Publications

We prove a stability theorem for the quadratic functional equation
f(x + y) + f(x + s(y)) = 2f(x) + 2f(y), x,y Î G,f(x + y) + f(x + \sigma (y)) = 2f(x) + 2f(y),\quad x,y \in G,
where G is an abelian group and σ is an involution of G. We also prove that for functions f from G to an inner product space E, the inequality
||2f(x) + 2f(y) - f(x + s(y))|| £ || f(x + y)||, x,y Î G.\|2f(x) + 2f(y) - f(x + \sigma (y))\| \leq \| f(x + y)\|,\quad x,y \in G.
implies that f is a solution to the equation.
KeywordsHyers–Ulam stability–Quadratic functional equation–Group homomorphisms–Unbounded Cauchy difference–Abelian group

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In this paper, using the Steklov function, we introduce the generalized continuity modulus and define the class of functions W p,φ r,k in the space L p . For this class, we prove an analog of the estimates in [V. A. Abilov and F. V. Abilova, Math. Notes 76, No. 6, 749–757 (2004); translation from Mat. Zametki 76, No. 6, 803–811 (2004; Zbl 1114.42001)] in the space L p .

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In this paper, we propose and study a new composite iterative scheme with certain control conditions for viscosity approximation of a zero of an accretive operator and the fixed points problems in a reflexive Banach space with weakly continuous duality mapping. Strong convergence of the sequence {x n } defined by the introduced iterative sequence is proved. The main results improve and complement the corresponding results of [K. Aoyama et al., Nonlinear Anal., Theory Methods Appl. 67, No. 8, A, 2350–2360 (2007; Zbl 1130.47045); R.-D. Chen and Z.-C. Zhu, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 4, A, 1356–1363 (2008; Zbl 1196.47045); A. Moudafi, J. Math. Anal. Appl. 241, No. 1, 46–55 (2000; Zbl 0957.47039)].

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Using the Hyers-Ulam-Rassias stability method, we investigate isomorphisms in Banach algebras and derivations on Banach algebras associated with the following generalized additive functional inequality ∥af(x)+bf(y)+cf(z)∥≤∥f(αx+βy+γz)∥·(1) Moreover, we prove the Hyers-Ulam-Rassias stability of homomorphisms in Banach algebras and of derivations on Banach algebras associated with the generalized additive functional inequality (1).

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The goal of this paper is to investigate the solution and stability in random normed spaces, in non-Archimedean spaces and also in p-Banach spaces and finally the stability using the alternative fixed point of generalized additive functions in several variables.

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W.-G. Park [J. Math. Anal. Appl. 376, No. 1, 193–202 (2011; Zbl 1213.39028)] proved the Hyers-Ulam stability of the Cauchy functional equation, the Jensen functional equation and the quadratic functional equation in 2-Banach spaces. One can easily see that all results of this paper are incorrect. Hence the control functions in all theorems of this paper are not correct. In this paper, we correct these results.

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Moslehian and Mirmostafaee, investigated the fuzzy stability problems for the Cauchy additive functional equation, the Jensen additive functional equation and the cubic functional equation in fuzzy Banach spaces. In this paper, we investigate the generalized Hyers-Ulam-Rassias stability of the generalized additive functional equation with n-variables, in fuzzy Banach spaces. Also, we show that there exists a close relationship between the fuzzy continuity behavior of a fuzzy almost additive function, control function and the unique additive function which approximate the almost additive function.

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It is shown that every almost linear bijection h:A→B of a unital C * -algebra A onto a unital C * -algebra B is a C * -algebra isomorphism when h(3 n uy)=h(3 n u)h(y) for all unitaries u∈A, all y∈A, and all n∈ℤ, and that almost linear continuous bijection h:A→B of a unital C * -algebra A of real rank zero onto a unital C * -algebra B is a C * -algebra isomorphism when h(3 n uy)=h(3 n u)h(y) for all u∈{v∈A∣v=v * ,∥v∥=1,visinvertible}, all y∈A, and all n∈ℤ. Assume that X and Y are left normed modules over a unital C * -algebra A. It is shown that every surjective isometry T:X→Y, satisfying T(0)=0 and T(ux)=uT(x) for all x∈X and all unitaries u∈A, is an A-linear isomorphism. This is applied to investigate C * -algebra isomorphisms in unital C * -algebras.

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Let K and X be compact plane sets such that K⊆X. Let P(K) be the uniform closure of polynomials on K. Let R(K) be the closure of rational functions K with poles off K. Define P(X,K) and R(X,K) to be the uniform algebras of functions in C(X) whose restriction to K belongs to P(K) and R(K), respectively. Let CZ(X,K) be the Banach algebra of functions f in C(X) such that f| K =0. In this paper, we show that every nonzero complex homomorphism φ on CZ(X,K) is an evaluation homomorphism e z for some z in X∖K. By considering this fact, we characterize the maximal ideal space of the uniform algebra P(X,K). Moreover, we show that the uniform algebra R(X,K) is natural.

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Using fixed-point methods, we investigate approximately higher Jordan ternary derivations in Banach ternary algebras via the functional equation D f (x 1 ,...,x m ):=∑ k=2 m ∑ i 1 =2 k ∑ i 2 =i 1 +1 k+1 ...∑ i m-k+1 =i m-k +1 m f∑ i=1,i≠i 1 ,...,i m-k+1 m x i -∑ r=1 m-k+1 x i r +f∑ i=1 m x i -2 m-1 f(x 1 )=0 where m≥2 is an integer number.

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In this paper we study the module contractibility of Banach algebras and characterize them in terms the concepts splitting and admissibility of short exact sequences. Also we investigate module contractibility of Banach algebras with the concept of the module diagonal.

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In this paper, using a generalized Dunkl translation operator, we obtain an analog of Titchmarsh’s theorem for the Dunkl transform for functions satisfying the Lipschitz-Dunkl condition in L 2,α =L α 2 (ℝ)=L 2 (ℝ,|x| 2α+1 dx), α>-1 2.

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Strong differential subordination and superordination properties are determined for some families analytic functions in the open unit disk which are associated with the Komatu operator by investigating appropriate classes of admissible functions. New strong differential sandwich-type results are also obtained.

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We show that higher derivations on a Hilbert C * -module associated with the Cauchy functional equation satisfy generalized Hyers-Ulam stability.

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We introduce a new iterative scheme for finding a common element of the solutions set of a generalized mixed equilibrium problem and the fixed points set of an infinitely countable family of nonexpansive mappings in a Banach space setting. Strong convergence theorems of the proposed iterative scheme are also established by the generalized projection method. Our results generalize the corresponding results in the literature.

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In this paper, we present recent results in integral inequality theory. Our results are based on the fractional integration in the sense of Riemann-Liouville.

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In this paper we investigate the generalized Hyers-Ulam stability of the following Cauchy-Jensen-type functional equation Qx+y 2+z+Qx+z 2+y+Qz+y 2+x=2[Q(x)+Q(y)+Q(z)] in non-Archimedean spaces.

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In this paper, we prove Hyers-Ulam stability of tribonacci functional equation f(x)=f(x-1)+f(x-2)+f(x-3) in the class of functions f:ℝ→X where X is a real non-Archimedean Banach space.

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In this paper, we prove the generalized Hyers-Ulam stability of the quadratic functional equation f(x+y)+f(x-y)=2f(x)+2f(y) in non-Archimedean ℒ-fuzzy normed spaces.

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The purpose of this paper is to study and give necessary and sufficient conditions for strong convergence of the multi-step iterative algorithm with errors for a finite family of generalized asymptotically quasi-nonexpansive mappings to converge to common fixed points in Banach spaces. Our results extend and improve some recent results in the literature.

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In this paper, we prove that an implicit iterative process with errors converges strongly to a common fixed point for a finite family of generalized asymptotically quasi-nonexpansive mappings on unbounded sets in a uniformly convex Banach space. Our results unify, improve and generalize the corresponding results of H. Fukhar-Ud-din and A. R. Khan [Int. J. Math. Math. Sci. 2005, No. 10, 1643–1653 (2005; Zbl 1096.47059)], Z. Sun [J. Math. Anal. Appl. 286, No. 1, 351–358 (2003; Zbl 1095.47046)], R. Wittmann [Lect. Notes Math. 1514, 229–233 (1992; Zbl 0786.47039)], H.-K. Xu and R. G. Ori [Numer. Funct. Anal. Optimization 22, No. 5–6, 767–773 (2001; Zbl 0999.47043)] and many others.

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In this paper, we introduce and study a new iterative scheme to approximate a common fixed point for a finite family of generalized asymptotically quasi-nonexpansive nonself-mappings in Banach spaces. Several strong and weak convergence theorems of the proposed iteration are established. The main results obtained in this paper generalize and refine some known results in the current literature.

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In this paper, we investigate the generalized Hyers-Ulam-Rassias stability for the quartic, cubic and additive functional equation f(x+ky)+f(x-ky)=k 2 f(x+y)+k 2 f(x-y)+(k 2 -1)[k 2 f(y)+k 2 f(-y)-2f(x)] (k∈ℤ-{0,±1}) in p-Banach spaces.

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We describe a variational problem on a surface under a constraint of geometrical character. Necessary and sufficient conditions for the existence of bifurcation points are provided. In local coordinates the problem corresponds to a quasilinear elliptic boundary value problem. The problem can be considered as a physical model for several applications referring to continuum medium and membranes.

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This paper is concerned with the study of the existence of positive solutions for a Navier boundary value problem involving the p-biharmonic operator; the right hand side of problem is a nonsmooth functional with variable parameters. The existence of at least three positive solutions is established by using nonsmooth version of a three critical points theorem for discontinuous functions. Our results also yield an estimate on the norms of the solutions independent of the parameters.

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We consider the bilinear Fourier integral operator S σ (f,g)(x)=∫ ℝ d ∫ ℝ d e iϕ 1 (x,ξ) e iϕ 2 (x,η) σ(x,ξ,η)f ^(ξ)g ^(η)dξdη, on modulation spaces. Our aim is to indicate this operator is well defined on S(ℝ d ) and to show the relationship between the bilinear operator and BFIO on modulation spaces.

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In this paper, we obtain a sufficient condition for boundedness of composition operators between weighted spaces of holomorphic functions on the upper half-plane whenever our weights are standard analytic weights, but they do not necessarily satisfy any growth condition.

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Let A=(a n,k ) n,k≥1 and B=(b n,k ) n,k≥1 be two non-negative matrices. Denote by L v,p,q,B (A), the supremum of those L, satisfying the following inequality: ∥Ax∥ v,B(q) ≥L∥x∥ v,B(p) , where x≥0 and x∈l p (v,B) and also v=(v n ) n=1 ∞ is an increasing, non-negative sequence of real numbers. In this paper, we obtain a Hardy-type formula for L v,p,q,B (H μ ), where H μ is the Hausdorff matrix and 0<q≤p≤1. Also for the case p=1, we obtain ∥A∥ w,B(1) , and for the case p≥1, we obtain L w,B(p) (A).

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In this paper, we study the convexity of the integral operator ∫ 0 z ∏ i=1 n (te f i (t) ) γ i dt where the functions f i , i∈{1,2,...,n} satisfy the condition |f i ' (z)z f i (z) μ i -1|<1-α i ,i∈{1,2,...,n}·

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A new class of function spaces on domains (i.e., open and connected subsets) of ℝ d , by means of the asymptotic behavior of modulations of functions and distributions, is defined. This class contains the classes of Lebesgue spaces and modulation spaces. Main properties of this class are studied, its applications in the study of function spaces and its relations to classical function spaces are discussed.

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In this paper, we use the Riemann-Liouville fractional integrals to establish some new integral inequalities related to Chebyshev’s functional in the case of two differentiable functions.

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A new class of nonlinear set-valued variational inclusions involving (A,η)-monotone mappings in a Banach space setting is introduced and then, based on the generalized resolvent operator technique associated with (A,η)-monotonicity, the existence and approximate solvability is investigated using an iterative algorithm and fixed point theory.

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We discuss the existence of a positive solution to the infinite semipositone problem -Δu=au-bu γ -f(u)-c u α ,x∈Ω,u=0,x∈∂Ω, where Δ is the Laplacian operator, γ>1, α∈(0,1), a, b and c are positive constants, Ω is a bounded domain in ℝ N with smooth boundary ∂Ω, and f:[0,∞)→ℝ is a continuous function such that f(u)→∞ as u→∞. Also we assume that there exist A>0 and β>1 such that f(s)≤As β , for all s≥0. We obtain our result via the method of sub- and supersolutions.

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We show that the variational inequality V I(C, A) has a unique solution for a relaxed (γ, r)-cocoercive, µ-Lipschitzian mapping A: C → H with r> γµ 2, where C is a nonempty closed convex subset of a Hilbert space H. From this result, it can be derived that, for example, the recent algorithms given in the references of this paper, despite their becoming more complicated, are not general as they should be.

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By using generalized Sălăgean differential operator a new class of univalent holomorphic functions with fixed finitely many coefficients is defined. Coefficient estimates, extreme points, arithmetic mean, and weighted mean properties are investigated.

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In M. A. Al-Thagafi and N. Shahzad [Acta Math. Sin., Engl. Ser. 24, No. 5, 867–876 (2008; Zbl 1175.41026)] introduced the notion of occasionally weakly compatible mappings (shortly, owc maps) which is more general than all the commutativity concepts. In the present paper, we prove common fixed point theorems for families of owc maps in Menger spaces. As applications to our results, we obtain the corresponding fixed point theorems in fuzzy metric spaces. Our results improve and extend the results of J. K. Kohli and S. Vashistha [Acta Math. Hung. 115, No. 1–2, 37–47 (2007; Zbl 1164.47057)], R. Vasuki [Indian J. Pure Appl. Math. 30, No. 4, 419–423 (1999; Zbl 0924.54010)], R. Chugh and S. Kumar [Bull. Calcutta Math. Soc. 94, No. 1, 17–22 (2002; Zbl 1065.54512)] and M. Imdad and J. Ali [Math. Commun. 11, No. 2, 153–163 (2006; Zbl 1152.54355)].

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Recently, Q.-N. Zhang and Y.-S. Song [Appl. Math. Lett. 22, No. 1, 75–78 (2009; Zbl 1163.47304)] proved a common fixed-point theorem for two maps satisfying generalized φ-weak contractions. In this paper, we prove a common fixed-point theorem for a family of compatible maps. In fact, a new generalization of Zhang and Song’s theorem is given.

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In this paper, fixed point and coincidence results are presented for two and three single-valued mappings. These results extend previous results given by B. E. Rhoades [Int. J. Math. Math. Sci. 2003, No. 63, 4007–4013 (2003; Zbl 1052.47052)] and A. Djoudi and F. Merghadi [J. Math. Anal. Appl. 341, No. 2, 953–960 (2008; Zbl 1151.54032)].

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The notion of a probabilistic metric space corresponds to the situations when we do not know exactly the distance. Probabilistic metric space was introduced by Karl Menger. In [Aequationes Math. 46, No. 1–2, 91–98 (1993; Zbl 0792.46062)] C. Alsina et al. gave a general definition of probabilistic normed space based on the definition of Menger. In this note we study the PN-spaces which are topological vector spaces and the open mapping and closed graph theorems in these spaces are proved.

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Let (X,d) be a complete metric space and let f,g:X→X be two mappings which satisfy a (ψ-φ)-weak contraction condition or generalized (ψ-φ)-weak contraction condition. Then f and g have a unique common fixed point. Our results extend previous results given by Ćirić (1971), Rhoades (2001), Branciari (2002), Rhoades (2003), Abbas and Ali Khan (2009), Zhang and Song (2009) and Moradi et al. (2011).

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In this paper, we introduce the concept of generalized ϕ-contractivity of a pair of maps w.r.t. another pair. We establish a common fixed-point result for two pairs of self-mappings, when one of these pairs is generalized ϕ-contraction w.r.t. the other and study the well-posedness of their fixed-point problem. In particular, our fixed-point result extends the main result of a recent paper of Qingnian Zhang and Yisheng Song.

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In this paper, seventh-order iterative methods for the solution of nonlinear equations are presented. The new iterative methods are developed by using weight function method and using an approximation for the last derivative, which reduces the required number of functional evaluations per step. Several examples are given to illustrate the efficiency and the performance of the new iterative methods.

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We consider the coupled system F(x,y)=G(x,y)=0, where F(x,y)=∑ k=0 m 1 A k (y)x m 1 -k andG(x,y)=∑ k=0 m 2 B k (y)x m 2 -k with entire functions A k (y), B k (y). We derive a priory estimates for the sums of the roots of the considered system and for the counting function of roots.

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In this paper, we investigate the generalized Hyers-Ulam stability of the functional equation 4(f(3x+y)+f(3x-y))=-12(f(x+y)+f(x-y)) +12(f(2x+y)+f(2x-y))-8f(y)-192f(x)+f(2y)+30f(2x) in fuzzy normed spaces.

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In this paper we are going to study the Hyers–Ulam–Rassias types of stability for nonlinear, nonhomogeneous Volterra integral equations with delay on finite intervals.

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For an arbitrary entire function f(z), let M(f,R)=max |z|=R |f(z)| and m(f,r)=min |z|=r |f(z)|. If P(z) is a polynomial of degree n having no zeros in |z|<k, k≥1, then for 0≤r≤ρ≤k, it is proved by Aziz et al. that M(P ' ,ρ)≤n ρ+k{ρ+k k+r n 1-k(k-ρ)(n|a 0 |-k|a 1 |)n (ρ 2 +k 2 )n|a 0 |+2k 2 ρ|a 1 |ρ-r k+ρk+r k+ρ n-1 M(P,r) -(n|a 0 |ρ+k 2 |a 1 |)(r+k) (ρ 2 +k 2 )n|a 0 |+2k 2 ρ|a 1 |×ρ+k r+k n -1-n(ρ-r)m(P,k)}· In this paper, we obtain a refinement of the above inequality. Moreover, we obtain a generalization of above inequality for M(P ' ,R), where R≥k.

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In this paper we study properties of symbols such that these belong to class of symbols sitting inside S ρ,φ m that we shall introduce as the following. So for because hypoelliptic pseudodifferential operators plays a key role in quantum mechanics we will investigate some properties of M-hypoelliptic pseudo differential operators for which define base on this class of symbols. Also we consider maximal and minimal operators of M-hypoelliptic pseudo differential operators and we express some results about these operators.

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