Singular right focal boundary value problem with given maximal values - Tian, Yu; Ge, Weigao (Tamkang Journal of Mathematics, 2006)
In this paper, we prove existence results for the singular problem (-1)^n-p(Φ_m x^(n-1))' (t)=μ f(t, x(t), …, x^(n-1)(t)), for 0<1, x^(i)(0)=0 , i=0, 1, …, p-1 , x^(i)(1)=0, i=p , p+1, …, n-1 , {x(t):t[0, 1]}=A . The paper presents conditions which guarantee that for any A>0 there exists μ_A>0 such that the above problem with μ=μ_A has a solution xC^n-1([0, 1]) with Φ_m(x^(n-1))AC([0, 1]) which is positive on (0, 1) . Here the positive Carath'edory function f may be singular at the zero value of all its phase variables. Proofs are based on the Leray-Schauder degree and Vitali's convergence theorem.