Asymptotic behavior of mild solutions for a class of abstract nonlinear difference equations of convolution type
Abstract We prove the existence and uniqueness of a weighted pseudo asymptotically mild solution to the following class of abstract semilinear difference equations: u(n+1)=A∑k=−∞na(n−k)u(k+1)+∑k=−∞nb(n−k)f(k,u(k)),n∈Z, $$ u(n+1)= A \sum_{k=-\infty }^{n} a(n-k)u(k+1)+ \sum _{k=-\infty }^{n} b(n-k)f\b...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2019-06-01
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| Series: | Advances in Difference Equations |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13662-019-2189-y |
| Summary: | Abstract We prove the existence and uniqueness of a weighted pseudo asymptotically mild solution to the following class of abstract semilinear difference equations: u(n+1)=A∑k=−∞na(n−k)u(k+1)+∑k=−∞nb(n−k)f(k,u(k)),n∈Z, $$ u(n+1)= A \sum_{k=-\infty }^{n} a(n-k)u(k+1)+ \sum _{k=-\infty }^{n} b(n-k)f\bigl(k,u(k)\bigr),\quad n\in \mathbb{Z}, $$ where A is the generator of a resolvent sequence {S(n)}n∈N0 $\{S(n)\}_{n\in \mathbb{N}_{0}}$ of bounded and linear operators defined in a Banach space X, the sequences a,b $a, b$ are complex-valued, and f∈l1(Z×X,X) $f\in l^{1}( \mathbb{Z}\times X, X)$. |
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| ISSN: | 1687-1847 |