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Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions

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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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Authors and Affiliations

  1. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

    Pengyu Chen & Yongxiang Li

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  1. Pengyu Chen
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  2. Yongxiang Li
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Correspondence to Pengyu Chen.

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Research supported by NNSF of China (11261053), NSF of Gansu Province (1208RJZA129), and Project of NWNULKQN-11-3.

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Chen, P., Li, Y. Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions. Z. Angew. Math. Phys. 65, 711–728 (2014). https://doi.org/10.1007/s00033-013-0351-z

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  • DOI: https://doi.org/10.1007/s00033-013-0351-z

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