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The main result of the paper is the following global implicit function
theorem.
Theorem Let \(X\), \(Y\)be real Banach spaces, \(H\) - a real Hilbert space. If \(F:X\times
Y\rightarrow H\) is continuously differentiable on \(X\times Y\)
and
- differential \(F_{x}^{\prime}(x,y):X\rightarrow H\) is bijective for any \((x,y)\in X\times Y\)
- for any fixed \(y\in Y\), the functional
\[
\varphi:X\ni x\longmapsto(1/2)\left\Vert F(x,y)\right\Vert ^{2}\in\mathbb{R}
\]
satisfies the Palais-Smale condition,
then there exists a unique function \(\lambda:Y\rightarrow
X\) such that \(F(\lambda(y),y)=0\) for any \(y\in Y\)
and this function is continuously differentiable on \(Y\) with
differential \(\lambda^{\prime}(y)\) at \(y\in Y\) given by
\[
\lambda^{\prime}(y)=-[F_{x}(\lambda(y),y)]^{-1}\circ F_{y}(\lambda(y),y).
\]
Some applications of the theorem to problems containing the integrals and
derivatives of fractional order are given.
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