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We deal with the problem of identifying a source term in the Rayleigh-Stokes type equation with a nonlinear perturbation, where the nonlinearity may have a superlinear growth and the additional measurement is given at final time and depends on the state. Our aim is to prove the unique solvability and stability of solution. Furthermore, we show that the obtained solution is differentiable and it is the strong one.

We prove the existence of ground state solutions by variational methods to the nonlinear Choquard equations with a nonlinear perturbation \[ -{\Delta}u+ u=\big(I_\alpha*|u|^{\frac{\alpha}{N}+1}\big)|u|^{\frac{\alpha}{N}-1}u+f(x,u)\qquad \text{ in } \mathbb{R}^N \] where $N\geq 1$, $I_\alpha$ is the Riesz potential of order $\alpha \in (0, N)$, the exponent $\frac{\alpha}{N}+1$ is critical with respect to the Hardy--Littlewood--Sobolev inequality and the nonlinear perturbation $f$ satisfies suitable growth and structural assumptions., Comment: 18 pages

Let Ω o and Ω i be open bounded regular subsets of R n such that the closure of Ω i is contained in Ω o . Let f o be a regular function on ∂ Ω o and let F and G be continuous functions from ∂ Ω i × R to R . By exploiting an argument based on potential theory and on the Leray–Schauder principle we show that under suitable and completely explicit conditions on F and G there exists at least one pair of continuous functions ( u o , u i ) such that { Δ u o = 0 in Ω o ∖ cl Ω i , Δ u i = 0 in Ω i , u o ( x ) = f o ( x ) for all x ∈ ∂ Ω o , u o ( x ) = F ( x , u i ( x ) ) for all x ∈ ∂ Ω i , ν Ω i ⋅ ∇ u o ( x ) − ν Ω i ⋅ ∇ u i ( x ) = G ( x , u i ( x ) ) for all x ∈ ∂ Ω i , where the last equality is attained in certain weak sense. A simple example shows that such a pair of functions ( u o , u i ) is in general neither unique nor locally unique. If instead the fourth condition of the problem is obtained by a small nonlinear perturbation of a homogeneous linear condition, then we prove the existence of at least one classical solution which is in addition locally unique.

For a linear integral equation x ( t ) = a ( t ) − ∫ 0 t B ( t , s ) x ( s ) d s there is a resolvent equation R ( t , s ) = B ( t , s ) − ∫ s t B ( t , u ) R ( u , s ) d u and a variation of parameters formula x ( t ) = a ( t ) − ∫ 0 t R ( t , s ) a ( s ) d s . It is assumed that B is a perturbed convex function and that a ( t ) may be badly behaved in several ways. When the first two equations are treated separately by means of a Liapunov functional, restrictive conditions are required separately on a ( t ) and B ( t , s ) . Here, we treat them as a single equation f ( t ) = S ( t ) − ∫ 0 t B ( t , u ) f ( u ) d u where S is an integral combination of a ( t ) and B ( t , s ) . There are two distinct advantages. First, possibly bad behavior of a ( t ) is smoothed. Next, properties of S needed in the Liapunov functional can be obtained from an array of properties of a ( t ) and B ( t , s ) yielding considerable flexibility not seen in standard treatment. The results are used to treat nonlinear perturbation problems. Moreover, the function y ( t ) = a ( t ) − ∫ 0 t B ( t , s ) a ( s ) d s is shown to converge pointwise and in L 2 [ 0 , ∞ ) to x ( t ) .

Several sufficient conditions are developed for controllability of the perturbed quasi-linear system x = A(t, x, u)x + B(t, x, u)u + f(t, x, u) . In particular, these reduce to conditions on the perturbation function f which guarantee that if the linear system x = A(t)x + B(t)u is controllable, then the system x = A(t)x + B(t)u + f(t, x, u) is controllable. These conditions are growth conditions in (x, u) and are obtained by solving a system of nonlinear integral equations.

This paper deals with the solvability of the nonlinear operator equations in normed spaces Lx = EGx + f, where L is a linear map with possible nontrivial kernel. Applications are given to the existence of periodic solutions for the third-order scalar differential equation x′′ + ax″ + bx′ + cx + g(t, x) = p(t) under various conditions on the interaction of g(t, x)x with spectral configurations of a, b, and C.

We derive some explicit sufficient conditions for the asymptotic stability of the zero solution in a general linear higher order difference equation, and compare our estimations with other related results in the literature. Our main result also applies to some nonlinear perturbations satisfying a kind of sublinearity condition.

Using the well-known Mountain Pass Theorem due to Ambrosetti and Rabinowitz, we establish conditions for the existence of a positive solution for a class of nonlinear elliptic problems in R N with critical growth.