Method of Guiding Functions in Finite-Dimensional Spaces
In this section we present the guiding functions method for studying the periodic problem for a differential inclusion in a finite-dimensional space.
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Method of Guiding Functions in Finite-Dimensional Spaces
2.1 Periodic Problem for a Differential Inclusion In this section we present the guiding functions method for studying the periodic problem for a differential inclusion in a finite-dimensional space. We start considering a differential inclusion in a finite-dimensional space Rn of the following form: x 0 .t/ 2 F .t; x .t// ;
a:e:
t 2 Œ0; T
(2.1)
where F W Œ0; T Rn ! Kv .Rn / is an L1 -upper Caratheodory K multimap. By a solution of inclusion (2.1) we mean an absolutely continuous function x W Œ0; T ! Rn satisfying (2.1) for a.e. t 2 Œ0; T . It is well known (see, e.g., [25, 80]) that the L1 -upper Caratheodory K condition implies the existence of a local solution to the Cauchy problem corresponding to (2.1), i.e., a solution defined on some interval Œ0; h; 0 < h T and satisfying the initial condition x.0/ D x0 2 Rn :
(2.2)
To guarantee the existence of a global solution .h D T / it is sufficient to strengthen the condition posed on the multimap F supposing that F is an L1 -upper Caratheordory K multimap with ˛-sublinear growth. More precisely, the following assertion holds (see, e.g., [24, 39, 64, 80]) K Proposition 2.1. If F W Œ0; T Rn ! Kv .Rn / is an L1 -upper Caratheodory multimap with ˛-sublinear growth, then for each x0 2 Rn , the solution set ˘F .x0 / of the Cauchy problem x 0 .t/ 2 F .t; x.t//
a:e:
t 2 Œ0; T
x.0/ D x0
V. Obukhovskii et al., Method of Guiding Functions in Problems of Nonlinear Analysis, Lecture Notes in Mathematics 2076, DOI 10.1007/978-3-642-37070-0 2, © Springer-Verlag Berlin Heidelberg 2013
(2.3) (2.4)
25
26
2 Method of Guiding Functions in Finite-Dimensional Spaces
is an Rı -set in the space C.Œ0; T I Rn / endowed with the usual norm of uniform convergence. Moreover, the multimap ˘F W Rn ! K.C.Œ0; T I Rn //, x ( ˘F .x/ is u.s.c. Now, we say that a solution x to differential inclusion (2.1) is T -periodic if it satisfies the following boundary value condition of periodicity x .0/ D x .T / :
(2.5)
It is clear that such function can be extended to a T -periodic solution defined on R provided that F is T -periodic, i.e., the multimap F W R Rn ! Kv.Rn / satisfies F .t C T; / D F .t; / for all t 2 Œ0; T . In order to study periodic problem (2.1), (2.5) we can introduce the translation multioperator along the trajectories of (2.1), (2.5) in the following way. For any t 2 Œ0; T let t W C.Œ0; T I Rn / ! Rn be the evaluation map defined as t .y/ D y.t/ : Then the multioperator PFt W Rn ( Rn given as the composition PFt .x/ D t ı ˘F .x/ ; is called the translation multioperator along the trajectories of problem (2.1), (2.5), or simply, the translation multioperator. The following assertion is evident Proposition 2.2. Periodic problem (2.1), (2.5) has a solution if and only if the corresponding translation multioperator PFT has a fixed point x 2 Rn , x 2 PFT .x /. As a direct consequence of Propositions 2.1 and 2.2 we get the following general existence result for problem (2.1), (2.5). Theor
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