795 research outputs found

    Multiple positive solutions for a class of Neumann problems

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    We study the existence of multiple positive solutions of the Neumann problem \begin{equation*} \begin{split} -u''(x)&=\lambda f(u(x)), \qquad x\in(0,1),\\ u'(0)&=0=u'(1), \end{split} \end{equation*} where λ\lambda is a positive parameter, f∈C([0,∞),R)f\in C([0,\infty),\mathbb{R}) and for some β>0\beta>0 such that f(0)=0f(0)=0, f(s)>0f(s)>0 for s∈(β,∞)s\in(\beta,\infty), f(s)β)f(s)\beta) is the unique positive zero of F(s)=∫0sf(t) dtF(s)=\int_0^sf(t)\,dt. In particular, we prove that there exist at least 2n+12n+1 positive solutions for λ∈(n2π2f′(β),∞)\lambda\in \big(\frac{n^2\pi^2}{f'(\beta)},\infty\big), where n∈Nn\in \mathbb{N}. The proof of our main result is based upon the bifurcation and continuation methods

    Existence of Positive Solutions for Fractional Differential Equation with Nonlocal Boundary Condition

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    By using the fixed point theorem, existence of positive solutions for fractional differential equation with nonlocal boundary condition D0+αu(t)+a(t)f(t,u(t))=0, 0<t<1, u(0)=0, u(1)=∑i=1∞αiu(ξi) is considered, where 1<α≤2 is a real number, D0+α is the standard Riemann-Liouville differentiation, and ξi∈(0,1),  αi∈[0,∞) with ∑i=1∞αiξiα-1<1, a(t)∈C([0,1],[0,∞)),  f(t,u)∈C([0,1]×[0,∞),[0,∞))
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