We propose another extension of Orlicz-Sobolev spaces to metric spaces based on the concepts of the $\Phi$-modulus and $\Phi$-capacity. The resulting space $N_{\Phi }^1$ is a Banach space. The relationship between $N_{\Phi }^1$ and $M_{\Phi }^1$ (the first extension defined in Aïssaoui (2002)) is studied. We also explore and compare different definitions of capacities and give a criterion under which $N_{\Phi }^1$ is strictly smaller than the Orlicz space ${\mathbf{L}}_{\Phi }$.

Local Lipschitz continuity of local minimizers of vectorial integrals ${\int }_{\Omega }f\left(x,Du\right)dx$ is proved when $f$ satisfies $p$-$q$ growth condition and $\xi \mapsto f\left(x,\xi \right)$ is not convex. The uniform convexity and the radial structure condition with respect to the last variable are assumed only at infinity. In the proof, we use semicontinuity and relaxation results for functionals with nonstandard growth.

Theorems on the Fredholm alternative and well-posedness of the linear boundary value problem $u^{\prime }\left(t\right)=\ell \left(u\right)\left(t\right)+q\left(t\right)$, $h\left(u\right)=c$, where $\ell :C\left(\left[a,b\right];ℝ\right)\to L\left(\left[a,b\right];ℝ\right)$ and $h:C\left(\left[a,b\right];ℝ\right)\to ℝ$ are linear bounded operators, $q\in L\left(\left[a,b\right];ℝ\right)$, and $c\in ℝ$, are established even in the case when $\ell$ is not a strongly bounded operator. The question on the dimension of the solution space of the homogeneous equation $u^{\prime }\left(t\right)=\ell \left(u\right)\left(t\right)$ is discussed as well.

We give new criterions to decide if some vector-valued function is a local Laplace transform and apply this to the theory of local Cauchy problems. This leads to an improvement of known results and new Hille-Yosida-type theorems for local convoluted semigroups.