Search

In the present paper, we investigate the existence and regularity of weak solutions for $$\psi $$ -Hilfer fractional boundary value problem in $$\mathbb {C}^{\alpha ,\beta ;\psi }_{2}$$ and $$\mathcal {H}$$ (Hilbert space) spaces, using extension of the Lax–Milgram theorem. In this sense, to finalize the paper, we discuss the integration by parts for $$\psi $$ -Riemann–Liouville fractional integral and $$\psi $$ -Hilfer fractional derivative.

Let $$(U(t))_ {t\ge 0}$$ be a strongly continuous semigroup of bounded linear operators on a Banach space X and B be a bounded operator on X. In this paper, we develop some aspects of the theory of semigroup for which U(t)B (respectively, BU(t), BU(t)B) is demicompact for some (respectively, every) $$t>0$$ . In addition, we study the demicompactness of similar, subspace and product semigroups. We also investigate the demicompactness of the resolvent. We close this paper by giving some conditions guaranteeing the demicompactness of a generator of a strongly continuous semigroup in a Hilbert space.

We introduce here the notion of conformal semi-Riemannian map between semi-Riemannian manifolds aiming to unify and generalize two geometric concepts. The first one is studied by Garcia-Rio and Kupeli (namely, semi-Riemannian map between semi-Riemannian manifolds). The second notion is defined by Aahin (namely, conformal Riemannian map between Riemannian manifolds) as an extension of the notion of Riemannian map introduced by Fischer. We support the main notion of this paper with several classes of examples, e.g. semi-Riemanninan submersions (see O'Neill's book and Falcitelli, Ianus and Pastore's book) and isometric immersions between semi-Riemannian manifolds. As a tool, we use the screen distributions (specific in semi-Riemannian geometry) of Duggal and Bejancu's book to obtain some characterizations and to give a semi-Riemannian version of Fischer's (resp. Aahin's) results, using the new map introduced here. We study the generalized eikonal equation and at the end relate the main notion of the paper with harmonicity.

This paper is devoted to introduce a numerical simulation with a theoretical study for flow of a Newtonian fluid over an impermeable stretching sheet which embedded in a porous medium with a power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by a non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing PDEs are transformed into a system of coupled non-linear ODEs which are using appropriate boundary conditions for various physical parameters. The proposed method is based on replacement of the unknown function by truncated series of well known shifted Legendre expansion of functions. An approximate formula of the integer derivative is introduced. Special attention is given to study the convergence analysis and derive an upper bound of the error of the presented approximate formula. The introduced method converts the proposed equation by means of collocation points to a system of algebraic equations with shifted Legendre coefficients. Thus, by solving this system of equations, the shifted Legendre coefficients are obtained. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, thermal conductivity parameter and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and Nusselt numbers are presented. Comparison of obtained numerical results is made with previously published results in some special cases, and excellent agreement is noted. The results attained in this paper confirm the idea that proposed method is powerful mathematical tool and it can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.

The present paper gives several subordination results involving a generalized Srivastava–Attiya operator (defined below). Among the results presented in this paper include also a sufficiency condition for the convexity of the convolution of certain functions and a sharp result relating to the convolution structure. We also mention various useful special cases of the main results including those which are related to the Zeta function.

This paper is devoted to the study of the error estimates in the periodic homogenization of elliptic equations in divergence form with Dirichlet boundary conditions. We are interested in the application of a two-scale asymptotic expansions method. We present the error estimates of the third-order with and without boundary layers terms.

In this paper, we introduce a new conjugation $$C_{\xi }$$ on the weighted Hardy space $$H_{\rho }(\mathbb {D})$$ , where $$C_{\xi }$$ is given by (2.1) in Theorem 2.2. In particular, we prove that $$C_{\xi }$$ and $$C_{\mu ,\lambda }$$ are unitarily equivalent where $$C_{\mu ,\lambda }$$ is given in Ko and Lee (J Math Anal Appl 434:20–34, 2016). Using this, we investigate a complex symmetric Toeplitz operator $$T_{\varphi }$$ with respect to the conjugation $$C_{\xi }$$ on the weighted Hardy space $$H_{\rho }(\mathbb {D})$$ . Finally, we consider $$C_{\mu ,\lambda }$$ -invariant of Berezin transform.

In this paper, we are concerned in establishing properties about the function $$\vartheta $$ and versions of the classical Keller–Osserman condition to prove existence of solutions to the Schrodinger quasilinear elliptic problem $$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \mathrm{div}\left( \vartheta (u)\nabla u\right) -\frac{1}{2}\vartheta '(u)|\nabla u|^2=a(x)g(u)~ \text{ in }~ \Omega ,\\ u\ge 0\ \text{ in }~\Omega ,\ u(x){\mathop {\longrightarrow }\limits ^{d(x)\rightarrow 0}} \infty , \end{array} \right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^N$$ , with $$N\ge 3$$ , is a bounded domain, $$a:{\bar{\Omega }} \rightarrow [0,\infty )$$ and $$g:[0,\infty ) \rightarrow [0,\infty )$$ are suitable nonnegative continuous functions, $$\vartheta :{\mathbb {R}}\rightarrow (0,\infty )$$ is a $$C^1$$ -function satisfying appropriated hypotheses, and $$d(x)=\mathrm{dist}(x,\partial \Omega )$$ stands for the distance function to the boundary of $$\Omega $$ . By exploring a dual approach and the relationship among the properties of $$\vartheta $$ with its corresponding Keller–Osserman condition, we were able to show existence of solutions for this problem.

This paper is devoted to the following p-Kirchhoff type of problems: $$\begin{aligned} \left\{ \begin{array}{ll} -(a+b\int _{\Omega }|\nabla u|^{p}\,\text{ d }x)\Delta _{p} u=-\lambda |u|^{q-2}u+f(x,u),x\in \Omega \\ u=0,x\in \partial \Omega . \end{array} \right. \end{aligned}$$ Without assuming the standard subcritical polynomial growth condition ensuring the compactness of a bounded (P.S.) sequence, we show that the Dirichlet boundary value problem has at least a weak nontrivial solution by using Morse theory.

In this paper, we shall be interested in solving Dirichlet-type problems for solutions of certain classes of Beltrami equations, to be called $$\beta -$$ analytic. Orthogonal decomposition of the corresponding Sobolev space as well as the ortho-projections onto the subspaces of theses decompositions are obtained. Analytic representation formulas for the underlying solutions in terms of integral operators are established.

In this paper, we present a set of important properties of the special relativistic Fourier transformation (SFT) on the complex space–time algebra $${\mathcal {G}}{(3,1)}$$ , such as inversion property, the Plancherel theorem, and the Hausdorff–Young inequality. The main objective of this article is to introduce the concept of the vector derivative in geometric algebra and using it together with the notion of the space–time split to derive the Heisenberg–Pauli–Weyl inequality. Finally, we apply the SFT properties for proving the Donoho–Stark uncertainty principle for $${\mathcal {G}}{(3,1)}$$ multi-vector functions.

The framework of the paper is the phase universe, described by the total space of the cotangent bundle of a manifold M, which is of interest for both mathematics and theoretical physics. When M carries a symmetric linear connection, then $$T^*M$$ is endowed with a semi-Riemannian metric, namely the classical Riemann extension, introduced by Patterson and Walker and then by Willmore. We consider here a generalization provided by Sekizawa and Kowalski of this metric, called the natural Riemann extension, which is also a metric of signature (n, n). We give the complete classification of conformal and Killing vector fields with respect to an arbitrary natural Riemann extension. Ricci soliton is a topic that has been increasingly studied lately. Necessary and sufficient conditions for the phase space to become a Ricci soliton (or Einstein) are given at the end.

This paper considers a class of fractional Schrodinger–Poisson type systems with doubly critical growth $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^su+V(x)u-\phi |u|^{2^*_s-3}u=K(x)|u|^{2^*_s-2}u,&{} \text{ in } {\mathbb {R}}^3,\\ (-\Delta )^s\phi =|u|^{2^*_s-1},&{} \text{ in } {\mathbb {R}}^3, \end{array}\right. \end{aligned}$$ where $$s\in (3/4,1)$$ , $$2^*_s=\frac{6}{3-2s}$$ , $$V\in L^{\frac{3}{2s}}({\mathbb {R}}^{3})$$ , $$K\in L^{\infty }({\mathbb {R}}^{3})$$ . By applying the concentration-compactness principle and variational method, the existence of ground state solutions to the systems is derived.

In this paper, we characterize finitely generated shift-invariant subspaces of $$L^2(G)$$ , where G is a locally compact abelian group. In particular, we give a formula for the coefficients in the known representation of the Fourier transform of the elements of finitely generated shift-invariant subspaces. Also, certain orthogonalization procedure for generators which is reminiscent of the Gram–Schmidt orthogonalization process is given.

In this paper, we propose a new iterative algorithm for solving the multiple-sets split feasibility problem and the split equality fixed point problem of firmly quasi-nonexpansive mappings in real Hilbert spaces. Under very mild conditions, we prove a weak convergence theorem for our algorithm using projection method and the properties of firmly quasi-nonexpansive mappings. Our result improves and extends the corresponding ones announced by some others in the earlier and recent literature.

This paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of the analysis operator associated with the function be dense. The study is done also with the help of the generalized frame operator associated with a weakly measurable function, which has better properties than the usual frame operator. A special attention is given to lower semi-frames: indeed, if the domain of the analysis operator is dense, then a lower semi-frame can be transformed into a Parseval frame with a (special) metric operator.

We introduce a pointwise definition of Lipschitz (also called Holder) spaces adapted to the parabolic Hermite operator $$\mathbb {H}= \partial _t- \Delta _x+|x|^2$$ on $$\mathbb {{R}}^{n+1}$$ . Also for every $$\alpha >0$$ , we define the following spaces by means of the Poisson semigroup of $$\mathbb {H}$$ , $$\mathcal {P}_y^{\mathbb {H}}=e^{-y\sqrt{\mathbb {H}}}$$ : $$\begin{aligned} \Lambda _\alpha ^{\mathcal {P}^\mathbb {H}}= & {} \left\{ f: \;f\in L^\infty (\mathbb {R}^{n+1})\, \mathrm{and} \, \left\| \partial _y^k e^{-y\sqrt{\mathbb {H}}} f \right\| _{L^\infty (\mathbb {R}^{n+1})}\right. \\&\left. \le C_k y^{-k+\alpha },\, \mathrm {for}\, k=[\alpha ]+1,\;y>0 \right\} , \end{aligned}$$ with the obvious norm. We prove that both spaces do coincide and their norms are equivalent. For the harmonic oscillator, $$\mathcal {{H}}=-\Delta _x+|x|^2$$ , Stinga and Torrea introduced in 2011 adapted Holder classes. Parallel to the parabolic case, we characterize these pointwise Holder spaces via the $$L^\infty $$ norm of the derivatives of the Poisson and heat semigroups, $$e^{-y\sqrt{\mathcal {{H}}}}$$ and $$e^{-\tau \mathcal {{H}}}$$ , respectively. As important applications of these semigroups characterizations, we get regularity results regarding the boundedness in these adapted Lipschitz spaces of operators related to $$\mathbb {H}$$ and $$\mathcal {{H}}$$ as fractional (positive and negative) powers, Bessel potentials, Hermite Riesz transforms, and Laplace transform multipliers, in a more direct way. The proofs use in a fundamental way the semigroup definition of the operators considered along the paper. The non-convolution structure of the operators produces an extra difficulty on the arguments.

The notions of (metric) hypersurface data were introduced in Mars (Gen Relativ Gravit 45:2175–2221, 2013) as a tool to analyze, from an abstract viewpoint, hypersurfaces of arbitrary signature in pseudo-Riemannian manifolds. In this paper, general geometric properties of these notions are studied. In particular, the properties of the gauge group inherent to the geometric construction are analyzed and the metric hypersurface connection and its corresponding curvature tensor are studied. The results set up the stage for various potential applications. The particular but relevant case of spherical symmetry is considered in detail. In particular, a collection of gauge invariant quantities and a radial covariant derivative is introduced, such that the constraint equations of the Einstein field equations with matter can be written in a very compact form. The general solution of these equations in the vacuum case and Lorentzian ambient signature is obtained, and a generalization of the Birkhoff theorem to this abstract hypersurface setting is derived.

In this paper, we consider the Cauchy problem for a generalized Boussinesq equation with double damping terms. By using improved convexity method combined with potential well method and Fourier transform, we show the finite time blow-up of the solution with arbitrarily high initial energy while many similar results require the corresponding energy to be less than some certain numbers.

We consider the nonlinear elliptic equation with quadratic convection $$ -\Delta u + g(u) |\nabla u|^2=\lambda f(u) $$ - Δ u + g ( u ) | ∇ u | 2 = λ f ( u ) in a smooth bounded domain $$ \Omega \subset {\mathbb {R}}^N $$ Ω ⊂ R N ($$ N \ge 3$$ N ≥ 3 ) with zero Dirichlet boundary condition. Here, $$ \lambda $$ λ is a positive parameter, $$ f:[0, \infty ):(0\infty ) $$ f : [ 0 , ∞ ) : ( 0 ∞ ) is a strictly increasing function of class $$C^1$$ C 1 , and g is a continuous positive decreasing function in $$ (0, \infty ) $$ ( 0 , ∞ ) and integrable in a neighborhood of zero. Under natural hypotheses on the nonlinearities f and g, we provide some new regularity results for the extremal solution $$u^*$$ u ∗ . A feature of this paper is that our main contributions require neither the convexity (even at infinity) of the function $$ h(t)=f(t)e^{-\int _0^t g(s)ds}$$ h ( t ) = f ( t ) e - ∫ 0 t g ( s ) d s , nor that the functions $$ gh/h'$$ g h / h ′ or $$ h'' h/h'^2$$ h ′ ′ h / h ′ 2 admit a limit at infinity.

Let $$(M,\langle \;,\;\rangle _{TM})$$ be a Riemannian manifold. It is well known that the Sasaki metric on TM is very rigid, but it has nice properties when restricted to $$T^{(r)}M=\{u\in TM,|u|=r \}$$ . In this paper, we consider a general situation where we replace TM by a vector bundle $$E\longrightarrow M$$ endowed with a Euclidean product $$\langle \;,\;\rangle _E$$ and a connection $$\nabla ^E$$ which preserves $$\langle \;,\;\rangle _E$$ . We define the Sasaki metric on E and we consider its restriction h to $$E^{(r)}=\{a\in E,\langle a,a\rangle _E=r^2 \}$$ . We study the Riemannian geometry of $$(E^{(r)},h)$$ generalizing many results first obtained on $$T^{(r)}M$$ and establishing new ones. We apply the results obtained in this general setting to the class of Euclidean Atiyah vector bundles introduced by the authors in Boucetta and Essoufi J Geom Phys 140:161–177, 2019). Finally, we prove that any unimodular three dimensional Lie group G carries a left invariant Riemannian metric, such that $$(T^{(1)}G,h)$$ has a positive scalar curvature.

Gosper in 2001 introduced the q-trigonometric functions and conjectured many interesting q-trigonometric identities. In this paper, we apply Riemann’s addition formula to deduce two Jacobi theta function identities. From these theta function identities, we confirm a q-trigonometric identity conjectured by Gosper and establish two other similar results. As an application, two theta function analogues for Ptolemy’s theorem are given.

In this paper, we define the Dunford–Henstock–Kurzweil and the Dunford–McShane integrals of Banach space-valued functions defined on a bounded Lebesgue measurable subset of m-dimensional Euclidean space $${\mathbb {R}}^{m}$$ . We will show that the new integrals are “natural” extensions of the McShane and the Henstock–Kurzweil integrals from m-dimensional closed non-degenerate intervals to m-dimensional bounded Lebesgue measurable sets. As applications, we will present full descriptive characterizations of the McShane and Henstock–Kurzweil integrals in terms of our integrals. Moreover, a relationship between new integrals will be proved in terms of the Dunford integral.

In this paper, we derive inequalities for the coefficients in generalized Fourier expansions of (m, n) convex functions in the sense of Popoviciu. Classical Fourier expansions as well as expansions relative to orthogonal polynomials are considered. The results presented here generalize the ones obtained by Niculescu and Rovenţa (Positivity 24(1):129–139, 2020). Some of the results obtained in deriving inequalities for these coefficients can be further used in obtaining Favard-type inequalities similar to the ones given in Wulbert (Math Comput Model 37(12–13):1383–1391, 2003). Favard type inequalities can be used in obtaining probabilistic inequalities which may be further used in fields such as statistical machine learning.

In this paper we study Riemannian manifolds $$(M^n, g)$$ admitting an m-quasi-Einstein metric with V as its potential vector field. We derive an integral formula for compact m-quasi-Einstein manifolds and prove that the vector field V vanishes under certain integral inequality. Next, we prove that if the metrically equivalent 1-form $$V^{\flat }$$ associated with the potential vector field is a harmonic 1-form, then V is an infinitesimal harmonic transformation. Moreover, if M is compact then it is Einstein. Some more results were obtained when (i) V generates an infinitesimal harmonic transformation, (ii) V is a conformal vector field.

In this paper, we study the existence of multiple positive solutions for a class of fractional Schrodinger–Poisson systems involving singular terms and critical nonlinearities. Using the Nehari manifold method, we obtain a multiplicity result for them.

In this paper, we study a class of Kirchhoff-type equation with asymptotically linear right-hand side and compute the critical groups at a point of mountain pass type under suitable Hilbert space. The existence results of three nontrivial solutions under the resonance and non-resonance conditions are established by using the minimax method and Morse theory.

This paper discusses the stability of a kind of spectral property called property (R) under compact perturbations in the setting of Hilbert space. Necessary and sufficient conditions are given for such a spectral property to be invariant under compact perturbations.

In recent years, research on a special class of frames, named K-frames where K is an operator, has become significant in theory and applications. Since the finite sum of K-frames may not be a K-frame for the Hilbert space, in this paper, we discuss the sum and stability of K-frames in Hilbert spaces. First, we obtain some sufficient conditions for the finite sum of a K-frame and a Bessel sequence to be a K-frame. Then we get the K-dual of the sum of K-frames by the dual of the original K-frames. In particular, we give some new results about the operator K and the analysis operator in the discussion. Moreover, we discuss the stability of K-frames and get some conclusions.

In this paper, we give several inequalities involving the Davis–Wielandt radius and the numerical radii of Hilbert space operators. In particular, we show that if T is a bounded linear operator on a complex Hilbert space, then $$\begin{aligned} dw(T) \le \Big (w\big (|T|^4 + |T|^8\big ) + 2w^2\big (|T|^2T\big )\Big )^{\frac{1}{4}}, \end{aligned}$$where $$dw(\cdot )$$ and $$w(\cdot )$$ are the Davis–Wielandt radius and the numerical radius, respectively.

In this paper, we study horizontally conformal submersions from CR-submanifolds of a locally conformal Kahler manifold onto almost Hermitian manifolds, generalizing the results obtained by Sahin (Kodai Math. J. 31, 2008), for horizontally conformal submersions of CR-submanifolds in Kahler ambient space. In particular, we show that any horizontally homothetic submersion of a CR-submanifold M of a locally conformal Kahler manifold with Lee vector field normal to M is a Riemannian submersion up to a scale. Moreover, we obtain that such a map is harmonic, provided that the CR-submanifold is mixed geodesic.

In this paper, we prove some results concerning the loxodromes on an invariant surface in a three-dimensional Riemannian manifold, a part of which generalizes classical results about loxodromes on rotational surfaces in $${{\mathbb {R}}}^3$$. In particular, we show how to parametrize a loxodrome on an invariant surface of $${\mathbb {H}}^2\times {{\mathbb {R}}}$$ and $${\mathbb {H}}_3$$, and we exhibit the loxodromes of some remarkable minimal invariant surfaces of these spaces. In addition, we give an explicit description of the loxodromes on an invariant surface with constant Gauss curvature.

Present paper is devoted to study a pair of linear canonical Hankel transformations of random order and its inverse. Some interesting properties of these transformations are given. Finally, these transformations are used to obtain the solution of some partial differential equations involving Bessel type differential operators.

In this paper, we first introduce a family of geometric constants of a real normed space X and give some results concerning these constants. Then, we give some characterizations of Hilbert spaces and uniformly non-square spaces and obtain sufficient conditions for normal structure related to these constants.

In this paper, we show that the Reeb sectional curvature of a locally symmetric almost coKahler manifold $$M^{2n+1}$$ is a constant if and only if $$M^{2n+1}$$ is locally isometric to the product of $$\mathbb {R}$$ and a locally symmetric almost Kahler manifold. Similar result in the framework of almost Kenmotsu manifolds is established. We give a characterization for a Ricci symmetric almost Kenmotsu manifold to be Einstein.

In this paper, we investigate solvability of fractional analogs of the Dirichlet and Neumann boundary-value problems for the Laplace equation. Operators of fractional differentiation in the Riemann–Liouville and Caputo sense are considered as boundary operators. The considered problems are solved by reducing them to Fredholm integral equations. Theorems on existence and uniqueness of solutions of the problems are proved.

In this paper, we consider the split null point problem and the fixed point problem for multivalued mappings in Hilbert spaces. We introduce a Halpern-type algorithm for solving the problem for maximal monotone operators and demicontractive multivalued mappings, and establish a strong convergence result under some suitable conditions. Also, we apply our problem of main result to other split problems, that is, the split feasibility problem, the split equilibrium problem, and the split minimization problem. Finally, a numerical result for supporting our main result is also supplied.

In this paper, we will develop a four-stage high algebraic order symmetric two-step method with vanished phase-lag and its first up to the fourth derivative. For the proposed method, we will study the following: the phase-lag analysis of the new method; the development of the new method; the local truncation error analysis which is based on the radial Schrodinger equation; the stability and the interval of periodicity analysis which is based on a scalar test equation with frequency different than the frequency of the scalar test equation used for the phase-lag analysis; the error estimation procedure which is based on the algebraic order; and the numerical results from our numerical tests for the examination of the efficiency of the new obtained method. The numerical tests are based on the numerical solution of the Schrodinger equation.

Let \({{\mathcal B}(H)}\) be the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space \({H}\). In this paper, we get a complete classification of surjective linear maps on \({{\mathcal B}(H)}\) that preserve quadratic operators in both directions. An analogue result in the setting of finite-dimensional Banach spaces is given.

Analytical solution of the two-dimensional integral equations are usually difficult. In many cases, approximate solutions are required. In this paper, we study the approximate solution for two-dimensional Fredholm integral equations of the first kind by two-dimensional wavelet. First, definition and the properties of one-dimensional Franklin wavelet must be presented. Next, integral equations converted via regularization method into the second kind, then, using the idea of wavelet Galerkin method, we will find an approximate solution. Finally, the convergence and efficiency of this method will be discussed with some examples which indicate the ability and accuracy of the method.

In this paper, by constructing a cone K 1 × K 2 in the Cartesian product space C[0, 1] × C[0, 1], and using spectral analysis of the relevant linear operator for the corresponding differential system, some properties of the first eigenvalue corresponding to the relevant linear operator are obtained, and the fixed-point index of nonlinear operator in the K 1 × K 2 is calculated explicitly and the existence of at least one positive solution or two positive solutions of the singular differential system with integral boundary conditions is established.

In this paper we study the invariant metrizability and projective metrizability problems for the special case of the geodesic spray associated to the canonical connection of a Lie group. We prove that such canonical spray is projectively Finsler metrizable if and only if it is Riemann metrizable. This result means that this structure is rigid in the sense that considering left-invariant metrics, the potentially much larger class of projective Finsler metrizable canonical sprays, corresponding to Lie groups, coincides with the class of Riemann metrizable canonical sprays. Generalisation of these results for geodesic orbit spaces are given., final version, accepted by MJOM

A Mann-type iterative scheme which converges strongly to a fixed point of a multi-valued nonexpansive non-self mapping T is constructed in a real Hilbert space H. We also constructed a Mann-type sequence which converges to a fixed point of a multi-valued quasi-nonexpansive non-self mapping under appropriate conditions. In addition, an Ishikawa-type iterative scheme which approximates the fixed points of multi-valued Lipschitz pseudocontractive non-self mappings is constructed in Banach spaces. The results obtained in this paper improve and extend the known results in the literature.

In this paper, we prove an analog of Younis’s result [Int J Math Math Sci 9(2): 301–312 1986, Theorem 5.2] on the image under the Fourier–Helgason transform of a set of functions satisfying the Dini–Lipschitz functions in \({L^{p} (1 < p \leq 2)}\) for functions on noncompact rank 1 Riemannian symmetric spaces.

[EN] Semilocal convergence for an iteration of order five for solving nonlinear equations in Banach spaces is established under second-order Fr,chet derivative satisfying the Lipschitz condition. It is done by deriving a number of recurrence relations. A theorem for the existence-uniqueness along with the estimation of error bounds of the solution is established. Its R-order is shown to be equal to five. Both efficiency and computational efficiency indices are given. A variety of examples are worked out to show its applicability. In comparison to existing methods having similar R-orders, improved results in terms of computational efficiency index and error bounds are found using our methodology., The authors thank the referees for their valuable comments which have improved the presentation of the paper. The authors thankfully acknowledge the financial assistance provided by Council of Scientific and Industrial Research (CSIR), New Delhi, India.

The fractal interpolation function (FIF) is a special type of continuous function on a compact subset of $${\mathbb{R}}$$ interpolating a given data set. They have been proved to be a very important tool in the study of irregular curves arising from financial series, electrocardiograms and bioelectric recording in general as an alternative to the classical methods. It is well known that Jacobi polynomials form an orthonormal system in $${\mathcal{L}^{2}(-1,1)}$$ with respect to the weight function $${\rho^{(r,s)}(x)=(1-x)^{r} (1+x)^{s}}$$ , $${r > -1}$$ and $${s > -1}$$ . In this paper, a fractal Jacobi system which is fractal analogous of Jacobi polynomials is defined. The Weierstrass type theorem providing an approximation for square integrable function in terms of $${\alpha}$$ -fractal Jacobi sum is derived. A fractal basis for the space of weighted square integrable functions $${\mathcal{L}_{\rho}^{2}(-1,1)}$$ is found. The Fourier–Jacobi expansion corresponding to an affine FIF (AFIF) interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is established. The closeness of the original function to the Fourier–Jacobi expansion of the AFIF is proved for certain scale vector. Finally, the Fourier–Jacobi expansion corresponding to a non-affine smooth FIF interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is investigated as well.

In this note, we deal with semilinear integro-differential equations subject to homogeneous Dirichlet boundary conditions given on the boundaries of the sections. Even if the differentiation will be taken only in some directions, it is not possible to see the main problem parameterized by the other coordinates because of the non-local terms which also obliged the problem to be degenerate. We establish the existence of solutions by employing the singular perturbations method as a natural tool. The perturbed problems are classical, non-local, semilinear elliptic problems and the limits of the subsequences of their solutions, in weighted Sobolev type spaces, are solutions of the main problem. Some improvement, concerning the existence of the solutions and the convergence results depending on the weights, will be established. The paper also gives an idea about the study of the anisotropic singular perturbations in the framework of weighted spaces.

In this paper, using variational methods and critical point theory, we study the existence of ground-state solutions for the following nonlinear Schrodinger–Maxwell equations $$\left\{\begin{array}{l@{\quad}l} -\triangle u + V(x)u + \phi u = f(x, u), & {\rm in}\, \mathbb{R}^{3},\\ -\triangle\phi = 4\pi u^{2}, & {\rm in} \, \mathbb{R}^{3},\end{array}\right. $$ (NSM) where f is asymptotically cubic, V 1-periodic in each of \({x_1, x_2, x_3}\) and \({\underline{V}:= {\rm inf}_{x\in\mathbb{R}^3}V(x) > 0}\). Under some more assumptions on V and f, we develop a direct and simple method to find ground-state solutions for \({(\mathrm{NSM})}\). The main idea is to find a minimizing (PS) sequence for the energy functional outside the Nehari manifold \({\mathcal{N}}\) using the diagonal method. This seems to be the first result for \({(\mathrm{NSM})}\) satisfying the assumptions (V) and (N).

Our aim in this paper is to deal with the boundedness of the Hardy–Littlewood maximal operator on non-homogeneous central Musielak–Orlicz–Morrey spaces. Further, we give Sobolev’s inequality for generalized Riesz potentials.

Garcia-Cuerva (J Lond Math Soc (2) 39:499–513, 1989) has introduced Herz spaces associated to \({A^p}\) and studied atomic decomposition and its duality, where the space \({A^p}\) is a special case of Herz space. In this paper, we extend the atomic decomposition and duality results to the variable exponent settings.