Fixed-point theory for the sum of two operators. (English) Zbl 0858.34049

The author presents different situations concerning the properties of \(F\), \(F_1\), \(F_2\) in fixed point theorems for the sum of two operators \(F=F_1+F_2\).
As an application, the author considers the second-order boundary value problem (1) \(y''+f(t,y)=0\) a.e. on \([0,1]\), \(y(0)=y(1)=0\), where \(E\) is a Banach space and \(f:[0,1]\times E\to E\) has a decomposition of the form \(f(t,y)=f_1(t,y)+f_2(t,y)\), \(f_1\), \(f_2\) having to satisfy some hypotheses such that solving (1) is equivalent to finding a fixed point of an operator \(N:{\mathcal C}([0,1],E)\to{\mathcal C}([0,1], E)\), \(N=N_1+N_2\), \(N_1\) completely continuous, compact and \(N\) a nonlinear contraction.


34G20 Nonlinear differential equations in abstract spaces
47H10 Fixed-point theorems
Full Text: DOI


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