Caratheodory Conditions Research Articles

Nonlinear Functional Differential Equation in Banach Space: a Study of Its Solution

Existence theorem for nonlinear functional differential equation in Banach space under the Lipschitz and Caratheodory conditions have been studied. The evidence of the existence theorem has been illustrated.

On the Compactness and Continuity of Uryson's Operator in Orlicz Spaces

Uryson's operators are very famous in the theory of fuzzy functional analysis. This paper is dedicated to studying and generalizing many results about the compactness and the continuity of Uryson's operator in two-variables defined with integral equation on G with the norm ‖u‖f= sup(ρ(v,f)≤) |∫u(x)v(x)dxG | ;v(x)∈L*g ,u(x)∈Lf*. Also, we study the convergence of Urysons' sequences Kn defined with the family of functions Kn (x, y; u) by using the convergence with respect to the defined measure and Caratheodory Condition.

Existence and Uniqueness Theorems for a Differential Equation with a Discontinuous Right-Hand Side

We consider new conditions for existence and uniqueness of a Caratheodory solution for an initial value problem with a discontinuous right-hand side. The method used here is based on:1) the representation of the solution as a Fourier series in a system of functions orthogonal in Sobolev sense and generated by a classical orthogonal system;2) the use of a specially constructed operator $A$ acting in $l_2$, the fixed point of which are the coefficients of the Fourier series of the solution.Under conditions given here the operator $A$ is contractive. This property can be employed to construct robust, fast and easy to implement spectral numerical methods of solving an initial value problem with discontinuous right-hand side.Relationship of new conditions with classical ones (Caratheodory conditions with Lipschitz condition) is also studied.Namely, we show that if in classical conditions we replace $L^1$ by $L^2$, then they become equivalent to the conditions given in this article.

Hybrid Initial Value Problems of Fractional Order with q‐Derivatives

In the present article, it has been tried to extend the theory of q‐fractional hybrid differential equations that entails Riemann–Liouville q‐differential operators. We have also attempted to prove an existence theorem for FHDEs under Lipschitz and Caratheodory conditions. We have also presented some fundamental fractional differential inequalities that can be employed to indicate that some external solutions exist for such problems. The required tools are taken into consideration and the principle for comparison is offered which can be used to conduct further research work regarding qualitative behavior of solutions.

A novel increment approach for optimal control problem of fractional‐order (1, 2] nonlinear systems

This paper deals with fractional optimal control governed by semilinear equations using the increment approach. We have considered controlled object as with the initial conditions: where α ∈ (1, 2], s(τ) is state variable in , B(τ) ∈ Lp, ϱ(0, T), and . Let the m‐dimensional control vector function be C(τ) and defined as . Assume the function η(τ, C(τ)) satisfies Caratheodory condition and defined on . We have obtained our results with the help of the adjoint equation and Pontryagin's maximum condition. For better understanding, we have included one example.

Existence results for a general class of sequential hybrid fractional differential equations

In this paper, we study a class of nonlinear boundary value problems (BVPs) consisting of a more general class of sequential hybrid fractional differential equations (SHFDEs) together with a class of nonlinear boundary conditions at both end points of the domain. The nonlinear functions involved depend explicitly on the fractional derivatives. We study the necessary conditions required for the unique solution to the suggested BVP under the Caratheodory conditions using the technique of measure of noncompactness and degree theory. We also develop conditions for uniqueness results and also on stability analysis.

Суперпозиционная измеримость многозначной функции при обобщенных условиях Каратеодори

For a multivalued mapping F:[a; b] × R^m → comp(R^m), the problem of superpositional measurability and superpositional selectivity is considered. As it is known, for superpositional measurability it is sufficient that the mapping F satisfies the Caratheodory conditions, for superpositional selectivity it is sufficient that F(•,x) has a measurable section and F(t; •) is upper semicontinuous. In this paper, we propose generalizations of these conditions based on the replacement, in the definitions of continuity and semicontinuity, of the limit of the sequence of coordinates of points in the images of multivalued mappings to a one-sided limit. It is shown that under such weakened conditions the multivalued mapping F possesses the required properties of superpositional measurability / superpositional selectivity. Illustrative examples are given, as well as examples of the significance of the proposed conditions. For single-valued mappings, the proposed conditions coincide with the generalized Caratheodory conditions proposed by I.V. Shragin (see [Bulletin of the Tambov University. Series: natural and technical sciences, 2014, 19:2, 476–478]).

Один метод исследования разрешимости краевых задач для неявного дифференциального уравнения

The article concernes a boundary value problem with linear boundary conditions of general form for the scalar differential equation f(t,x(t),x ̇(t))=y ̂(t), not resolved with respect to the derivative x ̇ of the required function. It is assumed that the function f satisfies the Caratheodory conditions, and the function 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,•): R→R and the Lipschitz set of the function f(t,•,x_2): R →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 y ̂ and the value of the boundary condition, are obtained.

Stability theory to a coupled system of nonlinear fractional hybrid differential equations

In the current manuscript, we investigate existence of solutions to a coupled system of fractional hybrid differential equations (FHDEs). With the help of mixed type Lipschitz and Caratheodory conditions, some conditions for the existence of solutions to the considered problem are established. Considering the tools of nonlinear analysis and hybrid fixed points theory, we establish our results. Further some new type results about stability including Ulam-Hyers (UH), generalized Ulam-Hyers (GUH) stability, Ulam-Hyers-Rassias (UHR) and generalized Ulam-Hyers-Rassias (GUHR) stability are developed. A test problem is given to demonstrate the establish results.

Solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type

In this paper, we consider the solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type. The expression of the solution for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is discussed based on the definition and the property of the Caputo differential operators. By the fixed point theorem in Banach algebra due to Dhage, an existence theorem for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is given under mixed Lipschitz and Caratheodory conditions. As an application, an example is presented to illustrate the main results. Our results in this paper extend and improve some well-known results. To some extent, our work fills the gap on some basic theory for the boundary value problems of fractional differential equations with mixed perturbations of the second type involving Caputo differential operator.

<i>G</i>-neutral stochastic differential equations with variable delay and non-Lipschitz coefficients

This paper has two parts. In part Ⅰ, existence and uniqueness theorem is established for solutions of neutral stochastic differential equations with variable delays driven by \begin{document}$ G $\end{document} -Brownian motion (VNSDDEGs in short) under global Caratheodory conditions. In part Ⅱ, a simplified VNSDDEGs for the original one is proposed. And the convergence both in \begin{document}$ L^p $\end{document} -sense and capacity between the solutions of the simplified and original VNSDDEGs are established in view of the approximation theorems. Two examples are conducted to justify the theoretical results of the approximation theorems.

Robustness of Hyperbolicity in Delay Equations

We establish the robustness of the notion of an exponential dichotomy for a nonautonomous linear delay equation. We consider the general cases of equations satisfying the Caratheodory conditions and of noninvertible evolution families defined by the linear equations. The stable and unstable spaces of the perturbed equations are obtained as images of linear operators that in their turn are fixed points of appropriate contraction maps.

О распространении теоремы Чаплыгина на дифференциальные уравнения нейтрального типа

We consider functional-differential equation x ̇((g(t) )= f(t; x(h(t) ) ),t ∈ [0; 1], where function f satisfies the Caratheodory conditions, but not necessarily guarantee the boundedness of the respective superposition operator from the space of the essentially bounded functions into the space of integrable functions. As a result, we cannot apply the standard analysis methods (in particular the fixed point theorems) to the integral equivalent of the respective Cauchy problem. Instead, to study the solvability of such integral equation we use the approach based not on the fixed point theorems but on the results received in [A.V. Arutyunov, E.S. Zhukovskiy, S.E. Zhukovskiy. Coincidence points principle for mappings in partially ordered spaces // Topology and its Applications, 2015, v. 179, № 1, 13–33] on the coincidence points of mappings in partially ordered spaces. As a result, we receive the conditions on the existence and estimates of the solutions of the Cauchy problem for the corresponding functional-differential equation similar to the well-known Chaplygin theorem. The main assumptions in the proof of this result are the non-decreasing function f(t; •) and the existence of two absolutely continuous functions v,w, that for almost each t ∈ [0; 1] satisfy the inequalities v ̇(g(t) )≥f(t; v(h(t) ) ),w ̇(g(t) )≤f(t;w(h(t) ) ). The main result is illustrated by an example.

Periodic and Nonnegative Periodic Solutions of Nonlinear Neutral Dynamic Equations on a Time Scale

Let T be a periodic time scale. We use Krasnoselskii--Burton's fixed point theorem to show new results on the existence of periodic and nonnegative periodic solutions of nonlinear neutral dynamic equation with variable delay of the form$x^{\Delta }(t)=-a(t)h(x^{\sigma }(t))+Q(t,x(t-\tau (t)))^{\Delta}+G(t,x(t),x(t-\tau (t))),\text{ }t\in \mathbb{T}.$We invert the given equation to obtain an equivalent integral equation from which we define a fixed point mapping written as a sum of a large contraction and a completely continuous map. The Caratheodory condition is used for the functions $Q$ and $G$. The results obtained here extend the work of Mesmouli, Ardjouni and Djoudi [16].

Some remarks on the absolute continuity of set-valued mappings

Recently, many authors have considered the existence, uniqueness and properties of solutions of set-valued differential and integral-differential equations, higher-order equations and investigated impulsive and control systems within the framework of the theory of set-valued equations. Obviously, obtaining all these results would be impossible without the development of the theory of set-valued analysis. In particular, when considering set-valued differential equations, when the right-hand side satisfies Caratheodory conditions, absolutely continuous set-valued mappings are considered as solutions. The article show that absolutely continuous set-valued mappings (under the existing concepts of the derivative and integral) do not satisfy those properties that are satisfied by single-valued absolutely continuous functions and therefore it is proposed to introduce additionally the concept of a integrally absolutely continuous set-valued mapping.

Positive solutions of a singular fractional boundary value problem with a fractional boundary condition

For \(\alpha\in(1,2]\), the singular fractional boundary value problem \[D^{\alpha}_{0^+}x+f\left(t,x,D^{\mu}_{0^+}x\right)=0,\quad 0\lt t\lt 1,\] satisfying the boundary conditions \(x(0)=D^{\beta}_{0^+}x(1)=0\), where \(\beta\in(0,\alpha-1]\), \(\mu\in(0,\alpha-1]\), and \(D^{\alpha}_{0^+}\), \(D^{\beta}_{0^+}\) and \(D^{\mu}_{0^+}\) are Riemann-Liouville derivatives of order \(\alpha\), \(\beta\) and \(\mu\) respectively, is considered. Here \(f\) satisfies a local Caratheodory condition, and \(f(t,x,y)\) may be singular at the value 0 in its space variable \(x\). Using regularization and sequential techniques and Krasnosel'skii's fixed point theorem, it is shown this boundary value problem has a positive solution. An example is given.

Existence of three solutions for equations of p ( x ) $p(x)$ -Laplace type operators with nonlinear Neumann boundary conditions

In this paper, we are concerned with nonlinear elliptic equations of the $p(x)$ -Laplace type operators $$\textstyle\begin{cases} -\operatorname{div}(a(x,\nabla u))+\vert u\vert ^{p(x)-2}u=\lambda f(x,u) &\mbox{in } \Omega, a(x,\nabla u)\frac{\partial u}{\partial n} = \lambda\theta g(x,u) &\mbox{on } \partial\Omega, \end{cases} $$ which are subject to nonlinear Neumann boundary conditions. Here the function $a(x,v)$ is of type $\vert v\vert ^{p(x)-2}v$ with a continuous function $p: \overline{\Omega} \to(1,\infty)$ and the functions $f, g$ satisfy a Caratheodory condition. The main purpose of this paper is to establish the existence of at least three weak solutions of the above problem by applying an abstract three critical points theorem which is inspired by the work of Ricceri (Nonlinear Anal. 74:7446-7454, 2011) Furthermore, we determine two intervals of λ’s precisely such that the first is where the given problem admits only the trivial solution, and the second is where the given problem has at least two nontrivial solutions as considering the positive principal eigenvalue for the $p(x)$ -Laplacian Neumann problems and an estimate of the Sobolev trace embedding’s constant.

Method of generalized integral guiding functions in the problem of the existence of periodic solutions for functional-differential inclusions

We suggest new methods for the solution of a periodic problem for a nonlinear object described by the differential inclusion x′(t) ∈ F(t, xt) under the assumption that the multimapping F has convex compact values and satisfies the upper Caratheodory conditions. We also study the case in which this multimapping is not convex-valued but is normal. The class of normal multimappings includes, for example, bounded almost lower semicontinuous multimappings with compact values and mappings satisfying the Caratheodory conditions. In both cases, a generalized integral guiding function is used to study the problem.

Optimal Controls for Fractional Stochastic Functional Differential Equations of Order $$\alpha \in (1, 2]$$ α ∈ ( 1 , 2

In this paper, we study optimal control problems for a class of fractional stochastic functional differential equations of order \(\alpha \in (1, 2]\) in Hilbert spaces. Firstly, a more appropriate concept for mild solutions is introduced. Secondly, existence of mild solutions are investigated by using the fractional calculus, stochastic analysis theory, and fixed point theorems with the strongly continuous \(\alpha \)-order cosine family. Then, the existence conditions of optimal pairs of systems governed by a fractional stochastic partial differential equations of order \(\alpha \in (1, 2]\) with infinite delay are presented. The results are obtained under the mixed Lipschitz and Caratheodory conditions. Finally, an example is presented to illustrate the main results.

Multiplicity results for nonlinear Neumann boundary value problems involving p-Laplace type operators

We consider the existence of at least two or three distinct weak solutions for the nonlinear elliptic equations $$ \textstyle\begin{cases} {-}\operatorname{div}(\varphi(x,\nabla u))+{|u|}^{p-2}u= \lambda f(x,u) &\mbox{in } \Omega, \varphi(x,\nabla u) \frac{\partial u}{\partial n}= \lambda g(x,u) & \mbox{on }\partial\Omega. \end{cases} $$ Here the function $\varphi(x,v)$ is of type $|v|^{p-2}v$ and the functions f, g satisfy a Caratheodory condition. To do this, we give some critical point theorems for continuously differentiable functions with the Cerami condition which are extensions of the recent results in Bonanno (Adv. Nonlinear Anal. 1:205-220, 2012) and Bonanno and Marano (Appl. Anal. 89:1-10, 2010) by applying Zhong’s Ekeland variational principle.