Necessary and sufficient conditions are obtained for a linear functional differential equation of the second order with a monotone operator to be everywhere solvable for a given norm of the monotone functional operator. If the found condition on the norm is not satisfied, there exists a monotone operator with the given norm for which the functional differential equation of the second order will not have a solution for some righthand sides. In the case of everywhere solvability, a set of boundary value problems for the given equation is indicated, at least one of which is uniquely solvable. To find necessary and sufficient conditions for everywhere solvability (surjectivity), a finite-dimensional representation of the functional operator on the null space of solutions of the functional differential equation is used. Previously, such a representation was considered only for a one-dimensional set of solutions of a homogeneous boundary value problem. The possibility of a finite-dimensional representation of a functional operator on a multidimensional subspace of solutions significantly expands the capabilities of the method for obtaining exact solvability conditions.
We consider a linear autonomous differential equation with aftereffect and coefficients of different signs whose fundamental solution is positive. For these equations we obtain two-sided estimates of the fundamental solution in the form of two exponential functions with exact efficiently calculated exponents and coefficients.
The paper is devoted to invertibility of the operator at the derivative in an autonomous neutral type differential equation in Lebesgue spaces Lp. For this purpose, the spectrum of the internal superposition operator is studied and the location of the roots of the characteristic equation is determined. The general form of the spectrum is established. It is also shown that there exists a right boundary of the roots, which is exact in case when the delays at the derivative are linearly independent, and the position of this boundary on the complex plane is determined. The stability conditions of different types in terms of the coefficients of the original neutral type equation are formulated.
For a class of fractional functional differential system, the control problem with respect to a target vector-functional is considered. Conditions for the solvability of the problem are obtained. The constructions used are based on the general theorems of the theory of abstract functional differential equation developed by professors N.V. Azbelev and L.F. Rakhmatullina.
In the present paper, we give conditions for the existence and uniqueness of the solution of the boundary problem
A functional differential equation with a discrete retarded argument and constant delay is considered. The problem of asymptotic stability of this equation is reduced to the problem of the location of the spectrum of the shift operator. Coefficient sufficient conditions for the asymptotic stability of this equation are obtained. A set of parameters of the equation such that these conditions are necessary.
We consider several issues concerning effective oscillation conditions and estimates of the length of nonoscillation intervals for solutions of delay differential equations. We refine and deepen several new results using approaches recognized as classical. The main goal of our work is to show that the potential of these approaches is far from being fully utilized.
Within the framework of a constructive approach to the study of functional differential equations, a way for exact constructing the Cauchy matrix for a system of linear differential equations with piecewise linear delay is proposed. An illustrative example is given.
In this paper, we consider a coupled system of Riemann-Liouville fractional differential equations with nonlocal boundary conditions. We have compared the first extremal points of a fractional differential equation with nonlocal boundary conditions, assuming that the functions are nonnegative.
This article presents a study where the tool measure of noncompactness (MNC) and Meir-Keeler condensing operator (MKCO) are employed to establish the existence of solutions to an infinite system of Hadamard-type fractional functional integral equation in the Banach sequence space lp, p > 1. Additionally, an illustrative example is provided to validate the obtained results. Furthermore, we employ a coupled semianalytical method based on the modified homotopy perturbation method (mHPM) and Adomian’s decomposition method (ADM), to approximate the solution for the given example.
In this paper, we propose nonlinear fractional Lane-Emden equations, Dαy(x) +λ|x Dβy(x) + f(y) = 0, 1 < α ≤ 2, 0 < β ≤ 1, 0 < x, with initial conditions, y(0) = A, y (0) = B, where Dα,Dβ are Caputo fractional derivatives, λ = 1,2 and A,B are constants, 0 < β ≤ 1, 1 < α ≤ 2 and f(y) is a nonlinear function of y. We developed two collocation methods, namely the uniform fractional Haar wavelet collocation method and the nonuniform fractional Haar wavelet collocation method, and computed the solutions. In this procedure, fractional Haar integrations are used to find out the nonlinear system, which, when solved, gives the required solution. Our findings indicate that when (α, β) approach (2, 1), the solutions of the fractional and classical Lane-Emden problems become arbitrarily close to each other.
We study the existence of eigen values yielding positive solutions to coupled systems. These results are obtained by using fixed point theorems in cones.
In this paper, we consider the second order three point boundary value problem with integral bounday conditions on a half-line of the form u”(t) + q(t)f(t, u(t), u'(t)) = 0, t∈ (0,+∞), u(0) − au’ (0) = λη 0 u(s)ds, lim t→+∞ u (t) = u (+∞) = λ η0 u(s)ds, where λ > 0, 0 < λη < 1, a > 0, q : (0,+∞) → (0,+∞), f : [0,+∞) × R × R → R is continuous and satisfies Nagumo’s condition. We employ the Schauder’s fixed point theorem, the upper and lower solution method and topological degree theories to show the existence of at least one solution and then existence of three solutions of the above boundary value problem. Finally, we have shown the importance of our results through an examples.
The W-transform in the form of the regularization techniques, based on inverse-positive matrices, is used to obtain verifiable global stability conditions for linear and nonlinear stochastic delay systems with random coefficients and impulses at random times. The results are conveniently formulated in terms of coefficients of the systems. Several examples illustrating the main results are offered.
The authors study the existence of positive solutions of nonlinear Sturm- Liouville boundary value problem: (p(t)y’ (t))’+ λf(t, y(t)) = 0, 0 < t < 1 with mixed boundary conditions ay(0) − bp(0)y'(0) = 0, cy(1) + dp(1)y'(1) = 0, for λ > 0 on a suitable interval. The methods involve the applications of a fixed point theorem for operators on a cone in a Banach space. An attempt has been made to establish the existence of a positive solutions of the above boundary value problem under certain sufficient conditions on f(t, y). Examples are given at the end of the paper to justify our results.
Thisworkisconcernedabout thenecessaryandsufficientconditions for oscillationofsolutionsof2-dimensionalnonlinearneutraldelaydifferentialsystemsofthe form:
Fortheneutralfunctionaldifferentialequationwithtwotypescontrolsand thediscontinuous initial condition,whoseright-handsideare linearwithrespect tothe prehistoryofthephasevelocity,theoremsonthecontinuousdependenceofasolutionwith respecttoperturbationsof the initialdataandtheright-handsideareproposed. Under theinitialdataweimplythecollectionofdelayparameters, initialfunctions, initialvector andcontrol functions. Inthework, twocasesareconsidered,whenperturbationof the nonlineartermintheright-handsidearesmall intheintegralandEuclideantopologies. Discontinuityattheinitialmomentmeansthatattheinitialmomentvaluesoftheinitial functionandtrajectory, ingeneral,donotcoincide, thissituationmayberelatedtothe instantchange inadynamicalprocess (changesof investment, environmentandsoon). Suchtypetheoremsplayanimportantroleinprovingofnecessaryconditionsofoptimality intheoptimizationproblem,inprovingvariationformulasofsolutionandinthesensitivity analysisofmathematicalmodels.
The linearisation of the Lotka-Volterra system with time delay is considered. Any type of interaction between two spices is under consideration. The sufficient conditions of asymptotic stability for this linear system are obtained
Thispaperdealswiththeoscillatory eha io ro sol tio stoaclasso e e order o li eardiffere tial e atio swithada cedar e ts. efi dasi le co ditio thate s restheoscillatio o thest diede atio . Ea plesarepro idedto de o stratethe ai fi di s. Keywords: da ceddiffere tiale atio ,Oscillatio ,No li ear,Ee order. MSC(2020)34 11,34 10.
