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The present work continues the study for the superstability and solution of the Pexider type functional equation , which is the mixed functional equation represented by sum of the sine, cosine, tangent, hyperbolic trigonometric, and exponential functions. The stability of the cosine (d'Alembert) functional equation and the Wilson equation was researched by many authors: Baker [7], Badora [5], Kannappan [14], Kim ([16, 19]), and Fassi, etc [11]. The stability of the sine type equations was researched by Cholewa [10], Kim ([18], [20]). The stability of the difference type equation for the above equation was studied by Kim ([21], [22]). In this paper, we investigate the superstability of the sine functional equation and the Wilson equation from the Pexider type difference functional equation , which is the mixed equation represented by the sine, cosine, tangent, hyperbolic trigonometric functions, and exponential functions. Also, we obtain additionally that the Wilson equation and the cosine functional eqaution in the obtained results can be represented by the composition of a homomorphism. In here, the domain (G; +) of functions is a noncommutative semigroup (or 2-divisible Abelian group), and A is an unital commutative normed algebra with unit 1A. The obtained results can be applied and expanded to the stability for the difference type's functional equation which consists of the (hyperbolic) secant, cosecant, logarithmic functions.

We have proved theHyers-Ulam stability and the hyperstability of the quadratic functional equation f (x + y + z) + f (x + y − z) + f (x − y + z) + f (−x + y + z) = 4[f (x) + f (y) + f (z)] in the class of functions from an abelian group G into a Banach space.

In this paper, by using fixed point theorem, we obtain the stability of the following functional equations \begin{align*} f(pr,qs)+g(ps,qr)&=\theta(p,q, r,s)f(p,q)h(r,s) \\ f(pr,qs)+g(ps,qr)&=\theta(p,q, r,s)g(p,q)h(r,s), \end{align*} where $G$ is a commutative semigroup, $\theta : G^{4} \rightarrow \mathbb{R}_{k}$ a function and $f,g,h$ are functionals on $G^{2}$.

In this paper, we investigate the generalized Hyers–Ulam stability for the generalized psi functional equation f ( x + p ) = f ( x ) + φ ( x ) by the direct method in the sense of P. Gǎvruta and the Hyers–Ulam–Rassias stability.

In this paper, we solve the Hyers-Ulam stability problem for the following cubic type functional equation f(rx + sy) + f(rx− sy) = rsf(x + y) + rsf(x− y) + 2r(r − s)f(x) in quasi-Banach space and non-Archimedean space, where r 6= ±1, 0 and s are real numbers.

In this study, the character amenability of Banach algebras is considered and some characterization theorems are established. Indeed, we prove that the character amenability of Lipschitz algebras is equivalent to that of Banach algebras.

This work aims to study of the stability of two generalizations of the functional equa- tion f(pr,qs)+f(ps,qr )= f(p,q) f(r,s), namely (i) f(pr,qs)+g(ps,qr )= h(p,q)h(r,s) ,a nd (ii) f(pr,qs)+g(ps,qr )= h(p,q)k(r,s) for all p,q,r,s ∈ G ,w hereG is a commutative semi- group. Thus this work is a continuation of our earlier works (15 )a nd (16), and the functional equations studied here arise in the characterizations of symmetrically compositive sum form distance measures.

The aim of this paper is to prove the superstability of the following functional equations $$f\left(\frac{x+y}{m}\right)^m \,=\, g(x)h(y),$$ where \({f, g, h: V^{2} \to A}\) are unknown mappings and m is a fixed positive integer. Here V is a vector space, and A is a unital normed algebra.

The present work extends the study on the stability of the functional equation f(pr,qs)+ f(ps,qr )= f(p,q) f(r,s), which arises in the characterization of symmetrically compositive sum-form distance measures, and as a products of some multiplicative functions. In this paper, we obtain the superstability of the functional equations f(pr,qs) − f(ps,qr )= f(p,q)g(r,s) f(pr,qs) − f(ps,qr )= g(p,q)f(r,s) f(pr,qs) − f(ps,qr )= g(p,q)g(r,s) f(pr,qs) − f(ps,qr )= g(p,q)h(r,s), for all p,q,r,s ∈ G ,w hereG is an Abelian group. These functional equations arise in the char- acterization of the nonsymmetrically compositive difference-form related to distance measures, products of some multiplicative functions. In reduction, they can be represented as exponential functional equations.

The aim of this paper is to investigate the superstability of the pexider type sine(hyperbolic sine) functional equation which is bounded by the unknown functions or . As a consequence, we have generalized the stability results for the sine functional equation by P. M. Cholewa, R. Badora, R. Ger, and G. H. Kim.

By using fixed point methods and direct method, we establish the generalized Hyers-Ulam stability of the following additive-quadratic functional equationf(x+ky)+f(x−ky)=f(x+y)+f(x−y)+(2(k+1)/k)f(ky)−2(k+1)f(y)for fixed integerskwithk≠0,±1in fuzzy Banach spaces.

Let n>1 be an integer, let A be an algebra, and X be an A-module. A quadratic function D:A→X is called a quadratic n-derivation if D(∏i=1nai)=D(a1)a22⋯an2+a12D(a2)a32⋯an2+⋯+a12a22⋯an−12D(an) for all a1,...,an∈A. We investigate the Hyers-Ulam stability of quadratic n-derivations from non-Archimedean Banach algebras into non-Archimedean Banach modules by using the Banach fixed point theorem.

Let 𝑛 be a fixed integer greater than 3 and let 𝜆 be a real number with 𝜆 ≠ ( 𝑛 2 − 𝑛 + 4 ) / 2 . We investigate the Hyers-Ulam stability of derivations on Banach algebras related to the following generalized Cauchy functional inequality ‖ ∑ 1 ≤ 𝑖 𝑗 ≤ 𝑛 1 ≤ 𝑘 𝑙 ≠ 𝑖 , 𝑗 ≤ 𝑛 𝑓 ( ( 𝑥 𝑖 + 𝑥 𝑗 ∑ ) / 2 + 𝑛 − 2 𝑙 = 1 𝑥 𝑘 𝑙 ∑ ) + 𝑓 ( 𝑛 𝑖 = 2 𝑥 𝑖 ) + 𝑓 ( 𝑥 1 ∑ ) ‖ ≤ ‖ 𝜆 𝑓 ( 𝑛 𝑖 = 1 𝑥 𝑖 ) ‖ .

We present a fixed point theorem for generalized contraction in partially ordered complete metric spaces. As an application, we give an existence and uniqueness for the solution of a periodic boundary value problem.

The present work aims to study the stability of the following three functional equations: (i) f ( p r , q s ) + f ( p s , q r ) = f ( p , q ) f ( r , s ) , (ii) f ( p r , q s ) + f ( p s , q r ) = f ( p , q ) g ( r , s ) , and (iii) f ( p r , q s ) + f ( p s , q r ) = g ( p , q ) f ( r , s ) for all p , q , r , s ∈ ( 0 , 1 ) . The first functional equation arises in the characterization of symmetrically compositive sum form distance measures.

The aim of this paper is to investigate the superstability of the Lobacevski equation f (x + y 2 )2 = f(x)f(y), which is bounded by the unknown functions φ(x) or φ(y). The obtained result is a generalization of P. Gǎvruta’s result in 1994.

We prove the generalized Hyers-Ulam stability of the 2nd-order linear differential equation of the form 𝑦   + 𝑝 ( 𝑥 ) 𝑦  + 𝑞 ( 𝑥 ) 𝑦 = 𝑓 ( 𝑥 ) , with condition that there exists a nonzero 𝑦 1 ∶ 𝐼 → 𝑋 in 𝐶 2 ( 𝐼 ) such that 𝑦 1   + 𝑝 ( 𝑥 ) 𝑦  1 + 𝑞 ( 𝑥 ) 𝑦 1 = 0 and 𝐼 is an open interval. As a consequence of our main theorem, we prove the generalized Hyers-Ulam stability of several important well-known differential equations.

‎The present work continues the study of the stability of the functional equations‎ ‎of the type $f(pr‎, ‎qs)‎ + ‎f(ps‎, ‎qr) = f(p,q)\‎, ‎f(r‎, ‎s)$ namely‎ ‎(i) $f(pr‎, ‎qs)‎ + ‎f(ps‎, ‎qr) = g(p,q)\‎, ‎g(r‎, ‎s)$‎, ‎and (ii)‎ ‎$f(pr‎, ‎qs)‎ + ‎f(ps‎, ‎qr) = g(p,q)\‎, ‎h(r‎, ‎s)$‎ ‎for all $p‎, ‎q‎, ‎r‎, ‎s \in G$‎, ‎where $G$ is an abelian group‎. ‎These‎ ‎functional equations arise in the characterization‎ ‎of symmetrically compositive sumform distance measures‎.

We will investigate the superstability of the (hyperbolic) trigonometric functional equation from the following functional equations: and , which can be considered the mixed functional equations of the sine function and cosine function, the hyperbolic sine function and hyperbolic cosine function, and the exponential functions, respectively.

We show that every unbounded approximate gamma–beta type function is of gamma–beta type. That is, we obtain the superstability of a gamma–beta type functional equation β ( x , y ) f ( x + y ) = f ( x ) f ( y ) and also investigate the stability in the sense of R . Ger of this equation in the following setting : | β ( x , y ) f ( x + y ) f ( x ) f ( y ) − 1 | ≤ φ ( x , y ) . From these results, we obtain stabilities of a generalized exponential functional equation and Cauchy’s gamma–beta functional equation.

In this work, we prove that approximate trigonometric functions are bounded. That is, if a non-zero function f satisfies the inequality | f ( x + y ) − f ( x − y ) − 2 f ( x ) f ( y ) | ≤ φ ( x ) or φ ( y ) , then f is bounded.

The aim of this paper is to study the stability problem of the generalized sine functional equations as follows: $$ \begin{array}{*{20}c} {g(x)f(y) = f\left( {\frac{{x + y}} {2}} \right)^2 - f\left( {\frac{{x - y}} {2}} \right)^2 ,} \\ {f(x)g(y) = f\left( {\frac{{x + y}} {2}} \right)^2 - f\left( {\frac{{x - y}} {2}} \right)^2 ,} \\ {g(x)g(y) = f\left( {\frac{{x + y}} {2}} \right)^2 - f\left( {\frac{{x - y}} {2}} \right)^2 .} \\ \end{array} $$ .

The aim of this paper is to investigate the superstability problem of the cosine(d’Alembert), the Wilson, and the sine functional equation from the pexiderized trigonometric functional equation ( T gh ) under the condition: | f ( x + y ) - f ( x - y ) - 2 g ( x ) h ( y ) | ⩽ φ ( x ) or φ ( y ) , and also will investigated the stability of the Jensen type functional equation ( J y ) under the condition: | f ( x + y ) - f ( x - y ) - 2 f ( y ) | ⩽ φ ( x , y ) .

The aim of this paper is to study the stability problem of the d'Alembert type and Jensen type functional equations: f ( x + y ) + f ( x + σ y ) = 2 g ( x ) f ( y ) , f ( x + y ) + f ( x + σ y ) = 2 f ( x ) g ( y ) , f ( x + y ) + f ( x + σ y ) = 2 f ( x ) .

We investigate a generalization of the Hyers-Ulam-Rassias stability for the gamma type functional equation f (x+p) = Γ(x)f (x)+ψ(x) and the stability in the sense of Ger for the functional equation of the form f (x + p) = Γ(x)f (x) . As a consequence, we obtain a stability result in the sense of Hyers-Ulam-Rassias for G -functional type equations. Mathematics subject classification (2000): 39B52, 39B72, 39B82.

The functional equation related to a distance measure $$f(pr, qs) + f(ps, qr) = f(p,q) f(r, s) $$ can be generalized as follows: $$\sum_{i=0}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=f(P)f(Q), $$ where f is an information measure, P and Q are in the set of n-ary discrete complete probability, and $\sigma_{i}$ is a permutation for each $i=0, 1, \ldots, n-1$ . In this paper, we investigate the superstability of the above functional equation and also four generalized functional equations: $$\begin{aligned}& \sum_{i=1}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=f(P)g(Q),\qquad \sum _{i=0}^{n-1} f\bigl(P\cdot\sigma_{i} (Q) \bigr)=g(P)f(Q), \\& \sum_{i=0}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=g(P)g(Q),\qquad \sum _{i=0}^{n-1} f\bigl(P\cdot\sigma_{i} (Q) \bigr)=g(P)h(Q). \end{aligned}$$

Let ( X , ⋄ ) be a square-symmetric groupoid, and ( Y , * , d ) a complete metric divisible square-symmetric groupoid. In this paper, we obtain the Hyers–Ulam stability problem of the functional inequality d ( g ( x ⋄ y ) , g ( x ) * g ( y ) ) ⩽ e ( x , y ) for approximate mapping g : X → Y of the functional equation f ( x ⋄ y ) = f ( x ) * f ( y ) differing by e : X 2 → [ 0 , ∞ ) . In particular, we have investigated the case of f ( x ) * f ( y ) = H ( f ( x ) 1 / t , f ( y ) 1 / t ) on some set Y.

Let ( X , ⋄ ) be a square-symmetric groupoid, and ( Y , * , d ) a complete metric divisible square-symmetric groupoid. In this paper, we obtain the Hyers–Ulam stability problem of functional inequality d ( g ( x ⋄ y ) , g ( x ) * g ( y ) ) ⩽ e ( x , y ) for approximate mapping g : X → Y of functional equation f ( x ⋄ y ) = f ( x ) * f ( y ) differing by e : X 2 → [ 0 , ∞ ) . In particular, we have investigated the case of f ( x ) * f ( y ) = H ( f ( x ) 1 / t , f ( y ) 1 / t ) on some set Y .

In this paper we investigate a generalization of the Hyers-Ulam-Rassias stability for a functional equation of the form f('(X)) = `(X)f(X), where X lie in n-variables. As a conse- quence, we obtain a stability result in the sense of Hyers, Ulam, Rassias, and Gavruta for many other equations such as the gamma, beta, Schroder, iterative, and G-function type's equations.

In this paper, we obtain the generalization of the Hyers-Ulam-Rassias stability in the sense of Gavruta and Ger of the generalized G-type functional equations of the form . As a consequence in the cases , we obtain the stability theorem of G-functional equation : the reciprocal functional equation of the double gamma function.

In this paper we investigate a generalization of the Hyers–Ulam–Rassias stability for a functional equation of the form f ( φ ( X ) ) = ϕ ( X ) f ( X ) + ψ ( X ) and the stability in the sense of Ger for the functional equation of the form f ( φ ( X ) ) = ϕ ( X ) f ( X ) , where X lie in n-variables. As a consequence, we obtain a stability result in the sense of Hyers–Ulam–Rassias, Gǎvruta, and Ger for some well-known equations such as the gamma, beta, and G-function type's equations.

With labor shortage and high-wage era, the construction cost is rising and the construction business is dull, demanding the construction environment of Korea to raise profitability through major cost savings and rationalization of construction management. However, although reinforcing bar(rebar) work, which greatly effects the building`s safety, endurance, and construction time, is an important phase in construction, it holds serious problem of quality and productivity deterioration due to its characteristic of intensive-labor and maintaining of old work methods resulting in poor management, and costs increase. Therefor in this study to investigate current situation and problems of rebar work and to find methods of betterment, a survey was conducted to site engineers and individuals in division of cost estimate of domestic construction company. The survey questions were on the subjects of (1) calculating rebar quantity, (2) ordering and procurement, (3) rebar cutting and bending, and (4) rebar work management. Method of improvement was sought by analyzing the results of the survey

This paper is concerned with a functional differential equation x^'(z)=1/x(az+bx(z)), where a, b are two complex numbers. By constructing a convergent power series solution y(z) of a auxiliary equation of the form b^2y^'(z)=(y(@m^2z)-ay(@mz))(@my^'(@mz)-ay^'(z)), analytic solutions of the form 1b(y(@my^-^1(z))-az) for the original differential equation are obtained.

In the present study, an experimental study of natural convection in a partitioned 2D square enclosure has been carried out. The square enclosure consist of two adiabatic vertical walls and the upper cold and the lower hot walls. A partition is positioned perpendicularly at the center of the left vortical insulated wall. The PIV measurements were performed with the variations of Rayleigh number, partition length and inclination of the enclosure. The working fluid is water with Prandtl number of 6.996 at 20. The captured images were analyzed by using a cross-correlation (two-frame/single-exposure) PIV method.

In this paper, we obtain the superstability of the functional equation for all , where G is an Abelian group, f a functional on , and θ a cocycle on . This functional equation is a generalized form of the functional equation , which arises in the characterization of symmetrically compositive sum-form distance measures, and as products of some multiplicative functions. In reduction, they can be represented as exponential functional equations. Also we investigate the superstability with following functional equations: , , , . MSC:39B82, 39B52.

A new coupled fixed point theorem related to the Pata contraction for mappings having the mixed monotone property in partially ordered complete metric spaces is established. It is shown that the coupled fixed point can be unique under some extra suitable conditions involving mid point lower or upper bound properties. Also the corresponding convergence rate is estimated when the iterates of our function converge to its coupled fixed point. MSC:47H10, 34B15, 54H25.

In this paper, we obtain the modified Hyers-Ulam-Rassias stability for the family of functional equations f (x ◦ y) = H(f (x), f (y)) (x, y ∈ S) , where H is a homogeneous function and ◦ is a square-symmetric operation on the set S . As a consequence we obtain the Hyers-Ulam stability of its functional equation. Mathematics subject classification (2000): 39B52, 39B72, 39B82.