The Douglas–Rachford method has been employed successfully to solve many kinds of nonconvex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary value problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well suited to parallelization. We explore the stability of the method by applying it to several BVPs, including cases where the traditional Newton’s method fails.

For the third-order differential equation y′″ = ƒ(t, y, y′, y″), where , questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

We discuss the multiplicity of nonnegative solutions of a parametric one-dimensional mean curvature problem. Our main effort here is to describe the configuration of the limits of a certain function, depending on the potential at zero, that yield, for certain values of the parameter, the existence of infinitely many weak nonnegative and nontrivial solutions. Moreover, thanks to a classical regularity result due to Lieberman, this sequence of solutions strongly converges to zero in $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}C^1([0,1])$. Our approach is based on recent variational methods.

Let $\Omega $ be a bounded open interval, and let $p\gt 1$ and $q\in (0, p- 1)$. Let $m\in {L}^{{p}^{\prime } } (\Omega )$ and $0\leq c\in {L}^{\infty } (\Omega )$. We study the existence of strictly positive solutions for elliptic problems of the form $- (\vert {u}^{\prime } \mathop{\vert }\nolimits ^{p- 2} {u}^{\prime } ){\text{} }^{\prime } + c(x){u}^{p- 1} = m(x){u}^{q} $ in $\Omega $, $u= 0$ on $\partial \Omega $. We mention that our results are new even in the case $c\equiv 0$.

We consider a nonlinear periodic problem driven by the scalar p-Laplacian and with a reaction term which exhibits a (p – 1)-superlinear growth near ±∞ but need not satisfy the Ambrosetti-Rabinowitz condition. Combining critical point theory with Morse theory we prove an existence theorem. Then, using variational methods together with truncation techniques, we prove a multiplicity theorem establishing the existence of at least five non-trivial solutions, with precise sign information for all of them (two positive solutions, two negative solutions and a nodal (sign changing) solution).

In this paper, employing a very recent local minimum theorem for differentiable functionals, the existence of at least one nontrivial solution for a class of systems of $n$ second-order Sturm–Liouville equations is established.

In this paper we study impulsive periodic solutions for second-order nonautonomous singular differential equations. Our proof is based on the mountain pass theorem. Some recent results in the literature are extended.

Existence results of positive solutions for a two point boundary value problem are established. No asymptotic condition on the nonlinear term either at zero or at infinity is required. A classical result of Erbe and Wang is improved. The approach is based on variational methods.

We study classes of higher-order singular boundary-value problems on a time scale with a positive parameter λ in the differential equations. A homeomorphism and homomorphism ø are involved both in the differential equation and in the boundary conditions. Criteria are obtained for the existence and uniqueness of positive solutions. The dependence of positive solutions on the parameter λ is studied. Applications of our results to special problems are also discussed. Our analysis mainly relies on the mixed monotone operator theory. The results here are new, even in the cases of second-order differential and difference equations.

By making use of Merle's general shooting method we investigate Dirac equations of the form

Here it is possible that F(0) = −∞ and that F(s) defined on (0,+∞) is not monotonously nondecreasing. Our results cover some known ones as a special case.

The nonlinear eigenvalue problem

for 0 ≤ x < ∞, fixed p ∈ (1, ∞), and with y′(0)/y(0) specified is studied under various conditions on the coefficients s and q, leading to either oscillatory or non-oscillatory situations.

We study the existence of multiple solutions for a two-point boundary-value problem associated with a planar system of second-order ordinary differential equations by using a shooting technique. We consider asymptotically linear nonlinearities satisfying suitable sign conditions. Multiplicity is ensured by assumptions involving the Morse indices of the linearizations at zero and at infinity.

The upper and lower solutions method and Leray–Schauder degree theory are employed to establish the existence result for a class of nonlinear third-order two-point boundary-value problems with a sign-type Nagumo condition.

In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem (BVP) where 1<α≤2, η∈(0,1),β∈ℝ=(−∞,+∞), βηα−1≠1, Dα is the Riemann–Liouville differential operator of order α, and f:[0,1]×ℝ→ℝ is continuous, q(t):[0,1]→[0,+∞) is Lebesgue integrable. We give some sufficient conditions for the existence of nontrivial solutions to the above boundary-value problems. Our approach is based on the Leray–Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity on f which was essential for the technique used in almost all existed literature.

We study a class of second-order nonlinear differential equations on a finite interval with periodic boundary conditions. The nonlinearity in the equations can take negative values and may be unbounded from below. Criteria are established for the existence of non-trivial solutions, positive solutions and negative solutions of the problems under consideration. Applications of our results to related eigenvalue problems are also discussed. Examples are included to illustrate some of the results. Our analysis relies mainly on topological degree theory.

In this paper, we investigate the existence of positive solutions to a second-order Sturm–Liouville boundary-value problem with impulsive effects. The ideas involve differential inequalities and variational methods.

This paper presents new existence results for the singular boundary-value problem

\begin{gather*} -u''+p(t)u=f(t,u),\quad t\in(0,1),\\ u(0)=0=u(1). \end{gather*}

In particular, our nonlinearity $f$ may be singular at $t=0,1$ and $u=0$.

We consider a class of even-order boundary-value problems with nonlinear boundary conditions and an eigenvalue parameter $\lambda$ in the equations. Sufficient conditions are obtained for the existence and non-existence of positive solutions of the problems for different values of $\lambda$.

The problem of finding a george joinning given points x0, x1 in a connected complete Riemannian manifold requires much more effort than determining a geodesic from initial data. Boundary value problems of this type are sometimes solved using shooting methods, which work best when good initial guesses are available expectually when x0, x1 are nearby. Galerkin methods have their drawbacks too. The situation is much more difficult with general variational problems, which is why we focus on the Riemannian case.

Our global algorithm is very simple to implement, and works well in practice, with no need for an initial guess. The proof of convergence to elementary and very carefully stated. with a view to possible generalizations latter on we have in mind the much larger class of interesting problems arising in optimal control especially from mechanical engineering.