Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions |
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Authors: | Pengyu Chen Yongxiang Li |
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Institution: | 1. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China
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Abstract: | In this paper, we are concerned with nonlocal problem for fractional evolution equations with mixed monotone nonlocal term of the form $$\left\{\begin{array}{ll}^CD^{q}_tu(t) + Au(t) = f(t, u(t), u(t)),\quad t \in J = 0, a],\\u(0) = g(u, u),\end{array}\right.$$ where E is an infinite-dimensional Banach space, \({^CD^{q}_t}\) is the Caputo fractional derivative of order \({q\in (0, 1)}\) , A : D(A) ? E → E is a closed linear operator and ?A generates a uniformly bounded C 0-semigroup T(t) (t ≥ 0) in E, \({f \in C(J\times E \times E, E)}\) , and g is appropriate continuous function so that it constitutes a nonlocal condition. Under a new concept of coupled lower and upper mild L-quasi-solutions, we construct a new monotone iterative method for nonlocal problem of fractional evolution equations with mixed monotone nonlocal term and obtain the existence of coupled extremal mild L-quasi-solutions and the mild solution between them. The results obtained generalize the recent conclusions on this topic. Finally, we present two applications to illustrate the feasibility of our abstract results. |
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