A is a continuous map with relatively compact image.
Proof It will be divided into four consecutive steps.Denote sup vi(t) + for a +/3.
k=l Hence, applying Lemmas 2.9, 2.11, 2.13 and 2.15, respectively,   we obtain q(7-) E re(/,/, {7-)) which contradicts the characteristic prop- erty of 7-.Since all assumptions of Proposition 2.18 are fulfilled, the existence of a solution _x(t E//is ensured. C. Recall that g(t, q(t)) =f(t, q(t)), for every q E cl Q and all E .S o, repeating definition (3.1.8)with A= and q=_xq), we can apply conditions (3.1.2),(3.1.3)to obtain x_(t E re(Q, [-l, l]).Since E 1 was chosen arbitrarily (f, Q) satisfies assumptions of Lemma 2.17 and we conclude that (1.1) admits a solution x__ cl Q.To finish the proof it is sufficient to confirm D. If there exists a solution x_ cl Q of(1.1) then x Q.
Therefore we can see that conditions (3.4.1) and (3.4.2) do not mean any restriction on O1,O2 in comparison with the assumptions of Theorem 3.1.