## 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.

### MSC:

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

Zbl 0656.47052
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### References:

 [1] J. Andres, Almost-periodic and bounded solutions of Carathéodory differential inclusions, Differential and Integral Equations, to appear. [2] Andres, J., Multiple bounded solutions of differential inclusions: the Nielsen theory approach, Journal of differential equations, 155, 285-310, (1999) · Zbl 0940.34008 [3] J. Andres, G. Gabor, L. Gorniewicz, Boundary value problems on infinite intervals, Trans. Amer. Math. Soc., to appear. · Zbl 0936.34023 [4] Andres, J.; Krajc, B., Unified approach to bounded, periodic and almost periodic solutions of differential systems, Ann. math. sil., 11, 39-53, (1997) · Zbl 0899.34029 [5] Cecchi, M.; Furi, M.; Marini, M., On continuity and compactness of some nonlinear operators associated with differential equations in non-compact intervals, Nonlinear anal. TMA, 9, 2, 171-180, (1985) · Zbl 0563.34018 [6] B.P. Demidovitch, Lectures on the Mathematical Stability Theory, Nauka, Moscow, 1967 (in Russian). [7] Fernandez, M.L.C.; Zanolin, F., On periodic solutions, in a given set, for differential systems, Riv. mat. pura appl., 8, 107-130, (1991) · Zbl 0725.34039 [8] Furi, M.; Pera, M.P., A continuation method on locally convex spaces and applications to ordinary differential equations on noncompact intervals, Ann. polon. math., 47, 331-346, (1987) · Zbl 0656.47052 [9] Gaines, R.E.; Santanilla, J.M., A coincidence theorem in convex sets with applications to periodic solutions of ordinary differential equations, Rocky mountain J. math., 12, 669-678, (1982) · Zbl 0508.34030 [10] M.A. Krasnosel’skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. [11] M.A. Krasnosel’skii, The Operator of Translation along the Trajectories of Differential Equations, Nauka, Moscow, 1966 (Russian). [12] Santanilla, J., Some coincidence theorems in wedges, cones and convex sets, J. math. anal. appl., 105, 357-371, (1985) · Zbl 0576.34018
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