Abstract
We study the exact and approximate solutions of a delay differential equation with various types of nonlocal history conditions. We establish the existence and uniqueness of mild, strong, and classical solutions for a class of such problems using the method of semidiscretization in time. We also establish a result concerning the global existence of solutions. Finally, we consider some examples and discuss their exact and approximate solutions.
Highlights
In our recent work [1, 2], we studied the functional differential equation (1.2) with the nonlocal history condition h(u[−τ,0]) = φ, where h is a Volterra-type operator from Ꮿ0 into itself and φ ∈ Ꮿ0
In this case the exact solution is obtained by solving the partial differential equation
Putting h = 0.1 in both cases, approximate as well as exact solutions are obtained. These approximate solutions uj, j = 1, 2, . . . , 10, corresponding to χ1 and χ2 along with their respective exact solutions are shown in Tables 5.3 and 5.4, respectively
Summary
We are concerned here with exact and approximate solutions of the following delay differential equation:. Most of them used semigroup theory and fixed point theorem to establish the unique existence and regularity of solution. In [7], Byszewski and Akca applied Schauder’s fixed point principle to prove the theorems for existence of mild and classical solutions of nonlocal Cauchy problem of the form u (t) + Au(t) = f t, u(t), u b1(t) , . In our recent work [1, 2], we studied the functional differential equation (1.2) with the nonlocal history condition h(u[−τ,0]) = φ, where h is a Volterra-type operator from Ꮿ0 into itself and φ ∈ Ꮿ0. We first use the method of semidiscretization to derive the existence of a unique strong solution, we prove that strong solution is a classical solution if additional conditions are assumed on the operator. The result of the paper consists, among other things, in that we obtain a solution of problem of much stronger regularity than in [1, 2]
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