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We investigate solutions to nonlinear operator equations which are difficult to investigate with variational methods and obtain some abstract existence results by topology degree methods. These results apply to ordinary differential systems with impulsive effects satisfying nonlinear boundary value conditions, and we obtain some new results.

In this paper, we study the existence of periodic solutions for a class of ordinary p-Laplacian systems. Our technique is based on the generalized mountain pass theorem of Rabinowitz.

By using the cone theory and the Banach contraction mapping principle, we study the existence and uniqueness of an iterative solution to the singular nth-order nonlocal boundary value problems.

This paper investigates some nonlinear third-order ordinary differential equations and inclusions with anti-periodic type integral boundary conditions and multi-strip boundary conditions. To establish the existence results for the given problems, we apply standard tools of fixed point theory for single-valued and multi-valued problems. The results are illustrated with the aid of examples.

In this article, we prove the existence of a nontrivial positive solution for the elliptic system

The paper studies the question of solution existence to a nonlinear equation in the degenerate case. This question is studied for three particular boundary value problems for ordinary and partial second-order differential equations. The so-called p-regularity theory is applied to these purposes as an effective apparatus to investigate many nonlinear mathematical, physical and numerical problems. All results obtained in the paper are based on the constructions of this theory whose basic concepts were described by Tret’yakov. We recall the main definitions and theorems of p-regularity theory and illustrate the results by examples including singular boundary value problems. In the first and second ones, the description of solutions by a tangent cone at an initial point are given. In the third example, we formulate a sufficient condition for p-regularity ( p = 2 ), which can be tested using the notion of resultant. MSC: 47J05, 34B15, 34B16.

The existence of anti-periodic solutions with symmetry for high-order Duffing equations and a high-order Duffing type p-Laplacian equation has been studied by using degree theory. The results obtained enrich some known works to some extent.

In this paper, by using the coincidence degree theory, we consider the following boundary value problem for fractional differential equation D 0 + α x ( t ) = f ( t , x ( t ) , x ′ ( t ) , x ″ ( t ) ) , t ∈ [ 0 , 1 ] , x ( 0 ) = x ( 1 ) , x ′ ( 0 ) = x ″ ( 0 ) = 0 , where D 0 + α denotes the Caputo fractional differential operator of order α, 2 < α ≤ 3. A new result on the existence of solutions for above fractional boundary value problem is obtained. Mathematics Subject Classification (2000): 34A08, 34B15.

We study periodic solutions for nonlinear second-order ordinary differential problem . By constructing upper and lower boundaries and using Leray-Schauder degree theory, we present a result about the existence and uniqueness of a periodic solution for second-order ordinary differential equations with some assumption.

We consider a second-order ordinary differential equation with antiperiodic boundary conditions and impulses. By using Schaefer's fixed-point theorem, some existence results are obtained.

Uniqueness implies uniqueness relationships are examined among solutions of the fourth-order ordinary differential equation, , satisfying -point, -point, and -point nonlocal boundary conditions.

Singular boundary value problems for ordinary differential equations model many real world phenomena ranging from different physics equations to biological, physiological, and medical processes 1–3 . This special issue places its emphasis on the study, theory, and applications of boundary value problems involving singularities. It includes some review articles such as 4 , equations with discontinuous nonlinearities 5 , boundary value problems with uncertainty 6 , fractional differential equations 7 , periodic or antiperiodic solutions 8 , and biological 9 or medical applications 10 . Different methods and techniques are used ranging from variational methods 11 to bifurcation techniques 12 . The editors aimed at a volume that may serve as a reference in the topic of the special issue and collect twenty five original and cutting-edge research articles by some of the top researchers in boundary value problems for ordinary differential equations worldwide and from many different countries Algeria, Austria, Bulgaria, China, Czech Republic, Greece, Iran, Ireland, Italy, Japan, New Zealand, Pakistan, Saudi Arabia, South Korea, Spain, USA . We would like to thank the authors for their contributions, the Editor-in-Chief of the journal, Professor Ravi P. Agarwal, and the Editorial Office of the journal for their support.

The aim of this paper is to study the asymptotic properties and oscillation of the nth order advanced differential equations (r(t) x (n–1) (t) γ ) � + q(t)x γ τ (t) =0 . The results obtained are based on the Riccati transformation. MSC: 34K11; 34C10

In this paper, a class of fourth-order impulsive differential equations depending on two control parameters is investigated. The existence and multiplicity of solutions are obtained by means of the variational methods and the critical point theory. Finally, an example which supports our theoretical results is also presented.

We present a new existence result for a second-order nonlinear ordinary differential equation with a three-point boundary value problem when the linear part is noninvertible.

In this paper, we study the existence of anti-periodic solutions for a second-order ordinary differential equation. Using the interaction of the nonlinearity with the Fucik spectrum related to the anti-periodic boundary conditions, we apply the Leray-Schauder degree theory and the Borsuk theorem to establish new results on the existence of anti-periodic solutions of second-order ordinary differential equations. Our nonlinearity may cross multiple consecutive branches of the Fucik spectrum curves, and recent results in the literature are complemented and generalized.

By using bifurcation techniques, this paper investigates the existence of nodal solutions for a class of fourth-order -point boundary value problems. Our results improve those in the literature.

We study the higher-order boundary value problems. The existence of symmetric positive solutions of the problem is discussed. Our results extend some recent work in the literature. The analysis of this paper mainly relies on the monotone iterative technique.

In this paper, by using the coincidence degree theory and the upper and lower solutions method, we deal with the existence of multiple solutions to three-point boundary value problems for second-order differential equation with impulses at resonance. An example is given to show the validity of our results.

In this paper, we investigate the existence of solutions for some second-order integral boundary value problems, by applying new fixed point theorems in Banach spaces with the lattice structure derived by Sun and Liu.

In the last decades, nonlocal boundary value problems have become a rapidly growing area of research. The study of this type of problems is driven not only by a theoretical interest, but also by the fact that several phenomena in engineering, physics and life sciences can be modelled in this way. For example, problems with feedback controls such as the steady-states of a thermostat, where a controller at one of its ends adds or removes heat, depending upon the temperature registered in another point, can be interpreted with a second-order ordinary differential equation subject to a three-point boundary condition.

In this article, we investigate the Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian { D 0 + β ( ϕ p ( D 0 + α u ( t ) ) ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , ξ u ( 0 ) − η u ′ ( 0 ) = 0 , γ u ( 1 ) + δ u ′ ( 1 ) = 0 , D 0 + α u ( 0 ) = 0 , where 1 < α ≤ 2 , 0 < β ≤ 1 , D 0 + α , D 0 + β are the standard Caputo fractional derivatives, ϕ p ( s ) = | s | p − 2 s , p > 1 , ϕ p − 1 = ϕ q , 1 / p + 1 / q = 1 , ξ , η , γ , δ ≥ 0 , ρ : = ξ γ + ξ δ + η γ > 0 , and f : [ 0 , 1 ] × [ 0 , + ∞ ) → [ 0 , + ∞ ) is continuous. By means of the properties of the Green’s function, Leggett-Williams fixed-point theorems, and fixed-point index theory, several new sufficient conditions for the existence of at least two or at least three positive solutions are obtained. As an application, an example is given to demonstrate the main result. MSC:34A08, 34B18, 35J05.