Invex monotonicity and pseudoinvex monotonicity of fuzzy mappings are introduced in this paper, and relations are discussed between invex monotonicity (pseudoinvex monotonicity) and invexity (pseudoinvexity) of fuzzy mappings. The existence of a solution to the fuzzy variational-like inequality is discussed, and the existence theorem can be achieved. Furthermore, some extended properties of the fuzzy variational-like inequality are researched. Finally, method of solution is discussed based on genetic algorithm.

We prove a theorem on the existence and uniqueness of a solution as well as on a sensitivity (i.e., differentiable dependence of a solution on a functional parameter) of a fractional integrodifferential Cauchy problem of Volterra type. The proof of this result is based on a theorem on diffeomorphism between Banach and Hilbert spaces. The main assumption is the Palais-Smale condition.

We use variational methods and iterative methods to investigate the solutions of impulsive differential equations with nonlinear derivative dependence. The conditions for the existence of solutions are established. The main results are also demonstrated with examples.

Consider a continuous function on a metric space. In the presence of linking between a compact pair and a closed set, depending on the different behaviors of the function on the linking sets, we establish minimax results guaranteeing existence of Palais-Smale sequences or providing gradient estimates. Our approach relies on deformation techniques.

We consider second-order $p$-Laplacian differential system. By using three critical points theorem, we establish the new criterion to guarantee that this $p$-Laplacian differential system has at least three homoclinic solutions. An example is presented to illustrate the main result.

This paper is concerned with the existence of homoclinic solutions for a class of the second order impulsive Hamiltonian systems. By employing the Mountain Pass Theorem, we demonstrate that the limit of a $2kT$-periodic approximation solution is a homoclinic solution of our problem.

By applying the fixed point theorem in a cone of Banach space, we obtain an easily verifiable necessary and sufficient condition for the existence of positive periodic solutions of two kinds of generalized $n$-species competition systems with multiple delays and impulses as follows: $x_i^{\prime }(t)=x_i(t)[a_i(t)-b_i(t)x_i(t)-{\sum }_{j=\mathrm{1}}^n{c}_{ij}(t)x_i(t-{\tau }_{ij}(t))-{\sum }_{j=\mathrm{1}}^n{d}_{ij}(t)x_j(t-{\gamma }_{ij}(t))-{\sum }_{j=\mathrm{1}}^n{e}_{ij}(t){\int }_{-{\sigma }_{ij}}^{\mathrm{0}}\mathrm{‍}{f}_{ij}(s)x_j(t+s)ds],\text{\hspace{0.17em}a}\text{.}\text{e}\text{.},\mathrm{\hspace{0.17em}t}>\mathrm{0},t\ne t_k,\mathrm{\hspace{0.17em}k}\in Z_{+},\mathrm{\hspace{0.17em}i}=\mathrm{1,2},\dots ,n;{\mathrm{\hspace{0.17em}x}}_i(t_k^{+})-x_i(t_k^{-})={\theta }_{ik}{x}_i(t_k),\mathrm{\hspace{0.17em}i}=\mathrm{1,2},\dots ,n,\mathrm{\hspace{0.17em}k}\in Z_{+};\text{and}{x}_i^{\prime }(t)=x_i(t)[a_i(t)-b_i(t)x_i(t)+{\sum }_{j=\mathrm{1}}^n{c}_{ij}(t)x_i(t-{\tau }_{ij}(t))-{\sum }_{j=\mathrm{1}}^n{d}_{ij}(t)x_j(t-{\gamma }_{ij}(t))-{\sum }_{j=\mathrm{1}}^n{e}_{ij}(t){\int }_{-{\sigma }_{ij}}^{\mathrm{0}}\mathrm{‍}{f}_{ij}(s)x_j(t+s)ds],\text{\hspace{0.17em}a}\text{.}\text{e}\text{.},\mathrm{\hspace{0.17em}t}>\mathrm{0},\mathrm{\hspace{0.17em}t}\ne t_k,\mathrm{\hspace{0.17em}k}\in Z_{+},\mathrm{\hspace{0.17em}i}=\mathrm{1,2},\dots ,n;\mathrm{}{\mathrm{\hspace{0.17em}x}}_i(t_k^{+})-x_i(t_k^{-})={\theta }_{ik}{x}_i(t_k),\mathrm{\hspace{0.17em}i}=\mathrm{1,2},\dots ,n,\mathrm{\hspace{0.17em}k}\in Z_{+}.$ It improves and generalizes a series of the well-known sufficiency theorems in the literature about the problems mentioned previously.

By using the standard scaling arguments, we show that the infimum of the following minimization problem: ${I}_{{\rho }^{2}}=\mathrm{inf}\{(1/2){\int }_{{\Bbb R}^{3}}^{}{|\nabla u|}^{2}dx+(1/4){\iint }_{{\Bbb R}^{3}}^{}({|u(x)|}^{2}{|u(y)|}^{2}/|x-y|)dx dy$ − $(1/p){\int }_{{\Bbb R}^{3}}^{}{|u|}^{p}dx:u\in {B}_{\rho }\}$ can be achieved for $p\in (\mathrm{2,3})$ and $\rho >\mathrm{0}$ small, where ${B}_{\rho }:=\{u\in {H}^{\mathrm{1}}({\Bbb R}^{\mathrm{3}}):{\parallel u\parallel }_{\mathrm{2}}=\rho \}$. Moreover, the properties of ${I}_{{\rho }^{\mathrm{2}}}/{\rho }^{\mathrm{2}}$ and the associated Lagrange multiplier ${\lambda }_{\rho }$ are also given if $p\in (\mathrm{2,8}/\mathrm{3}]$.

The purpose of this paper is to study a class of discrete nonlinear Schrödinger equations. Under a weak superlinearity condition at infinity instead of the classical Ambrosetti-Rabinowitz condition, the existence of standing waves of the equations is obtained by using the Nehari manifold approach.