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Existence of symmetric positive solutions for a boundary value problem with integral boundary conditions. (English) Zbl 1221.34062

The author studies the existence of symmetric positive solutions for the nonlinear nonlocal boundary problem
\[ (g(t)x'(t))'+w(t)f(t,x(t))=0, \]
\[ ax(0)-b\lim_{t\to 0^+} g(t)x'(t)=\int^1_0h(s)x(s)\,ds,\quad ax(1)+b\lim_{t\to 1^{-}} g(t)x'(t)=\int^1_0h(s)x(s)\,ds, \]
where \(a,b>0\), \(g\in C^1([0,1], (0, \infty))\), \(w\in L^p(0,1)\) and \(h\in L^1((0,1),(0,\infty))\) are symmetric on \([0,1]\), respectively; \(f:[0,1]\times [0,\infty)\to [0,\infty)\) is continuous and \(f(t, x)=f(1-t,x)\). The main tool is the theory of fixed point index.
Reviewer: Ruyun Ma (Lanzhou)

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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