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Multiple positive solutions for some multi-point boundary value problems with \(p\)-Laplacian. (English) Zbl 1141.34017

Summary: This paper deals with the existence of multiple positive solutions for the quasilinear second-order differential equation
\[ (\phi_p(u'(t)))'+a(t)f(t,u(t))=0,\quad t\in(0,1), \]
subject to one of the following boundary conditions:
\[ \phi_p(u'(0))=\sum^{m-2}_{i=1}a_i\phi_p(u'(\xi_i)),\quad u(1)=\sum^{m-2}_{i=1}b_iu(\xi_i), \]
or
\[ u(0)=\sum^{m-2}_{i=1}a_iu(\xi_i),\quad \phi_p(u'(1))=\sum^{m-2}_{i=1}b_i\phi_p(u'(\xi_i)), \]
where \(\phi_p(s)=|s|^{p-2}s\), \(p>1,0<\xi_1<\xi_2<\dots<\xi_{m-2}<1\), and \(a_i,b_i\) satisfy \(a_i,b_i\in [0,\infty)\), \((i=1,2,\dots,m-2)\), \(0<\sum^{m-2}_{i=1}a_i<1\), \(0<\sum^{m-2}_{i=1}b_i<1\). Using the five functionals fixed point theorem, we provide sufficient conditions for the existence of multiple (at least three) positive solutions for the above boundary value problems.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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References:

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