## 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.

### MSC:

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

### References:

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