In the first part of this paper, we consider the equation $$ \Big ( \frac{u'}{\sqrt{1-u'^2}} \Big )'+F'(u)=0 $$ modeling, if $F'(u)=\sin u$, the motion of the free relativistic planar pendulum. Using critical point theory for non-smooth functionals, we prove the existence of non-trivial $T$ periodic solutions provided $T$ is sufficiently large.

In the second part, we show the existence of periodic solutions to the free and forced relativistic spherical pendulum, where $F'$ is substituted by $$ F'(u)-h^2\, G'(u)\sim \sin u -h^2 \frac {\cos u}{\sin^3u} , \ \ \ h\in \mathbb R . $$

We consider reaction diffusion systems where some components react and diffuse on the surface, and others diffuse inside the domain and react with surface components through dynamic boundary conditions. Under reasonable hypotheses, we establish the existence of componentwise non-negative global solutions.

We discuss the existence of solutions to an optimal control problem for the mixed Dirichlet-Neumann boundary value problem for strongly non-linear elliptic equations with an exponential type of nonlinearity. A density of surface traction $u$ acting on a part of boundary of open domain is taken as a boundary control. The optimal control problem is to minimize the discrepancy between a given distribution $y_d\in L^2(\Omega)$ and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this, we make use of a variant of the classical Tikhonov regularization. We eliminate the differential constraints between control and state and allow such pairs run freely in their respective sets of feasibility by introducing some additional variable which plays the role of ``defect". We show that this special residual function can be determined in a unique way. We introduce a special family of regularized optimization problems and show that each of these problem is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we establish sufficient conditions of the existence of optimal solutions to the given class of nonlinear Dirichlet BVP and propose the way for their approximation.

In the current manuscript, we examine the existence and uniqueness of solution for generalized Langevin equation involving two distinct fractional orders with a three-point boundary value problem. We implement the notions of fractional calculus simultaneously with immutable point types to create the existence and uniqueness results. To explore our problem, we apply the Banach contraction principle, Krasnoselskii's fixed point theorem, nonlinear alternative Leray-Schauder theorem and Leray-Schauder degree theorem. Expository examples are rended to clarify the applicability of our main results.

In this paper, we introduce a new concept of dual anti-periodic boundary conditions. One of these conditions relates to the end points of an interval of arbitrary length, while the second one involves two nonlocal positions within the interval. Equipped with these conditions, we present the criteria for the existence of solutions for a fractional integro-differential equation involving two Caputo fractional derivatives of different orders and a Riemann-Liouville integral. Our study relies on the modern methods of functional analysis. Examples are constructed for illustrating the obtained results.