Carathéodory selections for multivalued mappings

We prove the existence of a Carathéodory selection for a set-valued mapping satisfying standard conditions. Also, an application to random fixed point for random operators is established.     

Controlled extensions of Carathéodory functions

In this paper we present two theorems on controlled extensions of Carathéodory functions. We obtain our results from corresponding ones for continuous functions by means of the Marczewski function.     

On bounded solutions of nonlinear first- and second-order equations with a Carathéodory function

In the paper we study the existence and uniqueness of bounded solutions for differential equations of the form: x^′−Ax=f(t,x), x^″−Ax=f(t,x), where A∈L(Rm), is a Carathéodory function and the homogeneous equations x^′−Ax=0, x^″−Ax=0 have nontrivial solutions bounded on R. Using a perturbation of the equations, the Leray-Schauder Topological Degree and Fixed Point Theory, we overcome the difficulty that the linear problems are non-Fredholm in any reasonable Banach space.     

On existence and asymptotic behaviour of solutions of a fractional integral equation

We study the solvability of a quadratic integral equation of fractional order with linear modification of the argument. This equation is considered in the Banach space of real functions, defined, bounded and continuous on an unbounded interval. Moreover, we will obtain some asymptotic characterization of solutions.     

On existence and asymptotic behaviour of solutions of a functional integral equation

The paper contains a result on the existence and asymptotic behaviour of solutions of a functional integral equation. That result is proved under rather general hypotheses. The main tools used in our considerations are the concept of a measure of noncompactness and the classical Schauder fixed point principle. The investigations of the paper are placed in the space of continuous and tempered functions on the real half-line. We prove an existence result which generalizes several ones concerning functional integral equations and obtained earlier by other authors. The applicability of our result is illustrated by some examples.     

Estimates of the Carathéodory metric on the symmetrized polydisc

Estimates for the Carathéodory metric on the symmetrized polydisc are obtained. It is also shown that the Carathéodory and Kobayashi distances of the symmetrized three-disc do not coincide.     

Existence of positive solutions of fractional‐order elastic beam equation with a non‐Carathéodory nonlinearity

In this article, the existence of positive solutions of a boundary value problem for nonlinear singular fractional‐order elastic beam equation is established. Here, f depends on t,x, and x′; f may be singular at t = 0 and t = 1; and f is a non‐Carathéodory function. The results obtained are based upon fixed‐point theorems in a cone in Banach space. An example is included to illustrate the main results. Copyright © 2015 John Wiley & Sons, Ltd.     

A note on integrable solutions of Hammerstein integral equations

We derive a set of sufficient conditions for the existence of solutions of a Hammerstein integral equation.     

含积分边界条件的分数阶微分方程边值问题的正解的存在性

研究了含积分边界条件的分数阶微分方程的边值问题,首先给出格林函数及性质,其次将问题转化为一个等价的积分方程,最后应用Krasnoselkii及Leggett-Williams不动点定理得到了一个及多个正解的存在性,推广了以往的结果．     

On the existence and multiplicity of positive periodic solutions for first-order vector differential equation

In this paper, we consider the existence and multiplicity of positive periodic solutions for first-order vector differential equation x^′(t)+f(t,x(t))=0, a.e. t∈[0,ω] under the periodic boundary value condition x(0)=x(ω). Here ω is a positive constant, and is a Carathéodory function. Some existence and multiplicity results of positive periodic solutions are derived by using a fixed point theorem in cones.     

Global asymptotic stability of solutions of a functional integral equation

Using the technique of measures of noncompactness we prove a theorem on the existence and global asymptotic stability of solutions of a functional integral equation. The investigations are placed in the Banach space of real functions defined, continuous and bounded on an unbounded interval. A few realizations of the result obtained are indicated.     

A class of evolution equations: Existence of solutions with functional boundary conditions

We consider the evolution equation whose right-hand side is the sum of a linear unbounded operator generating a compact strongly continuous semigroup and a continuous operator acting in function spaces. We prove the existence of a solution that stays within a given closed convex set and moreover, satisfies a functional boundary condition, particular cases of which are the Cauchy initial condition, periodicity condition, mixed condition including continuous transformations of spatial variables, etc. The main result is illustrated by using an example of the boundary-value problem for a partial operator-differential equation. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 48–60, January, 1999.     

Existence and global attractivity of solutions of a nonlinear functional integral equation

In this paper we prove a result on the existence and global attractivity of solutions of a nonlinear functional integral equations with deviating arguments. The investigations are placed in the Banach space of real functions defined, continuous and bounded on the real half-axis. The main tool used in consideration is a fixed point theorem of Krasnosel’skii type. A few examples illustrating the obtained results are also included.     

一类非线性抛物方程Cauchy问题解的存在性条件

研究了带源项的非线性抛物方程Cauchy问题解存在的必要条件以及解所应具有的性质,通过把文献中的线性算子推广到形式较一般的带源项的非线性抛物算子,利用其中处理线性问题的方法来处理非线性问题．     

On a mixed problem with integral conditions for one-dimensional parabolic equation

We consider an initial-boundary value problem for a one-dimensional parabolic equation with nonlocal boundary conditions. These nonlocal conditions are given in terms of integrals. Based on solution of the Dirichlet problem for the parabolic equation, we constructively establish the well-posedness for the nonlocal problem.     

Periodic solutions to the Cahn–Hilliard equation with constraint

This paper is concerned with the multidimensional Cahn–Hilliard equation with a constraint. The existence of periodic solutions of the problem is mainly proved under consideration by the viscosity approach. More precisely, with the help of the subdifferential operator theory and Schauder fixed point theorem, the existence of solutions to the approximation of the original problem is shown, and then the solution is obtained by using a passage‐to‐limit procedure based on a prior estimate. Copyright © 2015 John Wiley & Sons, Ltd.     

Monotonic solutions of a perturbed quadratic fractional integral equation

We present an existence theorem for monotonic solutions of a perturbed quadratic fractional integral equation in C[0,1]. The concept of a measure of noncompactness and a fixed point theorem due to Darbo are the main tools in carrying out our proof. Finally, we give an example for indicating the natural realizations of our abstract result presented in the paper.     

The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions

In this paper, first, we consider the existence of a positive solution for the nonlinear fractional differential equation boundary value problem where 0≤λ < 1,^CD^α is the Caputo's differential operator of order α, and f:[0,1] × [0,∞)→[0,∞) is a continuous function. Using some cone theoretic techniques, we deduce a general existence theorem for this problem. Then, we consider two following more general problems for arbitrary α, 1≤n < α≤n + 1: Problem 1: where , 0≤λ < k + 1; Problem 2: where 0≤λ≤α and D^α is the Riemann–Liouville fractional derivative of order α. For these problems, we give existence results, which improve recent results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Solvability of a functional integral equation of fractional order in the class of functions having limits at infinity

A functional integral equation of fractional order is investigated in the space of real functions which are defined, continuous and bounded on the real half-axis. Using the technique of measures of noncompactness and the fixed point theorem of Darbo type we prove that the equation in question has solutions in the mentioned function space. Moreover, a suitable choice of a measure of noncompactness enables us to prove that those solutions tend to limits at infinity. An example illustrating our result is also given.