In this paper, we discuss the asymptotically periodic problem for the abstract fractional evolution equation under order conditions and growth conditions. Without assuming the existence of upper and lower solutions, some new results on the existence of the positive S-asymptotically ω-periodic mild solutions are obtained by using monotone iterative method and fixed point theorem. It is worth noting that Lipschitz condition is no longer needed, which makes our results more widely applicable.
This paper considers the initial boundary value problem for the time-space fractional delayed diffusion equation with fractional Laplacian. By using the semigroup theory of operators and the monotone iterative technique, the existence and uniqueness of mild solutions for the abstract time-space evolution equation with delay under some quasimonotone conditions are obtained. Finally, the abstract results are applied to the time-space fractional delayed diffusion equation with fractional Laplacian operator, which improve and generalize the recent results of this issue.
In this paper, we devote to considering S -asymptotically periodic problem of fractional evolution equation with delay in ordered Banach space. Under some weaker assumptions, we construct monotone iterative method in the presence of the lower and upper solutions to the delayed fractional evolution equation, and obtain the existence of maximal and minimal S -asymptotically periodic mild solutions. Finally, we present two examples to illustrate the feasibility of our abstract results.
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