L. Malaguti^{∗}, C. Marcelli^{∗∗}, and N. Partsvania^{∗∗∗}

∗University of Modena and Reggio Emilia, Reggio Emilia, Italy ∗∗University of Ancona, Ancona, Italy

∗∗∗A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ON TRANSITIONAL SOLUTIONS OF SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS

We consider the differential equation

u′′ = f(t, u, u′),  (1)

where f : R × [0, 1] × R → R is a continuous function such that f(t, 0, 0) = f(t, 1, 0) = 0 for t ∈ R, and for this equation we study the problem on the existence of at least one so-called transitional solution, i. e. a solution satisfying the conditions

lim_{t→−∞} u(t) = 0, lim_{t→+∞} u(t) = 1, 0 ≤ u(t) ≤ 1 for t ∈ R.  (2)

On the basis of I. Kiguradze’s theorem from [1] new and in a certain sense optimal conditions are obtained for the solvability of problem (1), (2). Some of these results generalize the previous ones from [2] and [3].

References

1. I. T. Kiguradze and B. L. Shekhter, Singular boundary value problems for second-order ordinary differential equations. (Russian) Itogi Nauki i Tekhniki, Sovrem. Probl. Mat., Noveĭshie Dostizh. 30(1987), 105–201; translated in J. Soviet Math. 43(1988), No. 2, 2340–2417.

2. Yu. A. Klokov, Boundary value problems with a condition at infinity for equations of mathematical physics. (Russian) Riga, 1963.

3. L. Malaguti and C. Marcelli, Heteroclinic orbits in plane dynamical systems. Arch. Math. (Brno) 38(2002), No. 3, 183–200.