Let be an upper solution of (3.6) and let us show that for all.
Similarly, a function is said to be an upper solution of (1.1), (1.2) with if (6.2).
Let be an upper solution of (1.1), let, and for all let (3.23).
Let Hypothesis hold, and let (bar{u}(x)) be an upper solution of the following problem: begin{aligned} textstylebegin{cases} -mathcal{A}u=f x,u,K*u) &textit{in } Q, u=0 &textit{on } partialOmega.
A function (winOmega) is said to be an upper solution of (1.1) if there exist (Hgeq0) and (0< Lleq M_{tau}<1) such that left { textstylebegin{array}{l} Delta w k geq g k, w k), w(theta k)))+r_{w}, quad{k}neq{k}_{tau}, kin J', Delta w {k_{tau}} geq I_{tau}( w {k_{tau}}))+d_{w tau}, quadtau=1,2,ldots,m, end{array}displaystyle right.
A continuous function (varphi(t)) is said to be an upper solution of the problem (6) if it satisfies left { textstylebegin{array}l} -D_{0+}^{alpha}varphi(t geqlambda f t, muint_{0}^{1} G_{beta} t, s g s, varphi(s))),ds, varphi(0)leq0, varphi'(0)ldots ldots, varphi^{ n-2)}(0)leq0, qquad varphi^{ n-2int_{0}^{1}varphi(s),dH(s).
