三阶微分方程组非局部边值问题多个正解的存在性

THREE POSITIVE SOLUTIONS OF SINGULAR NONLOCAL BOUNDARY VALUE PROBLEMS FOR SYSTEMS OF NONLINEAR THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS

利用Leggett--Williams不动点定理,研究下列三阶微分方程组边值问题{u′″(t)+a_1(t)f_1(t,u(t),v(t))=0,0<t<1,v′″(t)+a_2(t)f_2(t,u(t),v(t))=0,0<t<1,u'(0)=u″(0)=0,u(1)=g_1(∫_0~1u(s)dф_1(s),∫_0~1v(s)dф_1(s)),v'(0)=v″(0)=0,v(1)=g_2(∫_0~1u(s)dф_2(s),∫_0~1v(s)dф_2(s))多个正解的存在性,其中a_i∈C((0,1),[0,+∞)),f_i,g_i∈C([0,1]×[0,+∞)×[0,+∞)→[0,+∞)),∫_0~1u(s)dфi(s),∫_0~1v(s)dфi(s)是Riemann-Stiltjes积分,i=1,2.

We consider the following system of third-order singular nonlocal boundary value problems {u′″（t）＋a_1（t）f_1（t,u（t）,v（t））=0,0t1,v′″（t）＋a_2（t）f_2（t,u（t）,v（t））=0,0t1,u＇（0）=u″（0）=0,u（1）=g_1（∫_0~1u（s）dф_1（s）,∫_0~1v（s）dф_1（s））,v＇（0）=v″（0）=0,v（1）=g_2（∫_0~1u（s）dф_2（s）,∫_0~1v（s）dф_2（s））.The proofs of our main results are based upon the Leggett-Williams fixed point theorem,we establish the existence of three positives solutions for the system.