Differentiability of solutions of boundary value problems with respect to data (Q5935533)

The author proves that the dependence of solutions to the boundary value problem (BVP) NEWLINE\[NEWLINEX'= f(t,X),\;L(X)= r,\quad\text{where }L: (C^0[a,b], \mathbb{R}^N)\to \mathbb{R}^N,NEWLINE\]NEWLINE on the boundary conditions is continuously differentiable.NEWLINENEWLINENEWLINESome results on the existence and uniqueness of solutions to nonlinear BVPs, where properties of the variational equations are involved, are established. Particularly, strongly nonlinear BVPs and those multipoint BVPs which admit a point, where all the derivative upto a fixed order \(p\) are given, are dealt with.