On the exceptional set for Diophantine inequality with unlike powers of primes

Let λ₂, λ₃, λ₄, λ₅ be nonzero real numbers, not all negative. Let \(\mathfrak{V}\) $\mathfrak{V}$ be a well-spaced sequence. Assume that λ₂/λ₃ is irrational and algebraic, and δ > 0. Let \(E\left(\mathfrak{V},N,\delta \right)\) $E\left(\mathfrak{V},,,N,,,\delta \right)$ be the number of \(\upsilon \in \mathfrak{V}\) $\upsilon \in \mathfrak{V}$ with \(\upsilon \le N\) $\upsilon \le N$ such that the Diophantine inequality \(\left|{\lambda }_{2}{p}_{2}^{2}+{\lambda }_{3}{p}_{3}^{3}+{\lambda }_{4}{p}_{4}^{4}+{\lambda }_{5}{p}_{5}^{5}-\upsilon \right|<{\upsilon }^{-\delta }\) $\left|{\lambda }_2,p_2^2,+,{\lambda }_3,p_3^3,+,{\lambda }_4,p_4^4,+,{\lambda }_5,p_5^5,-,\upsilon \right|<{\upsilon }^{-\delta }$ has no solution in primes p₂, p₃, p₄, p₅. In this paper, we prove that for any \(\varepsilon >0,E\left(\mathfrak{V},N,\delta \right)\ll {N}^{1-19/378+2\delta +\varepsilon },\) $\epsilon >0,E\left(\mathfrak{V},,,N,,,\delta \right)\ll N^{1-19/378+2\delta +\epsilon },$ which refines the previous result.