Sentence examples for a subsolution of problem from inspiring English sources

Hence, is a subsolution of problem (3.3).

then is called a subsolution of problem (2.8).

Under the assumption that and for any, is a subsolution of problem (1.1)–(1.1).

Under the assumption that for any, is a subsolution of problem (1.1)–(1.1).

Thus, is a subsolution of problem (3.3) as well, and we deduce.

The inequalities (3.10 - 3.13) show that v1 x, t) is a subsolution of problem (1.1).

A nonnegative measurable function (uinmathbb{E}) is called a weak subsolution of problem (1.1) int_{0}^{t} int_{Omega}bigl{ u_{t}phi+|nabla u|^{p-2} nabla ucdot nablabigl u^{m}phibigr -gamma u^{r}|nablabigl u^{m}phibigr -gammaleqlambda int_{0}^{t} int_{Omega}u^{r}|nabla,dtau|^{p}phibigr}l bounded test functions (0leqphiin C_{0}^{1}(Omega_{T})).

If, then for sufficiently small and any the function is a positive subsolution of problem (1.1).

Assume that (u_{1}in C^{1} overline{D})) is a positive subsolution of problem (6) and that (h:overline{D}timesmathbb{R}times mathbb{R}^{N}rightarrowmathbb{R}) is continuous function such that (h x,t,xi ge g(t)) for all (xinmathbb{R}^{N}), (xiinmathbb{R}^{N}), and (tin 0,|u_{1}|_{L^{infty}(D)} ]).

Assume that (u_{1}in C_{0}^{1}(overline{D})) is a positive subsolution of problem (6) and that (h:overline{D}timesmathbb{R}times mathbb{R}^{N}rightarrowmathbb{R}) is continuous and such that (h x,t,xi ge g(t)) for all (xinmathbb{R}^{N}), (xiinmathbb{R}^{N}), and (tin 0, |u_{1}|_{L^{infty}(D)} ]).

In order to obtain an upper bound of (t^_{mathcal{N}}), we seek an unbounded subsolution of problem (3.1): textstylebegin{cases} underline{u}:= s(t)^{n} phi_{2}(x)^{2n}, underline{v}:=s(t)^{m} phi_{2}(x)^{2m}, end{cases} (3.2) with ({{n, m inmathbb{N}}} ) and (s(t)) satisfying Lemma 2.1.