Let and be two differentiable functions ( is generalized differentiable as in Definition 3.2).
Let p be a non-negative function on ([0,infty)) and let f and g be two differentiable functions having the same monotony on ([0,infty )).
Let (f, g:mathbb{T}rightarrowmathbb{R}) be two differentiable functions at (tin{mathbb{T}}^{k}). Then the product (fg:mathbb {T}rightarrowmathbb{R}) is also differentiable at t with (fg)^{Delta}(t)=f^{Delta}(t g(t)+f^{sigma}(t g^{Delta }(t)=f(t g^{Delta}(t)+f^{Delta}(t g^{sigma}(t).
Suppose that f and g are two differentiable functions, (f', g'in L^{infty}((0,T);mathbb{R})), and p is a non-negative and integrable function on ([0,T]).
Suppose that f and g are two differentiable functions having the same monotony, (f', g'in L^{infty}((0,T);mathbb{R})), and p is a non-negative and integrable function on ([0,T]).
Let and be two twice differentiable functions in.
Assume there are two partially differentiable functions X μ = X μ ( x β ) of two variables x β for β, μ = 1, 2 on the Cartesian coordinates of the 2D plane.
Assume there are two partially differentiable functions X = X ( x, y ) and Y = Y ( x, y ) of the two variables x and y on the Cartesian coordinates of the 2D plane.
More specifically, we decompose the component H of the HJB equation yielding the optimal strategy as H=1 ψ≤ℓ H 1+1 ψ>ℓ H 2, where H 1 and H 2 are two continuously differentiable functions.
Now, let (Phi, Psi: XrightarrowBbb {R}) be two continuously Gâteaux differentiable functions; put I=Phi-Psi and fix (muin[-infty, +infty]), we say that the function I verifies the Cerami condition cut off upper at μ (in short, the ({(C ^{[mu]}} -condition) if any sequence ({u_{n} -conditiont ifsanysequencee Cerami condition and (Phi(u_{u_{n}u), for any (nin Bbb {N}), hasuchconvergenthatbsequence.
Let X be a real Banach space and let (Phi, Psi: Xrightarrow Bbb {R}) be two continuously Gâteaux differentiable functions with Φ bounded from below.
