Existence of mild solutions for fractional delay evolution systems. (English) Zbl 1242.34140
The authors consider the following nonlinear fractional evolution system with finite time delay
\[
{}^C\!D^q_tx(t)= -Ax(t)+f(t,x_t,x(t)), ~t\geq 0, ~q\in (0,1],
\]
\[ x(t) = \varphi (t),~~-r\leq t\leq 0, \] where \({}^C\!D^q_t\) denotes the Caputo fractional derivative, \(-A: D(A)\rightarrow X\) is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators \(\{T(t):t\geq 0\}\) on a Banach space \(X\), \(f\) is an \(X\)-valued function, \(x_t\) represents the history of the state from time \(-r\) up to the present time \(t\). By applying the generalized singular version of Gronwall’s inequality and weakly singular versions of a Bihari-type inequality, local and global existence results are obtained when \(f\) satisfies a local Lipschitz condition and if \(f\) is not necessarily a local Lipschitz mapping, respectively.
\[ x(t) = \varphi (t),~~-r\leq t\leq 0, \] where \({}^C\!D^q_t\) denotes the Caputo fractional derivative, \(-A: D(A)\rightarrow X\) is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators \(\{T(t):t\geq 0\}\) on a Banach space \(X\), \(f\) is an \(X\)-valued function, \(x_t\) represents the history of the state from time \(-r\) up to the present time \(t\). By applying the generalized singular version of Gronwall’s inequality and weakly singular versions of a Bihari-type inequality, local and global existence results are obtained when \(f\) satisfies a local Lipschitz condition and if \(f\) is not necessarily a local Lipschitz mapping, respectively.
Reviewer: Shaochun Ji (Huaian)
MSC:
| 34K37 | Functional-differential equations with fractional derivatives |
| 34K30 | Functional-differential equations in abstract spaces |
| 47D06 | One-parameter semigroups and linear evolution equations |
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