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The proof follows from similar steps in [11] and combines the argument we used in the above lemmas.

(v) The argument technique of our Theorem 3.1 combines the argument one in [[9], Theorem 3.1] and the argument one in [[11], Theorem 3.1].

Combining the argument presented above with the result in Appendix B, we get alpha leq frac{1}{d_{max}}.

Combining the argument mentioned above, we deduce that the function (phi (x,t)) satisfies (2.13) in Q.

Multiplying the above equality by v and integrating between 0 and kT, combining the argument of (2.4) and Definition 2.2, we obtain that ∑ j = 1 p e Q ( t j ) ( Δ ( P ( t j ) u ˙ ( t j ) ), v ( t j ) ) = ∑ j = 1 p e Q ( t j ) ( ∇ I j ( u ( t j ) ), v ( t j ) ).

By combining the arguments of left and right branch, the induction claim is proven.

By combining the arguments of [20, 23, 24], we get the existence result.

Combining the arguments of the above two cases, we obtain the equality condition of (2.1) as stated in Theorem 1.

where ([ t-1 ] ^=max { t-1,0 } ) and (w_{1}=ln ( frac{w_{0}}{w_{0}-1} ) ). Next, we state the local existence theorem that can be established by combining the arguments of [6, 17, 18].

Therefore, combining the arguments in Schneider [31], Corollaries 3.4 and 3.2, and the first part of the proof, we conclude that ((mathscr{D}_{beta, G}^{1, p}(mathbb{R}^{N}))^{2}) is compactly embedded in ((L^{q}(mathbb{R}^{N}, h(x)))^{2}) and the results follow.

end{aligned} (4.14) Combining the arguments of the uniqueness and the fact (4.7), (4.8), which implies that ({F_{m}}) and ({G_{m}}) are Cauchy sequences in (C [0,T] L^{2}(mathcal {O}timesOmega))), the uniqueness of the limit and (4.12 - 4.13) yield, for (mathbb {P} -a.s.