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A SINGULAR OR NONSINGULAR PERTURBATION PROBLEM FOR A SECOND ORDER NONLINEAR DIFFERENTIAL EQUATION WITH TWO FREE ENDPOINTS
Wang Junyu, Zheng Dawei, Wang Xuekong
Acta mathematica scientia,Series B. 1995, 15 (1):
39-47.
It is proved that for ε ≥ 0 and δ ≥ 0 the two -point boundary value problem -{y'(t)+f(t)+h(t)Ke(t,y(t))}=g(t)z(t),A≤t≤B, z'(t)=Ke(t,y(t)):={(k(t)+ε)/y(t)}1/N,A≤t≤B, y(A)=δ-Pz(A),y(B)=δ+Qz(B), has a unique solution (y(t,ε,δ),z(t,ε,δ)) under certain hypotheses with the aid of the appropriate Green's function integral operator.The unique solution (ξe,ηe,ve(s)) of the free boundary problem {(k(v)+ε)|v'|N-1v'}'+{sg(v)+f(v)}v'+h(v)=0,ε < s < η, v|s=ε=Ae-{(k(v)+ε)|N-1v'}|s=ε=Pε, v|s=η=Be=Be{(k(v)+ε)|v'|N-1v'}s=η=Qη, is constructed utilizing the solution (y(t,ε.0),z(t,ε,0)).The fine boundary problem is shown to be a singular perturbation problem when the function k(t) possesses intervals of degeneracy
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