We consider the estimation of the regression operator r in the functional model: Y= r(x)+ε, where the explanatory variable x is of functional fixed-design type, the response Y is a real random variable and the error process ε is a second order stationary process.
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By reducing the efficiency of unproductive splicing, the ESS would limit the impact of the cryptic 5′ss on the expression of functional FIX (35).
The well-known Avery-Henderson fixed point theorem and the five functionals fixed point theorem are adopted in the arguments.
Thus, we set out to verify that the operator T satisfies four functionals fixed point theorem, which will prove the existence of a fixed point of T. We first show that Q ( α, β, r, R ) is bounded, and T : Q ( α, β, r, R ) → P is completely continuous.
Thus we set out to verify that the operator T satisfies four functionals fixed point theorem, which will prove the existence of a fixed point of T. We first show that Q ( α, β, r, R ) is bounded and T : Q ( α, β, r, R ) → K is completely continuous.
Lemma 2.3 (four functionals fixed point theorem).
We are now in a position to present the Five functionals fixed point theorem (see [17]).
By using the Four functionals fixed point theorem and Five functionals fixed point theorem, we obtain the existence criteria of at least one positive solution and three positive solutions for the BVP (1.3).
By using the four functionals fixed point theorem and five functionals fixed point theorem, they obtained the existence criteria of at least one positive solution and three positive solutions.
