### Existence and asymptotic expansion of solutions to a nonlinear wave equation with a memory condition at the boundary.

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We first introduce new weighted Morrey spaces related to certain non-negative potentials satisfying the reverse Hölder inequality. Then we establish the weighted strong-type and weak-type estimates for the Riesz transforms and fractional integrals associated to Schrödinger operators. As an application, we prove the Calderón-Zygmund estimates for solutions to Schrödinger equation on these new spaces. Our results cover a number of known results.

By using Mawhin’s continuation theorem, we provide some sufficient conditions for the existence of solution for a class of high order differential equations of the form $${x}^{\left(n\right)}=f(t,x,{x}^{\text{'}},\cdots ,{x}^{(n-1)})\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}t\in [0,1]\phantom{\rule{0.166667em}{0ex}},$$ associated with the integral boundary conditions at resonance. The interesting point is that we shall deal with the case of nontrivial kernel of arbitrary dimension corresponding to high order differential operator which will cause some difficulties in constructing the generalized inverse operator.

We study a class of logarithmic fractional Schrödinger equations with possibly vanishing potentials. By using the fibrering maps and the Nehari manifold we obtain the existence of at least one nontrivial solution.

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