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This paper is concerned with the existence of solutions for a nonlinear multi-term impulsive fractional $ q $-integro-difference equation with nonlocal boundary conditions. The appropriated fixed point theorems are applied to accomplish the existence and uniqueness results for the given problem. We demonstrate the application of the obtained results with the aid of examples.

In this paper, we introduce and investigate a new class of nonlinear multi-term impulsive anti-periodic boundary value problems involving Caputo type fractional q -derivative operators of different orders and the Riemann–Liouville fractional q -integral operator. The uniqueness of solutions to the given problem is proved with the aid of Banach's fixed point theorem. Applying a Shaefer-like fixed point theorem, we also obtain an existence result for the problem at hand. Examples are constructed for illustrating the obtained results. The paper concludes with certain interesting observations concerning the reduction of the results proven in the paper to some new results under an appropriate choice of the parameters involved in the governing equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of solutions for a nonlinear multi-term impulsive fractional q -integro-difference equation with nonlocal boundary conditions. The appropriated fixed point theorems are applied to accomplish the existence and uniqueness results for the given problem. We demonstrate the application of the obtained results with the aid of examples. [ABSTRACT FROM AUTHOR]

In this paper we study a nonlocal boundary value problem for Gellerstedt equation with singular coefficient in an unbounded domain. The uniqueness of the solution to the problem is proved using the extremum principle. The existence of a solution to the problem is proved by the method of integral equations. The problem is equivalently reduced to solving a singular integral equation. We find its solution using the Carleman–Vekua method. Then, we get the Wiener–Hopf singular integral equation, and with the help of the Fourier transform it can be reduced to the Riemann boundary value problem. [ABSTRACT FROM AUTHOR]

In the paper, we consider a boundary value problem for a third-order mixed differential equation of parabolic-hyperbolic type with a fractional Gerasimov–Caputo operator. Under certain conditions on the data of a problem, applying the methods of the theory of integral equations and the Green function, we prove the unique solvability of the problem. The uniqueness of the solution is proved by the method of the extremum principle, and the existence of this problem is proved by reducing it to a boundary value problem for a fractional order differential equation, as well as to the Volterra integral equation of the second kind. [ABSTRACT FROM AUTHOR]

In the hyperbolic space, we consider the Cauchy problem for the aggregation equation with variable exponents. Non-negative initial function is bounded and summable. We prove the existence of a weak solution on a small time interval. In the case where the kernel of the integral operator is smooth and rapidly decreases at infinity, the existence of a bounded solution on an arbitrary time interval is proved. [ABSTRACT FROM AUTHOR]

In this paper we study a problem with local displacement conditions on a segment of the degeneration line and with displacements on pieces of boundary characteristics in an unbounded domain, the elliptical part of which is the upper half-plane, and the hyperbolic part is the characteristic triangle. The uniqueness of the solution of the problem is proved using the extremum principle. The proof of the existence is based on the theory of singular integral equations, Wiener–Hopf equations, and Fredholm integral equations. [ABSTRACT FROM AUTHOR]

In this article, a nonlocal boundary value problem for a fourth order partial differential equation is solved. The method of separation of variables is used. The solution is constructing in the form of Fourier series. Theorems on the existence and uniqueness of the solution to the problem are proved. [ABSTRACT FROM AUTHOR]

For the Gellerstedt equation with a singular coefficient in some mixed domain, when the ellipticity boundary coincides with the segment of the Oy axis and the normal curve of the equation, a problem with the Bitsadze–Samarskii conditions on the elliptic boundary and on the degeneration line is studied. The well-posedness of the formulated problem is proved. [ABSTRACT FROM AUTHOR]

In this article, a boundary value problem for a loaded partial differential equations of fourth order was considered on a rectangle domain. The solution of the problem is obtained in the form of Fourier series. Also, theorems about uniqueness and existence of the solution of the problem under consideration were proved by spectral method. [ABSTRACT FROM AUTHOR]

We consider the equation , where and and are metric spaces. This operator equation is compared with the "model" equation , where the function is continuous, nondecreasing in the first argument, and nonincreasing in the second argument. Conditions are obtained under which the existence of solutions of this operator equation follows from the solvability of the "model" equation. Conditions for the stability of the solutions under small variations in the mapping are established. The statements proved in the present paper extend the Kantorovich fixed-point theorem for differentiable mappings of Banach spaces, as well as its generalizations to coincidence points of mappings of metric spaces. [ABSTRACT FROM AUTHOR]

We consider a second-order differential equation with rapidly growing intermediate coefficients. We obtain a solvability result in the cases that the diffusion coefficient of equation is unbounded or it tends to zero at the infinity. Under additional conditions, we prove the Lp -- maximal regularity estimate for the solution of this equation. [ABSTRACT FROM AUTHOR]

We consider the -linear problem (also known as the Markushevich problem and the generalized Riemann boundary value problem) and the convolution integral equation of the second kind on a finite interval (also known as the truncated Wiener–Hopf equation). We find new conditions for correct solvability of the -linear problem and the truncated Wiener–Hopf equation. [ABSTRACT FROM AUTHOR]

We review earlier and recent results on the Markov moment problem and related polynomial approximation on unbounded subsets. Such results allow proving the existence and uniqueness of the solutions for some Markov moment problems. This is the first aim of the paper. Our solutions have a codomain space a commutative algebra of (linear) symmetric operators acting from the entire real or complex Hilbert space H to H; this algebra of operators is also an order complete Banach lattice. In particular, Hahn–Banach type theorems for the extension of linear operators having a codomain such a space can be applied. The truncated moment problem is briefly discussed by means of reference citations. This is the second purpose of the paper. In the end, a general extension theorem for linear operators with two constraints is recalled and applied to concrete spaces. Here polynomial approximation plays no role. This is the third aim of this work.

A class of integral equations with respect to one variable function as well as to multivariable function that are solvable for any right hand side of an equation has been singled out. A necessary and sufficient condition for existence of a solution has been obtained for the class of integral equations and the general form of their exact solutions has been found. Necessary and sufficient conditions for existence of solutions to the mentioned equations with a given right hand side are obtained by reducing them to solving an extremal problem. An algorithm for solving the extremal problem by constructing a minimizing sequence has been developed and a convergence rate estimation has been obtained. A solvability criterion as a requirement on infimum of functional has been formulated. A necessary and sufficient condition for solvability of an integral equation with parameter has been obtained and its general solution has been found.

We propose a cell-centered nonlinear finite volume scheme for the nonequilibrium three-temperature equations, where both the Dirichlet and Neumann boundary conditions are considered, and prove that the discrete solutions of the scheme satisfy the discrete maximum principle. In the construction of the flux we use the nonlinear combination of two single fluxes and the interpolation technique for the auxiliary unknowns. Some new interpolation methods are introduced, especially when the Neumann boundary condition is considered. Based on the bounded estimation of the discrete solution, we prove that there exists at least a solution for our scheme by using the Brouwer's fixed point theorem. Numerical results show that our scheme has second-order accuracy and good conservation and can preserve the discrete maximum principle. [ABSTRACT FROM AUTHOR]

We study solvability of the inverse problems of recovering an unknown function on the nonlinear right-hand side of a first order operator-differential equation in some Banach space. The equation is furnished with the Cauchy data, and the overdetermination condition is the value of some operator at a solution. The existence and uniqueness theorems local in time are established. [ABSTRACT FROM AUTHOR]

We study the first boundary value problem for the elliptic-hyperbolic type equation with two perpendicular lines of change of type and spectral parameter. We prove the existence and uniqueness of the solution. In the proof of the uniqueness of solution we use the completeness of biorthogonal system in space $L_2$ . When building a solution as the sum of a series there is a problem of small denominators. We obtained estimates of the denominators of the separation from zero.

A rational model of a dynamical contact of a viscoelastic body with its support is presented. It is assumed that the contact is frictionless and permits a limited interpenetration which is prescribed. A weak formulation of the problem is given and the existence of its solutions is proved. It is also shown that if the depth of the interpenetration tends to zero, the solutions converge to a solution of the Signorini contact (without interpenetration). [ABSTRACT FROM AUTHOR]

In this article the inverse problem of determining unknown coefficients on the right-hand side of a system of parabolic equations of the reaction-diffusion type in the case of the third boundary condition is considered. The existence of a generalized solution of the considered problem is proven by the method of successive approximations. [ABSTRACT FROM AUTHOR]

In this paper, we study and investigate an interesting Caputo fractional derivative and Riemann−Liouville integral boundary value problem (BVP): c D 0 + q u ( t ) = f ( t , u ( t ) ) , t ∈ [ 0 , T ] , u ( k ) ( 0 ) = ξ k , u ( T ) = ∑ i = 1 m β i R L I 0 + p i u ( η i ) , where n − 1 < q < n , n ≥ 2 , m , n ∈ N , ξ k , β i ∈ R , k = 0 , 1 , … , n − 2 , i = 1 , 2 , … , m , and c D 0 + q is the Caputo fractional derivatives, f : [ 0 , T ] × C ( [ 0 , T ] , E ) → E , where E is the Banach space. The space E is chosen as an arbitrary Banach space; it can also be R (with the absolute value) or C ( [ 0 , T ] , R ) with the supremum-norm. RL I 0 + p i is the Riemann−Liouville fractional integral of order p i > 0 , η i ∈ ( 0 , T ) , and ∑ i = 1 m β i η i p i + n − 1 Γ ( n ) Γ ( n + p i ) ≠ T n − 1 . Via the fixed point theorems of Krasnoselskii and Darbo, the authors study the existence of solutions to this problem. An example is included to illustrate the applicability of their results.

We consider a class of infinite-horizon optimal control problems with not necessarily bounded set of control constraints. Sufficient conditions for the existence of an optimal control are derived in the general nonlinear case by means of finite-horizon approximations and the tools of the Pontryagin maximum principle. Conditions guaranteeing the uniform local boundedness of optimal controls are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a generalization of the Dirichlet problem for the Poisson equation in a rectangular domain. We assume that the kth-order normal derivatives of an unknown function are given on lower and upper bases of the rectangle and that homogeneous boundary conditions of the first kind are given on the lateral sides. Under these conditions, we prove the existence of a unique regular solution of this problem. [ABSTRACT FROM AUTHOR]

Under natural conditions, we prove an existence theorem for stochastic differential equations with current velocities (mean derivatives) and with nonautonomous right-hand side. [ABSTRACT FROM AUTHOR]

We study a nonlinear fluid-structure interaction problem between an incompressible, viscous fluid in 3D and an elastic structure whose Lamé elastic parameters, thickness and density are all functions of space allowing jump discontinuities. This problem is motivated by studying the interaction between blood flow and arterial walls treated with vascular prostheses called stents. A stent is a metallic mesh-like tube used to prop the clogged arteries open. The Navier-Stokes equations for an incompressible, viscous fluid are used to model blood flow, and the cylindrical Koiter shell equations with discontinuous coefficients are used to model the elastic properties of arterial walls treated with stents. The fluid and structure are coupled via two coupling conditions evaluated at the moving fluid-structure interface. No assumption on axial symmetry is used in the model. We prove the existence of a weak solution to the underlying nonlinear 3D moving-boundary problem, and design a loosely-coupled partitioned scheme ( $$\beta $$ -scheme) for its solution. The numerical scheme was motivated by the main steps in the constructive existence proof. The existence proof shows that the proposed numerical $$\beta $$ -scheme converges to a weak solution of the nonlinear problem. This is the first convergence result for the proposed partitioned $$\beta $$ -scheme. Several numerical examples are presented where different stent configurations are considered. The numerical fluid-structure interaction solutions clearly show that the presence of a stent induces wave reflections in arterial walls, and significant flow disturbances, especially near the proximal site of the stent. [ABSTRACT FROM AUTHOR]

The paper considers quasi-variational inequalities with point to set operator. The existence of a solution, in the case when the operator of the quasi-variational inequality is semi-continuous and the feasible set is convex and compact, is proved.

The parabolic functional differential equation $\frac{{\partial u}} {{\partial t}} = D\frac{{\partial ^2 u}} {{\partial x^2 }} - u + K(1 + \gamma \cos u(x + \theta ,t - T)) $ is considered on the circle [0, 2π]. Here, D > 0, T > 0, K > 0, and γ ∈ (0, 1). Such equations arise in the modeling of nonlinear optical systems with a time delay T > 0 and a spatial argument rotated by an angle θ ∈ [0, 2π) in the nonlocal feedback loop in the approximation of a thin circular layer. The goal of this study is to describe spatially inhomogeneous rotating-wave solutions bifurcating from a homogeneous stationary solution in the case of a Andronov-Hopf bifurcation. The existence of such waves is proved by passing to a moving coordinate system, which makes it possible to reduce the problem to the construction of a nontrivial solution to a periodic boundary value problem for a stationary delay differential equation. The existence of rotating waves in an annulus resulting from a Andronov-Hopf bifurcation is proved, and the leading coefficients in the expansion of the solution in powers of a small parameter are obtained. The conditions for the stability of waves are derived by constructing a normal form for the Andronov-Hopf bifurcation for the functional differential equation under study. [ABSTRACT FROM AUTHOR]

Abstract: In this work a unique solvability of a class of mixed type partial differential equations with some unbounded coefficients is proved in . The estimates of the weighted norms of the solution and its partial derivatives and are derived. [Copyright &y& Elsevier]

Abstract: In this paper we establish a set of sufficient conditions for the controllability of nonlinear fractional dynamical systems. The results are obtained by using the recently derived formula for solution representation of systems of fractional differential equations and the application of the Schauder fixed point theorem. Examples are provided to illustrate the results. [Copyright &y& Elsevier]

Abstract: In this paper, an asymptotic one dimensional model for a clamped curved rod is rigorously derived as a limit of respective three dimensional models. The rod is made of isotropic elastic, heat conductive linearly responding material. Asymptotic analysis is used with respect to the thickness of domains. [Copyright &y& Elsevier]

We consider a nonlinear system of ordinary differential equations which is unsolved with respect to the derivative of the desired vector function and identically degenerate in the definition domain. We study the consistency manifold under assumptions that guarantee the existence of a solution. We prove an analog of the theorem on the continuous dependence of solutions on the initial data, assuming that the latter belong to the consistency manifold. [ABSTRACT FROM AUTHOR]

In this paper we study a problem with conditions analogous to Frankl' and Bitsadze-Samarskii ones for the Gellerstedt equation. We prove that the stated problem is well-posed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a damped Klein–Gordon equation in a noncylindrical domain. This work is devoted to proving the existence of global solutions and decay for the energy of solutions for a damped Klein–Gordon equation in a noncylindrical domain. [Copyright &y& Elsevier]

In this paper we study a nonlocal problem for a mixed-type equation in a domain whose elliptic part is the first quadrant of the plane and the hyperbolic part is the characteristic triangle. With the help of the method of integral equations and the principle of extremum we prove the unique solvability of the considered problem. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we prove the existence of solutions of certain kinds of nonlinear fractional integrodifferential equations in Banach spaces. Further, Cauchy problems with nonlocal initial conditions are discussed for the aforementioned fractional integrodifferential equations. At the end, an example is presented. [Copyright &y& Elsevier]

Abstract: We consider the immiscible two-phase mixture of water and hydrogen in a porous medium. The water phase is incompressible and the hydrogen phase is compressible. The hydrogen dissolves in the water. The flow is described by the system of non-linear evolution equations for the water saturation and the hydrogen pressure. Under non-degeneracy and slow oscillation assumptions on the diagonal coefficients and with small data for the hydrogen, we establish the existence of a weak solution. To cite this article: A. Mikelić, C. R. Mecanique 337 (2009). [Copyright &y& Elsevier]

Variational-like inequalities with set-valued mappings are very useful in economics and nonsmooth optimization problems. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational-like inequalities (VVLI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVLI. We investigate the existence of a solution for the generalized VVLI with a set-valued mapping by exploiting the existence of a solution of the VVLI with a single-valued function and a continuous selection theorem. [ABSTRACT FROM AUTHOR]

We consider a system of nonlinear ordinary differential equations that are not solved with respect to the derivative of the unknown vector function and degenerate identically in the domain of definition. We obtain conditions for the existence of an operator transforming the original system to the normal form and prove a general theorem on the solvability of the Cauchy problem. [ABSTRACT FROM AUTHOR]

Abstract: We consider a system of nonlinear coupled partial differential equations that models immiscible two-phase flow through a porous medium. A primary difficulty with this problem is its degenerate nature. Under reasonable assumptions on the data, and for appropriate boundary and initial conditions, we prove the existence of a weak solution to the problem, in a certain sense, using a compactness argument. This is accomplished by regularizing the problem and proving that the regularized problem has a unique solution which is bounded independently of the regularization parameter. We also establish a priori estimates for uniqueness of a solution. [Copyright &y& Elsevier]

In this paper, we prove an existence result for a solution to the vector equilibrium problems. Then, we establish variational principles, that is, vector optimization formulations of set-valued maps for vector equilibrium problems. A perturbation function [ABSTRACT FROM AUTHOR]