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Solutions for an operator equation under the conditions of pairs of paralleled lower and upper solutions. (English) Zbl 1165.47044

Let \(X\) and \(Z\) be ordered Banach spaces, \(L: D(L)\to Z\) be a linear operator and \(F\) a continuous and bounded operator. The paper discusses the multiplicity of solutions of the operator equation
\[ L x=F(x)\qquad\tag \(*\) \]
by the method of lower and upper solutions and fixed point index theory. When \((*)\) has two pairs of strict lower and upper solutions, the authors obtain a three solutions theorem, which is different to Amann’s three solutions theorem. Under three pairs of strict lower and upper solutions, they obtain existence results of five solutions and nine solutions. The abstract results are applied to a Sturm–Liouville boundary value problem.

MSC:

47J05 Equations involving nonlinear operators (general)
47H11 Degree theory for nonlinear operators
34B15 Nonlinear boundary value problems for ordinary differential equations
47H10 Fixed-point theorems
47N20 Applications of operator theory to differential and integral equations
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