Vol 26, No 136 (2021)
Original articles
On the existence and uniqueness of a positive solution to a boundary value problem for a nonlinear ordinary differential equation of even order
Abstract
In the article, we consider a boundary value problem for a nonlinear ordinary differential equation of even order which, obviously, has a trivial solution. Sufficient conditions for the existence and uniqueness of a positive solution to this problem are obtained. With the help of linear transformations of T. Y. Na [T. Y. Na, Computational Methods in Engineering Boundary Value Problems, Acad. Press, NY, 1979, ch. 7], the boundary value problem is reduced to the Cauchy problem, the initial conditions of which make it possible to uniquely determine the transformation parameter. It is shown that the transformations of T. Y. Na uniquely determine the solution of the original problem. In addition, based on the proof of the uniqueness of a positive solution to the boundary value problem, a sufficiently effective non–iterative numerical algorithm for constructing such a solution is obtained. A corresponding example is given.
341-347
On the Cauchy problem for implicit differential equations of higher orders
Abstract
The article is devoted to the study of implicit differential equations based on statements about covering mappings of products of metric spaces. First, we consider the system of equations
\begin{equation*}
\Phi_i(x_i,x_1,x_2,\ldots,x_n)=y_i, \ \ \ i=\overline{1,n},
\end{equation*}
where $\Phi_i: X_i \times X_1 \times \ldots \times X_n \to Y_i,$ $y_i \in Y_i,$ $X_i$ and $Y_i$ are metric spaces, $i=\overline{1,n}.$ It is assumed that the mapping $\Phi_i$
is covering in the first argument and Lipschitz in each of the other arguments starting from the second one. Conditions for the solvability of this system and estimates for the distance from an arbitrary given element $x_0 \in X$ to the set of solutions are obtained. Next, we obtain an assertion about the action of the Nemytskii operator in spaces of summable functions and establish the relationship between the covering properties of the Nemytskii operator and the covering of the function that generates it. The listed results are applied to the study of a system of implicit differential equations, for which a statement about the local solvability of the Cauchy problem with constraints on the derivative of a solution is proved. Such problems arise, in particular, in models of controlled systems. In the final part of the article, a differential equation of the $n$-th order not resolved with respect to the highest derivative is studied by similar methods. Conditions for the existence of a solution to the Cauchy problem are obtained.
348-362
On ring solutions of neural field equations
Abstract
The article is devoted to investigation of integro-differential equation with the Hammerstein integral operator of the following form:
\begin{equation*}
\begin{array}{c}
\partial_tu(t,x)=-\tau u(t,x,x_\mathrm{f})+\int\limits_{\mathbb{R}^2}
\omega(x-y)f(u(t,y)) dy, \
\ t\geq0,\ x\in \mathbb{R}^2.
\end{array}
\end{equation*}
The equation describes the dynamics of electrical potentials $u(t,x)$ in a planar neural medium and has the name of neural field equation. We study ring solutions that are represen\-ted by stationary radially symmetric solutions corresponding to the active state of the neural medium in between two concentric circles and the rest state elsewhere in the neural field. We suggest conditions of existence of ring solutions as well as a method of their numerical approximation. The approach used relies on the replacement of the probabilistic neuronal activation function $f$ that has sigmoidal shape by a Heaviside-type function. The theory is accompanied by an example illustrating the procedure of investigation of ring solutions of a neural field equation containing a typically used in the neuroscience community neuronal connectivity function that allows taking into account both excitatory and inhibitory inter\-neuronal interactions. Similar to the case of bump solutions (i.~e. stationary solutions of neural field equations, which correspond to the activated area in the neural field represented by the interior of some circle) at a high values of the neuronal activation threshold there coexist a broad ring and a narrow ring solutions that merge together at the critical value of the activation threshold, above which there are no ring solutions.
363-371
On the existence problem for a fixed point of a generalized contracting multivalued mapping
Abstract
We discuss the still unresolved question, posed in [S.~Reich, Some Fixed Point Problems, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 57:8 (1974), 194--198], of existence in a complete metric space $X$ of a fixed point for a generalized contracting multivalued map $\Phi: X \rightrightarrows X $ having closed values $ \Phi (x) \subset X$ for all $ x \in X. $ Generalized contraction is understood as a natural extension of the Browder--Krasnoselsky definition of this property to multivalued maps:
\begin{equation*}
\forall x, u \in X \ \ h \bigl(\varphi(x), \varphi(u) \bigr) \leq \eta \bigl(\rho(x, u) \bigr),
\end{equation*}
where the function $ \eta: \mathbb {R}_+\to\mathbb{R}_+$ is increasing, right continuous, and for all $d>0,$\linebreak $\eta(d)
372-381
On a necessary and sufficient condition for the negativeness of the Green’s function of a two-point boundary value problem for a functional differential equation
Abstract
Conditions of negativity for the Green's function of a two-point boundary value problem
\[
\Lc_\lambda u := u^{(n)}-\lambda\int_0^l u(s) d_s r(x,s)=f(x), \ \ \ x\in[0,l], \ \ \ B^k(u)=0,
\]
where $B^k(u)=(u(0),\ldots,u^{(n-k-1)}(0),u(l),-u'(l),\ldots,(-1)^{(k-1)}u^{(k-1)}(0)),$
$n\ge3,$ $0\!<\!k\!<\!n,$ $k$ is odd, are considered. The function $r(x,s)$ is assumed to be non-decreasing in the second argument.
A necessary and sufficient condition for the nonnegativity of the solution of this boundary value problem on the set $E$ of functions satisfying the conditions
\[
u(0)=\cdots=u^{(n-k-2)}(0)=0, \ \ \ u(l)=\cdots=u^{(k-2)}(l)=0,
\]
$u^{(n-k-1)}(0)\ge0,$ $u^{(k-1)}(l)\ge0,$ $f(x)\le 0$ is obtained.
This condition lies in the subcriticality of boundary value problems with vector functionals $B^{k-1}$ and $B^{k+1}.$ Let $k$ be even and $\lambda^k$ be the smallest positive value of $\lambda$ for which the problem $\Lc_\lambda u = 0,$ $B^ku = 0$ has a nontrivial solution.
Then the pair of conditions $\lambda <\lambda^{k-1}$ and $\lambda <\lambda^{k+1}$ is necessary and sufficient for positivity of the solution of the problem.
382-393
On an ill-posed boundary value problem for a metaharmonic equation in a circular cylinder
Abstract
In this paper, we consider a mixed problem for a metaharmonic equation in a domain in a circular cylinder. The cylindrical area is bounded on one side by an arbitrary surface on which the Cauchy conditions are set, i. e. the function and its normal derivative are set. The other border of the cylindrical area is free. On the lateral surface of the cylindrical domain, homogeneous boundary conditions of the first kind are given. The problem is illposed and its approximate solution, stable to errors in the Cauchy data, is constructed using regularization methods. The problem is reduced to a first kind Fredholm integral equation. Based on the solution of the integral equation obtained in the form of a Fourier series by the eigenfunctions of the first boundary value problem for the Laplace equation in a circle, an explicit representation of the exact solution of the problem is constructed. A stable solution of the integral equation is obtained by the method of Tikhonov regularization. The extremal of the Tikhonov functional is considered as an approximate solution. Based on this solution, an approximate solution of the problem as a whole is constructed. A theorem on convergence of the approximate solution of the problem to the exact one as the error in the Cauchy data tends to zero and the regularization parameter is matched with the error in the data, is given. The results can be used for mathematical processing of thermal imaging data in early diagnostics in medicine.
394-403
One method for investigating the solvability of boundary value problems for an implicit differential equation
Abstract
The article concernes a boundary value problem with linear boundary conditions of general form for the scalar differential equation
\begin{equation*}
f \big(t, x (t), \dot{x} (t) \big)= \widehat{y}(t),% \ \ t \ in [0, \ tau],
\end{equation*}
not resolved with respect to the derivative $\dot{x}$ of the required function. It is assumed that the function $f$ satisfies the Caratheodory conditions, and the function $\widehat{y}$ is measurable. The method proposed for studying such a boundary value problem is based on the results about operator equation with a mapping acting from a metric space to a set with distance (this distance satisfies only one axiom of a metric: it is equal to zero if and only if the elements coincide).
In terms of the covering set of the function $f(t, x_1, \cdot): \mathbb{R} \to \mathbb{R}$ and the Lipschitz set of the function $f (t,\cdot,x_2): \mathbb{R} \to \mathbb{R} $, conditions for the existence of solutions and their stability to perturbations of the function $f$ generating the differential equation, as well as to perturbations of the right-hand sides of the boundary value problem: the function $ \widehat{y} $ and the value of the boundary condition, are obtained.
404-413
Solving a second-order algebro-differential equation with respect to the derivative
Abstract
We consider a second-order algebro-differential equation. Equations and systems of second-order differential equations describe the operation of an electronic triode circuit with feedback, rotation of a rigid body with a cavity, reading information from a disk, etc. The highest derivative is preceded by an irreversible operator. This is a Fredholm operator with index zero, kernel of arbitrary dimension, and Jordan chains of arbitrary lengths. Equations with irreversible operators at the highest derivative are called algebro-differential. In this case, the solution to the problem exists under certain conditions on the components of the desired function. To solve the equation with respect to the derivative, the method of cascade splitting of the equation is used, which consists in the stepwise splitting of the equation into equations in subspaces of decreasing dimensions. Cases of one-step and two-step splitting are considered. The splitting uses the result on the solution of a linear equation with Fredholm operator. In each case, the corresponding result is formulated as a theorem. To illustrate the result obtained in the case of one-step splitting, an illustrative example of the Cauchy problem is given.
414-420