On stability of the general linear equation

We prove, using the fixed point approach, some stability results for the general linear functional equation. Namely we obtain sufficient conditions for the stability of a wide class of functional equations and control functions. Our results generalize a lot of the well known and recent outcomes concerning stability. In some examples we indicate how our method may be used to check if the particular functional equation is stable and we discuss the optimality of obtained bounding constants.

One of the ways of proving stability is the method of fixed points which is the most popular tool. Analyzing proofs of stability in particular cases of the functional equation (1.1), among others the above-mentioned (see [3,4]), we have found sufficient conditions for its stability.
What is more interesting, we present some examples how our criterion may be used as a tool to check the stability of particular functional equations of the linear type. Moreover, we discuss the optimality of obtained bounding constants in particular cases.
Some similar ideas can be found in [6], where the general method for proving stability is described. Our considerations based on the fixed point theorem lead to a simpler procedure and sufficient conditions which are easier to check.

Statement of the main result and applications
In this section we state the main result and we present its applications. The proof of the theorem will be given in the last section. The main theorem of this paper provides a criterion for the stability of the equation (1.1). It implies lots of the well known and recent results concerning particular cases of this functional equation. We describe a method of determining whether a functional equation of linear type is stable.
Our test method for stability of the equation is based on the use of the fixed point theorem. In proofs of numerous theorems concerning stability an appropriate substitution is used to obtain a contraction operator. The fixed point of the operator is a solution of the equation which is close to a given function.
Namely, considering a function g satisfying approximately equation (1.1) we look for a substitution x j = c j x, c j ∈ F, j ∈ {1, . . . , n} such that where T is a proper operator. For this purpose, the solution c 1 , . . . , c n ∈ F to the system consisting of at least one equation In the sequel R + = [0, +∞). A sum of numbers over an empty set is defined to be zero.
Assume that there exist ∅ = I ⊂ {1, . . . , m}, c 1 , . . . , c n ∈ F and ω 1 , . . . , ω n ∈ [0, +∞) such that Then there exists a unique solution G : X → Y of (1.1) such that Moreover G is a unique solution of (1.1) such that there exists a constant B ∈ (0, ∞) with It is easy to see that for each such function θ i there exists ω : F → R + fulfilling

p-Wright equation
which is a particular case of (1.1), with m = 4, n = 2, We look for solutions c 1 , c 2 of subsystems of the system of linear equations ⎡ which leads to the following two cases According to Theorem 2.1 applied to these solutions, we obtain estimations in the real case for a particular control function. To shorten the paper we omit the second conclusion of Theorem 2.1 (see condition (2.3)) in the statement of the following results.
Vol. 89 (2015) On stability of the general linear equation 1465 An analysis of the conditional minima of the above obtained bounding functions in the case k = 2 ensures the stability of this equation in this case. The detailed verification of the following corollary is left to the reader.
Then there exists a unique p-Wright affine function G : X → Y such that Remark 2.6. Applying the above result for p ∈ {− 1 2 , 1 2 } the optimal constant obtained by our method is equal to 1+ Observe that it is smaller than the constant in [12,Th. 1].
In the case p ∈ (0, 1) ⇐⇒ p 2 + (1 − p) 2 < 1 our estimation M is smaller than the estimate In the same manner we can study other cases in order to get the optimal constants.

Linear equation
Consider the linear equation x,y∈ X.
Therefore we obtain the results corresponding to each of the cases (number of each case is indicated in parentheses).

5)
then there exists a unique solution G : X → Y of (2.4) such that Moreover G is a unique solution of (2.4) such that there exists a constant K ∈ (0, ∞) with Kθ(x, x), x∈ X.
Assume that there exists ω ∈ R + such that ω < |A + B| and Then there exists a unique solution G : X → Y of (2.4) such that Proof. Applying Theorem 2.1 for c 1 = c 2 = 1, I = {2, 3} we obtain our claim.

Theorem 2.9 (3). Let Y be a Banach space,
Then there exists a unique solution G : X → Y of (2.4) such that Vol. 89 (2015) On stability of the general linear equation 1467

Theorem 2.10 (4). Let Y be a Banach space, ac
Then there exists a unique solution G : X → Y of (2.4) such that , x∈ X.
Then there exists a unique solution G : X → Y of (2.4) such that , x∈ X.
Note that interchanging a and b, A and B in the above three theorems, we obtain results for the cases (3')-(5').
According to Remark 2.3, in the case θ(x, y) := ε we can take ω ≡ 1. Consequently, combining all the above results in this case we deduce that the linear equation is stable and the bounding constant is the smallest one obtained by our method.
x, y ∈ X, then the linear equation (2.4) is stable. Namely, there exists its unique solution G : X → Y such that Proof. Take a, b, A, B ∈ R such that a + b = 1 and A + B = 1. Then at least the condition (1) holds, the conditions (2), (3), (3') are not satisfied. Define constants If M i > 0 for some i ∈ {1, 4, 5, 5 }, then there exists a unique soluton G i : X → Y of (2.4) such that see Theorems 2.7, 2.10, 2.11 and analogons of the last two ones.
It is easy to verify that M 1 = max{M 1 , M 4 , M 5 , M 5 }, therefore 1 M1 is the smallest bound obtained using linear substitutions. Moreover, according to the second assertion of Theorem 2.7 the solution G 1 is a unique one satisfying the inequality x ∈ X with any constant K ∈ R, which completes the proof.
The same, but more complicated reasoning, can be drawn for the case a + b = 1. To state our result in this case, define sets then the linear equation is stable. Namely, there exists its unique solution G : .
In the case A + B = 1 our criterion does not determine the stability of the equation (2.4) for the constant control function.
Remark 2.14. Among a lot of consequences of Theorems 2.7-2.13, we have the following results for Cauchy's equation (the case a = b = A = B = 1).
Applying Theorem 2.13 for θ(x, y) := ε and ω = 1 we have the well known stability result proved by Hyers in [9]. Observe that the constant M (A, B) = |A + B| − 1 obtained by our method is optimal. Theorem 2.8 used for the case θ(x, y) := C( x p + y p ), ω = |a + b| p leads to Aoki's outcome from [1].

Vol. 89 (2015)
On stability of the general linear equation 1469 It was proved in [2] that the estimation C x p |2 p−1 −1| is the optimum for p ≥ 0 and p = 1 in the general case.

Proof of the main result
In the sequel, T and Λ stand for maps of the forms for some α 1 , . . . , α k , β 1 , . . . , β k ∈ F, k ∈ N. Moreover we have the following lemmas, which will be needed to obtain our main result.
A. Bahyrycz, J. Olko AEM Lemma 3.1. Let X be a linear space over F, ε ∈ R + X and let Λ : R + X → R + X be given by (3.2). If there exist ω 1 , . . . , ω k ∈ R + such that Proof. Denote γ := k i=1 |α i |ω i . We prove that for every x ∈ X and l ∈ N 0 (nonnegative integers) (3.4) Obviously, the above inequality is fulfilled for l = 0. Take x ∈ X and l ∈ N 0 and assume (3.4). Thus and by induction the proof of the first assertion is completed. The second one is a consequence of the convergence of the power series.
Observe that operators (3.1) and (3.2) satisfy the assumptions of Theorem 1 in [3], therefore applying this version of the fixed point theorem and the above lemma we have the following result.

Lemma 3.2.
Let Y be a Banach space, ε ∈ R + X and let T : Y X → Y X be given by (3.1). Assume that there exist ω 1 , . . . , ω k ∈ R + such that γ := k i=1 |α i |ω i < 1 and the condition (3.3) holds. If g : X → Y satisfies the inequality