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We consider a perforated domain Ω(ε) of R² with a small hole of size ε and we study the behavior of the solution of a mixed Neumann-Robin problem in Ω(ε) as the size ε of the small hole tends to 0. In addition to the geometric degeneracy of the problem, the nonlinear ε-dependent Robin condition may degenerate into a Neumann condition for ε = 0 and the Robin datum may diverge to infinity. Our goal is to analyze the asymptotic behavior of the solutions to the problem as ε tends to 0 and to understand how the boundary condition affects the behavior of the solutions when ε is close to 0. The present paper extends to the planar case the results of [36] dealing with the case of dimension n ≥ 3. [ABSTRACT FROM AUTHOR]

We study the equivalences and the implications between linear (or homogeneous) nth order fractional differential equations (FDEs) and integral equations in the spaces L¹ (a, b) and C[a, b] when n ≥ 2. We establish the equivalence in C[a, b] of the IVP of the nth order FDE subject to the initial condition u(i) (a) = ui for i ∈ {0, 1, . . ., n − 1} when n ≥ 2. The difficulty is that the known conditions for such equivalence for the first order FDEs are not sufficient for equivalence in the nth order FDEs with n ≥ 2. In this article we provide additional conditions to ensure the equivalence for the nth order FDEs with n ≥ 2. In particular, we obtain conditions under which solutions of the integral equations are solutions of the linear nth order FDEs. These results are keys for further studying the existence of solutions and nonnegative solutions to initial and boundary value problems of nonlinear nth order FDEs. This is done via the corresponding integral equations by topological methods such as the Banach contraction principle, fixed point index theory, and degree theory. [ABSTRACT FROM AUTHOR]

We consider a reaction-diffusion equation with nonlocal anisotropic diffusion and a linear combination of local and nonlocal monostable-type reactions in a space of bounded functions on R^d. Using the properties of the corresponding semiflow, we prove the existence of monotone traveling waves along those directions where the diffusion kernel is exponentially integrable. Among other properties, we prove continuity, strict monotonicity and exponential integrability of the traveling wave profiles.

In this article, we considered two inverse source problems for fourth-order parabolic differential equation with fractional derivative in time. Determination of a space dependent source term from the data given at some time t=T is considered in one problem while other addresses the recovery of a time dependent source term from the integral type over-determination condition. Existence and uniqueness of the solution of both inverse source problems are proved. The stability results for the inverse problems are presented.

We consider a reaction-diffusion equation with nonlocal anisotropic diffusion and a linear combination of local and nonlocal monostable-type reactions in a space of bounded functions on ℝd. Using the properties of the corresponding semi ow, we prove the existence of monotone traveling waves along those directions where the diffusion kernel is exponentially integrable. Among other properties, we prove continuity, strict monotonicity and expo- nential integrability of the traveling wave profiles. [ABSTRACT FROM AUTHOR]

We prove the unique solvability of a boundary-value problems for a third-order loaded integro-differential equation with variable coefficients, by reducing the equation to a Volterra integral equation.

We establish the unique solvability for an inverse problem for semi-linear fractional telegraph equation $$ D^\alpha_t u+r(t)D^\beta_t u-\Delta u=F_0(x,t,u,D^\beta_t u), \quad (x,t) \in \Omega_0\times (0,T] $$ with regularized fractional derivatives $D^\alpha_t u, D^\beta_t u$ of orders $\alpha\in (1,2)$, $\beta\in (0,1)$ with respect to time on bounded cylindrical domain. This problem consists in the determination of a pair of functions: a classical solution $u$ of the first boundary-value problem for such equation, and an unknown continuous coefficient $r(t)$ under the over-determination condition $$ \int_{\Omega_0}u(x,t)\varphi(x)dx=F(t), \quad t\in [0,T] $$ with given functions $\varphi$ and $F$.

In this article we introduce a polyconvolution which related to the Hartley and Fourier cosine transforms. We prove some properties of this polyconvolution, and then solve a class of Toeplitz plus Hankel integral equations and systems of two Toeplitz plus Hankel integral equations.

In this article, we considered two inverse source problems for fourth-order parabolic differential equation with fractional derivative in time. Determination of a space dependent source term from the data given at some time t = T is considered in one problem while other addresses the recovery of a time dependent source term from the integral type over-determination condition. Existence and uniqueness of the solution of both inverse source problems are proved. The stability results for the inverse problems are presented. [ABSTRACT FROM AUTHOR]

In this note we give a generalization of a mean value problem which can be viewed as a problem related to Volterra operators. This problem can be seen as a generalization of a result concerning the zeroes of a Volterra operator in the Banach space of continuous functions with null integral on a compact interval.

In this article, we give sufficient conditions for the existence of solutions for an integral equation of fractional order with multiple time delays in Banach spaces. Our main tool is a fixed point theorem of Monch type associated with measures of noncompactness. Our results are illustrated by an example.

In this article, we prove the existence of positive nondecreasing solutions for a multi-term fractional-order functional differential equations. We consider Cauchy boundary problems with: nonlocal conditions, two-point boundary conditions, integral conditions, and deviated arguments.

Consider the second-order nonlinear dynamic equation $$ [r(t)x^Delta(ho(t))]^Delta+p(t)f(x(t))=0, $$ where $p(t)$ is the backward jump operator. We obtain a Kwong-Wong-type integral equation, that is: If $x(t)$ is a nonoscillatory solution of the above equation on $[T_0,infty)$, then the integral equation $$ frac{r^sigma(t)x^Delta(t)}{f(x^sigma(t))} =P^sigma(t)+int^infty_{sigma(t)}frac{r^sigma(s) [int^1_0f'(x_h(s))dh][x^Delta(s)]^2}{f(x(s)) f(x^sigma(s))}Delta s $$ is satisfied for $tgeq T_0$, where $P^sigma(t)=int^infty_{sigma(t)}p(s)Delta s$, and $x_h(s)=x(s)+hmu(s)x^Delta(s)$. As an application, we show that the superlinear dynamic equation $$ [r(t)x^{Delta}(ho(t))]^Delta+p(t)f(x(t))=0, $$ is oscillatory, under certain conditions.

We use a variation of Krasnoselskii fixed point theorem introduced by Burton to show that the nonlinear neutral differential equation $$ x'(t)=-a(t)x^3(t)+c(t)x'(t-g(t))+G(t,x^3(t-g(t)) $$ has a periodic solution. Since this equation is nonlinear, the variation of parameters can not be applied directly; we add and subtract a linear term to transform the differential into an equivalent integral equation suitable for applying a fixed point theorem. Our result is illustrated with an example.

In this article, we establish some fixed point results of Krasnoselskii type for the sum T+S, where S is weakly continuous and T may not be continuous. Some of the main results complement and encompass the previous ones. As an application, we study the existence of solution to one parameter operator equations. Finally, our results are used to prove the existence of solution for integral equations in reflexive Banach spaces.

In this article we derive existence and comparison results for discontinuous non-absolute functional integral equations of Volterra type in an ordered Banach space which has a regular order cone. The obtained results are then applied to first-order impulsive differential equations.

We give an existence theorem for some functional-integral equations which includes many key integral and functional equations that arise in nonlinear analysis and its applications. In particular, we extend the class of characteristic functions appearing in Chandrasekhar's classical integral equation from astrophysics and retain existence of its solutions. Extensive use is made of measures of noncompactness and abstract fixed point theorems such as Darbo's theorem.

We study the existence and uniqueness of periodic solutions and the stability of the zero solution of the nonlinear neutral differential equation $$ frac{d}{dt}x(t) = -a(t)x(t)+ frac{d}{dt}Q(t, x(t-g(t))) +G(t,x(t), x(t-g(t))). $$ In the process we use integrating factors and convert the given neutral differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii's fixed point theorem to show the existence of a periodic solution of this neutral differential equation. We also use the contraction mapping principle to show the existence of a unique periodic solution and the asymptotic stability of the zero solution provided that $Q(0,0)= G(t, 0,0) = 0$.

We use Krasnoselskii's fixed point theorem to show that the nonlinear neutral differential equation with functional delay $$ x'(t) = -a(t)x(t)+ c(t)x'(t-g(t))+ q(t, x(t), x(t-g(t)) $$ has a periodic solution. Also, by transforming the problem to an integral equation we are able, using the contraction mapping principle, to show that the periodic solution is unique.

We establish the unique solvability for an inverse problem for semi- linear fractional telegraph equation Dtα u + r(t) Dtβ u - Δu = Fo(x,t,u, Dtβ u), (x,t) ∈ Ω0 x (0,T] with regularized fractional derivatives Dtαu, Dtβu of orders α ∈ (1,2), β ∈ (0,1) with respect to time on bounded cylindrical domain. This problem consists in the determination of a pair of functions: a classical solution u of the first boundary-value problem for such equation, and an unknown continuous coefficient r(t) under the over-determination condition ... with given functions φ and F. [ABSTRACT FROM AUTHOR]

In this note we give a generalization of a mean value problem which can be viewed as a problem related to Volterra operators. This problem can be seen as a generalization of a result concerning the zeroes of a Volterra operator in the Banach space of continuous functions with null integral on a compact interval. [ABSTRACT FROM AUTHOR]

In this article, we give sufficient conditions for the existence of solutions for an integral equation of fractional order with multiple time delays in Banach spaces. Our main tool is a fixed point theorem of Mönch type associated with measures of noncompactness. Our results are illustrated by an example. [ABSTRACT FROM AUTHOR]

Consider the second-order nonlinear dynamic equation [r(t)xΔ(ρ(t))]Δ + p(t)f(x(t)) = 0, where p(t) is the backward jump operator. We obtain a Kwong-Wong-type integral equation, that is: If x(t) is a nonoscillatory solution of the above equation on [T0,∞), then the integral equation rσ(t)xΔ(t)/f(xσ(t)) = Ρσ(t) + ∫∞ σ(t) rσ(s)[∫¹⁰ f′(xh(s))dh][xΔ(s)]²/f(x(s))f(xσ(s)) Δs is satisfied for t ≥ T⁰, where Ρσ(t) = ∫∞σ(t) p(s)Δs, and xh(s) = x(s) + hµ(s)xΔ(s). As an application, we show that the superlinear dynamic equation [r(t)xΔ(ρ(t))]Δ + p(t)f(x(t)) = 0, is oscillatory, under certain conditions. [ABSTRACT FROM AUTHOR]

In this article, we prove the existence of positive nondecreasing solutions for a multi-term fractional-order functional differential equations. We consider Cauchy boundary problems with: nonlocal conditions, two-point boundary conditions, integral conditions, and deviated arguments. [ABSTRACT FROM AUTHOR]

We use a variation of Krasnoselskii fixed point theorem introduced by Burton to show that the nonlinear neutral differential equation x'(t) = -a(t)x³(t) + c(t)x'(t - g(t)) + G(t, x³(t - g(t)) has a periodic solution. Since this equation is nonlinear, the variation of parameters can not be applied directly; we add and subtract a linear term to transform the differential into an equivalent integral equation suitable for applying a fixed point theorem. Our result is illustrated with an example. [ABSTRACT FROM AUTHOR]

We study the existence of positive eigenvalues yielding nonnegative solutions to an integral equation. Also we study the positivity of solutions on specific sets. These results are obtained by using a fixed point theorem in cones and are illustrated by application to systems of boundary value problems. [ABSTRACT FROM AUTHOR]

In this article we derive existence and comparison results for discontinuous non-absolute functional integral equations of Volterra type in an ordered Banach space which has a regular order cone. The obtained results are then applied to first-order impulsive differential equations. [ABSTRACT FROM AUTHOR]

We give an existence theorem for some functional-integral equations which includes many key integral and functional equations that arise in nonlinear analysis and its applications. In particular, we extend the class of characteristic functions appearing in Chandrasekhar's classical integral equation from astrophysics and retain existence of its solutions. Extensive use is made of measures of noncompactness and abstract fixed point theorems such as Darbo's theorem. [ABSTRACT FROM AUTHOR]

In this paper, we present a modular version of Krasnoselskii's fixed point theorem. Then this result is applied to the existence of solutions to perturbed integral equations in modular function spaces. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of periodic solutions and the stability of the zero solution of the nonlinear neutral differential equation d/dt x(t) = -a(t)x(t) + d/dt Q(t, x(t - g(t))) + G(t, x(t), x(t - g(t))). In the process we use integrating factors and convert the given neutral differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii's fixed point theorem to show the existence of a periodic solution of this neutral differential equation. We also use the contraction mapping principle to show the existence of a unique periodic solution and the asymptotic stability of the zero solution provided that Q(0, 0) = G(t, 0, 0) = 0. [ABSTRACT FROM AUTHOR]

We use Krasnoselskii's fixed point theorem to show that the non-linear neutral differential equation with functional delay x'(t) = -a(t)x(t) + c(t)x'(t - g(t)) + q(t,x(t), x(t - g(t)) has a periodic solution. Also, by transforming the problem to an integral equation we are able, using the contraction mapping principle, to show that the periodic solution is unique. [ABSTRACT FROM AUTHOR]