i have this theorem 
in the paper they gives an example:
but here $H_1$ is not satisfied !
How to correct it please?
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1$\begingroup$ The paper is academia.edu/4591584/… $\endgroup$– ConderCommented Apr 19, 2014 at 7:18
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3$\begingroup$ This was simultaneously posted at MSE: math.stackexchange.com/questions/760139/theorem-with-an-example $\endgroup$– Martin SleziakCommented Apr 19, 2014 at 9:17
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1 Answer
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http://www.academia.edu/4591584/Multiple_solutions_of_some_nonlinear_fourth-order_beam_equations
I think you can change the example to $f(t,u)=81u+4015\sin u+tu$. Then H1 holds, while for H2, the integral is $$F(t,u)=({81+t\over 2})u^2+4015(1-\cos u)$$ and so is bounded as indicated with $\alpha={81+1\over 2}=41<\pi^4/2$ and $\beta=2\cdot 4015$. Finally, $$f'_u(t,u)=81+4015\cos u+t$$ so $f'_u(t,0)=4096+t$, for which $m=2$ gives the desired inequality in H3. At least if I got all the details right.
Whether or not this is what was intended, or if the modified example is of any relevance, is a different matter.
