Abstract
Sharp conditions are obtained for the unique solvability of focal boundary value problems for higher-order functional differential equations under integral restrictions on functional operators. In terms of the norm of the functional operator, unimprovable conditions for the unique solvability of the boundary value problem are established in the explicit form. If these conditions are not fulfilled, then there exists a positive bounded operator with a given norm such that the focal boundary value problem with this operator is not uniquely solvable. In the symmetric case, some estimates of the best constants in the solvability conditions are given. Comparison with existing results is also performed.
Highlights
We consider here boundary value problems (−1)(n−k)x(n) (t) + (Tx) (t) = f (t), t ∈ [0, 1], x(i) (0) = 0, i = 0, . . . , k − 1, (1.1) x(j) (1) =0, j = k, . . . , n − 1, where n ∈ {2, 3, . . .}, k ∈ {1, 2, . . . , n − 1}, T : C[0, 1] → L[0, 1] is a linear bounded operator, C[0, 1] and L[0, 1] are the space of real continuous and integrable functions with the standard norms, f ∈ L[0, 1]
The boundary value problems with such kind of boundary conditions are called focal ones. The solvability of such problems for linear and non-linear functional differential equations occupies a special place in many studies of physical, chemical, and biological processes
Tn,k, θ for every ε > 0 there exists a linear bounded operator T : C[0, 1] → L[0, 1] with T C→L = Tn,k + ε such that problem (1.1) isn’t uniquely solvable. It was shown in [24,25,26] that for certain monotone functional operators and for some boundary value problems, the solvability conditions based on contraction mapping principle can be essentially weakened
Summary
A real absolutely continuous function with absolutely continuous derivatives up to (n − 1)-th order which satisfies the boundary conditions from (1.1) and satisfies the functional differential equation from (1.1) almost everywhere on [0, 1] is called a solution to problem (1.1)
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