Let X be a real Banach space; (Phi Xtomathbb{R}) be a continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on (X^); (Psi:Xtomathbb{R}) be a continuously Gâteaux differentiable function whose Gâteaux derivative is compact.
Let X be a real Banach space; (Phi Xrightarrowmathbf{R}) be a sequentially weakly lower semicontinuous, coercive, and continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on (X^), (Psi:Xrightarrowmathbf{R}) be a continuously Gâteaux differentiable function whose Gâteaux derivative is compact.
Let X be a real Banach space; (Phi Xtomathbb{R}) be a sequentially weakly lower semicontinuous, coercive and continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on (X^{ast}); (Psi:Xtomathbb{R}) be a continuously Gâteaux differentiable function whose Gâteaux derivative is compact.
Let X be a reflexive real Banach space, (Phi X rightarrow mathbb{R}) be a convex, coercive and continuously Gâteaux differentiable functional whose derivative admits a continuous inverse on (X^{ast}), (Psi:X rightarrow mathbb{R}) be a continuously Gâteaux differentiable functional whose derivative is compact, such that 1. (inf_{X}Phi=Phi(0)=Psi(0)=0); 2.
We know from the definitions in (2.5) that is a nonnegative continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on, and is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact.
Let X be a real Banach space; Φ : X → R be a continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on X ∗ ; Ψ : X → R be a continuously Gâteaux differentiable function whose Gâteaux derivative is compact.
Let X be a reflexive real Banach space; Φ : X → R be a sequentially weakly lower semicontinuous, coercive, and continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on X ∗ ; Ψ : X → R be a continuously Gâteaux differentiable function whose Gâteaux derivative is compact.
Let Φ : X → R be a sequentially weakly lower semicontinuous, coercive and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on X ∗ ; let Ψ : X → R be a continuously Gâteaux differentiable functional whose Gáteaux derivative is compact.
Obviously, Φ is a nonnegative continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on (X^), and Ψ is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact.
By the definitions in (2.1), it is very clear that Φ is a nonnegative Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on (X^), and J is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact.
Let X be a reflexive real Banach space, (Phi XtoBbb {R}) be a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on (X^), (Psi:XtoBbb {R}) be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that inf_{uin X}Phi u)=Phi(0)=Psi(0)=0.
