半直线上三阶多点边值问题共振情形下可解性

Solvability for third-order multi-point boundary value problems at resonance on half-line

运用Mawhin重合度理论,讨论一类半直线上三阶多点边值问题(q(t)x″(t))′=f(t,x(t),x′(t),x″(t)),a.e.t∈[0,+∞);■(η■在dim Ker L=2共振情形下的可解性,获得了该边值问题至少存在一个解的充分条件.这里f:[0,1]×R^(3)→R满足L^(1)[0,+∞)-Carathéodory条件,αi,βj∈R(1≤i≤m,1≤j≤n),0<ξ_(1)<ξ_(2)<…<ξ_(m)<+∞,0<η_(1)<η_(2)<…<η_(n)<+∞(m,n∈Z+),q(t)>0,q(t)∈C[0,+∞)∩C^(2)(0,+∞),1/q(t)∈L^(1)[0,+∞).

By using the coincidence degree theory of Mawhin,the solvability for a class of the third-order multi-point boundary value problems(q(t)x″(t))′=f(t,x(t),x′(t),x″(t)),a.e.t∈[0,+∞);x(0)=∑m i=1αix(ξi),x′(0)=∑n j=1βjx′(ηj),lim t→+∞q(t)x″(t)=0 at resonance with dim Ker L=2 on the half-line is discussed,and a sufficient condition under which there exists at least one solution is obtained,where f:[0,1]×R 3→R satisfies L 1[0,+∞)-Carathéodory conditions,αi,βj∈R(1≤i≤m,1≤j≤n),0<ξ_(1)<ξ_(2)<…<ξ_(m)<+∞,0<η_(1)<η_(2)<…<η_(n)<+∞(m,n∈Z+),q(t)>0,q(t)∈C[0,+∞)∩C^(2)(0,+∞),1/q(t)∈L^(1)[0,+∞).