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Some nonlinear dynamic inequalities on time scales. (English) Zbl 1149.34308
In this paper, some continuous nonlinear inequalities are generalized to nonlinear dynamic inequalities on time scales. Two such inequalities are obtained. One of these runs as follows:
Let \(u, a,b, g,h\in C_{\text{rd}}\) and be nonnegative, where \(C_{\text{rd}}\) is the set of all rd-continuous functions. Then \[ (u(t))^p\leq a(t)+ b(t) \int^t_{t_0}[g(\tau)(u(\tau))^p+ h(\tau) u(\tau)]\,\Delta\tau, \] for \(t\in{\mathbf T}_0\), \[ u(t)\leq\Biggl\{a(t)+ b(t)\int^t_{t_0} \Biggl[a(\tau) g(\tau)+ h(\tau)\Biggl({p- 1+a(\tau)\over p}\Biggr)\Biggr] e_{bm}(t, \sigma(\tau))\,\Delta\tau\Biggr\}^{1/p} \] for \(t\in{\mathbf T}_0\), where \(m(t)= g(t)+ {h(t)\over p}\), \({\mathbf T}_0= [t_0,\infty)\cap{\mathbf T}\) and \({\mathbf T}\) is the time scale (for more details look at the paper). It may be noted that \(f^\Delta(t)= f'(t)\) if \({\mathbf T}= {\mathbf R}\) and \(f^\Delta(t)= \Delta f(t)\) if \({\mathbf T}={\mathbf Z}\), where \(\Delta f(t)\) denotes the forward difference operator defined by \(\Delta f(t)= f(t+ 1)- f(t)\).

MSC:
34A40 Differential inequalities involving functions of a single real variable
39A10 Additive difference equations
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