On the existence of stationary solutions for higher-order p-Kirchhoff problems

In this paper, we establish the existence of two nontrivial weak solutions of possibly degenerate nonlinear eigenvalue problems involving the p-polyharmonic Kirchhoff operator in bounded domains. The p-polyharmonic operators [Formula: see text] were recently introduced in [F. Colasuonno and P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal.74 (2011) 5962–5974] for all orders L and independently, in the same volume of the journal, in [V. F. Lubyshev, Multiple solutions of an even-order nonlinear problem with convex-concave nonlinearity, Nonlinear Anal.74 (2011) 1345–1354] only for L even. In Sec. 3, the results are then extended to non-degeneratep(x)-polyharmonic Kirchhoff operators. The main tool of the paper is a three critical points theorem given in [F. Colasuonno, P. Pucci and Cs. Varga, Multiple solutions for an eigenvalue problem involving p-Laplacian type operators, Nonlinear Anal.75 (2012) 4496–4512]. Several useful properties of the underlying functional solution space [Formula: see text], endowed with the natural norm arising from the variational structure of the problem, are also proved both in the homogeneous case p ≡ Const. and in the non-homogeneous case p = p(x). In the latter some sufficient conditions on the variable exponent p are given to prove the positivity of the infimum λ1of the Rayleigh quotient for the p(x)-polyharmonic operator [Formula: see text].

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Inverse multiparameter eigenvalue problems for matrices II

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500017788 ◽

1986 ◽

Vol 29 (3) ◽

pp. 343-348 ◽

Cited By ~ 3

Author(s):

Patrick J. Browne ◽

B. D. Sleeman

Keyword(s):

Fixed Point ◽

Eigenvalue Problems ◽

Symmetric Matrix ◽

Sufficient Conditions ◽

Degree Theory ◽

Main Tool ◽

Real Numbers ◽

Inverse Eigenvalue Problems ◽

Inverse Eigenvalue ◽

The One

This is a sequel to our previous paper [4] where we initiated a study of inverse eigenvalue problems for matrices in the multiparameter setting. The one parameter version of the problem under consideration asks for conditions on a given n × n symmetric matrix A and on n given real numbers s1≦s2≦…≦sn under which a diagonal matrix V can be found so that A + V has sl,…,sn as its eigenvalues. Our motivation for this problem and our method of attack on it in [4]p comes chiefly from the work of Hadeler [5] in which sufficient conditions were given for existence of the desired diagonal V. Hadeler's approach in [5] relied heavily on the Brouwer fixed point theorem and this was also our main tool in [4]. Subsequently, using properties of topological degree, Hadeler [6] gave somewhat different conditions for the existence of the diagonal V. It is our desire here to follow this lead and to use degree theory to give some results extending those in [6] to the multiparameter case.

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Multiple Solutions for Nonlocal Elliptic Systems Involving p(x)-Biharmonic Operator

Mathematics ◽

10.3390/math7080756 ◽

2019 ◽

Vol 7 (8) ◽

pp. 756

Author(s):

Qing Miao

Keyword(s):

Sobolev Space ◽

Elliptic System ◽

Multiple Solutions ◽

Sufficient Conditions ◽

Elliptic Systems ◽

Biharmonic Operator ◽

Variational Theorem ◽

Variational Structure ◽

Definition Of

This paper analyzes the nonlocal elliptic system involving the p(x)-biharmonic operator. We give the corresponding variational structure of the problem, and then by means of Ricceri’s Variational theorem and the definition of general Lebesgue-Sobolev space, we obtain sufficient conditions for the infinite solutions to this problem.

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Positive Solutions for a Class of Nonlinear Singular Fractional Differential Systems with Riemann–Stieltjes Coupled Integral Boundary Value Conditions

Symmetry ◽

10.3390/sym13010107 ◽

2021 ◽

Vol 13 (1) ◽

pp. 107

Author(s):

Daliang Zhao ◽

Juan Mao

Keyword(s):

Positive Solutions ◽

Boundary Value ◽

Sufficient Conditions ◽

Main Tool ◽

Integral Boundary ◽

Differential Systems ◽

Fractional Differential Systems ◽

Existence And Multiplicity ◽

Fractional Differential ◽

Multiplicity Of Positive Solutions

In this paper, sufficient conditions ensuring existence and multiplicity of positive solutions for a class of nonlinear singular fractional differential systems are derived with Riemann–Stieltjes coupled integral boundary value conditions in Banach Spaces. Nonlinear functions f(t,u,v) and g(t,u,v) in the considered systems are allowed to be singular at every variable. The boundary conditions here are coupled forms with Riemann–Stieltjes integrals. In order to overcome the difficulties arising from the singularity, a suitable cone is constructed through the properties of Green’s functions associated with the systems. The main tool used in the present paper is the fixed point theorem on cone. Lastly, an example is offered to show the effectiveness of our obtained new results.

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Blowing-up solutions of the time-fractional dispersive equations

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0153 ◽

2021 ◽

Vol 10 (1) ◽

pp. 952-971

Author(s):

Ahmed Alsaedi ◽

Bashir Ahmad ◽

Mokhtar Kirane ◽

Berikbol T. Torebek

Keyword(s):

Boundary Value Problems ◽

Boundary Value ◽

Sufficient Conditions ◽

Main Tool ◽

Initial Boundary ◽

Initial Boundary Value Problems ◽

Nonlinear Capacity ◽

Blowing Up ◽

De Vries ◽

Initial Boundary Value

Abstract This paper is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented. We also discuss the maximum principle and influence of gradient non-linearity on the global solvability of initial-boundary value problems for the time-fractional Burgers equation. The main tool of our study is the Pohozhaev nonlinear capacity method. We also provide some illustrative examples.

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Multiple solutions of fourth-order difference equations with different boundary conditions

Boundary Value Problems ◽

10.1186/s13661-019-1265-2 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 3

Author(s):

Yuhua Long ◽

Shaohong Wang ◽

Jiali Chen

Keyword(s):

Boundary Conditions ◽

Fourth Order ◽

Difference Equations ◽

Multiple Solutions ◽

Dirichlet Boundary ◽

Sufficient Conditions ◽

Dirichlet Boundary Conditions ◽

Invariant Sets ◽

Mountain Pass Lemma ◽

Order Difference

Abstract In the present paper, a class of fourth-order nonlinear difference equations with Dirichlet boundary conditions or periodic boundary conditions are considered. Based on the invariant sets of descending flow in combination with the mountain pass lemma, we establish a series of sufficient conditions on the existence of multiple solutions for these boundary value problems. In addition, some examples are provided to demonstrate the applicability of our results.

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Regular semigroups, fundamental semigroups and groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700021649 ◽

1980 ◽

Vol 29 (4) ◽

pp. 475-503 ◽

Cited By ~ 13

Author(s):

D. B. McAlister

Keyword(s):

Regular Semigroup ◽

Partial Order ◽

Direct Product ◽

Homomorphic Image ◽

Sufficient Conditions ◽

Main Tool ◽

Natural Partial Order ◽

Regular Semigroups ◽

Necessary And Sufficient ◽

Fundamental Semigroups

AbstractIn this paper we obtain necessary and sufficient conditions on a regular semigroup in order that it should be an idempotent separating homomorphic image of a full subsemigroup of the direct product of a group and a fundamental or combinatorial regular semigroup. The main tool used is the concept of a prehomomrphism θ: S → T between regular semigroups. This is a mapping such that (ab) θ ≦ aθ bθ in the natural partial order on T.

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Region of existence of multiple solutions for a class of Robin type four-point BVPs

Opuscula Mathematica ◽

10.7494/opmath.2021.41.4.571 ◽

2021 ◽

Vol 41 (4) ◽

pp. 571-600

Author(s):

Amit K. Verma ◽

Nazia Urus ◽

Ravi P. Agarwal

Keyword(s):

Multiple Solutions ◽

Nonlinear Boundary ◽

Source Term ◽

Lipschitz Constant ◽

Sufficient Conditions ◽

Existence Of Solution ◽

Monotone Iterative Technique ◽

Nonlinear Boundary Value Problems ◽

Existence Of A Solution ◽

Monotone Iterative

This article aims to prove the existence of a solution and compute the region of existence for a class of four-point nonlinear boundary value problems (NLBVPs) defined as \[\begin{gathered} -u''(x)=\psi(x,u,u'), \quad x\in (0,1),\\ u'(0)=\lambda_{1}u(\xi), \quad u'(1)=\lambda_{2} u(\eta),\end{gathered}\] where \(I=[0,1]\), \(0\lt\xi\leq\eta\lt 1\) and \(\lambda_1,\lambda_2\gt 0\). The nonlinear source term \(\psi\in C(I\times\mathbb{R}^2,\mathbb{R})\) is one sided Lipschitz in \(u\) with Lipschitz constant \(L_1\) and Lipschitz in \(u'\) such that \(|\psi(x,u,u')-\psi(x,u,v')|\leq L_2(x)|u'-v'|\). We develop monotone iterative technique (MI-technique) in both well ordered and reverse ordered cases. We prove maximum, anti-maximum principle under certain assumptions and use it to show the monotonic behaviour of the sequences of upper-lower solutions. The sufficient conditions are derived for the existence of solution and verified for two examples. The above NLBVPs is linearised using Newton's quasilinearization method which involves a parameter \(k\) equivalent to \(\max_u\frac{\partial \psi}{\partial u}\). We compute the range of \(k\) for which iterative sequences are convergent.

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AN ASYMPTOTIC STABILITY THEOREM FOR QUASILINEAR DELAY DIFFERENTIAL EQUATIONS WITH OSCILLATING COEFFICIENTS

International Journal of Mathematics ◽

10.1142/s0129167x13500924 ◽

2013 ◽

Vol 24 (11) ◽

pp. 1350092 ◽

Cited By ~ 1

Author(s):

NGUYEN TIEN DUNG

Keyword(s):

Asymptotic Stability ◽

Differential Equations ◽

Fixed Point Theory ◽

Sufficient Conditions ◽

Point Theory ◽

Asymptotic Behavior Of Solutions ◽

Variable Delays ◽

Nonlinear Anal ◽

Oscillating Coefficients ◽

Several Delays

In this paper, we provide new necessary and sufficient conditions of the asymptotic stability for a class of quasilinear differential equations with several delays and oscillating coefficients. Our results are established by means of fixed point theory and improve those obtained in [J. R. Graef, C. Qian and B. Zhang, Asymptotic behavior of solutions of differential equations with variable delays, Proc. London Math. Soc.81 (2000) 72–92; B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Anal.63 (2005) e233–e242].

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