Banach Spaces Of Type Research Articles

Nontrivial solutions for multiparameter k-Hessian systems

This paper is devoted to studying the coupled system of multiparameter k-Hessian equations where B = {x ∈ ℝ n : |x| < 1}, λ1, λ2 are positive parameters, ρi (i = 1, 2) is singular near the boundary ∂B, f i(i = 1, 2) satisfies a combined superlinear condition at ∞. Employing the fixed point theorem of Krasnosel’skii type in Banach space, the existence and multiplicity of nontrivial radial solutions are established. In particular, the dependence of the solutions on the parameters λ1, λ2 is also discussed. Finally, we study the nonexistence results of nontrivial radial solutions for the case λ1, λ2 ≫ 1.

Optimal Control of Functional-Differential Equations of Parabolic Type in Banach Spaces

Optimal Control of Functional-Differential Equations of Parabolic Type in Banach Spaces

Solvability of an infinite system of three point BVP in c0 and ℓ1 spaces using measure of noncompactness

Various existing works in the literature have addressed the solvability of different infinite systems of boundary value problems in different types of Banach spaces. In most of these works, the authors have restricted themselves in systems of differential equations with boundary conditions defined at two points. In this work, we have addressed the solvability of an infinite system of third-order differential equations with boundary conditions defined at three points in the sequence spaces [Formula: see text] and [Formula: see text]. We have also illustrated the results for both sequence spaces with suitable examples.

STABILITY OF FUNCTIONAL INEQUALITIES WITH 3K-VARIABLE BASED ON JORDAN-VON NEUMANN TYPE ADDITIVE FUNCTIONAL EQUATIONS IN BANACH SPACE

In this paper, we study to solve the Cauchy, Jensen and Cauchy-Jensen additive function inequalities with 3k-variables related to Jordan-von Neumann type in Banach space. These are the main results of this paper.

Fractional Linear Volterra Integro-Differential Equations in Banach Spaces

The paper presents the foundations of the theory of linear fractional Volterra integro-differential equations of convolution type in Banach spaces. It is established that the existence of a fractional resolvent operator for such equations is equivalent to the well-posedness of the formulation of the initial problem for them. Within the framework of this approach, a theorem of the Hille–Yosida type is proved.

Controllability of a Class of Impulsive ψ-Caputo Fractional Evolution Equations of Sobolev Type

In this paper, we investigate the controllability of a class of impulsive ψ-Caputo fractional evolution equations of Sobolev type in Banach spaces. Sufficient conditions are presented by two new characteristic solution operators, fractional calculus, and Schauder fixed point theorem. Our works are generalizations and continuations of the recent results about controllability of a class of impulsive ψ-Caputo fractional evolution equations. Finally, an example is given to illustrate the effectiveness of the main results.

On the inversion and admissibility for a class of Volterra integro-differential problems

Abstract We study the Volterra integro-differential problems of convolution kernel type from two perspectives: complex inversion formula and the admissibility in the Salamon–Weiss sense (which leads us to consider unbounded control operators). Due to the hybrid Cauchy–Volterra context of these problems, the control operators need to have the admissibility property. First, we show the validity of the inverse formula of the Laplace transform for the resolvent operators associated with some class of Volterra integro-differential problems of convolution type in Banach spaces with the leading operator generating a $C_{0}$-semigroup. This extends and improves the results in Hille &amp; Phillips (1957, 347–348, Theorem 11.6.1), Driouich &amp; El-Mennaoui (1999, 72, Theorem 1), Cioranescu &amp; Lizama (2003, 188–189, Proposition 2), Haase (2008, 84, 81–82, Theorems 4.2 and 4.1) and Fadili &amp; Bounit (2014, 7–8, Proposition 8), including the strong version for this class on UMD spaces. Second, the sufficient or/and necessary conditions for $L^{p}$-admissibility with $p\in (1,+\infty )$ of the control operators are given in terms of the UMD geometric property of its underlying control spaces for a wide class of Volterra integro-differential problems, which provides a generalization of a result known to hold for the standard Cauchy problems shown in Bounit et al. (2010, Proposition 3.2) and the standard Volterra integro-differential problems shown in Fadili &amp; Bounit (2014, 9–10, Theorem 12).

On the behaviour solutions of fractional and partial integro differential heat equations and its numerical solutions

Abstract In this paper, the semi-group method is used to discuss the existence and uniqueness of solutions for fractional and partial integro differential equations (F-PIDEs) of heat type in Banach space E. In addition, the stability of the solutions for F-PIDEs are discussed. Moreover, the Adomian decomposition method (ADM) is used to obtain the solutions numerically. Finally, numerical results of each case are obtained, and the difference of results between the fractional partial integro differential equation and partial integro differential equation are explained. Furthermore, the error is computed in each case.

Approximate controllability of fractional neutral evolution systems of hyperbolic type

&lt;p style='text-indent:20px;'&gt;In this paper, we deal with fractional neutral evolution systems of hyperbolic type in Banach spaces. We establish the existence and uniqueness of the mild solution and prove the approximate controllability of the systems under different conditions. These results are mainly based on fixed point theorems as well as constructing a Cauchy sequence and a control function. In the end, we give an example to illustrate the validity of the main results.&lt;/p&gt;

Nonlocal integro-differential equations of Sobolev type in Banach spaces involving $$\psi$$-Caputo fractional derivative

Nonlocal integro-differential equations of Sobolev type in Banach spaces involving $$\psi$$-Caputo fractional derivative

Simplification method for nonlinear equations of monotone type in Banach space

Abstract. In a Banach space, we study an operator equation with a monotone operator T. The operator is an operator from a Banach space to its conjugate, and T=AC, where A and C are operators of some classes. The considered problem belongs to the class of ill-posed problems. For this reason, an operator regularization method is proposed to solve it. This method is constructed using not the operator T of the original equation, but a more simple operator A, which is B-monotone, B=C−1. The existence of the operator B is assumed. In addition, when constructing the operator regularization method, we use a dual mapping with some gauge function. In this case, the operators of the equation and the right-hand side of the given equation are assumed to be perturbed. The requirements on the geometry of the Banach space and on the agreement conditions for the perturbation levels of the data and of the regularization parameter are established, which provide a strong convergence of the constructed approximations to some solution of the original equation. An example of a problem in Lebesgue space is given for which the proposed method is applicable.

To the theory of solvability of degenerate integro-differential equations in Banach spaces

The paper considers a new approach to constructing generalized solutions of degenerate integro-differential equations of convolution type in Banach spaces. The principal idea of the method proposed implies the refusal of the condition of existence of the full Jordan set for the Fredholm operator of the higher derivative with respect to the operator bundle formed by the rest of operator coefficients of the differential part and by the operator kernel of the integral component of the equation. The conditions are superimposed upon the values of the operator function specially constructed on the basis elements of the Fredholm operator kernel. Under such an approach, the differential part of the equation may include not only the higher derivative but also any combination of lower derivatives, what allows one to consider the convolution integral-differential equations from universal positions, without any special account of the structure of the operator bundle. The method proposed represents a form of generalization of the technique based on the application of Jordan sets of Fredholm operators, and, in the case of existence of the latter, the method coincides with this technique. A generalized solution is constructed in the form of a convolution of the fundamental operator function, which corresponds to the equation under investigation, and the function, which includes both the right-hand side of the equation and the initial data. The conditions, under which such a generalized solution does not contain any singular component, and the regular component converts the initial equation into an identity and satisfies the initial data will provide for the resolvability of the initial problem in the class of functions characterized by the respective smoothness. In this case, the generalized solution constructed will be classical. The theorem on the form of fundamental operator function has been proved. The abstract results have been illustrated via examples of initial-boundary value problems of applied character (from the theory electromagnetic fields, the theory of oscillations in visco-elastic media, the theory of vibrations of thermal-elastic plates).

On geometric properties of symmetric in some types of Banach spaces

In this paper, we -orthogonality and explore -symmetricity such as a -left-symmetric (-right-symmetric) of a vector x in a real Banach space (\U0001d54f, ‖·‖\U0001d569) and study the relation between a -right-symmetric (-left-symmetric) in ℐ(x). New results and proofs are include the notion of norm attainment set of a continuous linear functionals on a reflexive and strictly convex Banach space and using these results to characterize a smoothness of a vector in a unit sphere.

Dichotomy of Linear Partial Differential Equations of Neutral Type in Banach Spaces

This present paper is mainly devoted to investigate the property of spectral decomposition of neutral differential equations in infinite dimensional setting, that is the exponential dichotomy. In fact, we prove that the exponential dichotomy of the associated semigroup to such equations does not depend on that of their associated difference equations. Based on the regular linear systems and feedback theory, we introduce a new transformation of neutral-type equations which plays a key role in our investigation.

EXISTENCE, UNIQUENESS AND STABILITY SOLUTIONS FOR NEW NONLINEAR SYSTEM OF INTEGRO-DIFFERENTIAL EQUATIONS OF VOLTERRA TYPE

In this article, we established the existence, uniqueness and stability solutions for a nonlinear system of integro-differential equations of Volterra type in Banach spaces. Krasnoselskii Fixed point theorems and Picard approximation method are the main tool used here to establish the existence and uniqueness results. A simple example of application of the main result of this paper is presented.&#x0D;

Subordination principle for fractional diffusion-wave equations of Sobolev type

Abstract In this paper we study subordination principles for fractional differential equations of Sobolev type in Banach space. With the help of the theory of Sobolev type resolvent families (known also as propagation family) as well as these subordination principles, we obtain the existence of mild solutions for this kind of equations. We study simultaneously the case 0 &lt; α &lt; 1 and 1 &lt; α &lt; 2 for the Caputo and Riemann-Liouville fractional derivatives.

Mild Solution of Second-Order Impulsive Integro-Differential Evolution Equations of Volterra Type in Banach Spaces

In this paper, by using the cosine family theory, measure of non-compactness and the generalization of Darbo-fixed point theorem, we establish the existence of mild solutions of nonlinear second-order impulsive integro-differential evolution equations of Volterra type in Banach spaces. The result obtained herein generalizes and improves some known results.

О разрешимости в классе распределений вырожденных интегро-дифференциальных уравнений в банаховых пространствах

The paper considers a new approach to constructing generalized solutions of degenerate integro-differential equations convolution type in Banach spaces. The principal idea of the method proposed implies the refusal of the condition of existence of the full Jordan set for the Fredholm operator of the higher derivative with respect to the operator bundle formed by the rest of operator coefficients of the differential part and by the operator kernel of the integral component of the equation. The conditions are superimposed upon the values of the operator function specially constructed on the basis elements of the Fredholm operator kernel. Under such an approach, the differential part of the equation may include not only the higher derivative but also any combination lower derivatives, what allows one to consider the convolution integral-differential equations from universal positions, without any special account of the structure of the structure of the operator bundle. The method proposed represents a form of generalization of the technique based on the application of Jordan sets of Fredholm operators, and in the case of existence of the latter the method coincides with this technique. The generalized solutions are constructed in the form of a convolution of the fundamental operator function, which corresponds to the equation under investigation, and a function, which includes the right-hand side of the equation and the initial data. The conditions, under which such a generalized solution does not contain any singular component, and the regular component converts the initial equation into an identity and satisfies the initial data, and the result will provide for the resolvability of the initial problem in the class of functions of characterized by the respective smoothness. In this case, the generalized solution constructed will be classical. The theorem on the form of fundamental operator function has been proved, the abstract results have been illustrated via examples of initialboundary value problems of applied character from the theory electromagnetic fields, the theory of oscillations in visco-elastic media, the theory of vibrations of thermal-elastic plates.

Fractional differential equations of Sobolev type with sectorial operators

This paper treats the asymptotic behavior of resolvent operators of Sobolev type and its applications to the existence and uniqueness of mild solutions to fractional functional evolution equations of Sobolev type in Banach spaces. We first study the asymptotic decay of some resolvent operators (also called solution operators) and next, by using fixed point results, we obtain the existence and uniqueness of solutions to a class of Sobolev type fractional differential equation. We notice that, the existence or compactness of an operator $$E^{-1}$$ is not necessarily needed in our results.

Some fixed point results for generalized contractions of Suzuki type in Banach spaces

In this paper, we consider generalized contractions of Suzuki type. In particular, we give some geometrical properties of their fixed points. In addition, the convergence of some iterative processes and the problem of the existence of retractions associated with them are established.