Recent zbMATH articles in MSC 34Bhttps://www.zbmath.org/atom/cc/34B2022-05-16T20:40:13.078697ZWerkzeugSome new results for \(\psi\)-Hilfer fractional pantograph-type differential equation depending on \(\psi\)-Riemann-Liouville integralhttps://www.zbmath.org/1483.340132022-05-16T20:40:13.078697Z"Foukrach, Djamal"https://www.zbmath.org/authors/?q=ai:foukrach.djamal"Bouriah, Soufyane"https://www.zbmath.org/authors/?q=ai:bouriah.soufyane"Benchohra, Mouffak"https://www.zbmath.org/authors/?q=ai:benchohra.mouffak"Karapinar, Erdal"https://www.zbmath.org/authors/?q=ai:karapinar.erdalSummary: The aim of the present work is to study a large class of \(\psi\)-Hilfer fractional differential equation of Pantograph-type depending on \(\psi\)-Riemann-Liouville fractional integral operator associated with periodic-type fractional integral boundary conditions in a weighted space of continuous functions. We shall prove the existence and uniqueness results by means of Mawhin's coincidence degree theory. At the end, an illustrative example will be constructed to approve our findings.On impulsive boundary value problem with Riemann-Liouville fractional order derivativehttps://www.zbmath.org/1483.340152022-05-16T20:40:13.078697Z"Khan, Zareen A."https://www.zbmath.org/authors/?q=ai:khan.zareen-a-a|khan.zareen-abdulhameed"Gul, Rozi"https://www.zbmath.org/authors/?q=ai:gul.rozi"Shah, Kamal"https://www.zbmath.org/authors/?q=ai:shah.kamalA class of impulsive boundary value problems for Riemann-Liouville fractional differential equations is studied. Unfortunately, it is not taken into account that the lower limit of the Riemann-Liouville fractional order derivative is very important. According to the equation~(4), the Riemann-Liouville fractional order derivative has a lower limit at zero. Therefore, Lemma~5 and equation~(5) are true if both the fractional integral and the fractional derivative have one and the same lower limit. At the same time this Lemma is applied in the proof of the main Lemma~6 in order to obtain equality~(12) for the lower limit of the integral \(z_1\) and for the fractional derivative~$0$, which does not lead to~(12). It has a huge influence on the other results in this paper.
Reviewer: Snezhana Hristova (Plovdiv)Solutions of fractional differential equations with \(p\)-Laplacian operator in Banach spaceshttps://www.zbmath.org/1483.340182022-05-16T20:40:13.078697Z"Tan, Jingjing"https://www.zbmath.org/authors/?q=ai:tan.jingjing"Li, Meixia"https://www.zbmath.org/authors/?q=ai:li.meixiaSummary: In this paper, we study the solutions for nonlinear fractional differential equations with \(p\)-Laplacian operator nonlocal boundary value problem in a Banach space. By means of the technique of the properties of the Kuratowski noncompactness measure and the Sadovskii fixed point theorem, we establish some new existence criteria for the boundary value problem. As application, an interesting example is provided to illustrate the main results.Multiplicity results for impulsive fractional differential equations with \(p\)-Laplacian via variational methodshttps://www.zbmath.org/1483.340202022-05-16T20:40:13.078697Z"Zhao, Yulin"https://www.zbmath.org/authors/?q=ai:zhao.yulin"Tang, Liang"https://www.zbmath.org/authors/?q=ai:tang.liangSummary: In this paper, we apply critical point theory and variational methods to study the multiple solutions of boundary value problems for an impulsive fractional differential equation with \(p\)-Laplacian. Some new criteria guaranteeing the existence of multiple solutions are established for the considered problem.On coupled systems of Lidstone-type boundary value problemshttps://www.zbmath.org/1483.340232022-05-16T20:40:13.078697Z"de Sousa, Robert"https://www.zbmath.org/authors/?q=ai:de-sousa.robert"Minhós, Feliz"https://www.zbmath.org/authors/?q=ai:minhos.feliz-manuel"Fialho, João"https://www.zbmath.org/authors/?q=ai:fialho.joao-fSummary: This research concerns the existence and location of solutions for coupled system of differential equations with Lidstone-type boundary conditions. Methodology used utilizes three fundamental aspects: upper and lower solutions method, degree theory and nonlinearities with monotone conditions. In the last section an application to a coupled system composed by two fourth order equations, which models the bending of coupled suspension bridges or simply supported coupled beams, is presented.The influence function properties for a problem with discontinuous solutionshttps://www.zbmath.org/1483.340242022-05-16T20:40:13.078697Z"Kamenskii, Mikhail"https://www.zbmath.org/authors/?q=ai:kamenskii.mikhail-igorevich"Wen, Ching-Feng"https://www.zbmath.org/authors/?q=ai:wen.chingfeng"Zalukaev, Zhanna"https://www.zbmath.org/authors/?q=ai:zalukaev.zhanna"Zvereva, Margarita"https://www.zbmath.org/authors/?q=ai:zvereva.margarita-borisovnaSummary: We consider the boundary value problem describing deformations of a discontinuous Stieltjes string. Properties of the influence function (Green function) are investigated. The analysis is based on a refined Stieltjes integral.Simple numerical methods of second- and third-order convergence for solving a fully third-order nonlinear boundary value problemhttps://www.zbmath.org/1483.340262022-05-16T20:40:13.078697Z"Dang, Quang A"https://www.zbmath.org/authors/?q=ai:dang-quang-a."Dang, Quang Long"https://www.zbmath.org/authors/?q=ai:dang.quang-longThis paper is concerned with the following fully third-order nonlinear boundary value problem that is of great interest of many researchers
\begin{align*}
&u^{(3)}(t)=f\left(t, u(t), u^{\prime}(t), u^{\prime \prime}(t)\right), \quad 0<t<1, \\
&u(0)=c_{1}, u^{\prime}(0)=c_{2}, u^{\prime}(1)=c_{3}.
\end{align*}
First, the existence and uniqueness of solution are discussed. Next, the simple iterative methods on both continuous and discrete levels are proposed. The discrete methods are of second-order and third-order of accuracy due to the use of appropriate formulas for numerical integration. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative methods. A completely different method, specifically, an iterative method on both continuous and discrete levels for the considered fully third-order differential equations is developed. These methods are based on the popular trapezoidal rule and a modified Simpson rule for numerical integration. Further, an analysis of the total error of the solution is obtained. The obtained error includes the error of the iterative method on continuous level and the error arising in the numerical realization of this iterative method. The obtained total error estimate suggests how to choose a suitable grid size for discretization to get an approximate solution with a given accuracy. In order to justify the total error estimate, some results on existence and uniqueness of solution are established. Also, the applicability shows that, these methods can easily be extend to higher order nonlinear boundary value problems. Numerical results are also given.
Reviewer: Saurabh Tomar (Kharagpur)Inverse spectral problem of an anharmonic oscillator on a half-axis with the Neumann boundary conditionhttps://www.zbmath.org/1483.340292022-05-16T20:40:13.078697Z"Khanmamedov, Agil K."https://www.zbmath.org/authors/?q=ai:khanmamedov.agil-kh"Gafarova, Nigar F."https://www.zbmath.org/authors/?q=ai:gafarova.nigar-fThe paper under review deals with the half-line Neumann problem for the equation \[-y''+ x^2 y + q(x) y = \lambda y\] under some smoothness and integrability conditions on \(q\). The authors consider the inverse problem of recovering this boundary value problem by its spectrum and norming constants. They obtain a Gelfand-Levitan-Marchenko-type integral equation, prove its unique solvability, and indicate a constructive algorithm for recovering \(q\).
Reviewer: Namig Guliyev (Baku)Well ordered monotone iterative technique for nonlinear second order four point Dirichlet BVPshttps://www.zbmath.org/1483.340362022-05-16T20:40:13.078697Z"Verma, Amit K."https://www.zbmath.org/authors/?q=ai:verma.amit-kumar"Urus, Nazia"https://www.zbmath.org/authors/?q=ai:urus.naziaSummary: In this article, we develop a monotone iterative technique (MI-technique) with lower and upper (L-U) solutions for a class of four-point Dirichlet nonlinear boundary value problems (NLBVPs), defined as,
\[
-\psi^{\prime \prime}(x) = F (x, \psi, \psi^{\prime}), \quad 0 < x < 1, \text{ BCs}(i) \equiv \psi(i) - c_i\psi(\eta_i) = 0, \quad i = 0, 1,
\]
where \(0 < c_0 < 1\), \(c_1 > 0\), \(0 < \eta_0 \leq \eta_1 < 1\), \(\psi(x) \in C^2[0, 1]\), the non linear term \(F (x, \psi, \psi^{\prime})\) is continuous function in \(x\), one sided Lipschitz in \(\psi\) and Lipschitz in \(\psi^{\prime}\). To show the existence result, we construct Green's function and iterative sequences for the corresponding linear problem. We use quasilinearization to construct these iterative schemes. We prove maximum principle and establish monotonicity of sequences of lower solution \((l_m(x))_m\) and upper solution \((u_m(x))_m\) such that \(l_m(x) \leq u_m(x)\), \(\forall m \in \mathbb{N}\). Then under certain sufficient conditions we prove that these sequences converge uniformly to the solution \(\psi (x)\) in a specific region where \(\frac{\partial F}{\partial\psi} \neq 0\).Eigenvalues of a class of fourth-order boundary value problems with transmission conditions using matrix theoryhttps://www.zbmath.org/1483.340372022-05-16T20:40:13.078697Z"Ao, Ji-jun"https://www.zbmath.org/authors/?q=ai:ao.jijun"Sun, Jiong"https://www.zbmath.org/authors/?q=ai:sun.jiongThe authors study the differential equation \[ (p u'')'' + q u = \lambda w u \] on \(J = (a, c) \cup (c, b)\) for finite \(a < c < b\), together with boundary conditions \[ A U (a) + B U (b) = 0, \quad U = \begin{pmatrix} u \\
u' \\
p u'' \\
(p u'')' \end{pmatrix}, \quad A, B \in M_4 (\mathbb{R}), \] and transmission conditions \[ C U (c-) + D U (c+) = 0. \] Here \(C, D\) are real-valued \(4 \times 4\)-matrices with positive determinants and the coefficient functions \(1/p, q, w\) are integrable on \(J\). Moreover, conditions on the matrices \(A, B\) are imposed which make the problem self-adjoint. As the main result, sufficient conditions are provided under which this eigenvalue problem is equivalent to a matrix eigenvalue problem of the form \((\mathbb P + \mathbb Q) \mathbb Y = \lambda \mathbb W \mathbb Y\), where \(\mathbb {P, Q, W}\) are constructed explicitly.
Reviewer: Jonathan Rohleder (Stockholm)On the spectra of boundary value problems generated by some one-dimensional embedding theoremshttps://www.zbmath.org/1483.340382022-05-16T20:40:13.078697Z"Minarsky, A. M."https://www.zbmath.org/authors/?q=ai:minarsky.a-m"Nazarov, A. I."https://www.zbmath.org/authors/?q=ai:nazarov.alexander-iSummary: The spectra of boundary value problems related to one-dimensional high order embedding theorems are considered. It is proved that for some orders, the eigenvalues corresponding to even eigenfunctions of different problems cannot coincide.Global structure for a fourth-order boundary value problem with sign-changing weighthttps://www.zbmath.org/1483.340392022-05-16T20:40:13.078697Z"Ye, Fumei"https://www.zbmath.org/authors/?q=ai:ye.fumeiSummary: We study the fourth-order boundary value problem with a sign-changing weight function:
\[
\begin{cases}
u''''=\lambda m(t)u+f_1(t,u,u',u'',u''',\lambda)+f_2(t,u,u',u'',u''',\lambda),\quad t\in(0,1),\\
u(0)=u(1)=u''(0)=u''(1)=0,
\end{cases}
\]
where \(\lambda\in\mathbb{R}\) is a parameter, \(f_1,f_2\in C([0, 1] \times\mathbb{R}^5,\mathbb{R}),f_1\) is not differentiable at the origin and infinity. Under some suitable conditions on nonlinear terms, we prove the existence of unbounded continua of positive and negative solutions of this problem which bifurcating from intervals of the line of trivial solutions or from infinity, respectively.Dependence of eigenvalues of \(2m\)th-order spectral problemshttps://www.zbmath.org/1483.340402022-05-16T20:40:13.078697Z"Zheng, Zhaowen"https://www.zbmath.org/authors/?q=ai:zheng.zhaowen"Ma, Yujuan"https://www.zbmath.org/authors/?q=ai:ma.yujuanSummary: A regular \(2m\)th-order spectral problem with self-adjoint boundary conditions is considered in this paper. The continuous dependence of eigenvalues and normalized eigenfunctions on the problem is researched. The derivative formulas of eigenvalues with respect to the given parameters are obtained: endpoints, boundary conditions, coefficients and the weight function. These are of both theoretical and computational importance.Bound sets for a class of \(\varphi \)-Laplacian operatorshttps://www.zbmath.org/1483.340412022-05-16T20:40:13.078697Z"Feltrin, Guglielmo"https://www.zbmath.org/authors/?q=ai:feltrin.guglielmo"Zanolin, Fabio"https://www.zbmath.org/authors/?q=ai:zanolin.fabioThe authors provide an extension of the Hartman-Knobloch theorem for periodic solutions of vector differential systems to a general class of \(\phi\)-Laplacian differential operators. Their main tool is a variant of the Manásevich-Mawhin continuation theorem developed for this class of operator equations, together with the theory of bound sets. They also extend a classical theorem by Reissig for scalar periodically perturbed Liénard equations.
Reviewer: Alessandro Fonda (Trieste)The Dirichlet problem for the fourth order nonlinear ordinary differential equations at resonancehttps://www.zbmath.org/1483.340422022-05-16T20:40:13.078697Z"Mukhigulashvili, S."https://www.zbmath.org/authors/?q=ai:mukhigulashvili.sulkhan"Manjikashvili, M."https://www.zbmath.org/authors/?q=ai:manjikashvili.mariamThis paper discusses the solvability of the following fourth order boundary value problem \[u^{(4)}(t)=p(t)u(t)+f(t,u(t))+h(t),\;t\in I=[a,b],\tag{1}\] \[u^{(i)}(a)=0,\,u^{(i)}(b)=0,\,i=0,1,\tag{2}\] where \(h, p\in L(I, \mathbb R)\) and \(f:I\times \mathbb R\to \mathbb R\) is a Carathéodory function.
The authors suppose that the problem \[w^{(4)}(t)=p(t)w(t),\;t\in I,\tag{3}\] \[w^{(i)}(a)=0,\,w^{(i)}(b)=0,\,i=0,1\tag{4}\] has a nonzero solution \(w,\) introduce the set \({N_p:=\{t\in I: w(t)=0\}}\) and for a finite subset \(A=\{t_1,\dots, t_k\}\) of \(I\) introduce the set \(E(A)\) of all Carathéodory functions \(f : I\times \mathbb R\to \mathbb R\) such that for an arbitrary neighbourhood \(U(A)\) of \(A\) and a positive constant \(r\) there exists \(\alpha_1>0\) with the property \[\int\limits_{U'(A)\setminus U_\alpha}|f(s,x)|ds\, - \, \int\limits_{U_\alpha}|f(s,x)|ds\geq0\;\;\text{for}\;\;|x|\geq r,\;\alpha\leq\alpha_1,\] where \(U'(A)=I\cap U(A),\) and \(U_\alpha=I\cap\Bigl(\cup_{j=1}^k[t_j-\alpha, t_j+\alpha]\Bigr).\) Besides for an arbitrary \(r>0\) \(f^*(t,r)=\sup\{|f(t,x)|: |x|\leq r\}\in L(I, [0,+\infty))\) and \[[x(t)]_+=(|x(t)|+x(t))/2,\;[x(t)]_-=(|x(t)|-x(t))/2\] for a function \(x: I\to \mathbb R.\)
One of the main results guarantees at least one solution of (1), (2), i.e. at least one function \(u\in\widetilde{C}^3(I, \mathbb R)\) wich satisfies (1) almost everywhere on \(I\) and satisfies (2), under the assumptions that \(r>0\) and the functions \(f\in E(N_p), f^+, f^-\in L(I, [0,+\infty))\) are such that for \(j\in\{0,1\}\) \[(-1)^j f(t,x)\leq -f^-(t)\;\;\text{for}\;\;x\leq-r,\;t\in I,\] \[f^+(t)\leq(-1)^j f(t,x)\;\;\text{for}\;\;x\geq r,\;t\in I,\] \[\lim_{\rho\to+\infty} \frac{1}{\rho}\int\limits_a^b f^*(s,\rho)ds=0,\] and there exists \(\varepsilon>0\) such that for an arbitrary nonzero solution \(w\) of (3), (4) the following holds \[-\int\limits_a^b\Bigl( f^+(s)[w(s)]_-+f^-(s)[w(s)]_+\Bigr)ds+\varepsilon\gamma_r||w||_C\] \[\leq(-1)^{j+1}\int\limits_a^b h(s)w(s)ds\leq\int\limits_a^b\Bigl( f^-(s)[w(s)]_-+f^+(s)[w(s)]_+\Bigr)ds - \varepsilon\gamma_r||w||_C,\] where \(\gamma_r=\int\limits_a^b f^*(s,r)ds.\)
Reviewer: Petio S. Kelevedjiev (Sliven)Positive solutions to classes of infinite semipositone \((p,q)\)-Laplace problems with nonlinear boundary conditionshttps://www.zbmath.org/1483.340432022-05-16T20:40:13.078697Z"Sim, Inbo"https://www.zbmath.org/authors/?q=ai:sim.inbo"Son, Byungjae"https://www.zbmath.org/authors/?q=ai:son.byungjaeIn this interesting paper the authors study the existence, multiplicity and nonexistence of positive solutions for the one-dimensional \((p,q)\)-Laplacian problems:
\begin{gather*}
-(\varphi(u'))'=\lambda h(t)f(u),\qquad t\in (0,1),\\
u(0)=0=au'(1)+g(\lambda,u(1))u(1),
\end{gather*}
where \(\lambda>0\), \(a\geq 0\), \(\varphi(s)=|s|^{p-2}s+|s|^{q-2}s\) with \(1<p<q<+\infty\), \(h\in C((0,1),(0,+\infty))\), and \(f\in C((0,+\infty),\mathbb{R})\) may have a singularity at 0 of repulsive type. The proofs are based on a classical Krasnoselskii type fixed point theorem which is fit to overcome a lack of homogeneity.
Reviewer: Manuel Zamora (Concepción)New multiple positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite intervalhttps://www.zbmath.org/1483.340442022-05-16T20:40:13.078697Z"Zhang, Wei"https://www.zbmath.org/authors/?q=ai:zhang.wei.10|zhang.wei.19|zhang.wei.6|zhang.wei.5|zhang.wei.3|zhang.wei.17|zhang.wei.15|zhang.wei.16|zhang.wei.9|zhang.wei.7|zhang.wei.12|zhang.wei.4|zhang.wei.13|zhang.wei.2|zhang.wei.18"Ni, Jinbo"https://www.zbmath.org/authors/?q=ai:ni.jinboIn this paper, the authors consider nonlinear Hadamard-type fractional differential equations with nonlocal boundary conditions on an infinite interval. The existence of multiple positive solutions of the addressed problem is obtained by applying the generalized Avery-Henderson fixed point theorem. Finally, an example was given to show the effectiveness of the main result. This paper provides a new fixed point theorem to study multiple solutions.
Reviewer: Wengui Yang (Sanmenxia)Oscillation properties for non-classical Sturm-Liouville problems with additional transmission conditionshttps://www.zbmath.org/1483.340452022-05-16T20:40:13.078697Z"Mukhtarov, Oktay Sh."https://www.zbmath.org/authors/?q=ai:mukhtarov.oktay-sh"Aydemir, Kadriye"https://www.zbmath.org/authors/?q=ai:aydemir.kadriyeSummary: This work is aimed at studying some comparison and oscillation properties of boundary value problems (BVP's) of a new type, which differ from classical problems in that they are defined on two disjoint intervals and include additional transfer conditions that describe the interaction between the left and right intervals. This type of problems we call boundary value-transmission problems (BVTP's). The main difficulty arises when studying the distribution of zeros of eigenfunctions, since it is unclear how to apply the classical methods of Sturm's theory to problems of this type. We established new criteria for comparison and oscillation properties and new approaches used to obtain these criteria. The obtained results extend and generalizes the Sturm's classical theorems on comparison and oscillation.On a series representation for integral kernels of transmutation operators for perturbed Bessel equationshttps://www.zbmath.org/1483.340462022-05-16T20:40:13.078697Z"Kravchenko, V. V."https://www.zbmath.org/authors/?q=ai:kravchenko.vladislav-v"Shishkina, E. L."https://www.zbmath.org/authors/?q=ai:shishkina.elina-leonidovna"Torba, S. M."https://www.zbmath.org/authors/?q=ai:torba.sergii-mSummary: A representation for the kernel of the transmutation operator relating a perturbed Bessel equation to the unperturbed one is obtained in the form of a functional series with coefficients calculated by a recurrent integration procedure. New properties of the transmutation kernel are established. A new representation of a regular solution of a perturbed Bessel equation is given, which admits a uniform error bound with respect to the spectral parameter for partial sums of the series. A numerical illustration of the application of the obtained result to solve Dirichlet spectral problems is presented.Third-order generalized discontinuous impulsive problems on the half-linehttps://www.zbmath.org/1483.340472022-05-16T20:40:13.078697Z"Minhós, Feliz"https://www.zbmath.org/authors/?q=ai:minhos.feliz-manuel"Carapinha, Rui"https://www.zbmath.org/authors/?q=ai:carapinha.ruiSummary: In this paper, we improve the existing results in the literature by presenting weaker sufficient conditions for the solvability of a third-order impulsive problem on the half-line, having generalized impulse effects. More precisely, our nonlinearities do not need to be positive nor sublinear and the monotone assumptions are local ones. Our method makes use of some truncation and perturbed techniques and on the equiconvergence at infinity and the impulsive points. The last section contains an application to a boundary layer flow problem over a stretching sheet with and without heat transfer.On Pleijel's nodal domain theorem for quantum graphshttps://www.zbmath.org/1483.340482022-05-16T20:40:13.078697Z"Hofmann, Matthias"https://www.zbmath.org/authors/?q=ai:hofmann.matthias"Kennedy, James B."https://www.zbmath.org/authors/?q=ai:kennedy.james-b"Mugnolo, Delio"https://www.zbmath.org/authors/?q=ai:mugnolo.delio"Plümer, Marvin"https://www.zbmath.org/authors/?q=ai:plumer.marvinSummary: We establish metric graph counterparts of Pleijel's theorem on the asymptotics of the number of nodal domains \(\nu_n\) of the \(n\)th eigenfunction(s) of a broad class of operators on compact metric graphs, including Schrödinger operators with \(L^1\)-potentials and a variety of vertex conditions as well as the \(p\)-Laplacian with natural vertex conditions, and without any assumptions on the lengths of the edges, the topology of the graph, or the behaviour of the eigenfunctions at the vertices. Among other things, these results characterise the accumulation points of the sequence \(\left(\frac{\nu_n}{n}\right)_{n\in \mathbb{N}} \), which are shown always to form a finite subset of \((0, 1]\). This extends the previously known result that \(\nu_n\sim n\) \textit{generically}, for certain realisations of the Laplacian, in several directions. In particular, in the special cases of the Laplacian with natural conditions, we show that for graphs any graph with pairwise commensurable edge lengths and at least one cycle, one can find eigenfunctions thereon for which \({\nu_n}\not \sim{n} \); but in this case even the set of points of accumulation may depend on the choice of eigenbasis.S-shaped connected component of positive solutions for a Minkowski-curvature Dirichlet problem with indefinite weighthttps://www.zbmath.org/1483.340542022-05-16T20:40:13.078697Z"He, Zhiqian"https://www.zbmath.org/authors/?q=ai:he.zhiqian"Miao, Liangying"https://www.zbmath.org/authors/?q=ai:miao.liangyingSummary: In this paper, we investigate the existence of an S-shaped connected component in the set of positive solutions for a Minkowski-curvature Dirichlet problem with indefinite weight. By figuring the shape of unbounded continua of solutions, we show the existence and multiplicity of positive solutions with respect to the parameter \(\lambda\). In particular, we obtain the existence of at least three positive solutions for \(\lambda\) being in a certain interval.Asymptotics of the solution to a two-band two-point boundary value problemhttps://www.zbmath.org/1483.340762022-05-16T20:40:13.078697Z"Tursunov, D. A."https://www.zbmath.org/authors/?q=ai:tursunov.dilmurat-abdillazhanovich"Omaralieva, G. A."https://www.zbmath.org/authors/?q=ai:omaralieva.g-aThe authors constructed a complete asymptotic approximation of the solution with respect to a small parameter of the Dirichlet boundary value problem for a singularly perturbed linear inhomogeneous second-order ordinary differential equation
\[
\varepsilon^4 y''_{\varepsilon}(x) + x^2p(x)y'_{\varepsilon}(x)-\varepsilon q(x)y_{\varepsilon}(x) = f(x),\ x\in(0,1),
\]
\[
y_{\varepsilon}(0) = a,\ y_{\varepsilon}(1) = b,
\]
where \(0 < \varepsilon\ll 1,\) \(p(x)>0,\) \(q(x)>0\) and \(f(x)\) are smooth functions on \([0,1]\) and \(a,b\in\mathbb{R}\) are known constants.
The problem under consideration differs from the previously investigated problems in that in the vicinity of the left boundary point \(x = 0\) there is a two-layer boundary layer, and the solution of the corresponding unperturbed problem is not a smooth function. Therefore, it is impossible to solve the problem using the classical method of boundary functions. First, a formal asymptotic approximation of the problem under study was constructed by generalized and classical methods of boundary functions, then, using the maximum principle, an estimate was obtained for the residual function of the constructed series. The resulting series is asymptotic in the sense of Erdei.
Reviewer: Robert Vrabel (Trnava)Positive periodic solutions for \(p\)-Laplacian neutral differential equations with a singularityhttps://www.zbmath.org/1483.340972022-05-16T20:40:13.078697Z"Li, Zhiyan"https://www.zbmath.org/authors/?q=ai:li.zhiyan"Kong, Fanchao"https://www.zbmath.org/authors/?q=ai:kong.fanchaoSummary: In this paper, we study the positive periodic solutions of a kind of \(p\)-Laplacian neutral differential equation with a singularity. By applying the continuation theorem and some analytic techniques, we shall establish several new criteria for the existence of positive periodic solutions for the considered problem. Some recent results in the literature are generalized and improved. Three examples are given to illustrate the effectiveness of our results.Existence results for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary conditionshttps://www.zbmath.org/1483.341122022-05-16T20:40:13.078697Z"Zuo, Mingyue"https://www.zbmath.org/authors/?q=ai:zuo.mingyue"Hao, Xinan"https://www.zbmath.org/authors/?q=ai:hao.xinan"Liu, Lishan"https://www.zbmath.org/authors/?q=ai:liu.lishan"Cui, Yujun"https://www.zbmath.org/authors/?q=ai:cui.yujunSummary: In this paper, we are concerned with the existence and uniqueness of solutions for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary condition. Our results are based on the Banach contraction mapping principle and the Krasnoselskii fixed point theorem. Some examples are also given to illustrate our results.Inverse problem for the Sturm-Liouville equation with piecewise entire potential and piecewise constant weight on a curvehttps://www.zbmath.org/1483.341172022-05-16T20:40:13.078697Z"Golubkov, Andrey Alexandrovich"https://www.zbmath.org/authors/?q=ai:golubkov.andrey-alexandrovichSummary: A Sturm-Liouville equation with a piecewise entire potential and a non-zero piecewise constant weight function on a curve of an arbitrary shape lying on the complex plane is considered. For such equation, the inverse spectral problem is posed with respect to the ratio of elements of one column or one row of the transfer matrix along the curve. The uniqueness of the solution to the problem is proved with the help of the method of unit transfer matrix using the study of asymptotic solutions of the Sturm-Liouville equation for large values of the absolute value of the spectral parameter. The obtained results allowed to consider inverse problem for a previously unexplored class of Sturm-Liouville equations with three unknown coefficients on a segment of the real axis.Solutions of Painlevé II on real intervals: novel approximating sequenceshttps://www.zbmath.org/1483.341222022-05-16T20:40:13.078697Z"Bracken, Anthony J."https://www.zbmath.org/authors/?q=ai:bracken.anthony-jAn approximation method is proposed for a boundary value problem for the second Painlevé equation with Neumann boundary conditions
\[
y^{\prime\prime}(z) =2y(z)^3 +z y(z) +C, \quad y'(a)=0, \quad y'(b)=0, \quad (a<z<b).
\]
In the first step, a generalization of the second Painlevé equation is considered. The generalized equation on \(E(x)\) contains \(E(0), E(1)\) as nonlinear terms. In the expansion series \( E(x)=E_1(x) +E_2(x)+\cdots \), each \(E_n\) satisfies a nonhomogeneous Airy equation. By a suitable change of variables, a curious approximation series \(y_E^{(n)}\) is defined. \(y_E^{(n)}\) is a solution of a nonlinear equation with Neumann boundary conditions on an interval \([a_n, b_n\)], accompanied with a constant \(C_n\). When \(n \to \infty\), \(y_E^{(n)}\) converges to the solution \(y(z)\) and \(a_n, b_n\) and \(C_n\) also converges to \(a, b\) and \(C\), respectively, if the nonlinear term \(|y(z)|^3\) is small. A numerical example is also shown.
It is not clear why this method works well although the second Painlevé equation is closely related to the Airy function.
Reviewer: Yousuke Ohyama (Tokushima)Wavefronts for degenerate diffusion-convection reaction equations with sign-changing diffusivityhttps://www.zbmath.org/1483.351232022-05-16T20:40:13.078697Z"Berti, Diego"https://www.zbmath.org/authors/?q=ai:berti.diego"Corli, Andrea"https://www.zbmath.org/authors/?q=ai:corli.andrea"Malaguti, Luisa"https://www.zbmath.org/authors/?q=ai:malaguti.luisaSummary: We consider in this paper a diffusion-convection reaction equation in one space dimension. The main assumptions are about the reaction term, which is monostable, and the diffusivity, which changes sign once or even more than once; then, we deal with a forward-backward parabolic equation. Our main results concern the existence of globally defined traveling waves, which connect two equilibria and cross both regions where the diffusivity is positive and regions where it is negative. We also investigate the monotony of the profiles and show the appearance of sharp behaviors at the points where the diffusivity degenerates. In particular, if such points are interior points, then the sharp behaviors are new and unusual.Friedrichs extension of operators defined by even order Sturm-Liouville equations on time scaleshttps://www.zbmath.org/1483.470192022-05-16T20:40:13.078697Z"Zemánek, Petr"https://www.zbmath.org/authors/?q=ai:zemanek.petr"Hasil, Petr"https://www.zbmath.org/authors/?q=ai:hasil.petrSummary: In this paper we characterize the Friedrichs extension of operators associated with the 2\(n\)th order Sturm-Liouville dynamic equations on time scales with using the time reversed symplectic systems and its recessive system of solutions. A~nontrivial example is also provided.On the Bari basis properties of the root functions of non-self adjoint \(q\)-Sturm-Liouville problemshttps://www.zbmath.org/1483.470322022-05-16T20:40:13.078697Z"Allahverdiev, B. P."https://www.zbmath.org/authors/?q=ai:allahverdiev.bilender-pasaoglu"Tuna, H."https://www.zbmath.org/authors/?q=ai:tuna.huseyin|tuna.huseinSummary: This paper deals with the dissipative regular \(q\)-Sturm-Liouville problem. We prove that the system of root functions of this operator forms a Bari bases in the space \(L_q^2(I)\) by using the asymptotic behavior at infinity for its eigenvalues.Dilations, models and spectral problems of non-self-adjoint Sturm-Liouville operatorshttps://www.zbmath.org/1483.470432022-05-16T20:40:13.078697Z"Allahverdiev, Bilender P."https://www.zbmath.org/authors/?q=ai:allahverdiev.bilender-pasaogluSummary: In this study, we investigate the maximal dissipative singular Sturm-Liouville operators acting in the Hilbert space \(L_{r}^{2}(a,b)\)\ \( (-\infty \leq a<b\leq \infty)\), that [are] the extensions of a minimal symmetric operator\ with defect index (\(2,2\)) (in limit-circle case at singular end points \(a\)\ and \(b\)).\ We examine two classes of dissipative operators with separated boundary conditions and we establish, for each case, a self-adjoint dilation\ of the dissipative operator as well as its incoming and outgoing spectral representations, which enables us to define the scattering matrix of the dilation. Moreover, we construct a functional model of the dissipative operator and identify its characteristic function in terms of the Weyl function of a self-adjoint operator. We present several theorems on completeness of the system of root functions of the dissipative perators and verify them.On higher regularized traces of a differential operator with bounded operator coefficient given in a finite intervalhttps://www.zbmath.org/1483.470792022-05-16T20:40:13.078697Z"Sezer, Yonca"https://www.zbmath.org/authors/?q=ai:sezer.yonca"Bakşi, Özlem"https://www.zbmath.org/authors/?q=ai:baksi.ozlem"Karayel, Serpil"https://www.zbmath.org/authors/?q=ai:karayel.serpil-sengulSummary: In this work, we find a higher regularized trace formula for a regular Sturm-Liouville differential operator with operator coefficient.A robust pseudospectral method for numerical solution of nonlinear optimal control problemshttps://www.zbmath.org/1483.490102022-05-16T20:40:13.078697Z"Mehrpouya, Mohammad Ali"https://www.zbmath.org/authors/?q=ai:mehrpouya.mohammad-ali"Peng, Haijun"https://www.zbmath.org/authors/?q=ai:peng.haijunSummary: In the present paper, a robust pseudospectral method for efficient numerical solution of nonlinear optimal control problems is presented. In the proposed method, at first, based on the Pontryagin's minimum principle, the first-order necessary conditions of optimality which are led to the Hamiltonian boundary value problem are derived. Then, utilizing a pseudospectral method for discretization, the nonlinear optimal control problem is converted to a system of nonlinear algebraic equations. However, the need to have a good initial guess may lead to a challenging problem for solving the obtained system of nonlinear equations. So, an optimization approach is introduced to simplify the need of a good initial guess. Numerical findings of some benchmark examples are presented at the end and computational features of the proposed method are reported.An ODE reduction method for the semi-Riemannian Yamabe problem on space formshttps://www.zbmath.org/1483.530592022-05-16T20:40:13.078697Z"Fernández, Juan Carlos"https://www.zbmath.org/authors/?q=ai:fernandez.juan-carlos"Palmas, Oscar"https://www.zbmath.org/authors/?q=ai:palmas.oscarThe authors prove the existence of blowing-up and globally defined solutions of Yamabe-type partial differential equations on semi-Euclidean space and on the pseudosphere of dimension at least 3. In the proof they use isoparametric functions which allow the reduction to a generalized Emden-Fowler ordinary differential equation.
Reviewer: Hans-Bert Rademacher (Leipzig)Quantum graphs on radially symmetric antitreeshttps://www.zbmath.org/1483.810742022-05-16T20:40:13.078697Z"Kostenko, Aleksey"https://www.zbmath.org/authors/?q=ai:kostenko.aleksey-s"Nicolussi, Noema"https://www.zbmath.org/authors/?q=ai:nicolussi.noemaIn the present study the authors mainly focused their attention on antitrees from the perspective of quantum graphs and discussed a detailed spectral analysis of the Kirchhoff Laplacian on radially symmetric antitrees. Antitrees come into sight in the investigation of discrete Laplacians and attracted a noteworthy attention especially after the work of \textit{K.-T. Sturm} [J. Reine Angew. Math. 456, 173--196 (1994; Zbl 0806.53041)]. Also, Kostenko and Nicolussi considered the approach intorudced by [\textit{V. A. Mikhailets}, Funct. Anal. Appl. 30, No. 2, 144--146 (1996; Zbl 0874.34069); translation from Funkts. Anal. Prilozh. 30, No. 2, 90--93 (1996); \textit{B. Muckenhoupt}, Stud. Math. 44, 31--38 (1972; Zbl 0236.26015)] for radially symmetric trees and used some ideas from [\textit{J. Breuer} and \textit{N. Levi}, Ann. Henri Poincaré 21, No. 2, 499--537 (2020; Zbl 1432.05061)], where discrete Laplacians on radially symmetric ``weighted'' graphs have been analyzed. To summarize in general terms, in this paper, after recalling some necessary definitions and presenting an hypothesis, the authors studied characterization of self-adjointness and a complete description of self-adjoint extensions, spectral gap estimates and spectral types (discrete, singular and absolutely continuous spectrum). Next, they demonstrated their main results by considering two special classes of antitrees: (i) antitrees with exponentially increasing sphere numbers and (ii) antitrees with polynomially increasing sphere numbers.
Reviewer: Mustafa Salti (Mersin)Non-local imprints of gravity on quantum theoryhttps://www.zbmath.org/1483.830242022-05-16T20:40:13.078697Z"Maziashvili, Michael"https://www.zbmath.org/authors/?q=ai:maziashvili.michael"Silagadze, Zurab K."https://www.zbmath.org/authors/?q=ai:silagadze.zurab-kSummary: During the last two decades or so much effort has been devoted to the discussion of quantum mechanics (QM) that in some way incorporates the notion of a minimum length. This upsurge of research has been prompted by the modified uncertainty relation brought about in the framework of string theory. In general, the implementation of minimum length in QM can be done either by modification of position and momentum operators or by restriction of their domains. In the former case we have the so called soccer-ball problem when the naive classical limit appears to be drastically different from the usual one. Starting with the latter possibility, an alternative approach was suggested in the form of a band-limited QM. However, applying momentum cutoff to the wave-function, one faces the problem of incompatibility with the Schrödinger equation. One can overcome this problem in a natural fashion by appropriately modifying Schrödinger equation. But incompatibility takes place for boundary conditions as well. Such wave-function cannot have any more a finite support in the coordinate space as it simply follows from the Paley-Wiener theorem. Treating, for instance, the simplest quantum-mechanical problem of a particle in an infinite potential well, one can no longer impose box boundary conditions. In such cases, further modification of the theory is in order. We propose a non-local modification of QM, which has close ties to the band-limited QM, but does not require a hard momentum cutoff. In the framework of this model, one can easily work out the corrections to various processes and discuss further the semi-classical limit of the theory.