One approach, used in [5, 6], consists to reduce (1) (or, more generally, (2)) to a differential inclusion of the form u^{ k)}(t)inPhibigl t,u(t),ldots,u^{ k-1)}(t)bigr),u^{ k-1 'well-behaved' multifunction Φ, and then to apply existence results for differential inclusions or selection arguments.
In Section 3, we apply an existence result of maximal elements in noncompact Hadamard manifolds in order to prove an existence theorem of solutions to AGVQEP under some suitable conditions.
We show that ifRcoE=PcoE(and two other hypotheses, named the segment property and the extreme points property) and ifϕ∈C1(Ω; Rm) is such thatDϕ(x)∈E∪int PcoE, x∈Ωthen there exists (a dense set of u∈W1, ∞(Ω; Rm) such that[formula]We apply this existence theorem to some relevant examples studied in the literature, as well as to problems with (x, u) dependence.
For instance, in the following example, we apply the existence result to an operator Φ having exponential growth.
We apply an existence theorem of variational inclusion problem on metric spaces to study optimization problems, set-valued vector saddle point problems, bilevel problems, and mathematical programs with equilibrium constraint on metric spaces.
end{aligned} end{aligned} (13) Since the square root function is not Lipschitz continuous near 0, we cannot apply standard existence theorems to obtain a solution to (13) with the full multiplicative noise term.
We will develop and apply the existence theory for first-order differential equations (see [16, 17]) to study certain second-order differential equations associated with nonlinear maximal monotone operators in Hilbert spaces.
Proof Rewrite (1) as an equivalent system of the form y ′ = − G − 1 ( u ), u ′ = − p ( t ) F ( y ), where G − 1 is the inverse of G. Then we apply the existence result [[9], Theorem 1] to obtain DS ≠ ∅.
Here we will use some of recently results developed by H. Amann to investigate a specific boundary value problems and then apply the existence theorem to two nonlocal problems.
Ma and Wang [42] have proved that there exist at least two positive solutions by applying the existence of positive solutions under the fact that is either superlinear or sublinear on by employing the fixed point theorem of cone extension or compression.
