Abstract
This paper is concerned with the existence of mild and strong solutions for a class of semilinear evolution equations with nonlocal initial conditions. The linear part is assumed to be a (not necessarily densely defined) sectorial operator in a Banach spaceX. Considering the equations in the norm of some interpolation spaces betweenXand the domain of the linear part, we generalize the recent conclusions on this topic. The obtained results will be applied to a class of semilinear functional partial differential equations with nonlocal conditions.
Highlights
The purpose of this paper is to study the existence of mild and strong solutions to the following semilinear evolution problem with nonlocal initial conditions: u (t) = Au (t) + f (t, u (t)), t ∈ [0, T], (1)
For the importance of nonlocal Cauchy problems in applications, we refer to the papers [1, 2] and the references therein
For our main result we introduce in the following a family of nonlocal Cauchy problems
Summary
Journal of Function Spaces and Applications at t = 0 This gap can be filled by assuming complete continuity on the nonlocal term g, but it is too restrictive in applications. To take away this unsatisfactory condition, Liang et al [7] observed that in many studies of nonlocal Cauchy problems, for example [1, 6, 9], the nonlocal condition g is completely determined on [δ, T], for some δ > 0; that is, such a g ignores t = 0; for instance, in [9] the function g(u) is given by p g (u) = ∑ciu (ti) ,.
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