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Classically, solution theories for state-dependent delay equations are developed in spaces of continuous or continuously differentiable functions. The former can be technically challenging to apply in as much as suitably Lipschitz continuous extensions of mappings onto the space of continuous functions are required; whereas the latter approach leads to restrictions on the class of initial pre-histories. Here, we establish a solution theory for state-dependent delay equations for arbitrary Lipschitz continuous pre-histories and suitably Lipschitz continuous right-hand sides on the Sobolev space H 1. The provided solution theory is independent of previous ones and is based on the contraction mapping principle on exponentially weighted spaces. In particular, initial pre-histories are not required to belong to solution manifolds and the generality of the approach permits the consideration of a large class of functional differential equations even for which the continuity of the right-hand side has constraints on the derivative. [ABSTRACT FROM AUTHOR]

In this paper we establish uniqueness of the weak solutions of nonlinear stochastic functional differential equations of neutral type in Hilbert spaces. We also study the continuous dependence of such solutions on the initial data

This paper focuses on the oscillatory properties of the third-order semi-canonical nonlinear delay differential equation. By using the new canonical transform method, we transformed the studied equation into a canonical-type equation, which simplified the examination of the studied equation. The obtained oscillation results are new and complement the existing results mentioned in the literature. Examples are provided to illustrate the importance and novelty of the main results. [ABSTRACT FROM AUTHOR]

We study certain Volterra integral equations that extend and recover first order ordinary differential equations (ODEs). We formulate the former equations from the latter by replacing classical derivatives with nonlocal integral operators with anti-symmetric kernels. Replacements of spatial derivatives have seen success in fracture mechanics, diffusion, and image processing. In this paper, we consider nonlocal replacements of time derivatives which contain future data. To account for the nonlocal nature of the operators, we formulate initial "volume" problems (IVPs) for these integral equations; the initial data is prescribed on a time interval rather than at a single point. As a nonlocality parameter vanishes, we show that the solutions to these equations recover those of classical ODEs. We demonstrate this convergence with exact solutions of some simple IVPs. However, we find that the solutions of these nonlocal models exhibit several properties distinct from their classical counterparts. For example, the solutions exhibit discontinuities at periodic intervals. In addition, for some IVPs, a continuous initial profile develops a measure-valued singularity in finite time. At subsequent periodic intervals, these solutions develop increasingly higher order distributional singularities. [ABSTRACT FROM AUTHOR]

The equation G (x , x) = y ˜ , where G : X × X → Y , and X , Y are vector metric spaces (meaning that the values of a distance between the points in these spaces belong to some cones E + , M + of a Banach space E and a linear space M, respectively), is considered. This operator equation is compared with a "model" equation, namely, g (t , t) = 0 , where a continuous map g : E + × E + → M + is orderly covering in the first argument and antitone in the second one. The idea to study equations comparing them with "simpler" ones goes back to the Kantorovich fixed-point theorem for an operator acting in a Banach space. In this paper, the conditions under which the solvability of the "model" equation guarantees the existence of solutions to the operator equation are obtained. The statement proved extends the recent results about fixed points and coincidence points to more general equations in more general vector metric spaces. The results obtained for the operator equation are then applied to the study of the solvability, as well as to finding solution estimates, of the Cauchy problem for a functional differential equation. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of an elliptic-type functional differential equation containing the contraction transformation of the arguments of the unknown function in its leading part, and the contractions are different for different arguments. Some necessary and sufficient conditions for the validity of a Gå rding-type inequality, which is an analog of the strong ellipticity condition, are presented in an explicit form. The Fredholm solvability and the structure of the spectrum of the first boundary value problem in the Sobolev spaces are studied. Sufficient conditions for the solvability of the equation in the Kondrat'ev weighted spaces on the plane are given. In the course of the proof, sufficient conditions for the invertibility of a finite-difference operator with variable coefficients on a line are obtained. Some concrete examples illustrating the results thus obtained are presented. [ABSTRACT FROM AUTHOR]

In this work, new oscillation criteria are established for a second-order differential equation with several sublinear neutral terms and in the canonical case. To determine the oscillation conditions, we followed the Riccati approach and also compared the studied equation with a first-order delay equation. Obtaining the oscillation conditions required deducing some new relationships linking the solution to the corresponding function as well as its derivatives. The paper addresses some interesting analytical points in the study of the oscillation of equations with several sublinear neutral terms. These new findings complement some well-known findings in the literature. Furthermore, an example is provided to show the importance of the results. [ABSTRACT FROM AUTHOR]

Human cells have a natural cancerous defense system, and p53 protein is at the core. It is well known that p53 relies on oscillatory dynamic behavior to perform tumor suppressor functions. However, the mechanism of its oscillation remains unclear. To reveal the effect of time lag on the onset of p53 oscillations, we analytically and numerically investigate a possible scenario in which p53 oscillations are driven by a time lag. The model has only three nodes, two negative loops, and one delay. We focus on two types of time delays: discrete and distributed, use Hopf bifurcation theory to derive conditions for the occurrence of limit cycle oscillations, and confirm the properties of Hopf bifurcations by normal form and central manifold approaches. The analytical results are supported by numerical results. We suggest that discrete time lag promote p53 oscillations, while distributed time lag have a dual effect on p53 oscillations. These results imply that controlling the time lag in an appropriate range favors the inhibition of p53 on cancer cells.

Necessary and sufficient conditions for the existence of a valid Dirichlet solution were obtained. A method was developed to find Dirichlet solutions of the functional differential equation with non-delayed linear argument deviation.

In this paper, some new findings on the uniqueness and existence of positive periodic solutions to first-order functional differential equations are presented. These equations have wide applications in a variety of fields. The most important feature of our argument is that we use the theory of Hilbert's metric to prove the uniqueness of the positive periodic solution when q=−1 and −1

The aim of the present paper is to investigate of some properties of periodic solutions of a nonlinear autonomous parabolic systems with a periodic condition. We investigate parabolic systems of differential equations using an integral manifolds method of the theory of nonlinear oscillations. We prove the existence of periodic solutions in an autonomous parabolic system of differential equations with weak diffusion on the circle. We study the existence and stability of an arbitrarily large finite number of cycles for a parabolic system with weak diffusion. The periodic solution of parabolic equation is sought in the form of traveling wave. A representation of the integral manifold is obtained. We seek a solution of parabolic system with the periodic condition in the form of a Fourier series in the complex form and introduce a norm in the space of the coefficients in the Fourier expansion. We use the normal forms method in the general parabolic system of differential equations with retarded argument and weak diffusion. We use bifurcation theory for delay differential equations and quasilinear parabolic equations. The existence of periodic solutions in an autonomous parabolic system of differential equations on the circle with retarded argument and small diffusion is proved. The problems of existence and stability of traveling waves in the parabolic system with retarded argument and weak diffusion are investigated.

Various scholars have lately employed a wide range of strategies to resolve two specific types of symmetrical fractional differential equations. The evolution of a number of real-world systems in the physical and biological sciences exhibits impulsive dynamical features that can be represented via impulsive differential equations. In this paper, we explore some existence and controllability theories for the Caputo order q ∈ (1 , 2) of delay- and random-effect-affected fractional functional integroevolution equations (FFIEEs). In order to prove that random solutions exist, we must prove a random fixed point theorem using a stochastic domain and the mild solution. Then we demonstrate that our solutions are controllable. At the end, applications and example is illustrated which indicates the applicability of this manuscript. [ABSTRACT FROM AUTHOR]

We consider linear autonomous differential equations of neutral type. We obtain the exact estimation for the exponent of solutions of these equations. [ABSTRACT FROM AUTHOR]

Unprovable solvability conditions are obtained for the Cauchy problem for second order functional differential equations with both pointwise and integral restrictions on functional operators. [ABSTRACT FROM AUTHOR]

In order to improve the information quality of financial accounting reports, this paper puts forward a functional differential equation, establishes the determination model of fair value by mathematical analysis method and studies the correlation of financial accounting information of fair value through this model. The results show that according to the fair value determination model, in the free accounting environment, fair value financial accounting information has an impact.

The aim of the present paper is to investigate of some properties of periodic solutions of a nonlinear autonomous parabolic systems with a periodic condition. We investigate parabolic systems of differential equations using an integral manifolds method of the theory of nonlinear oscillations. We prove the existence of periodic solutions in an autonomous parabolic system of differential equations with weak diffusion on the circle. We study the existence and stability of an arbitrarily large finite number of cycles for a parabolic system with weak diffusion. The periodic solution of parabolic equation is sought in the form of traveling wave. A representation of the integral manifold is obtained. We seek a solution of parabolic system with the periodic condition in the form of a Fourier series in the complex form and introduce a norm in the space of the coefficients in the Fourier expansion. We use the normal forms method in the general parabolic system of differential equations with retarded argument and weak diffusion. We use bifurcation theory for delay differential equations and quasilinear parabolic equations. The existence of periodic solutions in an autonomous parabolic system of differential equations on the circle with retarded argument and small diffusion is proved. The problems of existence and stability of traveling waves in the parabolic system with retarded argument and weak diffusion are investigated. [ABSTRACT FROM AUTHOR]

The present study is devoted to finding the approximate solutions of a new design of a secondorder nonlinear functional system of Lane-Emden differential equation with boundary conditions. Our proposed matrix technique is based on the Chelyshkov functions together with collocation points to transform the nonlinear system into an algebraic fundamental matrix equation. The convergence of the spectral Chelyshkov approach is also proved. To show the efficiency and the accuracy of the presented technique, three test examples are solved numerically. Also, comparisons with the exact solutions and with an available method in the literature are performed. [ABSTRACT FROM AUTHOR]

Functional differential equations of neutral type are a class of differential equations in which the derivative of the unknown functions depends on the history of the function and its derivative as well. Due to this nature the explicit solutions of these equations are not easy to compute and sometime even not possible. Therefore, one must use some numerical technique to find an approximate solution to these equations. In this paper, we used a spectral collocation method which is based on Bernstein polynomials to find the approximate solution. The disadvantage of using Bernstein polynomials is that they are not orthogonal and therefore one cannot use the properties of orthogonal polynomials for the efficient evaluation of differential equations. In order to avoid this issue and to fully use the properties of orthogonal polynomials, a change of basis transformation from Bernstein to Legendre polynomials is used. An error analysis in infinity norm is provided, followed by several numerical examples to justify the efficiency and accuracy of the proposed scheme.

In this study, we examine the behavior of solutions of the neutral functional differential equations. Using a suitable real root of the corresponding characteristic equation, the asymptotic behavior of the solutions and the stability of the trivial solution are explained. Three examples are also provided to illustrate our results.

In this paper, we investigate the generalized Hopf bifurcation for semilinear functional differential equations in general Banach spaces by applying the Lyapunov–Schmidt reduction process. We not only obtain explicit expressions about the parameters of the original system, but also use these expressions to determine whether there are several periodic solutions when the parameters are changed. Moreover, an example is given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we study some existence and controllability results for two classes of second order functional differential equations with delay and random effects. To begin, we employ a random fixed point theorem with a stochastic domain to demonstrate the existence of mild random solutions. Next, we prove that our problems are controllable. Finally, an example is given to validate the theory part. [ABSTRACT FROM AUTHOR]

Let (X1,→) and (X2, ↬) be two L-spaces, U be a nonempty subset of X1 × X2 such that UX1:= {X2 ∈ X2 | (X1, X2) ∈ U} is nonempty, for each X1 ∈ X1. Let T1: X1 → X1, T2: U → X2 be two operators and T: U → X1 × X2 defined by T(X1,X2):= (T1(X1),T2(X1,X2)). If we suppose that T(U) ⊂ U, FT1 ≠ ∅ and FT2(X1,·) ≠ ∅ for each X1 ∈ X1, the problem is in which additional conditions T is a weakly Picard operator? In this paper we study this problem in the case when the convergence structures on X1 and X2 are defined by metrics. Some applications to the fixed point equations on spaces of continuous functions are also given. [ABSTRACT FROM AUTHOR]

In this work, we are concerned with weak and pseudo solutions for some initial value problems of fractional order and their corresponding functional integral equation of fractional order. These initial value problems includes many initial value problems that arise in nonlinear analysis and its applications. [ABSTRACT FROM AUTHOR]

We are concerned with the existence of extremal solutions to a large class of first–order functional differential problems under weak regularity assumptions. Our technique involves multivalued analysis and the method of lower and upper solutions in order to obtain a new existence result to a scalar Cauchy problem. As a consequence of this result and a monotone iterative method for discontinuous operators, we derive our main existence result which is illustrated by several examples concerning well–known models: a generalized logistic equation or second–order problems in the presence of dry friction. [ABSTRACT FROM AUTHOR]

In this research study, a novel computational algorithm for solving a second-order singular functional differential equation as a generalization of the well-known Lane–Emden and differential-difference equations is presented by using the Bessel bases. This technique depends on transforming the problem into a system of algebraic equations and by solving this system the unknown Bessel coefficients are determined and the solution will be known. The method is tested on several test examples and proves to provide accurate results as compared to other existing methods from the literature. The simplicity and robustness of the proposed technique drive us to investigate more of their applications to several similar problems in the future.

Abstract This paper deals with some existence results for two classes of coupled systems of Hilfer and Hilfer–Hadamard random fractional differential equations. The main tool used to carry out our results is Itoh’s random fixed point theorem.

Abstract The main purpose of this paper is to investigate the convergence and stability of stochastic parabolic functional differential equations. Firstly, a comparison theorem in the context of Lyapunov-like function together with differential inequality is established. Secondly, various criteria for the convergence and stability are obtained on the basis of the comparison theorem and stochastic analysis techniques. Finally, two examples are provided to illustrate the significance of the theoretical results.

We consider a system of equations describing automatic position control by echo. The system can be reduced to a single differential equation with state-dependent delay. The delayed terms come from the control mechanism and the reaction time. H.-O. Walther [Differ. Integral Equ. 15(2002), No. 8, 923–944] proved that stable periodic motion is possible for large enough reaction time. We show that, for sufficiently small reaction lag, the control is perfect, i.e., the preferred position of the system is globally asymptotically stable.

A nonlinear perturbation of a linear autonomous retarded functional differential equation is considered. According to a Perron type theorem, with the possible exception of small solutions the Lyapunov exponents of the solutions of the perturbed equation coincide with the real parts of the characteristic roots of the linear part. In this paper, we study those solutions which are positive in the sense that they lie in a given order cone in the phase space. The main result shows that if the Lyapunov exponent of a positive solution of the perturbed equation is finite, then it is a characteristic root of the unperturbed equation with a positive eigenfunction. As a corollary, a necessary and sufficient condition for the existence of a positive solution of a linear autonomous delay differential equation is obtained.

In this article, we give the existence of a smooth stable manifold which is invariant under the semiflows of the delay differential equation x ′ = A x (t) + L x t + f (t , x t , λ) , with the assumption that the corresponding linear differential equation admits a nonuniform exponential dichotomy and the perturbation f(t, xt, λ) is small and smooth enough. We also show that the obtained manifold is Lipschitz in the parameter λ. [ABSTRACT FROM AUTHOR]

Abstract This article deals with some existence and Ulam-Hyers-Rassias stability results for a class of functional differential equations involving the Hilfer-Hadamard fractional derivative. An application is made of a Schauder fixed point theorem for the existence of solutions. Next we prove that our problem is generalized Ulam-Hyers-Rassias stable.

In this paper we construct an $O(2)$-equivarint Hopf bifurcation normal form for a model of a nonlinear optical system with delay and diffraction in the feedback loop whose dynamics is governed by a system of coupled quasilinear diffusion equation and linear Schrödinger equation. The coefficients of the normal form are expressed explicitly in terms of the parameters of the model. This makes it possible to constructively analyze the phase portrait of the normal form and, based on the analysis, study the stability properties of the bifurcating rotating and standing waves.

In this paper, we obtain the exact rates of decay to the non-hyperbolic equilibrium of the solution of a functional differential equation with maxima and unbounded delay. We study the convergence rates for both locally and globally stable solutions. We also give examples showing how the rate of growth of decay of solutions depends on the rate of growth of the unbounded delay as well as the nonlinearity local to the equilibrium.

We establish the existence of fast positive semi-wavefront solutions to a delay integro-differential problem cu'(t)=J*u(t)-u(t) + f(u(t-h)), t ∈ R, u(-∞)=0; where the asymmetric kernel J is exponentially bounded, the nonlinearity f ∈ C1([0, + &#8734);R) is monostable, h ≥ 0, and c > 0. [ABSTRACT FROM AUTHOR]

In this paper, a delayed eco-epidemiological model with diffusion effects and homogeneous Neumann boundary conditions is proposed. Sufficient conditions for the occurrence of the Hopf-zero, Takens–Bogdanov and saddle-node bifurcations at several steady states are derived. By taking the delay as the bifurcation parameter, it was shown that spatially homogeneous and nonhomogeneous Hopf bifurcations occur at several steady states for a sequence of critical values of the delay parameter. In addition, by applying the normal form theory and center manifold theorem for partial functional differential equations, we present the explicit formula for determining the properties of spatial Hopf bifurcations. Some numerical simulations are carried out. [ABSTRACT FROM AUTHOR]

Human cells have a natural cancerous defense system, and p53 protein is at the core. It is well known that p53 relies on oscillatory dynamic behavior to perform tumor suppressor functions. However, the mechanism of its oscillation remains unclear. To reveal the effect of time lag on the onset of p53 oscillations, we analytically and numerically investigate a possible scenario in which p53 oscillations are driven by a time lag. The model has only three nodes, two negative loops, and one delay. We focus on two types of time delays: discrete and distributed, use Hopf bifurcation theory to derive conditions for the occurrence of limit cycle oscillations, and confirm the properties of Hopf bifurcations by normal form and central manifold approaches. The analytical results are supported by numerical results. We suggest that discrete time lag promote p53 oscillations, while distributed time lag have a dual effect on p53 oscillations. These results imply that controlling the time lag in an appropriate range favors the inhibition of p53 on cancer cells. • An atypical p53 oscillator model is concerned. • Two types of delay are considered respectively. • The requirements of oscillation are revealed. [ABSTRACT FROM AUTHOR]

Abstract We study the asymptotic properties of the solutions of a class of even-order damped differential equations with p-Laplacian like operators, delayed and advanced arguments. We present new theorems that improve and complement related contributions reported in the literature. Several examples are provided to illustrate the practicability, maneuverability, and efficiency of the results obtained. An open problem is proposed.

In this paper we consider the following problems:

We obtain sufficient conditions for oscillation of solutions to a linear differential equation with distributed delay. We construct examples showing that constants in the conditions are unimprovable.

In this article, we first establish some results on composition of Stepanov-like weighted pseudo almost automorphic functions so called class r and class infinity under a uniform continuity condition with respect to L^p-norm. And then, we study the existence and uniqueness of weighted pseudo almost automorphic solutions to an abstract partial neutral functional differential equation with infinite delay with a Stepanov-like nonlinear term.

This paper deals with a dynamical systems approach for studying nonlinear autonomous differential equations with bounded state-dependent delay. Starting with the semiflow generated by solutions of such an equation, we revisit the principles of linearized stability and instability enabling the local stability analysis of equilibria via linearization. In particular, we prove both principles in an elementary way by using only a quantitative version of continuous dependence of the semiflow on initial data together with basic properties of the discrete semi-dynamical system induced by iterations of some time-$t$-map.

This paper is a continuation of the recent study by Bohner et al [9] on oscillation properties of nonlinear third order functional differential equation under the assumption that the second order differential equation is nonoscillatory. We consider both the delayed and advanced case of the studied equation. The presented results correct and extend earlier ones. Several illustrative examples are included.

In this work, we establish the existence and uniqueness of antiperiodic solution for a class of nth-order functional differential equations with infinite delay. The main tool in our study is the coincidence degree theory. An example is presented to illustrate the results obtained.

We show the use of parametrization techniques and successive approximations for the effective construction of solutions of linear boundary value problems for differential systems with multiple argument deviations. The approach is illustrated with a numerical example.

This paper is devoted to the study of fractional differential equations with Riemann-Liouville fractional derivatives and infinite delay in Banach spaces. The weighted delay is developed to deal with the case of non-zero initial value, which leads to the unboundedness of the solutions. Existence and uniqueness results are obtained based on the theory of measure of non-compactness, Schaude’s and Banach’s fixed point theorems. As auxiliary results, a fractional Gronwall type inequality is proved, and the comparison property of fractional integral is discussed.

We present such a restriction on parameters of linear functional differential equations of retarded type that is sufficient for the uniform asymptotic stability of an equation to be equivalent to its exponential stability.

In this work we study the existence of mild solutions of a functional differential equation with delay and random effects. We use a random fixed point theorem with stochastic domain to show the existence of mild random solutions.

In this paper, by using the fixed point method, we prove two new results on the Hyers-Ulam-Rassias and the Hyers-Ulam stability for the first order delay differential equation of the form y′(t) = F(t, y(t), y(t −τ )). Our results improve some related results in the literature.

The Banach fixed point theorem and a nonlinear alternative of Leray-Schauder type are used to investigate the existence and uniqueness of solutions for fractional order Hadamard-type functional and neutral functional differential equations.