Homoclinic and quasi-homoclinic solutions for damped differential equations

Homoclinic and quasi-homoclinic solutions for damped differential equations

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We study the existence and multiplicity of homoclinic solutions for the second-order damped differential equation flex 4 heartworm test $$ ddot{u}+cdot{u}-L(t)u+W_u(t,u)=0, $$ where L(t) and W(t,u) are neither autonomous nor periodic in t.Under certain assumptions on L and W, we obtain infinitely many homoclinic solutions when the nonlinearity W(t,u) is sub-quadratic or super-quadratic by using critical point theorems.Some recent results in the literature are generalized, and the open problem proposed by Zhang and Yuan is solved.In addition, with the help of the Nehari manifold, we consider the Neuw Rebel Skinny Eternal Black Jeans case where W(t,u) is indefinite and prove the existence of at least one nontrivial quasi-homoclinic solution.