characteristic numbers of non-autonomous emden-fowler type ...

406 A. Gritsans, F. SadyrbaevThen the functionx(t) ={x1 (t), if −1 ≤ t ≤ 0,x 2 (t), if 0 ≤ t ≤ 1is a C 2 -solution of the problem (2.1), (2.2) if additionally the smoothnessconditionx ′ 1(0) = x ′ 2(0)is satisfied. The problem (2.3) can be explicitly resolved as( )x 1 (t, β 1 ) = β 1 21 (ht + η) · sl β 1 t + 121 ,ht + ηwhereandβ 1 = x ′ 1 (−1) > 0x 1 (0; β 1 ) = τ.The derivative is given by( )( )x ′ 1(t; β 1 ) = β 1 21 h · sl β 1 t + 1 −h + η21 + β 1ht + η ht + η · sl′ β 1 t + 121 .ht + ηSimilar formulas are valid for x 2 (t). Notice that x ′ 2 (1) = −β 2 < 0. In orderto get an explicit formula for a solution of the BVP (2.1), (2.2) one have tosolve a system of two equations with respect to (β 1 , β 2 )x 1 (0; β 1 ) = x 2 (0; β 2 ),x ′ 1(0; β 1 ) = x ′ 2(0; β 2 ).This system after replacements and simplifications looks as⎧ ( ) ( )⎪⎨ β 1 β 1 221 · sl 1= β 1 β 1 22η 2 · sl 2,η( ))( )⎪⎩ β 1 β 1 221 h · sl 1+ β 1(β 1 2η η · 1 sl′ = −β 1 β 1 22η2 h · sl 2− β 2(β 1 2η η · 2 sl′ η),where 0 < β 1 2 1η, β 2 1 2η< 2A. In new variables u := β 2 1 1η, v := β 1 2 2ηthe systemtakes the form{ u sl u = v sl v, 0 < u, v < 2A,hu sl u + u 2 sl ′ u = −hv sl v − v 2 sl ′ v, h > 0.(2.4)Notice that if a solution (ū, ¯v) of the system (2.4) exists, then a solution x(t)of the BVP (2.1), (2.2) can be constructed such thatx ′ (−1) = β 1 = ū 2 (h + 1) 2 , x ′ (1) = −β 2 = −¯v 2 (h + 1) 2 .