Solutions to Fractional Differential Equations with Nonlocal Initial Condition in Banach Spaces
© Zhi-Wei Lv et al. 2010
Received: 4 January 2010
Accepted: 8 February 2010
Published: 14 February 2010
A new existence and uniqueness theorem is given for solutions to differential equations involving the Caputo fractional derivative with nonlocal initial condition in Banach spaces. An application is also given.
Fractional differential equations have played a significant role in physics, mechanics, chemistry, engineering, and so forth. In recent years, there are many papers dealing with the existence of solutions to various fractional differential equations; see, for example, [1–6].
2. Basic Lemmas
Let be a real Banach space, and the zero element of . Denote by the Banach space of all continuous functions with norm . Let be the Banach space of measurable functions which are Lebesgue integrable, equipped with the norm . Let , function is called a solution of (1.1) if it satisfies (1.1).
Recall the following defenition
Clearly, . For details on properties of the measure, the reader is referred to .
Lemma (see ).
Lemma (see ).
3. Main Results
If (H1) and (H2) hold, then the initial value problem (1.1) has at least one solution.
In the same way as in Example in , we can prove that is relatively compact in .
By a direct computation, we get
Moreover, we have
This work was supported partially by the NSF of China (10771202), the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805).
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