Let X be a real reflexive Banach space with $X^{⁎}$ its dual space and G be a nonempty and open subset of X. Let $A:X\supseteq D(A)\to 2^{X^{⁎}}$ be a strongly quasibounded maximal monotone operator and $T:X\supseteq D(T)\to 2^{X^{⁎}}$ be an operator of class ${\mathcal{A}}_G(S_{+})$ introduced by Kittilä. We develop a topological degree theory for the operator $A+T$. The theory generalizes the Browder degree theory for operators of type $(S_{+})$ and extends the Kittilä degree theory for operators of class ${\mathcal{A}}_G(S_{+})$. New existence results are established. The existence results give generalizations of similar known results for operators of type $(S_{+})$. Applications to strongly nonlinear problems are included.

令X成为一个真实的自反Banach空间$X^{⁎}$它的对偶空间G是X的一个非空且开放的子集。让$\mathrm{一种}：X\supseteq d（\mathrm{一种}）\to 2^{X^{⁎}}$ 成为强拟界最大单调算子 $Ť：X\supseteq d（Ť）\to 2^{X^{⁎}}$ 成为班级的经营者 ${\mathcal{一种}}_G（{\mathrm{小号}}_{+}）$由Kittilä引入。我们为运营商开发拓扑度理论$\mathrm{一种}+Ť$。该理论将Browder度理论推广到类型算子$（{\mathrm{小号}}_{+}）$ 并将Kittilä学位理论扩展到班级经营者 ${\mathcal{一种}}_G（{\mathrm{小号}}_{+}）$。建立了新的存在结果。存在结果为类型的算子给出了相似的已知结果的概括$（{\mathrm{小号}}_{+}）$。包括对强非线性问题的应用。