Uniquely solvable problems for abstract Legendre equation
Let \(A\) be a closed operator in a Banach space \(E\) with dense domain \(D(A)\) and suppose that \(A\) is the generator of a cosine operator family. Consider in \(E\) an abstract weakly loaded Legendre equation under the form \[ u''(t)+ k\coth(t)\left(u'(t)-\frac{\cosh^{2-k}(t/2)}{\cosh t} u'(0)\right)+\frac{k^2}{4}u(t)=Au(t) \] for \(t>0\). It is proved that for all \(u_0,u_1\in D(A)\) the Cauchy problem for the Legendre equation with the initial conditions \(u(0)=u_0,u'(0)=u_1\) has a unique solution and the explicit formula for the soluion is given. Under some additional conditions on the spectrum of \(A\) the solvability of the Legendre equation with the final condition \(u(1)=u_2,u'(1)=u_2\) on the interval \([0,1]\) is proved. The explicit formula for the solution is given. A result on the solvability of Legendre equation with non-local intergral condition and \(u'(0)=0\) is obtained as well.
scientific article; zbMATH DE number 6954488
scientific article