0$. Here for establishing Gr$\ddot{u}$ss inequality in fractional calculus the classical method of proof has been adopted also related results with Gr$\ddot{u}$ss inequality have been discussed. This work contributes in the current research by providing mathematical results along with their verifications. ]]>
0$ with modulus $\mu>0$ via general fractional integrals are established. Also, using the second lemma, some new estimates with respect to trapezium type integral inequalities for strongly $(h_{1},h_{2})$--preinvex functions of order $\sigma>0$ with modulus $\mu>0$ via general fractional integrals are obtained. It is pointed out that some new special cases can be deduced from main results. Some applications to special means for different real numbers and new approximation error estimates for the trapezoidal are provided as well. These results give us the generalizations of some previous known results. The ideas and techniques of this paper may stimulate further research in the fascinating field of inequalities.]]>
\mathcal{A}(x,t,u,\nabla u)-\mbox{div}\>\Phi(x,t,u)=f &\mbox{ in }\Omega\times (0,T)\\b(x,u)(t=0)=b(x,u_0) & \mbox{ in } \Omega\\u=0 &\mbox{ on } \partial\Omega\times (0,T).\end{array}\right.$$is given in the non reflexive Musielak spaces, where $b(x,\cdot)$ is a strictly increasing $C^1$-function for every $x\in\Omega$ with $b(x,0)=0$, the lower order term $\Phi$ is a non coercive Carath\'{e}odory function satisfying only a natural growth condition described by the appropriate Musielak function $\varphi$ and $f$ is an integrable data.]]>