([27]) Let $d:\mathcal{Y}\times\mathcal{Y}\rightarrow[0,\infty)$ be a $w^{0}$-distance on $\mathcal{Y}$ and a ceiling distance of $\rho$. Suppose that $V:\mathcal{Y}\rightarrow\mathcal{Y}$ is a continuous $w$-generalized weak contraction. Then, the operator $V$ has a unique fixed point in $\mathcal{Y}$. Moreover, for each $y_{0}\in\mathcal{Y}$, the successive approximation sequence $\{y_{n}\}_{n\in\mathbb{N}},$ defined by $y_{n}=V^{n}(y_{0}),$ for all $n\in\mathbb{N}$ converges to the unique fixed point of the operator $V$.

([27]) Let $d:\mathcal{Y}\times\mathcal{Y}\rightarrow[0,\infty)$ be a continuous $w^{0}$-distance on $\mathcal{Y}$ and a ceiling distance of $\rho$. Suppose that $V:\mathcal{Y}\rightarrow\mathcal{Y}$ is a $w$-generalized weak contraction. Then, the operator $V$ has a unique fixed point in $\mathcal{Y}$. Moreover, for each $y_{0}\in\mathcal{Y}$, the successive approximation sequence $\{y_{n}\}_{n\in\mathbb{N}},$ defined by $y_{n}=V^{n}(y_{0}),$ for all $n\in\mathbb{N}$ converges to the unique fixed point of the operator $V$.

([27]) Let $d:\mathcal{Y}\times\mathcal{Y}\rightarrow[0,\infty)$ be a continuous $w$-distance on $\mathcal{Y}$ and a ceiling distance of $\rho$. Suppose that $V:\mathcal{Y}\rightarrow\mathcal{Y}$ is a continuous operator such that, for all $y_{1},y_{2}\in\mathcal{Y}$

where $\psi:[0,\infty)\rightarrow[0,\infty)$ is an altering distance function, and $\phi:[0,\infty)\rightarrow[0,\infty)$ is a continuous function with $\phi(x)=0$ if and only if $x=0$. Then, the operator $V$ has a unique fixed point in $\mathcal{Y}$. Moreover, for each $y_{0}\in\mathcal{Y}$, the successive approximation sequence $\{y_{n}\}_{n\in\mathbb{N}},$ defined by $y_{n}=V^{n}(y_{0}),$ for all $n\in\mathbb{N}$ converges to the unique fixed point of the operator $V$.

(Rus [19, 20]) (Characterization theorem) Let $(\mathcal{Y},\rho)$ be a metric space and $V:\mathcal{Y}\rightarrow\mathcal{Y}$ an operator. Then $V$ is WPO if and only if there exists a partition of $\mathcal{Y}$, $\mathcal{Y}=$ $\underset{\lambda\in\Lambda}{\cup}\mathcal{Y}_{\lambda}$, such that

• (i)

  $\mathcal{Y}_{\lambda}\in I(V)$, for all $\lambda\in\Lambda;$

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• (ii)

  $V|_{\mathcal{Y}_{\lambda}}:\mathcal{Y}_{\lambda}\rightarrow\mathcal{Y}_{\lambda}$ is PO, for all $\lambda\in\Lambda.$

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([20]) (Abstract Gronwall Lemma) Let $(\mathcal{Y},\rho,\leq)$ be an ordered metric space and $V:\mathcal{Y}\rightarrow\mathcal{Y}$ be an increasing WPO. Then we have the following:

• j)

  for $y\in\mathcal{Y},y\leq V(y)\Rightarrow y\leq V^{\infty}(y);$

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• (jj)

  for $y\in\mathcal{Y},y\geq V(y)\Rightarrow y\geq V^{\infty}(y).$

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([20]) (Abstract Comparison Lemma) Let $(\mathcal{Y},\rho,\leq)$ be an ordered metric space and $V_{1},V_{2},V_{3}:\mathcal{Y}\rightarrow\mathcal{Y}$ be such that:

• h)

  $V_{1}\leq V_{2}\leq V_{3};$

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• hh)

  the operators $V_{1},V_{2},V_{3}$ are WPO;

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• hhh)

  the operator $V_{2}$ is increasing.

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Then, for $y_{1},y_{2},y_{3}\in\mathcal{Y},y_{1}\leq y_{2}\leq y_{3}\Rightarrow V_{1}^{\infty}(y_{1})\leq V_{2}^{\infty}(y_{2})\leq V_{3}^{\infty}(y_{3})$.

We consider the integral equation (1.1) where $a_{0},a_{1},a_{2}\in\mathbb{R},\ a_{0}<a_{1}<a_{2},$ and $\varphi\in C\left([a_{1},a_{2}],\mathbb{R}\right),\ F\in C\left([a_{1},a_{2}]\times[a_{1},a_{2}]\times\mathbb{R},\mathbb{R}\right),\ g\in C([a_{1},a_{2}],[a_{0},a_{2}])$ with$\ g(x)\leq x,$ are given functions. We suppose the following:

• (i)

  the mapping $V:C([a_{0},a_{2}],\mathbb{R})\rightarrow C([a_{0},a_{2}],\mathbb{R})$ defined by (3.2) is continuous;

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• (ii)

  the altering distance function $\psi:[0,\infty)\rightarrow[0,\infty)$ satisfies $\psi(x)<x,$ for all $x>0,$ and the continuous function $\phi:[0,\infty)\rightarrow[0,\infty)$ satisfies $\phi(x)=0,$ if and only if $x=0;$

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• (iii)

  	$\displaystyle\left|F(x,s,y_{1}(g(s)))\right|+\left|F(x,s,y_{2}(g(s)))\right|\leq$
  	$\displaystyle\leq\frac{1}{a_{2}-a_{1}}\left[\psi\Big(\left|y_{1}(g(s))\right|+\left|y_{2}(g(s))\right|\Big)-\phi\Big(\underset{t\in[a_{0},a_{2}]}{\sup}\left|y_{1}(g(t))\right|+\underset{t\in[a_{0},a_{2}]}{\sup}\left|y_{2}(g(t))\right|\Big)-2\left|\varphi(x)\right|\right],$

  for all$\ y_{1},y_{2}\in C([a_{0},a_{2}],\mathbb{R}),x,s\in[a_{1},a_{2}].$

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Then the integral equation (1.1) has a unique solution;

Moreover, for each $y_{0}\in C([a_{0},a_{2}],\mathbb{R}),$ the sequence of Picard iterations $\{y_{n}\}_{n\in\mathbb{N}}$, defined by $y_{n}=V^{n}(y_{0}),$ for all $n\in\mathbb{N}$, converges to the unique solution of the integral equation (1.1).

We consider the integral equation (1.1) where $a_{1},a_{2}\in\mathbb{R},\ a_{1}<a_{2},$ and the functions $\varphi\in C\left([a_{1},a_{2}],\mathbb{R}\right),\ F\in C\left([a_{1},a_{2}]\times[a_{1},a_{2}]\times\mathbb{R},\mathbb{R}\right),\ g\in C([a_{1},a_{2}],[a_{0},a_{2}])$ with$\ g(x)\leq x$, are given. We assume that the conditions (i)-(iii) from Theorem 3.2 hold. Furthermore, we suppose that

• (iv)

  $F(x,s,\cdot):\mathbb{R}\rightarrow\mathbb{R}$ is an increasing function with respect to the last argument, for all $x,s\in[a_{1},a_{2}].$

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Let $y^{\ast}\in C([a_{0},a_{2}],\mathbb{R})$ be the unique solution of the system. Then, the following implications hold:

1. 1)

   for all $y\in C([a_{0},a_{2}],\mathbb{R})$ with

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   $$y(x)\leq\varphi(x)+{\displaystyle\int\nolimits_{a_{1}}^{x}}F(x,s,y(g(s)))ds,$$

   for all $y\in[a_{1},a_{2}]$, we have $y\leq y^{\ast};$

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2. 2)

   for all $y\in C([a_{0},a_{2}],\mathbb{R})$ with

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   $$y(x)\geq\varphi(x)+{\displaystyle\int\nolimits_{a_{1}}^{x}}F(x,s,y(g(s)))ds,$$

   for all $x\in[a_{1},a_{2}]$, we have $y\geq y^{\ast}.$

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We consider the integral equation (1.1) where $a_{1},a_{2}\in\mathbb{R},\ a_{1}<a_{2},$ and the functions $\varphi_{i}\in C\left([a_{1},a_{2}],\mathbb{R}\right),\ F_{i}\in C\left([a_{1},a_{2}]\times[a_{1},a_{2}]\times\mathbb{R},\mathbb{R}\right)$ and $g_{i}\in C([a_{1},a_{2}],[a_{0},a_{2}]),\ i=1,2,3$ are given. We assume that the conditions (i)-(iii) from Theorem 3.2 hold. Furthermore, we suppose that

• (i)

  $\varphi_{1}\leq\varphi_{2}\leq\varphi_{3},\ F_{1}\leq F_{2}\leq F_{3},\ g_{1}\leq g_{2}\leq g_{3}$;

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• (ii)

  $\varphi_{2},\ F_{2},\ g_{2}$ are increasing;

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• (iii)

  $S_{\varphi_{1}}=S_{\varphi_{2}}=S_{\varphi_{3}}.$

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Let $y_{i}\in C([a_{1},b],\mathbb{R})$ be a solution of the equation

If $y_{1}(x)\leq y_{2}(x)\leq y_{3}(x)$, $x\in[a_{1},a]$ then $y_{1}(x)\leq y_{2}(x)\leq y_{3}(x),\ x\in[a_{1},a_{2}].$