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In the present paper, we discuss the approximation properties of Jain–Baskakov operators with parameter c. The paper deals with the modified forms of the Baskakov basis functions. Some direct results are established, which include the asymptotic formula, error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct the King modification of these operators, which preserves the test functions e 0 {e_{0}} and e 1 {e_{1}} . It is shown that these King type operators provide a better approximation order than some Baskakov–Durrmeyer operators for continuous functions defined on some closed intervals.

The basic problem of satisfaction of boundary conditions is considered when the generalized stress vector is given on the surfaces of elastic plates and shells. This problem has so far remained open for both refined theories in a wide sense and hierarchic type models. In the linear case, it was formulated by I. N. Vekua for hierarchic models. In the nonlinear case, bending and compression-expansion processes do not split and in this context the exact structure is presented for the system of differential equations of von Kármán–Mindlin–Reisner (KMR) type, constructed without using a variety of ad hoc assumptions since one of the two relations of this system in the classical form is the compatibility condition, but not the equilibrium equation. In this paper, a unity mathematical theory is elaborated in both linear and nonlinear cases for anisotropic inhomogeneous elastic thin-walled structures. The theory approximately satisfies the corresponding system of partial differential equations and the boundary conditions on the surfaces of such structures. The problem is investigated and solved for hierarchic models too. The obtained results broaden the sphere of applications of complex analysis methods. The classical theory of finding a general solution of partial differential equations of complex analysis, which in the linear case was thoroughly developed in the works of Goursat, Weyl, Walsh, Bergman, Kolosov, Muskhelishvili, Bers, Vekua and others, is extended to the solution of basic nonlinear differential equations containing the nonlinear summand, which is a composition of Laplace and Monge–Ampére operators.

In the paper, we derive a formula of integration by parts for the left-sided Hilfer derivative. Moreover, we present a formula of such a type for the right-sided Hilfer derivative. These results generalize the well-known rules of integration by parts for the Riemann–Liouville and Caputo derivatives. Finally, we use these rules to investigate some boundary value problems involving mixed fractional derivatives.

In the present paper the sufficient conditions are obtained for the generalized r-absolute convergence ( 0 < r < 2 {0 ) of the single Fourier trigonometric series in terms of the modulus of δ-variation of a function. It is proved that these conditions are unimprovable in a certain sense. The classical results of Berstein, Szasz, Zygmund and others, related to the absolute convergence of single trigonometric Fourier series, were previously generalized by [L. Gogoladze and R. Meskhia, On the absolute convergence of trigonometric Fourier series, Proc. A. Razmadze Math. Inst. 141 2006, 29–40].

The paper establishes the basic oscillational properties of differential equations of fractional order. It is also shown that the so-called fractional oscillational equation does not have the basic oscillational properties.

The paper is concerned with the Cauchy problem for a nonlinear generalized heat equation which is related to the generalized Gauss–Weierstrass semigroup via Duhamel’s principle. For the initial data we assume that they belong to some fractional Sobolev spaces. We study the existence and uniqueness of mild and strong solutions which are local in time. Moreover, they are smooth functions and belong to Lebesgue spaces with respect to the space variable. We use both fixed point arguments and mapping properties of the generalized Gauss–Weierstrass semigroup. Finally, we study the well-posedness of the problem.

In this paper, a new computational method based on the Chebyshev wavelets (CWs) is proposed for solving nonlinear stochastic Itô–Volterra integral equations. In this way, a new stochastic operational matrix (SOM) for the CWs is obtained. By using these basis functions and their SOM, such problems can be transformed into nonlinear systems of algebraic equations which can be simply solved. Moreover, a new technique for computation of nonlinear terms in such problems is presented. Further error analysis of the proposed method is also investigated and the efficiency of this method is shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient.

In this paper, we determine the forbidden set, introduce an explicit formula for the solutions and discuss the global behavior of solutions of the difference equation x n + 1 = a ⁢ x n ⁢ x n - k b ⁢ x n - c ⁢ x n - k - 1 , n = 0 , 1 , … , x_{n+1}=\frac{ax_{n}x_{n-k}}{bx_{n}-cx_{n-k-1}},\quad n=0,1,\ldots, where a , b , c {a,b,c} are positive real numbers and the initial conditions x - k - 1 , x - k , … , x - 1 , x 0 {x_{-k-1},x_{-k},\ldots,x_{-1},x_{0}} are real numbers. We show that when a = b = c {a=b=c} , the behavior of the solutions depends on whether k is even or odd.

The purpose of this paper is to study a nondifferentiable multiobjective fractional programming problem (MFP) in which each component of objective functions contains the support function of a compact convex set. For a differentiable function, we introduce the class of second-order ( C , α , ρ , d ) - V {(C,\alpha,\rho,d)-V} -type-I convex functions. Further, Mond–Weir- and Wolfe-type duals are formulated for this problem and appropriate duality results are proved under the aforesaid assumptions.

The aim of this paper is to consider a generalized mixed variational-like inequality problem and prove a Minty-type lemma for its related auxiliary problems in a real Banach space. We prove the existence of a solution of these auxiliary problems. Further, we prove some properties of a solution set of generalized mixed variational-like inequality problems. Furthermore, we use a hybrid projection method to find a common element of a solution set of a system of unrelated generalized mixed variational-like inequality problems for generalized relaxed α-monotone mappings and the set of fixed points of a common fixed point problem for a family of generalized asymptotically quasi-ϕ-nonexpansive mappings in a reflexive, uniformly smooth and strictly convex Banach space.

In this paper, we consider a Neumann problem for a linear elliptic equation with lower-order terms. A comparison result for solutions of the problem is proved by using Steiner symmetrization.

In this paper, we are interested in the study of the existence of solutions of a class of nonlinear fractional differential equations coupled with Dirichlet boundary conditions. We make an exhaustive study of the sign of Green’s function related to the linear problem and obtain the exact values for which it has the constant sign on its whole square of definition. This constant sign is equivalent to the validity of anti-maximum principles. The existence of solutions is deduced from a recent fixed point theorem valid for operators defined on suitable cones of Banach spaces. Moreover, the method of lower and upper solutions is developed for the non-homogeneous Dirichlet problem.

In this paper, we first establish some refinements of the weighted generalized trapezoid inequality for functions of bounded variation in terms of a cumulative variation function, and then obtain some perturbed versions of the weighted generalized trapezoid inequality. Several applications for selfadjoint operators on complex Hilbert spaces are also given.

The present paper deals with the bivariate ( p , q ) {(p,q)} -Baskakov–Kantorovich operators and their approximation properties. First we construct the operators and obtain some auxiliary results such as calculations of moments and central moments, etc. Our main results consist of uniform convergence of the operators via the Korovkin theorem and rate of convergence in terms of modulus of continuity.

In this paper we prove a stability result for the functional equation 1 L ⁢ ∑ λ ∈ K F ⁢ ( x + λ ⁢ y ) = α ⁢ ( y ) ⁢ G ⁢ ( x ) + H ⁢ ( y ) , x , y ∈ S , \frac{1}{L}\sum_{\lambda\in K}F(x+\lambda y)=\alpha(y)G(x)+H(y),\quad x,y\in S, for the functions F , G , H : S → X {F,G,H\colon S\to X} , α : S → 𝕂 {\alpha\colon S\to\mathbb{K}} , where ( S , + ) {(S,+)} is an abelian semigroup, K is a finite subgroup of the automorphism group of S, L := | K | {L:=|K|} and ( X , ∥ ⋅ ∥ ) {(X,\|\cdot\|)} is the Banach space over a field 𝕂 ∈ { ℝ , ℂ } {\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}} .

In this paper, we present an asymptotic formula for Bateman’s G-function G ⁢ ( x ) ${G(x)}$ and deduce the double inequality 1 2 ⁢ x 2 + 3 / 2 < G ⁢ ( x ) - 1 x < 1 2 ⁢ x 2 , x > 0 . $\frac{1}{2x^{2}+3/2}0.$ We apply this result to find estimates for the error term of the alternating series ∑ k = 1 ∞ ( - 1 ) k - 1 k + h $\sum_{k=1}^{\infty}\kern-2.0pt\frac{(-1)^{k-1}}{k+h}$ , h ≠ - 1 , - 2 , - 3 , … $h\kern-1.0pt\neq\kern-1.0pt-1,\kern-0.5pt-2,\kern-0.5pt-3,\ldots\,$ . Also, we study the monotonicity of some functions involving the function G ⁢ ( x ) ${G(x)}$ . Finally, we propose a sharp double inequality for the function G ⁢ ( x ) ${G(x)}$ as a conjecture.

In this paper, we are interested in the study of the asymptotic analysis of a dynamical problem in elasticity with nonlinear friction of Tresca type. The Lamé coefficients of a thin layer are assumed to vary with respect to the thin layer parameter ε and to depend on the temperature. We prove the existence and uniqueness of a weak solution for the limit problem. The proof is carried out by the use of the asymptotic behavior when the dimension of the domain tends to zero.

In [15] we obtained the Hyers–Ulam stability of the functional equation ∫ K ∫ G f ( x t k · y ) d μ ( t ) d k = f ( x ) g ( y ) , x , y ∈ G , $ \int _{K}\int _{G} f(xtk\cdot y)\,d\mu (t)\,dk=f(x)g(y), \quad x, y\in G, $ where G is a Hausdorff locally compact topological group, K is a compact subgroup of morphisms of G, μ is a K-invariant complex measure with compact support, provided that the continuous function f satisfies some Kannappan type condition. The purpose of this paper is to remove this restriction.