Recent zbMATH articles in MSC 34Bhttps://www.zbmath.org/atom/cc/34B2021-04-16T16:22:00+00:00WerkzeugHereditary circularity for energy minimal diffeomorphisms.https://www.zbmath.org/1456.300432021-04-16T16:22:00+00:00"Koh, Ngin-Tee"https://www.zbmath.org/authors/?q=ai:koh.ngin-teeSummary: We reveal some geometric properties of energy minimal diffeomorphisms defined on an annulus, whose existence was established in works by
\textit{T. Iwaniec} et al. [Invent. Math. 186, No. 3, 667--707 (2011; Zbl 1255.30031); J. Am. Math. Soc. 24, No. 2, 345--373 (2011; Zbl 1214.31001)]
and \textit{D. Kalaj} [Calc. Var. Partial Differ. Equ. 51, No. 1--2, 465--494 (2014; Zbl 1296.30052)].Existence of homoclinic orbits for a singular differential equation involving \(p\)-Laplacian.https://www.zbmath.org/1456.340452021-04-16T16:22:00+00:00"Yin, Honghui"https://www.zbmath.org/authors/?q=ai:yin.honghui"Du, Bo"https://www.zbmath.org/authors/?q=ai:du.bo"Yang, Qing"https://www.zbmath.org/authors/?q=ai:yang.qing"Duan, Feng"https://www.zbmath.org/authors/?q=ai:duan.fengIn the present manuscript, the authors are concerned with the existence of {homoclinic} solutions for the following singular ODE
\[
\Big(\Phi_p\big(x'(t)\big)\Big)'+f\big(x'(t)\big) + g\big(x(t)\big) + \frac{h(t)}{1-x(t)} = e(t), \tag{1}
\]
where $\Phi_p(s) = |s|^{p-2}s$ (for some $p > 1$), $f,g,h,e\in C(\mathbb{R};\mathbb{R})$ and, moreover, $h$ is a strictly positive $T$-periodic function.
As usual, a \textit{homoclinic solution} of (1) is a solution $x\in C(\mathbb{R};\mathbb{R})$ satisfying
\[
\text{$x(t)\to\infty$ as $|t|\to\infty$}.
\]
Due to their relevance in several contexts, homoclinic solutions for general differential systems have been studied by many authors and with different techniques (variational methods, critical-point theory, method of lower/upper solutions and fixed-point theorems, etc.); however, since equation (1) is strongly nonlinear, these traditional techniques are no-longer applicable.
Using a new continuation theorem due to Manásevich and Mawhin, the authors obtain the following theorem, which is the main result of the paper.
Theorem 1.
Assume that the following assumptions are satisfied:
\begin{itemize}
\item[{(H.1)}] $f:\mathbb{R}\to\mathbb{R}$ is continuous, bounded and non-negative;
\item[{(H.2)}] $g:\mathbb{R}\to\mathbb{R}$ is strictly monotone increasing and there are positive constants $\sigma$ and $n$ such that
\[
xg(x)\geq \sigma|x|^{n+1}\quad\text{for all $x\in\mathbb{R}$};
\]
\item[{(H.3)}] $\rho_1 := \sup_{t\in\mathbb{R}}|e(t)| < \infty$ and
\[
\rho_2 := \int_{\mathbb{R}}|e(t)|^{1+1/n}\,\mathrm{d} t < \infty.
\]
\end{itemize}
Then, if $\rho_1 > f(0)$ and $h_l/\rho_1 - f(0) < 1$ (with $h_l := \min_{t\in\mathbb{R}}h(t)$), there exists at least one positive homoclinic solution $\omega_0$, further satisfying
\[
|\omega_0'(t)|\to 0\quad\text{as $|t|\to\infty$}.
\]
Thought it is based on the continuation theorem by Manásevich and Mawhin, the proof of Theorem 1 is sophisticated and it requires some preliminary lemmas of independent interest. On the other hand, a couple of examples at the end of the paper show the wide range of applicability of this result.
Reviewer: Stefano Biagi (Milano)Various spectral problems with the same characteristic determinant.https://www.zbmath.org/1456.340202021-04-16T16:22:00+00:00"Akhtyamov, A. M."https://www.zbmath.org/authors/?q=ai:akhtyamov.azamat-moukhtarovich|akhtyamov.azamat-mukhtarovichSummary: We show that there exist whole classes of various boundary value problems having the same characteristic determinant, with the respective problems allowed to have differing orders of the differential equations and to be defined both on intervals and on geometric graphs.Spectral expansion for singular conformable fractional Sturm-Liouville problem.https://www.zbmath.org/1456.340032021-04-16T16:22:00+00:00"Allahverdiev, Bilender P."https://www.zbmath.org/authors/?q=ai:allahverdiev.bilender-pasaoglu"Tuna, Hüseyin"https://www.zbmath.org/authors/?q=ai:tuna.huseyin"Yalçinkaya, Yüksel"https://www.zbmath.org/authors/?q=ai:yalcinkaya.yukselSummary: With this study, the spectral function for singular conformable fractional Sturm-Lioville problem is demonstrated. Further, we establish a Parseval equality and spectral expansion formula by terms of the spectral function.Implicit differential inclusions with acyclic right-hand sides: an essential fixed points approach.https://www.zbmath.org/1456.340152021-04-16T16:22:00+00:00"Andres, Jan"https://www.zbmath.org/authors/?q=ai:andres.jan"Górniewicz, Lech"https://www.zbmath.org/authors/?q=ai:gorniewicz.lechSummary: Effective criteria are given for the solvability of initial as well as boundary valueproblems to implicit ordinary differential inclusions whose right-hand sides are governed by compactacyclic maps. Cauchy and periodic implicit problems are also considered on proximate retracts. Ournew approach is based on the application of the topological essential fixed point theory. Implicitproblems for partial differential inclusions are only indicated.Response to: ``Comment on `The asymptotic iteration method revisited'''.https://www.zbmath.org/1456.340132021-04-16T16:22:00+00:00"Ismail, Mourad E. H."https://www.zbmath.org/authors/?q=ai:ismail.mourad-el-houssieny"Saad, Nasser"https://www.zbmath.org/authors/?q=ai:saad.nasserSummary: This response concers the Comment by \textit{F. M. Fernández} [J. Math. Phys. 61, No. 6, 064101, 2 p. (2020; Zbl 1456.34012)].
{\copyright 2020 American Institute of Physics}Comment on: ``The asymptotic iteration method revisited'''.https://www.zbmath.org/1456.340122021-04-16T16:22:00+00:00"Fernández, Francisco M."https://www.zbmath.org/authors/?q=ai:fernandez.francisco-mSummary: This comment concerns [\textit{M. E. H. Ismail} and \textit{N. Saad}, ibid. 61, No. 3, 033501, 12 p. (2020; Zbl 1443.34021)]. In this comment, we show that the eigenvalues of a quartic anharmonic oscillator obtained recently by means of the asymptotic iteration method may not be as accurate as the authors claim them to be.
{\copyright 2020 American Institute of Physics}An introduction to the mathematical theory of inverse problems. 3rd updated edition.https://www.zbmath.org/1456.350012021-04-16T16:22:00+00:00"Kirsch, Andreas"https://www.zbmath.org/authors/?q=ai:kirsch.andreasPublisher's description: This graduate-level textbook introduces the reader to the area of inverse problems, vital to many fields including geophysical exploration, system identification, nondestructive testing, and ultrasonic tomography. It aims to expose the basic notions and difficulties encountered with ill-posed problems, analyzing basic properties of regularization methods for ill-posed problems via several simple analytical and numerical examples. The book also presents three special nonlinear inverse problems in detail: the inverse spectral problem, the inverse problem of electrical impedance tomography (EIT), and the inverse scattering problem. The corresponding direct problems are studied with respect to existence, uniqueness, and continuous dependence on parameters. Ultimately, the text discusses theoretical results as well as numerical procedures for the inverse problems, including many exercises and illustrations to complement coursework in mathematics and engineering.
This updated text includes a new chapter on the theory of nonlinear inverse problems in response to the field's growing popularity, as well as a new section on the interior transmission eigenvalue problem which complements the Sturm-Liouville problem and which has received great attention since the previous edition was published.
See the review of the first edition in [Zbl 0865.35004]. For the second edition see [Zbl 1213.35004].Positive solutions of the \(p\)-Laplacian dynamic equations on time scales with sign changing nonlinearity.https://www.zbmath.org/1456.340842021-04-16T16:22:00+00:00"Dogan, Abdulkadir"https://www.zbmath.org/authors/?q=ai:dogan.abdulkadirSummary: This article concerns the existence of positive solutions for \(p\)-Laplacian boundary value problem on time scales. By applying fixed point index we obtain the existence of solutions. Emphasis is put on the fact that the nonlinear term is allowed to change sign. An example illustrates our results.On conditional stability of Inverse scattering problem on a Lasso-shaped graph.https://www.zbmath.org/1456.340812021-04-16T16:22:00+00:00"Mochizuki, Kiyoshi"https://www.zbmath.org/authors/?q=ai:mochizuki.kiyoshi"Trooshin, Igor"https://www.zbmath.org/authors/?q=ai:trooshin.igor-yuSummary: We investigate the conditional stability of the inverse scattering problem on a lasso-shaped graph using the fundamental equation of inverse scattering theory.
For the entire collection see [Zbl 1415.35004].Solutions for a singular Hadamard-type fractional differential equation by the spectral construct analysis.https://www.zbmath.org/1456.340082021-04-16T16:22:00+00:00"Zhang, Xinguang"https://www.zbmath.org/authors/?q=ai:zhang.xinguang"Yu, Lixin"https://www.zbmath.org/authors/?q=ai:yu.lixin"Jiang, Jiqiang"https://www.zbmath.org/authors/?q=ai:jiang.jiqiang"Wu, Yonghong"https://www.zbmath.org/authors/?q=ai:wu.yonghong.1"Cui, Yujun"https://www.zbmath.org/authors/?q=ai:cui.yujunSummary: In this paper, we consider the existence of positive solutions for a Hadamard-type fractional differential equation with singular nonlinearity. By using the spectral construct analysis for the corresponding linear operator and calculating the fixed point index of the nonlinear operator, the criteria of the existence of positive solutions for equation considered are established. The interesting point is that the nonlinear term possesses singularity at the time and space variables.Study of a boundary value problem for fractional order \(\psi\)-Hilfer fractional derivative.https://www.zbmath.org/1456.340052021-04-16T16:22:00+00:00"Harikrishnan, S."https://www.zbmath.org/authors/?q=ai:harikrishnan.sugumaran"Shah, Kamal"https://www.zbmath.org/authors/?q=ai:shah.kamal"Kanagarajan, K."https://www.zbmath.org/authors/?q=ai:kanagarajan.kana|kanagarajan.kuppusamySummary: This manuscript is devoted to the existence theory of a class of random fractional differential equations (RFDEs) involving boundary condition (BCs). Here we take the corresponding derivative of arbitrary order in \(\psi\)-Hilfer sense. By utilizing classical fixed point theory and nonlinear analysis we establish some basic results of the qualitative theory such as existence, uniqueness and stability of solutions to the considered boundary value problem of RFDEs. Further, for the justification of our analysis we provide two examples.Asymptotics of the solution to the boundary-value problems when limited equation has singular point.https://www.zbmath.org/1456.340662021-04-16T16:22:00+00:00"Kozhobekov, K. G."https://www.zbmath.org/authors/?q=ai:kozhobekov.kudaiberdi-gaparalievich"Erkebaev, U. Z."https://www.zbmath.org/authors/?q=ai:erkebaev.ulukbek-zairbekovich"Tursunov, D. A."https://www.zbmath.org/authors/?q=ai:tursunov.dilmurat-abdillazhanovichIn this paper, the authors deal with the two-point boundary value problem for a linear second-order ordinary differential equation
\[
\varepsilon y_{\varepsilon}''(x)-x^np(x)y_{\varepsilon}'(x)-q(x)y_{\varepsilon}(x)=f(x),\ p(x),q(x)>0,\ 0\leq x\leq1,
\]
with boundary condition of one of the three following conditions types:
\[
y_{\varepsilon}(0)=a,\ y_{\varepsilon}(1)=b \text{ (Dirichlet problem)},
\]
\[
y_{\varepsilon}'(0)=a,\ y_{\varepsilon}'(1)=b \text{ (Neumann problem)},
\]
\[
y_{\varepsilon}'(0)-h_1y_{\varepsilon}'(0)=a, \ y_{\varepsilon}'(1)+h_2y_{\varepsilon}'(1)=b,\ h_1,h_2>0,\ n\geq2\ \mathrm{(Robin\ problem)},
\]
wherethe corresponding reduced problem (Eq. with \(\varepsilon=0\)) has an irregular singular point \(x=0.\)
The goal of the paper is to construct a complete asymptotic expansion of the solution \(y_{\varepsilon}(t)\) on the interval \([0,1]\) (as \(\varepsilon\to 0^+\)) of the singularly perturbed problems with irregular singular points.
Reviewer: Robert Vrabel (Trnava)A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation.https://www.zbmath.org/1456.340192021-04-16T16:22:00+00:00"Dzhumabaev, Dulat"https://www.zbmath.org/authors/?q=ai:dzhumabaev.dulat-syzdykbekovich"Bakirova, Elmira"https://www.zbmath.org/authors/?q=ai:bakirova.elmira-a"Mynbayeva, Sandugash"https://www.zbmath.org/authors/?q=ai:mynbayeva.sandugash-tabyldyevnaIn this paper, the authors study a nonlinear loaded differential equation with a parameter on a finite interval
\[
\frac{dx}{dt}=f_0(t,x,\mu)+f_1(t,x(\theta_0),x(\theta_1),\dots,x(\theta_{m-1}),x(\theta_m),\mu),\ t\in(0,T),x\in \mathbb{R}^n,\mu\in \mathbb{R}^l
\]
subjected to the boundary value conditions
\[
g[x(0),x(T),\mu]=0,
\]
where \(f_0:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^l\to \mathbb{R}^n\), \(f_1:[0,T]\times\mathbb{R}^{n(m+1)}\times\mathbb{R}^l\to \mathbb{R}^n\), and \(g:\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^l\to \mathbb{R}^{n+l}\) are continuous functions, \(0=\theta_0<\theta_1<\cdots<\theta_{m-1}<\theta_m=T\), \(\|x\|=\max_{i=1,\dots,n}|x_i|\).
The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, the authors obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, they compose a system of nonlinear algebraic equations in parameters. They develop a method of solving the boundary value problem with a parameter for the above loaded differential equation which is based on finding the solution to the system of nonlinear algebraic equations by an iterative process.
Reviewer: Yanqiong Lu (Lanzhou)Evolution of autocatalytic sets in a competitive percolation model.https://www.zbmath.org/1456.921592021-04-16T16:22:00+00:00"Zhang, Renquan"https://www.zbmath.org/authors/?q=ai:zhang.renquan"Pei, Sen"https://www.zbmath.org/authors/?q=ai:pei.sen"Wei, Wei"https://www.zbmath.org/authors/?q=ai:wei.wei.5|wei.wei.7|wei.wei.2|wei.wei.3|wei.wei.6|wei.wei.4"Zheng, Zhiming"https://www.zbmath.org/authors/?q=ai:zheng.zhimingOn the existence of three solutions for some classes of two-point semi-linear and quasi-linear differential equations.https://www.zbmath.org/1456.340232021-04-16T16:22:00+00:00"Saiedinezhad, Somayeh"https://www.zbmath.org/authors/?q=ai:saiedinezhad.somayehThe paper under review deals with the semi-linear boundary-value problem which consists the differential equation
\[
u''(t)+\lambda f(u)=0\, t\in(0,1),
\]
and Dirichlet type boundary conditions \(u(0)=u(1)=0\).
The author prove the existence of three solutions for this two-point boundary value problem in an appropriate Sobolev space. Furthermore, some existence results for a quasi-linear Dirichlet problem is obtained.
The results in the paper can be considered as an extension and generalizations of results of the paper [\textit{G. Bonannao}, Appl. Math. Lett. 13, No. 5, 53--57 (2000; Zbl 1009.34019)].
Reviewer: Erdogan Sen (Tekirdağ)Boundary value problems for Caputo fractional differential equations with nonlocal and fractional integral boundary conditions.https://www.zbmath.org/1456.340042021-04-16T16:22:00+00:00"Derbazi, Choukri"https://www.zbmath.org/authors/?q=ai:derbazi.choukri"Hammouche, Hadda"https://www.zbmath.org/authors/?q=ai:hammouche.haddaSummary: In this paper, we study the existence and uniqueness of solutions for fractional differential equations with nonlocal and fractional integral boundary conditions. New existence and uniqueness results are established using the Banach contraction principle. Other existence results are obtained using O'Regan fixed point theorem and Burton and Kirk fixed point. In addition, an example is given to demonstrate the application of our main results.Solvability and stability of the inverse Sturm-Liouville problem with analytical functions in the boundary condition.https://www.zbmath.org/1456.340142021-04-16T16:22:00+00:00"Bondarenko, Natalia P."https://www.zbmath.org/authors/?q=ai:bondarenko.natalia-pThe paper deals with the boundary value problem
\[
-y''(x)+q(x)y(x)=\lambda y(x),\; 0<x<\pi,
\]
\[
y(0)=0,\; f_1(\lambda)y'(\pi)+f_2(\lambda)y(\pi)=0,
\]
where \(f_k(\lambda)\) are entire functions in \(\lambda.\) The author studies the inverse problem of recovering the potential \(q(x)\) from a part of the spectrum. Local and global solvability are established for the solution of this nonlinear inverse problem.
Reviewer: Vjacheslav Yurko (Saratov)Some theorems of existence of solutions for fractional hybrid \(q\)-difference inclusion.https://www.zbmath.org/1456.340062021-04-16T16:22:00+00:00"Samei, Mohammad Esmael"https://www.zbmath.org/authors/?q=ai:samei.mohammad-esmael"Ranjbar, Ghorban Khalilzadeh"https://www.zbmath.org/authors/?q=ai:ranjbar.ghorban-khalilzadehSummary: The purpose of this study is to obtain the existence of solutions for the fractional hybrid \(q\)-differential inclusions with the boundary conditions. Besides, we give the solution set of these \(q\)-differential inclusions with boundary values. Also, we investigate dimension of the results set for second fractional \(q\)-differential inclusions. Lastly, we present an example to elaborate our results and present the obtained results of close to mathematical calculations.Degenerate band edges in periodic quantum graphs.https://www.zbmath.org/1456.811942021-04-16T16:22:00+00:00"Berkolaiko, Gregory"https://www.zbmath.org/authors/?q=ai:berkolaiko.gregory"Kha, Minh"https://www.zbmath.org/authors/?q=ai:kha.minhSummary: Edges of bands of continuous spectrum of periodic structures arise as maxima and minima of the dispersion relation of their Floquet-Bloch transform. It is often assumed that the extrema generating the band edges are non-degenerate. This paper constructs a family of examples of \({\mathbb{Z}}^3\)-periodic quantum graphs where the non-degeneracy assumption fails: the maximum of the first band is achieved along an algebraic curve of co-dimension 2. The example is robust with respect to perturbations of edge lengths, vertex conditions and edge potentials. The simple idea behind the construction allows generalizations to more complicated graphs and lattice dimensions. The curves along which extrema are achieved have a natural interpretation as moduli spaces of planar polygons.Extremal problems of the density for vibrating string equations with applications to gap and ratio of eigenvalues.https://www.zbmath.org/1456.340222021-04-16T16:22:00+00:00"Qi, Jiangang"https://www.zbmath.org/authors/?q=ai:qi.jiangang"Li, Jing"https://www.zbmath.org/authors/?q=ai:li.jing.13"Xie, Bing"https://www.zbmath.org/authors/?q=ai:xie.bingIn this paper, the authors obtain the infimum of the densities for vibrating string equations
\[
-y''=\lambda w y,\, y=y(x),\, x\in (0,1)
\]
together with Dirichlet-type boundary conditions \(y(0)=y(1)=0\) in terms of the gap and ratio of the first two eigenvalues. Here, the density \(w\) is a nonnegative integrable function on \([0, 1]\). As a main result of this investigation the authors prove a generalized version of the Lyapunov inequality involving the first two eigenvalues. Furthermore, they find some new estimates of the gap and ratio for the first and second eigenvalues of the above-mentioned boundary-value problem.
Reviewer: Erdogan Sen (Tekirdağ)Edge connectivity and the spectral gap of combinatorial and quantum graphs.https://www.zbmath.org/1456.811932021-04-16T16:22:00+00:00"Berkolaiko, Gregory"https://www.zbmath.org/authors/?q=ai:berkolaiko.gregory"Kennedy, James B."https://www.zbmath.org/authors/?q=ai:kennedy.james-b"Kurasov, Pavel"https://www.zbmath.org/authors/?q=ai:kurasov.pavel-b"Mugnolo, Delio"https://www.zbmath.org/authors/?q=ai:mugnolo.delioApproximate solution of Bagley-Torvik equations with variable coefficients and three-point boundary-value conditions.https://www.zbmath.org/1456.651832021-04-16T16:22:00+00:00"Huang, Q. A."https://www.zbmath.org/authors/?q=ai:huang.qiongao|huang.qiu-an"Zhong, X. C."https://www.zbmath.org/authors/?q=ai:zhong.xichang|zhong.xiaochun|zhong.xian-ci"Guo, B. L."https://www.zbmath.org/authors/?q=ai:guo.baolin|guo.boling|guo.baolongSummary: The fractional Bagley-Torvik equation with variable coefficients is investigated under three-point boundary-value conditions. By using the integration method, the considered problems are transformed into Fredholm integral equations of the second kind. It is found that when the fractional order is \(1< \alpha <2\), the obtained Fredholm integral equation is with a weakly singular kernel. When the fractional order is \(0< \alpha <1\), the given Fredholm integral equation is with a continuous kernel or a weakly singular kernel depending on the applied boundary-value conditions. The uniqueness of solution for the obtained Fredholm integral equation of the second kind with weakly singular kernel is addressed in continuous function spaces. A new numerical method is further proposed to solve Fredholm integral equations of the second kind with weakly singular kernels. The approximate solution is made and its convergence and error estimate are analyzed. Several numerical examples are computed to show the effectiveness of the solution procedures.On the Riesz basisness of root functions of a Sturm-Liouville operator with conjugate conditions.https://www.zbmath.org/1456.340242021-04-16T16:22:00+00:00"Cabri, O."https://www.zbmath.org/authors/?q=ai:cabri.olgun"Mamedov, K. R."https://www.zbmath.org/authors/?q=ai:mamedov.khanlar-rIn this study, authors consider the discontinuous Sturm-Liouville operator
\[
l(y) =
\begin{cases}
l_1 (y_1), \quad & x \in (-1,0) \\
l_2 (y_2), \quad & x \in (0,1)
\end{cases}
\]
where
\[
l_1(y_1)= y_1''+q_1(x)y_1,\, l_2(y_2)=y_2''+q_2(x)y_2,
\]
\(q_1(x) \in C^1[-1,0)\) and \(q_1(x) \in C^1(0,1]\) are complex-valued functions and have finite limits \(\lim_{x \to \mp 0}q_k(x)\) for \(k=1, 2\).
The authors deal with the problem of the operator \(l(y)\) with the periodic boundary conditions and with conjugate conditions. Both conjugate conditions have different finite one-sided limits at the point zero. By using the fundamental solution of problem, the asymptotic expression of eigenvalues and eigenfunctions are obtained. By means of asymptotic formulas of eigenfunctions and Bessel properties of eigenfunctions, Riesz basisness of the root functions the boundary value problem is proved. Similar spectral properties are studied for same operator with anti-periodic boundary conditions and with the same conjugate conditions.
Reviewer: Rakib Efendiev (Baku)Positive solutions for nonlinear problems involving the one-dimensional \(\phi\)-Laplacian.https://www.zbmath.org/1456.340212021-04-16T16:22:00+00:00"Kaufmann, Uriel"https://www.zbmath.org/authors/?q=ai:kaufmann.uriel"Milne, Leandro"https://www.zbmath.org/authors/?q=ai:milne.leandroSummary: Let \({\Omega} : = (a, b) \subset \mathbb{R}\), \(m \in L^1(\Omega)\) and \(\lambda > 0\) be a real parameter. Let \(\mathcal{L}\) be the differential operator given by \(\mathcal{L} u : = - \phi(u^\prime)^\prime + r(x) \phi(u)\), where \(\phi : \mathbb{R} \rightarrow \mathbb{R}\) is an odd increasing homeomorphism and \(0 \leq r \in L^1(\Omega)\). We study the existence of \textit{positive} solutions for problems of the form
\[
\begin{cases} \mathcal{L} u = \lambda m(x) f(u) & \text{in } \Omega, \\ u = 0 & \text{on } \partial \Omega, \end{cases}
\]
where \(f : [0, \infty) \rightarrow [0, \infty)\) is a continuous function which is, roughly speaking, sublinear with respect to \(\phi\). Our approach combines the sub and supersolution method with some estimates on related nonlinear problems. We point out that our results are new even in the cases \(r \equiv 0\) and/or \(m \geq 0\).Fractional differential equations involving Hadamard fractional derivatives with nonlocal multi-point boundary conditions.https://www.zbmath.org/1456.340072021-04-16T16:22:00+00:00"Subramanian, Muthaiah"https://www.zbmath.org/authors/?q=ai:subramanian.muthaiah"Manigandan, Murugesan"https://www.zbmath.org/authors/?q=ai:manigandan.murugesan"Gopal, Thangaraj Nandha"https://www.zbmath.org/authors/?q=ai:gopal.thangaraj-nandhaSummary: In this paper, we investigate the existence and uniqueness of solutions for the Hadamard fractional boundary value problems with nonlocal multipoint boundary conditions. By using Leray-Schauder nonlinear alternative, Leray Schauder degree theory, Krasnoselskii fixed point theorem, Schaefer fixed point theorem, Banach fixed point theorem, Nonlinear Contractions, the existence and uniqueness of solutions are obtained. As an application, two examples are given to demonstrate our results.A Green's function iterative approach for the solution of a class of fractional BVPs arising in physical models.https://www.zbmath.org/1456.650522021-04-16T16:22:00+00:00"Hadid, S."https://www.zbmath.org/authors/?q=ai:hadid.samir-b"Khuri, S. A."https://www.zbmath.org/authors/?q=ai:khuri.suheil-a"Sayfy, A."https://www.zbmath.org/authors/?q=ai:sayfy.ali-m-sSummary: The aim of this study is to present an alternative approach for the numerical solution of a wide class of fractional boundary value problems (FBVPs) that arise in various physical applications. Examples of such FBVP include but not limited to Bagley-Torvik, Riccati, Bratu, and Troesch's problems that appear in applied mathematics and mechanics. The method is based on first constructing an integral operator that is given in terms of the Green's function associated with the linear differential term of the fractional differential equation. Fixed point iterative procedures, such as Picard's and Mann's, are then applied to the operator to generate an iterative scheme that yields a convergent semi-analytical solution. Numerical examples are reported to confirm the efficiency, reliability, accuracy and fast convergence of the scheme.Spectral inclusion and pollution for a class of dissipative perturbations.https://www.zbmath.org/1456.811822021-04-16T16:22:00+00:00"Stepanenko, Alexei"https://www.zbmath.org/authors/?q=ai:stepanenko.alexeiSummary: Spectral inclusion and spectral pollution results are proved for sequences of linear operators of the form \(T_0 + i \gamma s_n\) on a Hilbert space, where \(s_n\) is strongly convergent to the identity operator and \(\gamma > 0\). We work in both an abstract setting and a more concrete Sturm-Liouville framework. The results provide rigorous justification for a method of computing eigenvalues in spectral gaps.
{\copyright 2021 American Institute of Physics}Uniform convergence of Fourier series expansions for a fourth-order spectral problem with boundary conditions depending on the eigenparameter.https://www.zbmath.org/1456.340182021-04-16T16:22:00+00:00"Namazov, Faiq Mirzali"https://www.zbmath.org/authors/?q=ai:namazov.faiq-mirzaliSummary: In this paper, we consider the eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter contained in two of boundary conditions. We obtain refined asymptotic formulas for eigenvalues and eigenfunctions, and study uniform convergence of Fourier series expansions of continuous functions in the system of eigenfunctions of this problem.