Griffith energies as small strain limit of nonlinear models for nonsimple brittle materials

1.   Introduction and preliminaries

In 1940, Ulam ^[20] suggested a problem of stability on group homomorphisms in metric groups. Hyers is the first mathematician who answered the question of Ulam in 1941. He demonstrated the following theorem in ^[8].

Theorem 1.1. Let $X$ and $Y$ be two Banach spaces and $f:X\to Y$ be a mapping such that

for some $\delta > 0$ and all $x, y\in X$. Next there is an exclusive additive mapping $A:X\to Y$ such that

for all $x\in X$.

The Hyers' stability theorem was developed by other mathematicians. Recently, numerous consequences concerning the stability of various functional equations in different normed spaces and various control functions have been obtained. The problem of stability of some functional equations have been widely explored by direct methods and there are numerous exciting outcomes regarding this problem (^{[4,7,9,10,11,13,15,17]}). The fixed point approach has been used in other Hyers-Ulam stability investigations. The relationship between Hyers-Ulam stability and fixed point theory has been defined in (^[5,6,14,16]).

The first definition of Riesz spaces was done by Riesz in 1930. In ^[18], Riesz introduced the vector lattice spaces and their properties. A Riesz space (vector lattice) is a vector space which is also a lattice, so that the two structures are compatible in a certain natural way. If, in addition, the space is a Banach space (and, again, a certain natural compatibility axiom is satisfied), it is a Banach lattice. We present some of the terms and concepts of the Riesz spaces used in this article, concisely. However, we relegate the reader to ^[1,12,19,24], for the fundamental notions and theorems of Riesz spaces and Banach lattices.

A real vector space $X$ is supposed to be a partially orderly vector space or an ordered vector space, if it is equipped with a partial ordering $"\leq"$ that satisfies

$1.$ $x \leq x$ for every $x\in X.$

$2.$ $x \leq y$ and $y \leq x$ implies that $x = y.$

$3.$ $x\leq y$ and $y\leq z$ implies that $x \leq z.$

A Riesz space (or vector lattice) is an ordered vector space in which for all $x, y\in X$ the infimum and supremum of $\{x, y\}$, denoted by $x\wedge y$ and $x\vee y$ respectively, exist in $X$. The negative part, the positive part, and the absolute value of $x\in X$, are defined by $x^{-}: = -x\vee 0, \; x^{+}: = x\vee 0, \; \text{and}\; |x|: = x\vee -x$, respectively. Let $X$ be a Riesz space. X is called Archimedean if $\inf\{\frac{x}{n}: n\in \mathbb{N}\} = 0$ for all $x\in X^+$. If $|x|\leq |y|$ implies $\|x\|\leq\|y\|$ for all $x, y\in X$, then $\|.\|$ is called a lattice norm or Riesz norm on $X$.

$1.$ $x+y = x\vee y + x\wedge y\; \; \; ,\; \; \; -(x\vee y) = -x\wedge -y.$

$2.$ $x+(y\vee z) = (x+y)\vee (x+z)\; \; \; ,\; \; \; x+(y\wedge z) = (x+y)\wedge (x+z).$

$3.$ $\vert x\vert = x^{+}+x^{-}\; \; \; ,\; \; \; \vert x+y\vert \leq \vert x \vert+\vert y\vert.$

$4.$ $x\leq y$ is equivalent to $x^{+}\leq y^{+}$ and $y^{-}\leq x^{-}$.

$5.$ $(x\vee y)\wedge z = (x\wedge y)\vee (y \wedge z)\; \; \; ,\; \; \; (x\wedge y)\vee z = (x\vee y)\wedge (y \vee z)$.

Let $X$ be a Riesz space. The sequence $\{x_{n}\}$ is called uniformly bounded if there exists an element $e\in X^{+}$ and sequence $\{a_{n}\}\in l^{1}$ such that $x_{n}\leq a_{n}\cdot e$. A Riesz space $X$ is called uniformly complete if $\sup\{\sum_{i = 1}^{n} x_{i}\; :\; n\in \mathbb{N}\}$ exists for every uniformly bounded sequence $x_{n}\subset X^{+}.$ Let $X$ and $Y$ be Banach lattices. Then the function $F:X\to Y$ is called a cone-related function if $F(X^{+}) = \{F(|x|): x\in X \}\subset Y^{+}$.

Theorem 1.2. ^[1] For a mapping $F:X\to Y$ defined between two Riesz spaces, the following statements are equivalent:

$1.$ $F$ is a lattice homomorphism.

$2.$ $F(x)^{+} = F(x^{+})$ for all $x\in X$.

$3.$ $F(x)\wedge F(y) = F(x\wedge y)$, for all $x, y\in X$.

$4.$ If $x\wedge y = 0$ in $X$, then $F(x)\wedge F(y) = 0$ in $Y.$

$5.$ $F(\vert x\vert) = \vert F(x)\vert$, for every $x\in X.$

Definition 1.3. ^[23] The (real) vector lattice (Riesz space) $X$ is named a lattice ordered algebra (Riesz algebra, concisely, $l$-algebra) if it is a linear algebra (not essentially associative) so that if $a, b\in X^+$, then $ab\in X^+$. The latter property is equal to every of the following declarations:

(ⅰ) $\vert ab\vert \leq |a||b|$ for all $a, b\in X$;

(ⅱ) $(ab)^+\leq a^+b^+ +a^-b^-$ for all $a, b\in X$;

(ⅱ) $(ab)^-\leq a^+b^- +a^-b^+$ for all $a, b\in X$.

Definition 1.4. ^[2] Assume that $X$ be an $l$-algebra.

(ⅰ) If for all $a, b\in X$, $a\wedge b = 0$ implies $ab = 0$, then $X$ is named an almost $f$-algebra.

(ⅱ) If $c(a\vee b) = ca\vee cb$ and $(a\vee b)c = ac\vee ab$ for all $a, b\in X$ and $c\in X^+$, then $X$ is named a $d$-algebra.

(ⅲ) If $a\wedge b = 0$ implies $ca\wedge b = ac\wedge b = 0$ for all $a, b\in X$ and $c\in X^ +$, then $X$ is named an $f$-algebra.

Proposition 1.5. ^[2] For an l-algebra $X$, the following statements are equivalent:

(i) $X$ is a d-algebra;

(ii) $c|a| = |ca|$ and $|a|c = |ac|$ for all $a\in X$ and $c\in X^+$;

(iii) for all $a, b\in X$ and $c\in X^+$, $c(a\vee b) = ca\vee cb$ and $(a\wedge b)c = ac\wedge bc$.

Definition 1.6. ^[21] Any lattice ordered algebra $X$ that is meantime a Banach lattice is named a Banach lattice algebra when $\|ab\|\leq \|a\|\|b\|$ keeps for all $a, b\in X^+$. Moreover, if $X$ is an $f$-algebra next it is named a Banach lattice $f$-algebra, explicitly, $X$ is next a (real) Banach algebra.

It is explicit that every $f$-algebra is an almost $f$-algebra and a $d$-algebra. Every Archimedean $f$-algebra is commutative and associative. It turns out that every Archimedean almost $f$-algebra is commutative but not necessarily associative ^[2].

Theorem 1.7. For an $l$-algebra $A$ with unit element $e > 0$, the following are equivalent:

(i) $A$ is an $f$-algebra.

(ii) $A$ is a $d$-algebra.

(iii) $A$ is an almost $f$-algebra.

(iv) $e$ is a weak order unit (i.e., $a\perp e$ implies $a = 0$).

(v) For all $a\in A$, $aa^+\geq 0$.

(vi) For all $a\in A$, $a^2 \geq 0$.

We gather several simple $f$-algebra properties. Let $A$ be an $f$-algebra. In this case, for every $a, b\in A$, we have the following:

$(1)$ $|ab| = |a||b|$.

$(2)$ $a\perp b$ implies that $ab = 0$.

$(3)$ $a^2 = (a^+)^2+(a^-)^2\geq 0$.

$(4)$ $0\leq (a^+)^2 = aa^+$.

$(5)$ $(a\vee b)(a\wedge b) = ab$.

$(6)$ If $a^2 = 0$, then $ab = 0$.

$(7)$ If $A$ is semiprime (i.e., the only nilpotent element in $A$ is 0 or, equivalently, $a^2 = 0$ in $A$ implies $a = 0$), then $a^2 \leq b^2$ if and only if $|a|\leq |b|$.

$(8)$ If $A$ is semiprime, then $a\perp b$ if and only if $ab = 0$.

$(9)$ Every unital $f$-algebra is semiprime.

See ^[3,22,23] for consider more properties of $f$-algebras.

Definition 1.8. Let $X$ and $Y$ be Banach $f$-algebras and $F:X\to Y$ be a cone-related function. We describe the following:

$(P_{1})$ Supremum preserving functional equation:

$(P_{2})$ Multiplication preserving functional equation:

$(P_{3})$ Semi-homogeneity:

$(P_{4})$ Positive Cauchy additive functional equation:

Definition 1.9. ^[16] A function $d:X\times X \to [0, \infty]$ is named a generalized metric on set $X$ if $d$ satisfies the following conditions:

$(a)$ for each $x, y\in X$, $d(x, y) = 0$ iff $x = y$;

$(b)$ for all $x, y\in X$, $d(x, y) = d(y, x)$;

$(c)$ for all $x, y, z\in X$, $d(x, z)\leq d(x, y)+d(y, z)$.

Notice that the just generalized metric significant difference from the metric is that the generalized metric range contains the infinite.

Theorem 1.10. ^[16] Let $(X, d)$ be a complete generalized metric space and $J:X\to X$ be a contractive mapping with Lipschitz constant $L < 1$. Then for every $x\in X$, either

for all nonnegative integers $n$ or there exists an integer $n_0 > 0$ such that

$(a)$ For all $n\geq n_0$, $d(J^n x, J^{n+1}x) < \infty$;

$(b)$ $J^n (x)\to y^*$, where $y^*$ is a fixed point of $J$;

$(c)$ $y^*$ is the unique fixed point of $J$ in the set $Y = \{y\in X\; :\; d(J^{n_0}(x), y) < \infty\}$;

$(d)$ For each $y\in Y$, $d(y^*, y)\leq \frac{1}{1-L}d(Jy, y)$.

2.   Main results

In this part, we will investigate the stability of lattice multiplication functional equation in Banach $f$-algebra by using the fixed point method.

Lemma 2.1. Let $X$ and $Y$ be Riesz spaces and $F:X\to Y$ be a cone-related mapping such that

or

for all $x, y, z\in X$. Then $F$ satisfies $(P1)$.

Proof. It is easy to indicate that $F(0) = 0$. Replacing $z$ by $x$ in (2.1), we get

which means that $F$ satisfies $(P1)$. By the same reasoning, we can show that if (2.2) holds, then $F$ satisfies $(P1)$.

Lemma 2.2. Let $X$ and $Y$ be Riesz spaces. If $F:X\to Y$ is a cone-related mapping, which satisfies

or

or

for all $x, y, z\in X$, then $F:X\to Y$ satisfies ($P4$).

Proof. Letting $y = x$ in (2.3), we have

for all $x, y, z\in X$ and so $F$ is a positive Cauchy additive mapping. Letting $x = y = z = 0$ in (2.4), we get $F(0) = 0$. Now putting $z = y$ in (2.4), we show that $F$ satisfies $(P2)$. Finally, letting $y = x$ in (2.5), we obtain that $F$ is a positive Cauchy additive mapping.

Theorem 2.3. Let $X$ and $Y$ be two Banach lattices. Consider a cone-related function $F:X\to Y$ with $F(0) = 0$, that is,

for all $x, y, z\in X$ and $\tau, \eta, \nu\in [1, \infty)$. Suppose that a function $\varphi: X^3\to [0, \infty)$ satisfies

for all $x, y, z\in X$, $\tau, \eta, \nu\in [1, \infty)$ and there is a number $\alpha\in [0, \frac{1}{3})$. Then there is an individual cone-related mapping $T:X\to Y$ which satisfies the properties $(P1)$, $(P3)$ and

for all $x\in X$ and $\tau\in [1, \infty)$.

Proof. Letting $x = y = z$ and $\tau = \eta = \nu$ in (2.6), we get

By (2.7), we have

Dividing by $\tau$ in the above inequality, we get

for all $x\in X$, $\tau\geq 1$ and $\alpha\in [0, \tfrac{1}{3})$.

Consider the set

and the generalized metric on $\Delta$ defined by

where, as common, $\inf\emptyset = \infty$. It is easy to demonstrate that $(\Delta, d)$ is a complete generalized metric space (see [6,Theorem 3.1]).

Now, we define the operator $J:\Delta\to \Delta$ by

Given $G, H\in \Delta$, let $c\in [0, \infty)$ be a desired constant with $d(G, H)\leq c$, that means

Then we have

By Theorem 1.10, there is a mapping $T:X\to Y$ such that the following hold.

$(1)$ $T$ is a fixed point of $J$, that means

for all $x\in X$. Also the mapping $T$ is an individual fixed point of $J$ in the set

this implies that $T$ is an individual mapping satisfying (2.9) such that there exists $c\in (0, \infty)$ satisfying

for all $x\in X$.

$(2)$ $d\left(J^n F, T\right)\to 0$, as $n\to \infty$. This implies the equality

for all $x\in X.$

$(3)$ $d(F, T)\leq \frac{1}{1-L}d(JF, F)$, which implies the inequality

So we obtain

for all $x\in X$, $\tau\geq 1$ and $\alpha\in [0, \tfrac{1}{3})$. Accordingly the inequality (2.8) holds.

Next, we show that ($P1$) is satisfied. Putting $\tau = \eta = \nu: = \tau^n$ in (2.6), we get

and so

Substituting $x, y, z$ by $\tau^{n}y, \tau^{n}y, \tau^{n}z$, respectively, in the above inequality, we get

and thus

for all $x, y, z\in X$, $\tau\geq 1$ and $\alpha\in[0, \tfrac{1}{3})$. Dividing both sides of (2.10) by $\tau^{2n}$, we obtain

Since lattice operations are continuous, by (1.2), as $n\to\infty$, we get

Therefore,

for all $x, y, z\in X$. Accordingly, by Lemma 2.1, $T$ satisfies ($P1$).

Theorem 2.4. Let $F: X\to Y$ be a cone-related mapping with $F(0) = 0$ in two uniformly complete Banach $f$-algebras $X$ and $Y$. Assume that

and

where $\varphi: X^3\to [0, \infty)$ and $\psi: X^2\to [0, \infty)$ satisfy

and

for all $x, y, z\in X$, $\tau, \eta, \nu\in [1, \infty)$, $\alpha\in [0, \tfrac{1}{3})$. Then there exists an individual cone-related mapping $T:X\to Y$ satisfying $(P1), (P2), (P3)$ and

Proof. In the previous theorem, we proved that there exists an individual cone-related mapping $T:X\to Y$ such that

for all $x\in X$ and $\tau\in [1, \infty)$. Moreover, we showed that $T$ satisfies $(P1)$ and $(P3)$. It follows from (2.11), (2.12) and (2.13) that

Thus $T$ satisfies $(P2)$.

Corollary 2.5. Suppose that $\tau > 1$, $\alpha < \tfrac{1}{3}$ and $\theta < 1$ are nonnegative real numbers. Let $X$ and $Y$ be two uniformly complete Banach $f$-algebras and $F:X\to Y$ be a cone-related mapping with $F(0) = 0$. If $F$ satisfies

and

for all $x, y, z\in X$ and $\tau, \eta, \nu\in [0, \infty)$, then there exists an individual cone-related mapping $T:X\to Y$ satisfying $(P1)$, $(P2)$, $(P3)$ and

for all $x\in X.$

Proof. Letting

in the above theorem, we get the desired result.

Theorem 2.6. Suppose that $r, \theta, \tau, \eta$ and $\nu$ are nonnegative real numbers with $\theta > 0$, $r\in [0, \frac{1}{3})$ and $\tau, \eta, \nu \in [1, \infty)$. If $X$ is a Banach lattice and $Y$ is Banach $f$-algebra and $F:X\to Y$ is a mapping such that

for all $x, y, z\in X$, then there exists an individual mapping $T:X\to Y$ such that the next properties hold:

(1) For all $x\in X$,

(2) $T:X\to Y$ satisfies ($P4$).

Proof. By induction on $n$, we can easily show that the following inequality holds.

Moreover, for all $x\in X$, we have

for all $m, n\in\mathbb{N}$ with $n\geq m$ and all $x\in X$. As $m\to \infty$, the right side term in the above inequality tends to 0 and so the sequence $\bigg\{\tfrac{1}{\tau^n}F(\tau^n |x|)\bigg\}$ is a Cauchy sequence. Hence it converges since $Y$ is complete. Therefore we can define an operator $T:X\to Y$ by

for all $x\in X$. Then we obtain

by (2.15).

Next, we show that (2) is satisfied. Putting $\tau = \nu = \eta = \tau^n$ in (2.14), we have

Replacing $x$ by $\tau^n x$, $y$ by $\tau^n y$ and $z$ by $\tau^n z$ in the above inequality, we obtain

Dividing the last inequality by $\tau^{2n}$, we have

Since lattice operations are continuous, as $n\to \infty$ we obtain

by (2.16). Therefore, by Lemma 2.2, $T$ is a positive Cauchy additive mapping.

Corollary 2.7. Suppose that $r < \frac{1}{3}$ and $\theta$ are nonnegative real numbers and that $X, Y$ are Banach $f$-algebras. If $F: X\to Y$ is a cone related mapping satisfying (2.14) such that

for all $x, y\in X$, then there is an individual cone-related mapping $T:X\to Y$ such that satisfies $(P2)$ and $(P4)$.

Proof. In the above proof, we demonstrate that $T:X\to Y$ exists and satisfies ($P4$). It follows from (2.16) and (2.17) that

Thus $T(|x||y|) = T(|x|)T(|y|)$ and so $T$ satisfies $(P2)$.

3.   Conclusion

In this paper, we have proved the Hyers-Ulam stability of supremum, infimum and multiplication preserving functional equations in Banach $f$-algebras.

Conflict of interest

The authors equally conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

The authors declare that they have no competing interests.