By putting for each, we have By (W4), we have.
(1) By putting for each, we have By (W4), we have.
Putting for each, from (2.3) and the monotonicity of, we have (3.4).
It follows that the multifunction (T:h(X to2^{Y}) defined by putting, for each (sin h(X)), T s)=h^{-1}(s cap Y is lower semicontinuous in (h(X)) with nonempty values.
For each (nin{mathbf{N}}), let (h_{n}:Ttimes Wto[0,+infty[ ) be defined by putting, for each ((t,x in Ttimes W), h_{n} t,x =-d_{S}bigl(s_{n},x =-d_{S}biglBy as_{n}tion (i) and Theorem 3.3 oF t9],x bigrunction (h_{n}) is ({mathcal{T}}_{mu}otimes{mathcal{B}}(W))-measurable.
} tin[a,b], u(a)=u(b)=0, end{cases} (5) where (F:[a,b]timesmathbf{R}timesmathbf{R}tomathbf{R}) is the (single-valued) function defined by putting, for each ((x,y inmathbf{R}^{2}), F t,x,y)= textstylebegin{cases} 0&mbox{if }xinmathbf{Q}mbox{ or }yinmathbf{Q}, 1&mbox{otherwise}.
Let Y be an ordered q-starshaped subset of X and f, g : Y → Y. Put, Set, for each x in X comparable with q in Y,.
