Bounded solutions in a given set of differential systems. (English) Zbl 0944.34012

The authors deal with systems of ordinary differential equations of the form \[ \dot y= Ay+ g(t,y,z),\quad \dot z= h(t,y,z),\tag{1} \] where \(A\) is a hyperbolic \(m\times m\)-matrix (i.e. a real constant matrix with all eigenvalues having nonzero real parts). \(g\) and \(h\) are supposed to be continuous vector functions. Using the continuation method, which was developed by M. Furi and P. Pera [Ann. Pol. Math. 47, 331-346 (1987; Zbl 0656.47052)], the authors prove the existence of at least one bounded (on \(\mathbb{R}\)) solution to (1) lying in a given set. This set is defined by means of strict bounded (on \(\mathbb{R}\)) lower and upper functions to (1), whose existence is assumed. Under the existence of more families of such strict lower and upper functions to (1) a multiplicity result is formulated.


34B15 Nonlinear boundary value problems for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations


Zbl 0656.47052
Full Text: DOI


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