Two-point boundary value problem of integro-differential equations of mixed type in Banach spaces (Q1431062)

Let \(E\) be a real Banach space with cone \(P\). Denote \(I=[0,1]\) and \(C(I,E)=\{u(t):I\to E\}, u(t)\) is continuous, and \(C^2(I,E)=\{u(t):I\to E\}, u(t)\) possesses a continuous second-order derivative. In Banach space \(E\) the author studies the two-point boundary value problem of the following second-order integro-differential equation \[ -u^{\prime\prime}=f(t,u,T_1u,T_2u), \quad t\in [0,1]; \quad u(0)=x_0,\;u(1)=x_1, \] where \(T_ju (t)=g_j(t)+K_ju (t)\), and \(K_j\), \(j=1,2\) are linear integral operators on \([0,1]\) with kernels \(K_j(t,s)\) moreover \(K_1(t,s)=0, t<s\), and \(x_m\in E, m=0,1\). Under some conditions on \(f\), the existence and uniqueness of the solution for the boundary value problem above is studied.