In case i≠j, the result follows just as above by using the already established existence of 〈S k,S k 〉(t) for all k and t and by noting that (sum _{k,ell in {i,j}}{langle }S_{k},S_{ell }{rangle }={left langle S_{i}+S_{j},S_{i}+S_{j} right rangle }).
Then, these results will be used to establish existence and uniqueness criteria, and the convergence of Picard's, approximate Picard's, quasilinearization, and approximate quasilinearization iterative methods for the complementary Lidstone boundary value problems which consist of a th order differential equation and the complementary Lidstone boundary conditions.
The main results are then applied to optimal design, primarily the stress-constrained minimum weight problem, to illustrate how they can be used to establish existence of solutions and validity of "ε-perturbations" that are common in computational topology optimization.
Then a linking theorem is used to establish existence.
A version of degree theory known as topological degree theory was importantly introduced by Mawhin [31] and later on extended by Isaia [32]; it has been used to establish existence theory of nonlinear differential and integral equations.
Using only the axioms, we establish existence and (perhaps more surprisingly) uniqueness of stationary monetary equilibrium.
In Sections 3 5, we will use inequalities (1.5) to establish existence and uniqueness criteria, and the convergence of Picard's, approximate Picard's, quasilinearization, and approximate quasilinearization iterative methods for the complementary Lidstone boundary value problem (1.1), (1.2).
Later on, many articles used the monotone iterative technique to establish existence and comparison results for nonlinear problems.
Using the theory of coincidence degree, we establish existence results of positive solutions for higher-order multi-point boundary value problems at resonance for ordinary differential equation, with one of the following boundary conditions:, and,, where is a continuous function,, all the have not the same sign.
Also, the limit of viscous approximations for the equation is used to establish the existence of the global weak solution.
