Some results of solutions to neutral stochastic functional operator-differential equations

Abstract In this article, we shall consider the solution to neutral stochastic functional operator differential equations with finite delay <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>r</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> <m:mo>:</m:mo> <m:mi mathvariant="normal">d</m:mi> <m:mi>D</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:mi>t</m:mi> <m:mo>+</m:mo> <m:mi>g</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:mi>B</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width="0.33em"/> <m:msub> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>≤</m:mo> <m:mi>t</m:mi> <m:mo>≤</m:mo> <m:mi>T</m:mi> </m:math> r\gt 0:{\rm{d}}D\left(t,{x}_{t})=f\left(t,{x}_{t}){\rm{d}}t+g\left(t,{x}_{t}){\rm{d}}B\left(t),\hspace{0.33em}{t}_{0}\le t\le T . By using Banach fixed point theorem, we obtain the existence of the solutions. We also obtain some properties of the solution.