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The quantity of 2(2k+1)-periodic solutions to a specific differential delay system with 2k lags is studied and resolved by variational methods. Several results are revealed and two examples are given to illustrate the application of the main results.

In this paper, we discuss the existence of solutions for a nonlocal boundary value problems for hybrid ϕ-Caputo fractional integro-differential equations. Our main result is based on a hybrid fixed point theorem due to Dhage. Finally, we give an example to illustrate our main result.

Abstract In this paper, we consider differential delay systems of the form x′(t)=−∑s=12k−1(−1)s+1∇F(x(t−s)), $$x'(t)=-\sum_{s=1}^{2k-1}(-1)^{s+1} \nabla F \bigl(x(t-s) \bigr), $$ in which the coefficients of the nonlinear terms with different hysteresis have different signs. Such systems have not been studied before. The multiplicity of the periodic orbits is related to the eigenvalues of the limit matrix. The results provide a theoretical basis for the study of differential delay systems.

Abstract In this paper, we study the periodic solutions of high order differential delay equations with 2k−1 $2k-1$ lags. The 4k-periodic solutions are obtained by using the variational method and the method of Kaplan–Yorke coupling system. These are new types of differential delay equations compared with all previous research. And it provides a precise counting method for the number of periodic solutions. Two examples are given to demonstrate our main results.

Abstract This paper is concerned with the existence of positive periodic solutions for the prescribed mean curvature Rayleigh equations with a singularity. Our results are based on the Manásevich-Mawhin continuation theorem. The results to be obtained here extend the existing ones in the literature. Moreover, an example is given to illustrate the applicability of our results.

In this paper, we study the periodic orbits of a type of odd order differential delay system with 2k−1 lags via the S1 index theory and the variational method. This type of system has not been studied by others. Our results provide a new and more accurate method for counting the number of periodic orbits.

In this article we show the existence of solutions to a mixed boundary-value problem of second-order differential systems with a p-Laplacian. The associated Hamiltonian actions are indefinite and the discussion of the existence of solutions is due to the application of duality principle.

Using critical point theory, we prove the existence of multiple positive solutions for second-order discrete boundary value problems with p-Laplacian.

In this article, we study the self-adjoint second-order boundary-value problem with integral boundary conditions, $$displaylines{ (p(t)x'(t))'+ f(t,x(t))=0,quad tin (0,1),cr p(0)x'(0)=p(1)x'(1),quad x(1)=int_0^1x(s)g(s)ds, }$$ which involves an integral boundary condition. We prove the existence of positive solutions using a new tool: the Leggett-Williams norm-type theorem for coincidences.

In this paper, we deal with the following singular four-point boundary value problem with $p$-Laplacian $$ \left\{\begin{aligned} &(\phi_{p}(u'(t)))'+q(t)f(t,u(t))=0,\ t\in(0,1),\\ &u(0)-\alpha u'(\xi)=0,\ u(1)+\beta u'(\eta)=0, \end{aligned}\right. $$ where $f(t,u)$ may be singular at $u=0$ and $q(t)$ may be singular at $t=0$ or $1$. By imposing some suitable conditions on the nonlinear term $f$, existence results of at least two positive solutions are obtained. The proof is based upon theory of Leray-Schauder degree and Krasnosel'skii's fixed point theorem.

A multi-point boundary value problem involving the one dimensional $p$-Laplacian and depending on a parameter is studied in this paper and existence of positive solutions is established by means of a fixed point theorem for operators defined on Banach spaces with cones.

This paper investigates the solvability of the second-order boundary value problems with the one-dimensional $p$-Laplacian at resonance on a half-line $$\left\{\begin{array}{llll} (c(t)\phi_{p}(x'(t)))'=f(t,x(t),x'(t)),~~~~01$. The arguments are based upon an extension of Mawhin's continuation theorem due to Ge. And examples are given to illustrate our results.

In this work, we consider the second order multi-point boundary-value problem with a p-Laplacian $$displaylines{ ( ho(t)Phi_p(x'(t)))'+f(t, x(t), x'(t))=0,quad tin [0,+infty),cr x(0)=sum_{i=1}^{m}alpha_i x(xi_i), quad lim_{toinfty}x(t)=0,. }$$ By applying a nonlinear alternative theorem, we establish existence and uniqueness of solutions on the half-line. Also a uniqueness result for positive solutions is discussed when $f$ depends on the first-order derivative. The emphasis here is on the one dimensional p-Laplacian operator.

In this paper, we study the three-point boundary-value problem for a fourth-order one-dimensional $p$-Laplacian differential equation $$ ig(phi_p(u''(t))ig)''+ a(t)fig(u(t)ig)=0, quad tin (0,1), $$ subject to the nonlinear boundary conditions: $$displaylines{ u(0)=xi u(1),quad u'(1)=eta u'(0),cr (phi _{p}(u''(0))' =alpha _{1}(phi _{p}(u''(delta))', quad u''(1)=sqrt[p-1]{eta _{1}}u''(delta), }$$ where $phi_{p}(s)=|s|^{p-2}s$, $p>1$. Using the five functional fixed point theorem due to Avery, we obtain sufficient conditions for the existence of at least three positive solutions.

In this paper, we investigate the periodic solutions of Hamiltonian system with the p-Laplacian. By using Mountain Pass Theorem the existence of at least one periodic solution is obtained, Furthermore, under suitable assumptions, we obtain the existence of infinitely many solutions via $Z_2$-symmetric version of the Mountain Pass Theorem.

In this paper, we discuss the existence, nonexistence, and multiplicity of positive solutions for a class of singular eigenvalue problems. Some of our theorems are new, while others extend earlier results obtained by Zhang and Kong [12]. The interesting point is that the authors obtain the relation between the existence of solutions and the parameter $lambda$. The arguments are based on the fixed point index theory and the upper and lower solutions method.

In this paper, some nonexistence, existence and multiplicity of positive solutions are established for a class of singular boundary value problem. The authors also obtain the relation between the existence of the solutions and the parameter $\lambda$. The arguments are based upon the fixed point index theory and the upper and lower solutions method.

We prove existence results for solutions of periodic boundary value problems concerning the nth-order differential equation with p-Laplacian [φ(x(n−1)(t))]'=f(t,x(t),x'(t),...,x(n−1)(t)) and the boundary value conditions x(i)(0)=x(i)(T), i=0,...,n−1. Our method is based upon the coincidence degree theory of Mawhin. It is interesting that f may be a polynomial and the degree of some variables among x0,x1,...,xn−1 in the function f(t,x0,x1,...,xn−1) is allowed to be greater than 1.

A new fixed point theorem on double cones is applied to obtain the existence of at least two positive solutions to $$displaylines{ (Phi_p(y''(t))''-a(t)f(t,y(t),y''(t))=0,quad 0 less than t less than 1,cr y(0)=y(1)=0=y''(0)=y''(1), }$$ where $f:[0,1]imes[0,infty)imes(-infty,0]o R, ain L^1([0,1],(0,infty)) $. We also give some examples to illustrate our results.

A new fixed point theorem on cones is applied to obtain the existence of at least two positive solutions of a higher-order three-point boundary value problem for the differential equation subject to a class ofboundary value conditions. The associated Green's function is given. Some results obtained recently are generalized.

We obtain sufficient conditions for the existence of at least three positive solutions for the equation $ x''(t) + q(t)f(t, x(t), x'(t)) = 0 $ subject to some boundary conditions. This is an application of a new fixed-point theorem introduced by Avery and Peterson [6].

In this article, we study the differential equation $$ (-1)^{n-p} x^{(n)}(t)=f(t,x(t),x'(t),dots,x^{(n-1)}(t)), $$ subject to the multi-point boundary conditions $$displaylines{ x^{(i)}(0)=0 quad hbox{for }i=0,1,dots,p-1,cr x^{(i)}(1)=0 quad hbox{for }i=p+1,dots,n-1,cr sum_{i=1}^malpha_ix^{(p)}(xi_i)=0, }$$ where $1le ple n-1$. We establish sufficient conditions for the existence of at least one solution at resonance and another at non-resonance. The emphasis in this paper is that $f$ depends on all higher-order derivatives. Examples are given to illustrate the main results of this article.

In this paper, we consider multi-point boundary-value problems for the p-laplacian at resonance. Applying an extension of Mawhin's continuation theorem, we show the existence of at least one solution.

This article concerns the existence of positive solutions to the differential equation $$ (-1)^m x^{(2m)}(t)=f(t,x(t),x'(t),dots,x^{(m)}(t)), quad 0

Upper and lower solutions theories are established for a kind of m-point impulsive boundary value problems with p-Laplacian. By using such techniques and the Schauder fixed point theorem, the existence of solutions and positive solutions is obtained. Nagumo conditions play an important role in the nonlinear term involved with the first-order derivatives.

We consider a kind of Sturm-Liouville boundary value problems. Using variational techniques combined with the methods of upper-lower solutions, the existence of at least one positive solution is established. Moreover, the upper solution and the lower solution are presented.

This paper investigates the existence and multiplicity of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions. The results are established by converting the problem into an equivalent integral equation and applying Krasnoselskii's fixed-point theorem in cones. The nonexistence of positive solutions is also studied.

The fixed point theorem of cone expansion and compression of norm type for a strict set contraction operator is generalized by replacing the norms with a convex functional satisfying certain conditions. We then show how to apply our theorem to prove the existence of a positive solution to a second-order differential equation with integral boundary conditions in an ordered Banach space. An example is worked out to demonstrate the main results.

By constructing an available integral operator and combining Krasnosel'skii-Zabreiko fixed point theorem with properties of Green's function, this paper shows the existence of multiple positive solutions for a class of m-point second-order Sturm-Liouville-like boundary value problems on time scales with polynomial nonlinearity. The results significantly extend and improve many known results for both the continuous case and more general time scales. We illustrate our results by one example, which cannot be handled using the existing results.

By constructing available upper and lower solutions and combining the Schauder's fixed point theorem with maximum principle, this paper establishes sufficient and necessary conditions to guarantee the existence of Cld[0,1]𝕋 as well as CldΔ[0,1]𝕋 positive solutions for a class of singular boundary value problems on time scales. The results significantly extend and improve many known results for both the continuous case and more general time scales. We illustrate our results by one example.

This paper investigates the existence of positive solutions for a class of second-order singular m-point Sturm-Liouville-type boundary value problems by using fixed point theorem in cones. The results significantly extend and improve many known results even for nonsingular cases.

We consider the following impulsive boundary value problem, x″(t)=f(t,x,x′), t∈J\{t1,t2,…,tk}, Δx(ti)=Ii(x(ti),x′(ti)), Δx′(ti)=Ji(x(ti),x′(ti)), i=1,2,…,k, x(0)=(0), x′(1)=∑j=1m−2αjx′(ηj). By using the coincidence degree theory, a general theorem concerning the problem is given. Moreover, we get a concrete existence result which can be applied more conveniently than recent results. Our results extend some work concerning the usual m-point boundary value problem at resonance without impulses.