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In this paper we investigate the existence of nontrivial solutions for the following fractional boundary value problem: ddt(120Dt-β(u′(t))+12tDT-β(u′(t)))+∇F(t,u)=0,a.e.t∈[0,T],u(0)=u(T)=0,where 0Dt-β and tDT-β are the left and right Riemann-Liouville fractional integrals of order 0≤β<1, respectively, and ∇F(t,u) is the gradient of F(t, u) at u. The novelty of this paper is that, when the nonlinearity F(t, u) involves a combination of superquadratic and subquadratic terms, we present some reasonable assumptions and establish one new criterion to guarantee the existence of at least two nontrivial solutions. Recent results in the literature are generalized and significantly improved. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the spectral properties of one eigenvalue problem for the fourth-order ordinary differential equations with a spectral parameter contained in two of the boundary conditions. This spectral problem arises when the Fourier method is applied to a boundary value problem for partial differential equations describing small bending vibrations of a homogeneous rod under the action of a longitudinal force in cross sections. The left end of the rod is either free or freely supported, and the inertial mass is concentrated on the right end. We find the arrangement of the eigenvalues on the real axis, determine the multiplicities of all eigenvalues, and study the asymptotic behavior of the eigenvalues and eigenfunctions of this problem. Moreover, sufficient conditions were found under which the system of root functions with two removed functions is a basis in the space L p , 1 < p < ∞ . [ABSTRACT FROM AUTHOR]

In this paper, we give an existence theorem about positive solutions for the Dirichlet boundary value problem of one dimensional Minkowski curvature equations. We apply the theorem to one parameter family of problems to investigate a constructive method for numerical range of parameters where positive solutions exist. Moreover, we establish a nonexistence theorem of positive solutions for the corresponding one parameter family of problems. The coefficient function may be singular at the boundary and nonlinear term satisfies a sublinear growth condition. Main argument for the proof of existence theorem is employed by Krasnoselskii's theorem of cone expansion and compression. We give a numerical algorithm and various examples to illustrate numerical information about ranges of the existence and nonexistence parameters which have been given only in a theoretical manner so far. [ABSTRACT FROM AUTHOR]

In this paper, we develop an efficient explicit method based on the spectral parametric iteration method and quasilinearization scheme, which can be used for the efficient numerical solution of nonlinear stiff/nonstiff two-point boundary value problems. The method derived here has the advantage that it does not require the solution of nonlinear systems of equations. We derive the method, which requires one evaluation of the Jacobian and one LU decomposition per step. Some numerical experiments on nonlinear stiff/nonstiff problems show the efficiency and accuracy of the method. Moreover, the method provides us a simple way to control and modify the convergence rate of the solution. [ABSTRACT FROM AUTHOR]

This paper investigates the analytical approximate solutions of singular fourth order four-point boundary value problems using reproducing kernel method (RKM). The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the RKM can not be used directly to solve singular fourth order four-point boundary value problems (BVPs), since there is no method of obtaining reproducing kernel (RK) satisfying four-point boundary conditions. The aim of this paper is to fill this gap. A method for obtaining RK satisfying four-point boundary conditions is proposed so that RKM can be used to solve singular fourth order four-point BVPs. Results of numerical examples demonstrate that the method is quite accurate and efficient for singular fourth order four-point BVPs. [ABSTRACT FROM AUTHOR]

In this paper, we present some new results concerning positive solutions for the singular fractional boundary value problem with p-Laplacian. By imposing some suitable conditions on the nonlinear term f, existence results of positive solutions are obtained. The proof is based upon theory of Leray-Schauder degree. The interesting point is the nonlinear term f( t, u) may be singular at $$u=0$$ . [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of multiple solutions for the following biharmonic problem Δ 2 u = f (x , u) + g (x , u) in Ω , u = Δ u = 0 on ∂ Ω , where Ω ⊂ R N , (N > 4) is a smooth bounded domain and f (x , ξ) is odd in ξ , g (x , ξ) is a perturbation term. By using the variant of Rabinowitz's perturbation method, under some growth conditions on f and g, we show that there are infinitely many weak solutions to the problem. [ABSTRACT FROM AUTHOR]

In the present paper, we establish the range of two parameters for which a non-homogeneous boundary value problem admits at least three weak solutions. The proof of the main results relies on recent variational principles due to Ricceri. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to prove a geometric property for analytic functions p in the open unit disk with $$p(0)=1$$ by using Nunokawa's result, which is a generalized form of well-known Jack's lemma. This property concerns a boundary behavior of the functions p. As the applications of the main result, we obtain a few corollaries where several sufficient conditions for p to be of the real part greater than a given number $$\beta \ (0\le \beta <1)$$ are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, an initial boundary value problem for a system of semi-linear hyperbolic equations with damped term in a bounded domain is considered. We prove the global existence, uniquenes, blow up of solutions and give some estimates for the lifespan of solutions. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a new class of nonlocal boundary value problems of fractional differential equations and inclusions with a new integral boundary condition. This new boundary condition states that the value of the unknown function at an arbitrary (local) point $$\xi $$ is proportional to the contribution due to a sub-strip of arbitrary length $$(1-\eta ),$$ that is, $$x(\xi )=a \int _{\eta }^{1}x(s)\mathrm{d}s,$$ where $$0< \xi < \eta <1$$ and a is constant of proportionality. The existence of solutions for the given problems is shown by means of contraction mapping principle, a fixed point theorem due to O'Regan and nonlinear alternative for multivalued maps. The results are well illustrated with the aid of examples. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of nonlinear diffusion equations coupled by the nonlinear boundary sources on the exterior domain of the unit ball in $$\mathbb R^N$$ . We are interested in the critical curves which can describe the large time behavior of the solutions. It is shown that the critical global existence curve coincides with the critical Fujita curve for the system we considered. [ABSTRACT FROM AUTHOR]

This paper mainly deals with extinction and non-extinction of weak solutions for a non-Newton polytropic filtration system coupled with non-local source terms under homogeneous Dirichlet boundary condition by comparing the exponents of nonlinear terms. Particularly, precise decay estimates of extinction solutions in the whole dimensional space are also established. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce a method to handle some boundary value problems generated by fifth-order formally symmetric differential equation and separated, real-coupled and complex-coupled boundary conditions. Moreover, the continuity properties of the eigenvalues of these problems on some data are studied and some Frechet derivatives of the eigenvalues are introduced. [ABSTRACT FROM AUTHOR]

In this paper, the weak solutions to a class of nonlinear system of fractional boundary value problems are studied. Each equation of the system does have two kinds of nonlinear terms; one of them is in terms of gradient, and the other one is a nonlinear source term with respect to dependent variable. It is proved that there exists at least three distinct weak solutions by using the critical point theory and the variational method. To this aim, we apply the well-known theorem on the construction of the critical set of functionals with a weak compactness condition. Moreover, the main result is demonstrated by some examples to show its validity. [ABSTRACT FROM AUTHOR]

In this paper, we consider the boundary value problem for the one-dimensional Dirac system with spectral parameter in the boundary condition. We completely study the oscillatory properties of components of the eigenvector functions of this problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider some regular boundary value problems generated by a third-order differential equation and some boundary conditions. In particular, we construct maximal self-adjoint, maximal dissipative and maximal accumulative extensions of the minimal operator. Further using Lax–Phillips scattering theory and Sz.-Nagy–Foiaş characteristic function theory we prove a completeness theorem. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the existence of solutions, in infinite- dimensional Banach spaces, for impulsive differential inclusions and equations of order q ∈ (1 , 2) , with anti-periodic conditions and involving the Caputo derivative whether in the generalized sense (via the Riemann–Liouville fractional derivative,) or in the normal sense and whether the lower limit on each impulsive subinterval (t i , t i + 1 ] , i = 0 , 1 , ... , m is keep at zero or is set at t i . Using the technique of fixed point and the properties of the measure of noncompactness, existence results are obtained. Moreover, we derive in reflexive Banach spaces, by using a new version weakly convergent criteria in the space of piecewise continuous functions, an existence result of solutions without assuming any condition on the multivalued function in terms of measure of noncompactness. Some examples will be given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and uniqueness of solutions of a boundary value problem for a fractional differential equation of order α ∈ (2 , 3) , involving a general form of fractional derivative. First, we prove an equivalence between the Cauchy problem and the Volterra equation. Then, two results on the existence of solutions are proven, and we end with some illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, by applying the cone theory in ordered Banach spaces associated with the characters of increasing φ - (h , e) -concave operators, we investigate the existence and uniqueness of nontrivial solutions for a nonlinear fractional q-difference equation boundary value problem. The main results show that we can construct an iterative scheme approximating the unique nontrivial solution. Relying on an example, we show the efficiency and applicability of the main result. [ABSTRACT FROM AUTHOR]

In this paper, we study a coupled system of Caputo fractional differential equations with nonlinearities depending on the unknown functions as well as their derivatives, equipped with new kinds of integral and multi-point (discrete) boundary conditions. In fact, we have introduced the idea of unification of coupled strip and multi-point boundary conditions with their different variants in the present work. Though we apply the standard tools of the fixed point theory to develop the existence criteria for the solutions of given problems, the obtained results are new in the given scenario. Some examples illustrating the main results are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of sign-changing solutions and positive solutions for second-order nonlinear difference equations on a finite discrete segment with periodic boundary condition provided that the nonlinearity is asymptotically linear at infinity. The critical point theory and variational methods are employed to discuss this problem. [ABSTRACT FROM AUTHOR]

In this paper, we use the reproducing kernel Hilbert space method for solving a boundary value problem for the second order Bratu's differential equation. Convergence analysis of presented method is discussed. The numerical approximations to the exact solution are computed and compared with other existing methods. Our presented method produces more accurate results in comparison with those obtained by Adomian decomposition, Laplace decomposition, B-spline, non-polynomial spline and Lie-group shooting methods. Our yardstick is absolute error. The comparison of the results with exact ones is made to confirm the validity and efficiency. [ABSTRACT FROM AUTHOR]

We study a boundary value problem of sequential fractional differential equations equipped with nonlocal integral boundary conditions (strip conditions of finite arbitrary size) involving the first-order derivative of the unknown function. As a variant problem, we discuss a case when nonlocal integral boundary conditions are governed by the unknown function. The existence of solutions for the given problems is investigated by means of some standard tools of fixed point theory. We emphasize that existence results obtained in this paper are new and enrich the existence theory for sequential fractional differential equations. The obtained results are well illustrated with the aid of examples. [ABSTRACT FROM AUTHOR]

In this paper, by using the coincidence degree theory, we consider the following Neumann boundary value problem for a coupled system of fractional differential equations ... where D0+α, D0+β are the standard Caputo fractional derivative, 1 < α ≤ 2, 1 < β ≤ 2. A new result on the existence of solutions for above fractional boundary value problem is obtained. [ABSTRACT FROM AUTHOR]

This paper investigates the existence and uniqueness of C[0, 1] positive solutions for a second order integral boundary value problem. We mainly use the method of lower and upper solutions and the maximal principle. Our nonlinearity f(t, u) may be singular at u = 0, t = 0, 1. [ABSTRACT FROM AUTHOR]

The paper studies the posedness of the convection diffusion equation with homogeneous boundary condition and with the initial value $$u_0(x)\in L^{q-1+\frac{1}{m}}(\Omega )$$ . By considering its regularized problem, using Moser iteration technique, the local bounded properties of the $$L^{\infty }$$ -norm of $$u_{k}$$ and that of the $$L^{p}$$ -norm of the gradient $$\nabla u_{k}$$ are got, where $$u_{k}$$ is the solution of the regularized problem of the equation. By the compactness theorem, the existence of the solution of the equation itself is obtained. By using some techniques in Zhao and Yuan (Chin Ann Math A 16(2):179-194, 1995), the stability of the solutions is obtained too. [ABSTRACT FROM AUTHOR]

A matching in a graph is a set of edges no two of which share a common vertex. A matching is an induced matching if no two edges in the matching have a third edge in the graph connecting them. The problem of finding a maximum induced matching or shortly MIM is known to be NP-hard in general, and it remains so even when the input graph is bipartite. The decision problem of MIM is NP-complete in general, and it remains NP-complete even if restricted to several classes of graphs. On the other hand, the problem has been shown to be polynomial for some special sets of graphs. In this paper, we give tight upper and lower bounds on maximum induced matching in special subset of planar graphs, called hexagonal graphs. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of boundary value problems of fractional differential equations with integral and anti-periodic boundary conditions, which is a new type of mixed boundary condition. Using the contraction mapping principle, Krasnosel'skii fixed point theorem, and Leray-Schauder degree theory, we obtain some results of existence and uniqueness. Finally, several examples are provided for illustrating the applications of our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider the solvability of a third-order three-point boundary value problem on a half-line of the form: where $$\alpha \ne 1$$ and $$\eta \in (0,+\infty )$$ , while $$f:[0,+\infty )\times {\mathbb {R}}^3 \rightarrow {\mathbb {R}}$$ is $$S^2$$ -Carathéodory function. The existence and uniqueness of solutions for the boundary value problems are obtained by the Leray-Schauder continuation theorem. As an application, an example is given to demonstrate our results. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and multiplicity of positive solutions for a system of second-order discrete boundary value problem. The main results are obtained via Jensen's inequalities, properties of concave and convex functions, and the Krasnosel'skii-Zabreiko fixed point theorem. Furthermore, concave and convex functions are employed to emphasize the coupling behaviors of nonlinear terms $$f$$ and $$g$$ and we provide two explicit examples to illustrate our main results and the coupling behaviors. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the following nonlinear singular $$m$$ -point boundary value problem with fractional $$q$$ -differences where $$0 < \xi _1<\xi _2<\cdots <\xi _{m-2} < 1$$ and $$0 < \sum \nolimits _{i=1}^{m-2}\beta _i\xi _i^{\alpha -2} <1$$ , $$0 < q < 1$$ . $$f: (0, 1]\times [0, +\infty ) \rightarrow [0, +\infty )$$ with $$\lim \nolimits _{t\rightarrow 0^+}f(t,\cdot ) = \infty $$ , i.e., $$f$$ is singular at $$t=0$$ . By using the fixed point theorem in partially ordered sets, some new existence and uniqueness of positive solutions to the above boundary value problem are established. As application, an example is presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we use the Faedo-Galerkin method, compactness method and monotone method in order to study a nonlinear Love's equation with mixed nonhomogeneous conditions. The results obtained here are existence of a weak solution, uniqueness, regularity and asymptotic behavior of solutions. [ABSTRACT FROM AUTHOR]

In this work, we apply the extension of Mawhin's coincidence degree theory by Ge and Ren to show existence of at least one solution to the nonlinear higher order p-Laplacian boundary value problems with integral boundary conditions on the half-line. The results obtained generalizes and compliments existing results in the literature. An example will be used to validate our result. [ABSTRACT FROM AUTHOR]

Boundary value problems for fractional elliptic equations with parameter in Banach spaces are studied. Uniform L p -separability properties and sharp resolvent estimates are obtained for elliptic equations in terms of fractional derivatives. Particularly, it is proven that the fractional elliptic operators generated by these equations are positive and also are generators of the analytic semigroups. Moreover, maximal regularity properties of the fractional abstract parabolic equation are established. As an application, the parameter-dependent anisotropic fractional differential equations and the system of fractional differential equations are studied. [ABSTRACT FROM AUTHOR]

In this article, we establish criteria for the existence of at least three solutions of a fractional differential equation with Dirichlet boundary condition using variational methods and critical point theory. We discuss some consequences of our main results. Our work complements existing results in the literature. [ABSTRACT FROM AUTHOR]

In this article, we propose new spectral solutions for odd-order two-point boundary value problems. A numerical algorithm based on the use of collocation methods is implemented and analyzed. A new matrix of derivatives of certain Legendre basis polynomials is derived to serve in deriving the proposed algorithm. Our algorithm has the advantage that it is applicable to both linear and nonlinear odd-order two-point boundary value problems. The convergence and error analysis of the suggested method are investigated. Five illustrative examples supported with comparisons are displayed to ensure the efficiency and applicability of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The main aim of this work is to investigate the properties of scattering solutions and the scattering function of an impulsive Sturm–Liouville boundary value problem (ISBVP). It is important that the boundary condition depend on an eigenvalue parameter. After getting the Jost solution of this ISBVP, we find the scattering function and resolvent operator. Also, we discuss the discrete spectrum of this boundary value problem in detail. Finally, we present examples on an unperturbated ISBVPs to demonstrate the application of our results. [ABSTRACT FROM AUTHOR]

We prove the uniqueness of solutions for the boundary value problem of harmonic maps in the setting: given any continuous data f on the harmonic boundary of a complete Riemannian manifold with image within a regular geodesic ball, there exists a unique harmonic map, which is a limit of a sequence of harmonic maps with finite total energy in the sense of the supremum norm, from the manifold into the ball taking the same boundary value at each harmonic boundary point as that of f. [ABSTRACT FROM AUTHOR]

Fourth-order boundary value problems with general real self-adjoint boundary conditions are investigated. It is obtained that the eigenvalues of the problem depend not only continuously on all parameters of the problem but also smoothly on the boundary conditions. Furthermore, the derivatives of the eigenvalues with respect to boundary conditions are given in the sense of the fundamental canonical forms of fourth-order self-adjoint boundary conditions including each type of separated, mixed and coupled cases. [ABSTRACT FROM AUTHOR]

We study the homogeneous Dirichlet boundary value problem of generalized Laplacian equations with a singular weight which may not be integrable. Some existence, multiplicity and nonexistence results of positive solutions under two different asymptotic behaviors of the nonlinearity at 0 and ∞ are established in terms of different ranges of a parameter. Our approach is based on the fixed point theorem of expansion/compression of a cone and Schauder's fixed point theorem. [ABSTRACT FROM AUTHOR]

We consider an impulsive boundary value problem consisting of a quantum difference equation and impulsive boundary conditions. Discussing the Jost solution and scattering function of this problem, we study the properties of scattering function and relations between scattering solutions. Besides, we present an example that informs us about the eigenvalues of a special case of this problem. [ABSTRACT FROM AUTHOR]

We study some properties of Hermitian-2-monogenic circulant matrix function (H -2-monogenic function for short) and Hermitian singular integral matrix operator, i.e., Hermitian Hilbert transform. Using these properties and fixed-point theorem, we investigate a kind of generalized Riemann-type problem for H -2-monogenic function, where the constant circulant matrix in the transmission condition is changed into a circulant matrix function. Under some assumptions, we prove that the boundary value problem has a unique solution. [ABSTRACT FROM AUTHOR]

We study the homogeneous Dirichlet boundary value problem of a singular (p 1 , p 2) -Laplacian system with multiparameters. The global structure of multiparameters for existence, nonexistence and multiplicity of positive solutions can be obtained. Proofs are mainly based on the upper and lower solution method, the fixed point index theory in a cone, generalized Picone identity, global continuum theorem and degree arguments. As an application, we can establish the global result of the positive radial and decaying solutions for a class of quasilinear elliptic systems in exterior domains. [ABSTRACT FROM AUTHOR]

We consider a semilinear Robin problem driven by the negative Laplacian plus an indefinite, unbounded potential. The reaction term is a Caratheodory function of arbitrary structure outside an interval [-c,c] (c>0), odd on [-c,c] and concave near zero. Using a variant of the symmetric mountain pass theorem, together with truncation, perturbation and comparison techniques, we show that the problem has a whole sequence {un}n≥1 of distinct nodal solutions converging to zero in C1(Ω¯). [ABSTRACT FROM AUTHOR]

In this article, we introduce and study α-tall modules. We show that an α-tall module, where α≥0, is a tall module, i.e. M contains a submodule N such that N and MN are both non-Noetherian. We observe that every submodule of α-tall modules is countably generated, where α is countable. It is shown that if M is a β-atomic module, where β=α+2, for some ordinal α, then M is α-tall. It is also proved that if M is an α-atomic module, where α is a limit ordinal, then M is both an α-tall and α-short module. [ABSTRACT FROM AUTHOR]

The initial-boundary value problem for a system of higher-order nonlinear Kirchhoff-type equations with damping and source terms in bounded domain is studied. We prove a global nonexistence result for certain solutions under positive initial energy. [ABSTRACT FROM AUTHOR]

We consider the existence of positive solutions for a class of discrete second-order four-point boundary value problem with p-Laplacian. Using the well known Krasnosel'skii's fixed point theorem, some new existence criteria for positive solutions of the boundary value problem are presented. [ABSTRACT FROM AUTHOR]

A new comparison theorem is proved and then used to investigate the solvability of a third-order two-point boundary value problem {u‴(t) + f(t,u(t), u'(t), u″(t)) = 0, u(0) = u'(2π) = 0, u″(0) = u″(2π). Some existence results are established for this problem via upper and lower solutions method and fixed point theory. [ABSTRACT FROM AUTHOR]