Equations In Banach Spaces Research Articles

EXISTENCE AND APPROXIMATE CONTROLLABILITY OF HILFER FRACTIONAL EVOLUTION EQUATIONS IN BANACH SPACES

<abstract> This paper is concerned with the existence of mild solutions as well as approximate controllability for Hilfer fractional evolution equations in Banach spaces. Firstly, we give an appropriate definition of mild solutions for this type of fractional equations. The definition of mild solutions for studied problem was given based on a cosine family generated by the operator <italic>A</italic> and probability density function. Secondly, we discuss the existence results of the mild solutions for our concerned problem under the case sine family is compact. Moreover, we establish the approximate controllability when the corresponding linear system is approximately controllable. At last, as an application, two examples are presented to illustrate the abstract results. </abstract>

Asymptotically almost periodic solutions of fractional evolution equations

We study the asymptotic behavior of solutions of nonlinear fractional evolution equations in Banach spaces. Asymptotically almost periodic solutions on half line are obtained by establishing a sharp estimate on the resolvent operator family of evolution equations. In particular, when the semigroup generated by A is exponentially stable then the conditions of the existence asymptotically almost periodic solutions is satisfied. An application to a fractional partial differential equation with initial boundary condition is also considered.

Convolutions in (µ, &amp;#957;)-pseudo-almost periodic and (µ, &amp;#957;)-pseudo-almost automorphic function spaces and applications to solve integral equations

In this paper we give sufficient conditions on \(k\in L^1(\mathbb{R})\) and the positive measures \(\mu\), \(\nu\) such that the doubly-measure pseudo-almost periodic (respectively, doubly-measure pseudo-almost automorphic) function spaces are invariant by the convolution product \(\zeta f=k\ast f\). We provide an appropriate example to illustrate our convolution results. As a consequence, we study under Acquistapace-Terreni conditions and exponential dichotomy, the existence and uniqueness of \(\left( \mu,\nu\right)\)- pseudo-almost periodic (respectively, \(\left( \mu,\nu\right)\)- pseudo-almost automorphic) solutions to some nonautonomous partial evolution equations in Banach spaces like neutral systems.

On the controllability of Hilfer-Katugampola fractional differential equations

By employing Kuratowski's measure of noncompactness together with Sadovskii's fixed point theorem, sufficient conditions for controllability results of Hilfer-Katugampola fractional differential equations in Banach spaces are derived.

ON A PROBLEM FOR AN ELASTIC INFINITE SHEET STRENGTHENED BY TWO PARALLEL STRINGERS WITH FINITE LENGTHS THROUGH ADHESIVE SHEAR LAYERS

The article considers the problem for an elastic infinite sheet (plate), which is strengthened on two parallel finite parts of its upper surface by two parallel finite stringers with different elastic properties. The parallel stringers are located asymmetrically with respect to the horizontal axis of the sheet and deform under the action of horizontal forces. The interaction between the infinite sheet and stringers takes place through thin elastic adhesive layers. The problem of determining unknown shear stresses acting between the infinite sheet and stringers is reduced to a system of Fredholm integral equations of second kind with respect to unknown functions, which are specified on two parallel finite intervals. It is shown that in the certain domain of the change of the characteristic parameters of the problem this system of integral equations in Banach space can be solved by the method of successive approximations. Particular cases are considered, the character and behaviour of unknown shear stresses are investigated.

On the best Ulam constant of a first order linear difference equation in Banach spaces

We obtain some results on Ulam stability for the linear difference equation $$x_{n+1}=a_nx_n+b_n,\, n\geq0$$ , in a Banach space X. If there exists $$\lim_{n\to\infty}|a_n|=\lambda$$ , then the equation is Ulam stable if and only if $$\lambda\neq 1$$ . Moreover if $$(|a_n|)_{n\geq 0}$$ is a monotone sequence, then the best Ulam constant of the equation is $$\frac{1}{|\lambda-1|}$$ .

A New Algorithm for the Common Solutions of a Generalized Variational Inequality System and a Nonlinear Operator Equation in Banach Spaces

We study a new algorithm for the common solutions of a generalized variational inequality system and the fixed points of an asymptotically non-expansive mapping in Banach spaces. Under some specific assumptions imposed on the control parameters, some strong convergence theorems for the sequence generated by our new viscosity iterative scheme to approximate their common solutions are proved. As an application of our main results, we solve the standard constrained convex optimization problem. The results here generalize and improve some other authors’ recently corresponding results.

STABILITY ANALYSIS IN BANACH SPACES FOR A GENERAL ADDITIVE FUNCTIONAL EQUATION

In this paper, we present the Hyers-Ulam stability of generalized additive functional equation in Banach spaces and stability results have been obtained by a classical direct method by various general control functions.

Existence and continuous dependence of mild solutions for impulsive fractional integrodifferential equations in Banach spaces

In the present paper, we investigate the existence, uniqueness, and continuous dependence of mild solutions of impulsive neutral integrodifferential equations with nonlocal conditions in Banach spaces. We use Banach contraction principle and the theory of fractional power of operators to obtain our results. An example is given to illustrate our results.

On norm continuity, differentiability and compactness of perturbed semigroups

The main purpose of this paper is to treat semigroup properties like norm continuity, compactness and differentiability for perturbed semigroups in Banach spaces. In particular, we investigate three large classes of perturbations: Miyadera–Voigt, Desch–Schappacher and Staffans–Weiss perturbations. Our approach is mainly based on feedback theory of Salamon–Weiss systems. Our results are applied to abstract boundary integro-differential equations in Banach spaces.

Duotrigintic functional equation and its stability in Banach spaces

In this paper, we introduce a duotrigintic functional equation. Furthermore, we study the Hyers-Ulam stability of a duotrigintic functional equation in Banach spaces by using the direct method.

An accelerated Kaczmarz type method for nonlinear inverse problems in Banach spaces with uniformly convex penalty

In this paper, we propose and analyze a novel Kaczmarz type method for solving inverse problems which can be written as systems of nonlinear operator equations in Banach spaces. The proposed method is formulated by combining homotopy perturbation iteration and Kaczmarz approach with uniformly convex penalty terms. The penalty term is allowed to be non-smooth, including the L1 and the total variation like penalty functionals, to reconstruct special features of solutions such as sparsity and piecewise constancy. To accelerate the iteration, we introduce a sophisticated rule to determine the step sizes per iteration. Under certain conditions, we present the convergence result of the proposed method in the exact data case. When the data is given approximately, together with a suitable stopping rule, we establish the stability and regularization properties of the method. Finally, some numerical experiments on parameter identification in partial differential equations by boundary as well as interior measurements are provided to validate the effectiveness of the proposed method.

Approximate controllability of a non-autonomous evolution equation in Banach spaces

In this paper, we consider a class of non-autonomous nonlinear evolution equations in separable reflexive Banach spaces. First, we consider a linear problem and establish the approximate controllability results by finding a feedback control with the help of an optimal control problem. We then establish the approximate controllability results for a semilinear differential equation in Banach spaces using the theory of linear evolution systems, properties of resolvent operator and Schauder's fixed point theorem. Finally, we provide an example of a non-autonomous, nonlinear diffusion equation in Banach spaces to validate the results we obtained.

Kuratowski measure of noncompactness and integro-differential equations in Banach spaces

Kuratowski measure of noncompactness and integro-differential equations in Banach spaces

Compact almost automorphic solutions for semilinear parabolic evolution equations

ABSTRACT In this paper, using the subvariant functional method due to Favard [Favard J. Sur les équations différentielles linéaires á coefficients presque-périodiques. ActaMathematica. 1928;51(1):31–81.], we prove the existence and uniqueness of compact almost automorphic solutions for a class of semilinear evolution equations in Banach spaces provided the existence of at least one bounded solution on the right half line. More specifically, we improve the assumptions in [Cieutat P, Ezzinbi K. Almost automorphic solutions for some evolution equations through the minimizing for some subvariant functional, applications to heat and wave equations with nonlinearities. J Funct Anal. 2011;260(9):2598–2634.], we show that the almost automorphy of the coefficients in a weaker sense (Stepanov almost automorphy of order ) is enough to obtain solutions that are almost automorphic in a strong sense (Bochner almost automorphy). For that purpose we distinguish two cases, and . The main difficulty in this work, is to prove the existence of at least one solution with relatively compact range while the forcing term is not necessarily bounded. Moreover, we propose to study a large class of reaction-diffusion problems with unbounded forcing terms.

Identifying an unknown source term of a parabolic equation in Banach spaces

A new regularization method for an inverse source problem for a parabolic equation in a Banach space is proposed. Hölder-type error estimates for the regularized solutions are proved for both a priori and a posteriori regularization parameter choice rules. Some numerical examples are presented for illustrating the efficiency of the method.

A class of inverse problems for evolution equations with the Riemann‐Liouville derivative in the sectorial case

We investigate the unique solvability of a class of linear inverse problems with a time‐independent unknown coefficient in an evolution equation in Banach space, which is resolved with respect to the fractional Riemann‐Liouville derivative. We assume that the operator in the right‐hand side of the equation generates a family of resolving operators for the corresponding homogeneous equation, which is exponentially bounded and analytic in a sector containing the positive semiaxis. It is shown that the inverse problem is well‐posed with respect to the graph norm of the generating operator only. A well‐posedness criterion is found. The obtained abstract results are applied to the unique solvability study of an inverse problem for a class of time‐fractional partial differential equations. An example, in particular, shows that in the case of an unbounded generating operator, the inverse problem can be ill‐posed with respect to the norm of the whole space.

Iterative Solution for Systems of a Class of Abstract Operator Equations in Banach Spaces and Application

In this paper, by using the partial order method, the existence and uniqueness of a solution for systems of a class of abstract operator equations in Banach spaces are discussed. The result obtained in this paper improves and unifies many recent results. Two applications to the system of nonlinear differential equations and the systems of nonlinear differential equations in Banach spaces are given, and the unique solution and interactive sequences which converge the unique solution and the error estimation are obtained.

A Newton-type midpoint method with high efficiency index

In this paper we present a Newton-type two-step method to approximate a solution of a nonlinear equation in Banach spaces. We establish a semilocal convergence theorem under Newton-Kantorovich-type conditions, both the convergence order and the efficiency index of the developed method are found. The iterative procedure presented here is a simple modification of the Newton-Kantorovich method, however, with the same number of function and derivative evaluations at each iteration, it is improved in two important aspects: Firstly, the convergence order is increased from 2 for the Newton-Kantorovich method to 1+2≈2.414 for the new method. Secondly, the efficiency index (convergence order per function evaluation) is improved from 2≈1.414 to (1+2)1/2≈1.553. We illustrate some of our results with a numerical example.

The Parabolic Transform and Some Singular Integral Evolution Equations

Some singular integral evolution equations with wide class of closed operators are studied in Banach space. The considered integral equations are investigated without the existence of the resolvent of the closed operators. Also, some non-linear singular evolution equations are studied. An abstract parabolic transform is constructed to study the solutions of the considered ill-posed problems. Applications to fractional evolution equations and Hilfer fractional evolution equations are given. All the results can be applied to general singular integro-differential equations. The Fourier Transform plays an important role in constructing solutions of the Cauchy problems for parabolic and hyperbolic partial differential equations. This means that the Fourier transform is suitable but under conditions on the characteristic forms of the partial differential operators. Also, the Laplace transform plays an important role in studying the Cauchy problem for abstract differential equations in Banach space. But in this case, we need the existence of the resolvent of the considered abstract operators. This note is devoted to exploring the Cauchy problem for general singular integro-partial differential equations without conditions on the characteristic forms and also to study general singular integral evolution equations. Our approach is based on applying the new parabolic transform. This transform generalizes the methods developed within the regularization theory of ill-posed problems.