Multiple solutions to a boundary value problem for an n-th order nonlinear difference equation
We seek multiple solutions to the n-th order nonlinear difference equation $$Delta^n x(t)= (-1)^{n-k} f(t,x(t)),quad t in [0,T]$$ satisfying the boundary conditions $$x(0) = x(1) = cdots = x(k - 1) = x(T + k + 1) = cdots = x(T+ n) = 0,.$$ Guo's fixed point theorem is applied multiple times to a...
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| Format: | Article |
| Language: | English |
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Texas State University
1998-11-01
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| Series: | Electronic Journal of Differential Equations |
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| Online Access: | http://ejde.math.txstate.edu/conf-proc/01/l1/abstr.html |
| Summary: | We seek multiple solutions to the n-th order nonlinear difference equation $$Delta^n x(t)= (-1)^{n-k} f(t,x(t)),quad t in [0,T]$$ satisfying the boundary conditions $$x(0) = x(1) = cdots = x(k - 1) = x(T + k + 1) = cdots = x(T+ n) = 0,.$$ Guo's fixed point theorem is applied multiple times to an operator defined on annular regions in a cone. In addition, the hypotheses invoked to obtain multiple solutions to this problem involves the condition (A) $f:[0,T] imes {mathbb R}^+ o {mathbb R}^+$ is continuous in $x$, as well as one of the following: (B) $f$ is sublinear at $0$ and superlinear at $infty$, or (C) $f$ is superlinear at $0$ and sublinear at $infty$. |
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| ISSN: | 1072-6691 |