We study a semilinear fractional order differential inclusion in a separable Banach space E of the form $$ {}^{C}D^{q}x(t)\in Ax(t)+ F\bigl(t,x(t)\bigr),\quad t\in [0,T], $$ where ${}^{C}D^{q}$ is the Caputo fractional derivative of order $0 < q < 1$ , $A \colon D(A) \subset E \rightarrow E$ is a generator of a $C_{0}$ -semigroup, and $F \colon [0,T] \times E \multimap E$ is a nonlinear multivalued map. By using the method of the generalized translation multivalued operator and a fixed point theorem for condensing multivalued maps, we prove the existence of a mild solution to this inclusion satisfying the nonlocal boundary value condition: $$ x(0)\in \Delta (x), $$ where $\Delta : C([0,T];E) \multimap E$ is a given multivalued map. The semidiscretization scheme is developed and applied to the approximation of solutions to the considered nonlocal boundary value problem.

中文翻译：

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分数阶微分包含问题的非局部边值问题的解的存在和近似

我们研究形式为$$ {} ^ {C} D ^ {q} x（t）\ in Ax（t）+ F \ bigl（t，x（t）中的形式的可分离Banach空间E中的半线性分数阶微分包含）\ bigr），\ quad t \ in [0，T]，$$其中$ {} ^ {C} D ^ {q} $是阶数$ 0 <q <1 $的Caputo分数导数，$ A \冒号D（A）\子集E \ rightarrow E $是$ C_ {0} $-半群的生成器，而$ F \冒号[0，T] \ times E \ multimap E $是非线性多值映射。通过使用广义平移多值算子的方法和不动点定理来压缩多值映射，我们证明存在满足非局部边界值条件的该包含项的温和解：$$ x（0）\ in \ Delta（x ），$$，其中$ \ Delta：C（[0，T]; E）\ multimap E $是给定的多值映射。