Some properties of solutions of α-conformable differential equations with piecewise constant arguments: existence and uniqueness, asymptotic stability, oscillation and periodicity

So far, although there have been several articles on partial differential equations with piecewise constant arguments, as far as we know, there is no article on neither a heat equation with piecewise constant mixed arguments that includes three extra diffusion terms, delayed arguments $$[t-1], [t]$$ and an advanced argument $$[t+1],$$ or exploring qualitative properties of the equation. With the motivation to investigate elaborate and well-established qualitative properties of such an equation, in this paper, we deal with a problem involving a heat equation with piecewise constant mixed arguments and initial, boundary conditions. By using the separation of variables method, we obtain the formal solution of this problem. Because of the piecewise constant arguments, we get a differential equation and then a difference equation. With the help of qualitative properties of the solutions of the differential equation and with the behavior of the solutions of the difference equation, we investigate the existence of solutions and qualitative properties of the solutions of the problem such as the convergence of the solutions to zero, the unboundedness of the solutions and oscillations of them. In addition, two examples are given to illustrate the application of the results in particular cases.

I. Third Order Linear Homogeneous Differential Equations in Normal Form.- 1. Fundamental Properties of Solutions of the Third Order Linear Homogeneous Differential Equation.- 1. The Normal Form of a Third Order Linear Homogeneous Differential Equation.- 2. Adjoint and Self-adjoint Third Order Linear Differential Equations.- 3. Fundamental Properties of Solutions.- 4. Relationship between Solutions of the Differential Equations (a) and (b).- 5. Integral Identities.- 6. Notion of a Band of Solutions of the First, Second and Third Kinds.- 7. Further Properties of Solutions of the Differential Equation (a) Implied by Properties of Bands.- 8. Weakening of Property (v) for the Laguerre Invariant.- 2. Oscillatory Properties of Solutions of the Differential Equation (a).- 1. Basic Definitions.- 2. Sufficient Conditions for the Differential Equation (a) to Be Disconjugate.- 3. Sufficient Conditions for Oscillatoricity of Solutions of the Differencial Equation (a).- 4. Further Conditions Concerning Oscillatoricity or Non-oscillatoricity of Solutions of the Differential Equation (a).- 5. Relation between Solutions without Zeros and Oscillatoricity of the Differential Equation (a).- 6. Sufficient Conditions for Oscillatoricity of Solutions of the Differential Equation (a) in the Case A(x) ? 0, x ? (a, ?).- 7. Conjugate Points, Principal Solutions and the Relationship between the Adjoint Differential Equations (a) and (b).- 8. Criteria for Oscillatoricity of the Differential Equations (a) and (b) Implied by Properties of Conjugate Points.- 9. Further Criteria for Oscillatoricity of the Differential Equation (b).- 10. The Number of Oscillatory Solutions in a Fundamental System of Solutions of the Differential Equation (a).- 11. Criteria for Oscillatoricity of Solutions of the Differential Equation (a) in the Case that the Laguerre Invariant Does Not Satisfy Condition (v).- 12. The Case, When the Laguerre Invariant Is an Oscillatory Function of x.- 13. The Differential Equation (a) Having All Solutions Oscillatory in a Given Interval.- 3. Asymptotic Properties of Solutions of the Differential Equations (a) and (b).- 1. Asymptotic Properties of Solutions without Zeros of the Differential Equations (a) and (b).- 2. Asymptotic Properties of Oscillatory Solutions of the Differential Equation (b).- 3. Asymptotic Properties of All Solutions of the Differential Equation (a).- 4. Boundary Value Problems.- 1. The Green Function and Its Applications.- 2. Further Applications of Integral Equations to the Solution of Boundary-value Problems.- 3. Generalized Sturm Theory for Third Order Boundary-value Problems.- 4. Special Boundary-value Problems.- II. Third Order Linear Homogeneous Differential Equations with Continuous Coefficients.- 5. Principal Properties of Solutions of Linear Homogeneous Third Order Differential Equations with Continuous Coefficients.- 1. Principal Properties of Solutions of the Differential Equation (A).- 2. Bands of Solutions of the Differential Equation (A).- 3. Application of Bands to Solving a Three-point Boundary-value Problem.- 6. Conditions for Disconjugateness, Non-oscillatoricity and Oscillatoricity of Solutions of the Differential Equation (A).- 1. Conditions for Disconjugateness of Solutions of the Differential Equation (A).- 2. Solutions without Zeros and Their Relation to Oscillatoricity of Solutions of the Differential Equation (A).- 3. Conditions for the Existence of Oscillatory Solutions of the Differential Equation (A).- 4. On Uniqueness of Solutions without Zeros of the Differential Equation (A).- 5. Some Properties of Solutions of the Differential Equation (A) with r(x) ? 0.- 7. Comparison Theorems for Differential Equations of Type (A) and Their Applications.- 1. Comparison Theorems.- 2. A Simple Application of Comparison Theorems.- 3. Remark on Asymptotic Properties of Solutions of the Differential Equation (A).- III. Concluding Remarks.- 1. Special Forms of Third Order Differential Equations.- 2. Remark on Mutual Transformation of Solutions of Third Order Differential Equations.- IV. Applications of Third Order Linear Differential Equation Theory.- 8. Some Applications of Linear Third Order Differential Equation Theory to Non-linear Third Order Problems.- 1. Application of Quasi-linearization to Certain Problems Involving Ordinary Third Order Differential Equations.- 2. Three-point Boundary-value Problems for Third Order Non-linear Ordinary Differential Equations.- 3. On Properties of Solutions of a Certain Non-linear Third Order Differential Equation.- 9. Physical and Engineering Applications of Third Order Differential Equations.- 1. On Deflection of a Curved Beam.- 2. Three-layer Beam.- 3. Survey of Some Other Applications of Third Order Differential Equations.- References.

In this paper, we present some existence theorems for pseudo-almost periodic solutions of differential equations with piecewise constant argument by means of pseudo-almost periodic solutions of relevant difference equations.

It is well known that differential equations with piecewise constant arguments is a class of functional differential equations, which has fascinated many scholars in recent years. These delay differential equations have been successfully applied to diverse models in real life, especially in biology, physics, economics, etc. In this work, we are interested in the existence and uniqueness of asymptotically almost periodic solution for certain differential equation with piecewise constant arguments. Due to the particularity of the equations, we cannot use the traditional method to convert it into the difference equation with exponential dichotomy. Through constructing Cauchy matrix of the investigated system to find the corresponding Green matrix of the difference equation, we need the concept of exponential dichotomy and the Banach contraction fixed point theorem of the corresponding system. Then we give some sufficient conditions to obtain the existence and uniqueness of asymptotically almost periodic solutions for these systems.

In this paper, we give some sufficient conditions for the existence and uniqueness of S-asymptotically ω-periodic (mild) solutions for a differential equation with piecewise constant argument, when ω is an integer. An example is also given in order to illustrate the result.

Numerical approximation of the solutions of delay differential equations on an infinite interval using piecewise constant arguments

We introduce a new class of functional differential equations with functional response on piecewise constant argument, FDEPCA. It contains functional differential equations with continuous time as well as differential equations with piecewise constant argument which were introduced in Akhmet (On the integral manifolds of the differential equations with piecewise constant argument of generalized type. In: Proceedings of the conference on differential and difference equations at the Florida institute of technology, August 1–5, 2005, Melbourne, Florida, ed. by R.P. Agarval, K. Perera (Hindawi Publishing Corporation, London, 2006), pp. 11–20; Nonlinear Anal 66:367–383, 2007; Discontinuity Nonlinearity Complexity 1:1–6,2012; J Math Anal Appl 336:646–663, 2007; Nonlinear Anal Hybrid Syst 2:456–467, 2008), and then developed in many papers and books. This chapter considers equations which are introduced and developed in our papers. We concentrate only on retarded equations, but one can easily extend the discussion to any type of piecewise constant argument and functional differential equations. Nonlinear and quasilinear systems are under consideration. At the end of the chapter, we suggest how one can apply the systems for solution of real-world problems, provided more general systems for future investigations.

Comparison principle and stability of differential equations with piecewise constant arguments

On Ryabov’s Asymptotic Characteization of the Solutions of Quasi-Linear Differential Equations with Small Delays

By using Mawhin's continuation theorem, the existence of periodic solutions for a neutral differential equation with piecewise constant argument is studied.

We establish existence and uniqueness criteria for the periodic solutions of a first order differential equation with piecewise constant arguments.

Differential inequalities for delay differential equations with piecewise constant argument

In this paper, we study the existence of almost periodic solutions of neutral differential difference equations with piecewise constant arguments via difference equation methods.

By using the Gronwall Bellman inequality we prove some limit relations between the solutions of delay differential equations with continuous arguments and the solutions of some related delay differential equations with piecewise constant arguments(EPCA). EPCA are strongly related to some discrete difference equations arising in numerical analysis, therefore the results can be used to compute numerical solutions of delay differential equations. We also consider the delay differential equations of neutral type by applying a generalization of the Gronwall Bellman inequality.

We summarize the conditions discovered for the existence of new ergodic type solutions (asymptotically almost periodic, pseudo almost periodic, …) of differential equations with piecewise constant arguments. Their existence is characterized by introducing a new tool, the ergodic sequences.