Problems For Functional Differential Equations Research Articles

Global Hopf Bifurcation of State-Dependent Delay Differential Equations

We apply the [Formula: see text]-equivariant degree method to a Hopf bifurcation problem for functional differential equations with a state-dependent delay. The formal linearization of the system at a stationary state is extracted and translated into a bifurcation invariant by using the homotopy invariance of [Formula: see text]-equivariant degree. As a result, the local Hopf bifurcation is detected and the global continuation of periodic solutions is described.

On the solvability of the Cauchy problem for functional differential equations with mixed restrictions on functional operator

On the solvability of the Cauchy problem for functional differential equations with mixed restrictions on functional operator

Cattmul-Rom spline approach and the order of convergence of Green’s functional method for functional differential equations

The purpose of this work is to investigate the convergence properties of Green’s function method applied to boundary value problems for functional differential equations. Recently, involving Picard and Mann iterations, a Green’s function technique was developed (in Int. J. Computer Math. 95, no. 10 (2018) 1937-1949) for third order functional differential equations, but without specifying the order of convergence of the proposed method. In order to improve this aspect, here we establish the maximal order of convergence of Green’s function method applied to two-point boundary value problems associated to second and third order functional differential equations. In this context, by using suitable quadrature rule and appropriate spline interpolation procedure, the Picard iterations are approximated by a sequence of cubic splines on uniform mesh. Some numerical experiments are presented in order to test the theoretical results and to illustrate the accuracy of the method.

Unique solvability of boundary value problem for functional differential equations with involution

In this paper we consider a boundary value problem for systems of Fredholm type integral-differential equations with involutive transformation, containing derivative of the required function on the right-hand side under the integral sign. Applying properties of an involutive transformation, original boundary value problem is reduced to a boundary value problem for systems of integral-differential equations, containing derivative of the required function on the right side under the integral sign. Assuming existence of resolvent of the integral equation with respect to the kernel K˜2(t, s) (this is the kernel of the integral equation that contains the derivative of the desired function) and using properties of the resolvent, integral-differential equation with a derivative on the right-hand side is reduced to a Fredholm type integral-differential equation, in which there is no derivative of the desired function on the right side of the equation. Further, the obtained boundary value problem is solved by the parametrization method created by Professor D. Dzhumabaev. Based on this method, the problem is reduced to solving a special Cauchy problem with respect to the introduced new functions and to solving systems of linear algebraic equations with respect to the introduced parameters. An algorithm to find a solution is proposed. As is known, in contrast to the Cauchy problem for ordinary differential equations, the special Cauchy problem for systems of integral-differential equations is not always solvable. Necessary conditions for unique solvability of the special Cauchy problem were established. By using results obtained by Professor D. Dzhumabaev, necessary and sufficient conditions for the unique solvability of the original problem were established.

A singular problem for functional differential equations with decreasing non‐linearities

We consider a weighted initial value problem for first‐order functional differential equations with monotone non‐linearities which may have a non‐integrable singularity with respect to the independent variable. Conditions for the existence of solutions in a suitable weighted space of locally absolutely continuous functions are given.

ON THE CONSTRUCTION OF A CLASS OF COMPUTABLE OPERATORS

Within the framework of a constructive approach to the study of linear boundary value problems for functional differential equations, a way for constructing one class of so-called computable operators is proposed.

A qualitative investigation for some singular functional differential eqation

A singular Cauchy problem for functional differential equations of a certain type is considered, solved for the derivative of the unknown function. Solutions are sought in the class of continuously differentiable functions. It is proved that there exists a nonempty set of continuously differentiable solutions having certain asymptotic properties in a sufficiently small neighborhood of the singular point. Construction of the asymptotic behavior of solutions is as important result as proof of the existence of solutions. To study the task, a technique was used that combines elements of the theory of functions and the qualitative theory of differential equations. Moreover, a qualitative analysis was applied not only in constructing a certain nonlinear operator, but also in proving that this operator satisfies the conditions of the fixed-point theorem. This technique, in our opinion, can be used for a wide range of problems of the theory of nonlinear ordinary differential equations.

ON THE PERIODIC SOLUTIONS OF THE LIENARD EQUATION

Mathematical modeling of many problems of natural science leads to the need to study quasi-linear boundary value problems for functional differential equations with a linear part that is not uniquely solvable for all right-hand parts. The specificity of such problems is that the corresponding linear operator is not reversible. In the literature, such boundary value problems are usually called resonant. Since the 70s of the last century, the development of methods for studying resonant boundary value problems considered as a single operator equation has begun. A very important area of research from the point of view of applications is the application of General statements to the study of periodic boundary value problems for functional differential equations. The existence problem is considered ω - a periodic solution of the Lienard equation with a deviating argument of the form It is assumed that the function p ( t ) is measurable and Using an approach based on the application of theoretical existence for a quasilinear operator equation, sufficient conditions can be obtained in the work, at least one ω - a periodic solution must correspond to the equations. The obtained result refines some well-known results for the Lienard equations. Execution conditions decisions do not affect the existence of solutions.

Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays

An algorithm using the differential transformation which is convenient for finding numerical solutions to initial value problems for functional differential equations is proposed in this paper. We focus on retarded equations with delays which in general are functions of the independent variable. The delayed differential equation is turned into an ordinary differential equation using the method of steps. The ordinary differential equation is transformed into a recurrence relation in one variable using the differential transformation. Approximate solution has the form of a Taylor polynomial whose coefficients are determined by solving the recurrence relation. Practical implementation of the presented algorithm is demonstrated in an example of the initial value problem for a differential equation with nonlinear nonconstant delay. A two-dimensional neutral system of higher complexity with constant, nonconstant, and proportional delays has been chosen to show numerical performance of the algorithm. Results are compared against Matlab function DDENSD.

О разрешимости каузальных функциональных включений с бесконечным запаздыванием

In the present article we develop the results of works devoted to the study of problems for functional differential equations and inclusions with causal operators, in case of infinite delay. In the introduction of the article we substantiates the relevance of the research topic and provides links to relevant works A. N. Tikhonov, C. Corduneanu, A. I. Bulgakov, E. S. Zhukovskii, V. V. Obukhovskii and P. Zecca. In section two we present the necessary information from the theory of condensing multivalued maps and measures of noncompactness, also introduced the concept of a multivalued causal operator with infinite delay and illustrated it by examples. In the next section we formulate the Cauchy problem for functional inclusion, containing the composition of multivalued and single-valued causal operators; we study the properties of the multiopera-tor whose fixed points are solutions of the problem. In particular, sufficient conditions under which this multioperator is condensing on the respective measures of noncompacness. On this basis, in section four we prove local and global results and continuous dependence of the solution set on initial data. Next the case of inclusions with lower semicontinuous causal multioperators is considered. In the last section we generalize some results for semilinear differential inclusions and Volterra integrodifferential inclusions with infinite delay.

Asymptotic problems for functional differential equations via linearization method

We study the existence of positive decreasing solutions (the so-called Kneser solutions) for a class of second-order functional differential equations with a damping term. A linearization approach based on a general fixed point theorem is used to achieve this goal. The existence of zero-decaying Kneser solutions is also proved. Finally, the role of the deviating argument to the asymptotic behavior of solutions is illustrated together with some discrepancies between equations with or without delay.

A new reproducing kernel method for variable order fractional boundary value problems for functional differential equations

Based on reproducing kernel theory, a numerical method is proposed for solving variable order fractional boundary value problems for functional differential equations. In the previous works, piecewise polynomial reproducing kernels were employed to solve fractional differential equations. However, the computational cost of fractional order operator acting on such kernel functions is high. In this paper, reproducing kernels with polynomial form will be constructed and applied to solve variable order fractional functional boundary value problems. The method can reduce computation cost and provide highly accurate global approximate solutions.

Erratum to: Center Manifold Method in the Asymptotic Integration Problem for Functional Differential Equations with Oscillatory Decreasing Coefficients. II (in Russian)

In this paper, we study the asymptotic integration problem in the neighborhood of infinity for a certain class of linear functional differential systems. We construct the asymptotics for solutions of the considered systems in the critical case. Using the ideas of the center manifold method, we show the existence of the so-called critical manifold that is positively invariant for trajectories of the initial system. We establish that the asymptotics for solutions of the system on this manifold defines the asymptotics for all solutions of the initial system. In the first part of this work, we propose an algorithm for an approximate construction of the critical manifold. Moreover, we establish the unique solvability for auxiliary algebraic problems that occur within the algorithm implementation.

Boundary value problem for the second order impulsive delay differential equations

Abstract We present some existence and uniqueness result for a boundary value problem for functional differential equations of second order with impulses at fixed points.

Nontrivial solutions of boundary value problems for second-order functional differential equations

In this paper we present a theory for the existence of multiple nontrivial solutions for a class of perturbed Hammerstein integral equations. Our methodology, rather than to work directly in cones, is to utilize the theory of fixed point index on affine cones. This approach is fairly general and covers a class of nonlocal boundary value problems for functional differential equations. Some examples are given in order to illustrate our theoretical results.

Sheaf fuzzy problems for functional differential equations

Abstract In this paper, we present the studies on two kinds of solutions to fuzzy functional differential equations (FFDEs) and sheaf fuzzy functional differential equations (SFFDEs). The different types of solutions to FFDEs and SFFDEs are generated by the usage of generalized Hukuhara derivative concepts of fuzzy derivative in the formulation of a differential problem. Some examples are given to illustrate these results.

Periodic boundary value problems for second-order functional differential equations with impulse

In this paper, we study the existence of multiple positive solutions of the second-order periodic boundary value problems for functional differential equations with impulse. The proof of our main results is based upon the fixed point index theorem in cones.

Center Manifold Method in the Asymptotic Integration Problem for Functional Differential Equations with Oscillatory Decreasing Coefficients. I

In this paper, we study the asymptotic integration problem in the neighborhood of infinity for a certain class of linear functional differential systems. We construct the asymptotics for solutions of the considered systems in the critical case. Using the ideas of the center manifold method, we show the existence of the so-called critical manifold that is positively invariant for trajectories of the initial system. We establish that the asymptotics for solutions of the system on this manifold defines the asymptotics for all solutions of the initial system. In the first part of this work, we propose an algorithm for an approximate construction of the critical manifold. Moreover, we establish the unique solvability for auxiliary algebraic problems that occur within the algorithm implementation.

Center Manifold Method in the Asymptotic Integration Problem for Functional Differential Equations with Oscillatory Decreasing Coefficients. II

In this paper we study the asymptotic integration problem in the neighborhood of infinity for a certain class of linear functional differential systems. We construct the asymptotics for the solutions of the considered systems in a critical case. In the second part of the work we establish the existence of a critical manifold for the considered class of systems and study its main properties. We also investigate the asymptotic integration problem for a reduced system. We illustrate the proposed method with an example of constructing the asymptotics for the solutions of a certain scalar delay differential equation.

On constructive investigation of a class of non-linear boundary value problems for functional differential equations

We consider a two-point boundary value problem for a non-linear system of functional differential equations, for which a scheme of efficient investigation using certain iteration process is constructed. A new convergence condition is established in the case where a certain additional property of the Lipschitz operator is present, which may lead one to weaker assumptions.