On boundary value problems for a class of functional-differential equations
Let \(q> 1\) and \(f\in L_2(0,T)\) be a complex-valued function. The author considers the functional-differential equations \[ -\Biggl(\sum^{N_1}_{m= -N_2} a_m u'(q^{-m}t)\Biggr)= f(t),\tag{1} \] where \(a_m\in \mathbb{C}\), \(N_2> 0\), \(a_{N_1}, a_{N_2}\neq 0\) and \[ -\Biggl(\sum^l_{m=0} b_m(t) u'(q^{-m}t)\Biggr)'+ (A_1u)(t)= f(t),\tag{2} \] where \(b_m\in C^1[0,T]\) are complex-valued functions and \(A_1: AC^1(J)\to L_2(0,T)\) is a linear bounded operator together with the homogeneous boundary conditions \[ u(t)= 0\quad\text{for }t\not\in (0,T)\tag{3} \] and \[ u(0)= u(T)= 0.\tag{4} \] The paper presents conditions for the solvability of BVPs (1), (3) and (2), (4) in the class \(AC^1[0,T]\) (functions having absolutely continuous derivatives on \([0,T]\)). The proofs are based on properties of functional operators.
scientific article; zbMATH DE number 1761021