To show our results, we recall below two lemmas [15] under the assumption that \(x,y\in H^{1}(0,1)\).
Lemma 3.1
[15]
If there exists a real number
\(K_{1}\)
such that
\(\vert x(p)-y(p) \vert \leq K_{1}\)
for all
\(p\in[0,1]\), then
\(\vert {}^{\mathrm{CF}}C^{\alpha}x(p)-{{}^{\mathrm{CF}}}C^{\alpha}y(p) \vert \leq \frac{2-\alpha}{(1-\alpha)^{2}}K_{1}\)
for all
\(p \in[0,1]\).
Lemma 3.2
[15]
Assume that
\(x(0)=y(0)\)
and there exists a real number
\(K_{1}\)
such that
\(\vert x(p)-y(p) \vert \leq K_{1}\)
for
\(p\in[0,1]\). Then
\(\vert {}^{\mathrm{CF}}C^{\alpha}x(p)-{{}^{\mathrm{CF}}}C^{\alpha}y(p) \vert \leq\frac{1}{(1-\alpha)^{2}}K_{1}\)
for all
\(p\in[0,1]\).
Let \(x,y\in C_{\mathbb{R}}[0,1]\).
Lemma 3.3
[15]
If there is
\(K_{1}\geq0\)
such that
\(\vert x(p)-y(p) \vert \leq K_{1}\)
for all
\(p\in[0,1]\), then
\(\vert {}^{\mathrm{CF}}I^{\alpha}x(p)-{}^{\mathrm{CF}}I^{\alpha}y(p) \vert \leq K_{1}\)
for
\(p\in[0,1]\).
Now we are ready to show our main results. Using Lemmas 3.1 and 3.2, we obtain the next key results.
Lemma 3.4
Let
\(x,y\in H^{1}\). If there exists a real number
\(K_{1}\)
such that
\(\vert x(p)-y(p) \vert \leq K_{1}\)
for all
\(p\in[0,1]\), then
\(\vert {}^{\mathrm{CF}}\mathbb{C}_{\infty}^{(\alpha,\alpha )}x(p)-{{}^{\mathrm{CF}}}\mathbb{C}_{\infty}^{(\alpha,\alpha)}y(p) \vert \leq\frac{2-\alpha}{(1-\alpha)^{2}}K_{1}\)
for all
\(p\in[0,1]\).
Lemma 3.5
Let
\(x,y\in H^{1}\)
with
\(x(0)=y(0)\)
and
\(K_{1}\in\mathbb{R}\). If
\(\vert x(p)-y(p) \vert \leq K_{1}\)
for
\(p\in[0,1]\), then
\(\vert {}^{\mathrm{CF}}\mathbb{C}_{\infty}^{(\alpha,\alpha )}x(p)-{{}^{\mathrm{CF}}}\mathbb{C}_{\infty}^{(\alpha,\alpha)}y(p) \vert \leq\frac{1}{(1-\alpha)^{2}}K_{1}\)
for all
\(p\in[0,1]\).
Using Lemmas 3.4 and 3.5 and (*), we get the following results.
Lemma 3.6
Let
\(x,y\in H^{1}\). If there exists a real number
\(K_{1}\)
such that
\(\vert x(p)-y(p) \vert \leq K_{1}\)
for all
\(p\in[0,1]\), then
\(\vert {}^{\mathrm{CF}}\mathbb{C}_{(\mu,\infty)}^{(\alpha,\alpha )}x(p)-{{}^{\mathrm{CF}}}\mathbb{C}_{(\mu,\infty)}^{(\alpha,\alpha)}y(p) \vert \leq\frac{(2-\alpha)}{(2-\mu)(1-\alpha)^{2}} K_{1}\)
for all
\(p\in [0,1]\).
Lemma 3.7
Let
\(x,y\in H^{1}\)
with
\(x(0)=y(0)\)
and
\(K_{1}\in\mathbb{R}\). If
\(\vert x(p)-y(p) \vert \leq K_{1}\)
for all
\(p\in[0,1]\), then
\(\vert {}^{\mathrm{CF}}\mathbb{C}_{(\mu,\infty)}^{(\alpha,\alpha )}x(p)-{{}^{\mathrm{CF}}}\mathbb{C}_{(\mu,\infty)}^{(\alpha,\alpha)}y(p) \vert \leq\frac{1}{(2-\mu)(1-\alpha)^{2}} K_{1}\)
for all
\(p\in[0,1]\).
Lemma 3.8
Let
\(x,y\in C_{\mathbb{R}}[0,1]\). Let
\(K_{1}\)
be a real number such that
\(\vert x(p)-y(p) \vert \leq K_{1}\)
for all
\(p\in[0,1]\), then
\(\vert {}^{\mathrm{CF}}\mathbb{I}_{\infty}^{(\alpha,\alpha )}x(p)-{{}^{\mathrm{CF}}}\mathbb{I}_{\infty}^{(\alpha,\alpha)}y(p) \vert \leq K_{1}\)
for all
\(p\in[0,1]\).
Using Lemma 2.1, we can prove the next key result.
Lemma 3.9
Let
\(\alpha\in(0,1)\)
and
\(c\in\mathbb{R}\). The unique solution of the problem
$$ {}^{\mathrm{CF}}\mathbb{C}_{\infty}^{(\alpha,\alpha)}x(p)=y(p) $$
with boundary condition
\(x(0)=c\)
is given by
\(x(p)=c+a_{\alpha}(y(p)-y(0))+b_{\alpha}\int_{0}^{t} y (s)\,ds\).
Also, using Lemma 2.1 and (*), we can prove the next key result.
Lemma 3.10
Let
\(\alpha\in(0,1)\)
and
\(c\in\mathbb{R}\). The unique solution of the problem
$$ {}^{\mathrm{CF}}\mathbb{C}_{(\mu,\infty)}^{(\alpha,\alpha)}x(p)=y(p) $$
with boundary condition
\(x(0)=c\)
is given by
$$ x(p)=c+a_{\alpha}(2-\mu) \bigl(y(p)-y(0) \bigr)+b_{\alpha}(2-\mu) \int_{0}^{p} y (s)\,ds. $$
Let \(I=[0,1]\), and let \(\gamma,\lambda:[0,1] \times[0,1]\to [0,\infty)\) be two continuous maps such that \(\sup_{p\in I} \vert \int_{0}^{p} \lambda(p,r) \,dr \vert <\infty\) and \(\sup_{p\in I} \vert \int_{0}^{p} \gamma(p,r) \,dr \vert <\infty\). We introduce the following maps ϕ and φ defined by \((\phi u)(p)= \int_{0}^{p} \gamma(p,r)u(r)\,dr \) and \((\varphi u)(p)= \int_{0}^{p} \lambda(p,r)u(r)\,dr \), respectively. Let us consider \(\gamma_{0}=\sup \vert \int_{0}^{p} \gamma(p,r) \,dr \vert \) and \(\lambda_{0}=\sup \vert \int_{0}^{p} \lambda(p,r) \,dr \vert \), respectively. Let \(\eta(p)\in L^{\infty}(I)\) with \(\eta^{\ast}=\sup_{p\in I} \vert \eta(p) \vert \). We further are going to investigate the infinite CF fractional integro-differential problem, namely
$$ \begin{aligned}[b] {}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\alpha )}_{\infty}u_{1}^{\prime}(r)&= \mu{ \bigl(} ^{\mathrm{CF}}\mathbb{C}^{(\beta,\beta )}_{\infty}u_{1}^{\prime}(r)+^{\mathrm{CF}} \mathbb{C}^{(\gamma,\gamma)}_{\infty}u_{1}^{\prime}(r) { \bigr)} \\ &\quad{}+f^{\prime} \bigl(r,u_{1}^{\prime}(r), \bigl(\phi u_{1}^{\prime} \bigr) (r), \bigl(\varphi u_{1}^{\prime} \bigr) (r), ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta )}_{\infty} u_{1}^{\prime}(r),^{\mathrm{CF}}\mathbb{C}^{ (\delta,\delta)}_{\infty}u_{1}^{\prime}(r) \bigr) \end{aligned} $$
(1)
with \(u_{1}^{\prime}(0)=0\). Here \(\alpha,\beta,\gamma,\theta ,\delta\in (0,1)\), and \(\mu\geq0\).
Theorem 3.11
Let
\(f^{\prime}:[0,1]\times\mathbb{R}^{5}\rightarrow\mathbb{R}\)
be a continuous function satisfying
$$\begin{aligned}& \bigl\vert f^{\prime}(r,x_{1},y_{1},w_{1},u_{1},u_{2})-f^{\prime} \bigl(r,x_{1}^{\prime},y_{1}^{\prime},w_{1}^{\prime},v_{1},v_{2} \bigr) \bigr\vert \\& \quad \leq\eta(r) \bigl( \bigl\vert x_{1}-x_{1}^{\prime} \bigr\vert + \bigl\vert y_{1}-y_{1}^{\prime} \bigr\vert + \bigl\vert w_{1}-w_{1}^{\prime} \bigr\vert + \vert u_{1}-v_{1} \vert + \vert u_{2}-v_{2} \vert \bigr) \end{aligned}$$
for all
\(r\in I\)
and
\(x_{1},y_{1},w_{1},x_{1}^{\prime},y_{1}^{\prime },w_{1}^{\prime},u_{1}, u_{2},v_{1},v_{2} \in\mathbb{R}\). If
\(\Delta= [\eta^{*}(2+\gamma _{0} + \lambda_{0} + \frac{1}{(1-{\delta})^{2}})+{ \mu}( \frac{1}{(1-{\gamma})^{2} }+\frac{1}{(1-{\beta})^{2}})]<1\), then problem (1) possesses an approximate solution.
Proof
Let \(H^{1}\) be equipped with \(d(u_{1}^{\prime},v_{1}^{\prime })= \Vert u_{1}^{\prime}-v_{1}^{\prime} \Vert \), where \(\Vert u_{1}^{\prime} \Vert =\sup_{t\in I} \vert u_{1}^{\prime}(t) \vert \). Now, consider the selfmap \(F:H^{1}\to H^{1}\) defined by
$$\begin{aligned} \bigl(Fu_{1}^{\prime} \bigr) (r) &=a_{ \alpha} \bigl[ \mu{ \bigl(} ^{\mathrm{CF}}\mathbb{C}^{(\beta,\beta)}_{\infty }u_{1}^{\prime}(r)+^{\mathrm{CF}} \mathbb{C}^{(\gamma,\gamma)}_{\infty}u_{1}^{\prime}(r) { \bigr)} \\ &\quad{} {}+f^{\prime} \bigl(r,u_{1}^{\prime}(r), \bigl(\phi u_{1}^{\prime} \bigr) (r), \bigl(\varphi u_{1}^{\prime} \bigr) (r), ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta)}_{\infty} u_{1}^{\prime}(r),^{\mathrm{CF}}\mathbb{C}^{(\delta,\delta )}_{\infty} u_{1}^{\prime}(r) \bigr) \bigr] \\ &\quad {}+b_{\alpha} \int_{0}^{r} \bigl[ \mu{ \bigl(} ^{\mathrm{CF}} \mathbb{C}^{(\beta,\beta)}_{\infty} u_{1}^{\prime}(s)+^{\mathrm{CF}} \mathbb{C}^{(\gamma,\gamma)}_{\infty}u_{1}^{\prime}(s) { \bigr)} \\ &\quad {}+ f^{\prime} \bigl(s,u_{1}^{\prime}(s), \bigl(\phi u_{1}^{\prime} \bigr) (s), \bigl(\varphi u_{1}^{\prime} \bigr) (s), ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta )}_{\infty} u_{1}^{\prime}(r),^{\mathrm{CF}}\mathbb{C}^{(\delta ,\delta)}_{\infty} u_{1}^{\prime}(s) \bigr) \bigr]\,ds \end{aligned}$$
for all \(r\in I\) and \(u_{1}^{\prime},v_{1}^{\prime}\in H^{1}\), where \(a_{\alpha}\) and \(b_{\alpha}\) have the meaning given in Lemma 3.9. Now, utilizing Lemmas 3.5 and 3.8, we get
$$\begin{aligned} & \bigl\vert \bigl(Fu_{1}^{\prime} \bigr) (r) - \bigl(F v_{1}^{\prime} \bigr) (r) \bigr\vert \\ &\quad\leq a_{\alpha}\bigl( \mu \bigl\vert \bigl( ^{\mathrm{CF}} \mathbb{C}^{(\beta,\beta )}_{\infty}u_{1}^{\prime}(r)+ ^{\mathrm{CF}}\mathbb{C}^{(\gamma,\gamma)}_{\infty}u_{1}^{\prime}(r) \bigr)- \bigl( ^{\mathrm{CF}}\mathbb{C}^{(\beta,\beta )}_{\infty}v_{1}^{\prime}(r)+^{\mathrm{CF}} \mathbb{C}^{(\gamma,\gamma)}_{\infty}v_{1}^{\prime}(r) \bigr) \bigr\vert \\ &\quad\quad {}+ \bigl\vert f^{\prime} \bigl(r,u_{1}^{\prime}(r), \bigl(\phi u_{1}^{\prime} \bigr) (r), \bigl(\varphi u_{1}^{\prime} \bigr) (r), ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta)}_{\infty} u_{1}^{\prime }(r),^{\mathrm{CF}}\mathbb{C}^{(\delta,\delta)}_{\infty} u_{1}^{\prime}(r) \bigr) \\ &\quad\quad {}-f^{\prime} \bigl(r,v_{1}^{\prime}(t), \bigl( \phi v_{1}^{\prime} \bigr) (r), \bigl(\varphi v_{1}^{\prime} \bigr) (r), ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta)}_{\infty} v_{1}^{\prime}(r), ^{\mathrm{CF}}\mathbb{C}^{(\delta,\delta )}_{\infty} v_{1}^{\prime}(r) \bigr) \bigr\vert \bigr) \\ &\quad\quad {}+b_{\alpha} \int_{0}^{r} \bigl[ \mu \bigl\vert \bigl( ^{\mathrm{CF}}\mathbb{C}^{(\beta,\beta)}_{\infty}u_{1}^{\prime }(s)+^{\mathrm{CF}} \mathbb{C}^{(\gamma,\gamma)}_{\infty}u_{1}^{\prime}(s) \bigr)- \bigl( ^{\mathrm{CF}}\mathbb{C}^{(\beta,\beta)}_{\infty }v_{1}^{\prime}(s)+^{\mathrm{CF}} \mathbb{C}^{(\gamma,\gamma)}_{\infty}v_{1}^{\prime}(s) \bigr) \bigr\vert \\ &\quad\quad {}+ \bigl\vert f^{\prime} \bigl(s,u_{1}^{\prime}(r), \bigl( \phi u_{1}^{\prime} \bigr) (s), \bigl(\varphi u_{1}^{\prime} \bigr) (s), ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta)}_{\infty} u_{1}^{\prime}(s),^{\mathrm{CF}}\mathbb{C}^{(\delta,\delta )}_{\infty} u_{1}^{\prime}(s) \bigr) \\ &\quad\quad {}-f^{\prime} \bigl(s,v_{1}^{\prime}(s), \bigl( \phi v_{1}^{\prime} \bigr) (s), \bigl(\varphi v_{1}^{\prime} \bigr) (s), ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta)}_{\infty} v_{1}^{\prime}(s), ^{\mathrm{CF}}\mathbb{C}^{(\delta,\delta )}_{\infty} v_{1}^{\prime}(s) \bigr) \bigr\vert \bigr] \,ds \\ &\quad \leq a_{ \alpha} \mu \bigl[ \bigl\vert ^{\mathrm{CF}} \mathbb{C}^{(\beta,\beta)}_{\infty} \bigl(u_{1}^{\prime}(r)-v_{1}^{\prime}(r) \bigr) \bigr\vert \\ &\quad\quad {}+ \bigl\vert ^{\mathrm{CF}}\mathbb{C}^{(\gamma,\gamma)}_{\infty} \bigl(u_{1}^{\prime}(r)-v_{1}^{\prime}(r) \bigr) \bigr\vert \bigr]+a_{ \alpha} \bigl\vert \eta(r) \bigr\vert \bigl[ \bigl\vert u_{1}^{\prime}(r)-v_{1}^{\prime}(r) \bigr\vert + \bigl\vert \bigl(\phi u_{1}^{\prime} \bigr) (r)- \bigl( \phi v_{1}^{\prime} \bigr) (r) \bigr\vert \\ &\quad\quad {}+ \bigl\vert \bigl(\varphi u_{1}^{\prime} \bigr) (r)- \bigl( \varphi v_{1}^{\prime} \bigr) (r) \bigr\vert + \bigl\vert ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta)}_{\infty} u_{1}^{\prime}(r)- ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta)}_{\infty} v_{1}^{\prime}(r) \bigr\vert \\ &\quad\quad{} + \bigl\vert ^{\mathrm{CF}}\mathbb{C}^{(\delta,\delta)} _{\infty} u_{1}^{\prime}(r)-^{\mathrm{CF}} \mathbb{C}^{(\delta,\delta)}_{\infty} v_{1}^{\prime}(r) \bigr\vert \bigr] \\ &\quad\quad {}+b_{\alpha} \int_{0}^{r} \bigl[ \mu \bigl( \bigl\vert ^{\mathrm{CF}}\mathbb{C}^{(\beta,\beta)}_{\infty} \bigl(u_{1}^{\prime}(s)-v_{1}^{\prime}(s) \bigr) \bigr\vert + \bigl\vert ^{\mathrm{CF}}\mathbb{C}^{(\gamma ,\gamma)}_{\infty} \bigl(u_{1}^{\prime}(s)-v_{1}^{\prime}(s) \bigr) \bigr\vert \bigr) \\ &\quad\quad{} + \bigl\vert \eta(s) \bigr\vert \bigl( \bigl\vert u_{1}^{\prime}(s)-v_{1}^{\prime}(s) \bigr\vert \\ &\quad\quad {}+ \bigl\vert \bigl(\phi u_{1}^{\prime} \bigr) (s)- \bigl( \phi v_{1}^{\prime} \bigr) (s) \bigr\vert + \bigl\vert \bigl( \varphi u_{1}^{\prime} \bigr) (s)- \bigl(\varphi v_{1}^{\prime} \bigr) (s) \bigr\vert + \bigl\vert ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta)}_{\infty} u_{1}^{\prime}(s) - ^{\mathrm{CF}}I^{(\theta,\theta)}_{\infty} v_{1}^{\prime}(s) \bigr\vert \\ &\quad\quad {}+ \bigl\vert ^{\mathrm{CF}}\mathbb{C}^{(\delta,\delta)}_{\infty} u_{1}^{\prime}(s)-^{\mathrm{CF}}\mathbb{C}^{(\delta,\delta )}_{\infty} v_{1}^{\prime}(s) \bigr\vert \bigr) \bigr] \,ds \\ &\quad \leq \biggl[\eta^{*} \biggl(2+\gamma_{0} + \lambda _{0} + \frac{1}{(1-{\delta})^{2}} \biggr)+{ \mu}\biggl( \frac{1}{(1-{\gamma})^{2}}+ \frac{1}{(1-{\beta})^{2}}\biggr) \biggr] [a_{\alpha}+b_{\alpha}] \bigl\Vert u_{1}^{\prime}-v_{1}^{\prime} \bigr\Vert \end{aligned}$$
for all \(r\in I\) and \(u_{1}^{\prime},v_{1}^{\prime}\in H^{1}\). Hence,
$$ \bigl\Vert Fu_{1}^{\prime}-Fv_{1}^{\prime} \bigr\Vert \leq \biggl[\eta^{*} \biggl(2+\gamma_{0} + \lambda_{0} + \frac{1}{(1-{\delta})^{2}} \biggr)+{ \mu} \biggl( \frac{1}{(1-{\gamma})^{2} }+\frac{1}{(1-{\beta})^{2}} \biggr) \biggr] \bigl\Vert u_{1}^{\prime}-v_{1}^{\prime} \bigr\Vert $$
for all \(u_{1}^{\prime},v_{1}^{\prime}\in H^{1}\). Consider the mappings \(j:[0,\infty)^{5} \to[0,\infty) \) and \(\alpha :H^{1}\times H^{1}\to[0,\infty)\) defined by \(j(t_{1},t_{2},t_{3},t_{4},t_{5})= \Delta t_{1}\) and \(\alpha(t,s) =1\) for all \(t,s\in H^{1}\). We can check that \(j \in\mathcal{R}\) and F is a generalized α-contraction. From Theorem 2.2 we conclude that F possesses an approximate fixed point, which is an approximate solution for problem (1). □
Let c be a real number, and k, s, and q bounded functions on \(I=[0,1]\) with \(M_{1}=\sup_{p\in I} \vert k(p) \vert <\infty\), \(M_{2}=\sup_{p\in I} \vert s(p) \vert <\infty\), and \(M_{3}=\sup_{t\in I} \vert q(p) \vert <\infty\). We investigate the infinite coefficient-symmetric CF fractional integro-differential problem
$$ \begin{aligned}[b] ^{\mathrm{CF}}\mathbb{C}_{(\mu,\infty)}^{(\alpha ,\alpha)}x(p)&= \lambda k(p)^{\mathrm{CF}}\mathbb{C}_{\infty}^{(\delta,\delta )}x(p) +{\rho }s(p)^{\mathrm{CF}}\mathbb{I}_{\infty}^{(\theta,\theta)}x(p) \\ &\quad{}+ \int_{0}^{p} f \bigl(w,x(w),(\varphi x) (w),q(w)^{\mathrm{CF}}\mathbb{C}_{(m,\infty)}^{(\gamma, \gamma )}x(w) \bigr)\,dw \end{aligned} $$
(2)
with \(x(0)=c\), where \(\lambda,\rho\geq0\) and \(\alpha, \gamma ,\delta,\theta\in(0,1)\).
Theorem 3.12
Let
\(\xi_{1},\xi_{2},\xi_{3}\geq0\), and let
\(f:[0,1]\times\mathbb {R}^{3}\rightarrow\mathbb{R}\)
be a bounded integrable function satisfying
\(\vert f(p,x_{1},y_{1},w_{1})-f(p,x_{1}^{\prime },y_{1}^{\prime},w_{1}^{\prime}) \vert \leq \xi_{1} \vert x_{1} -x_{1}^{\prime} \vert +\xi_{2} \vert y_{1} -y_{1}^{\prime} \vert +\xi_{3} \vert w_{1}-w_{1}^{\prime} \vert \)
for all
\(p \in I\)
and
\(x_{1},y_{1},w_{1},v_{1},x_{1}^{\prime },y_{1}^{\prime},w_{1}^{\prime}\in\mathbb{R}\). If
\(\Delta= \vert 2-\mu \vert [\lambda\frac{M_{1}}{(1-\delta )^{2}}+\rho M_{2}+ \xi_{1} + \xi_{2} \gamma_{0} +\xi_{3} \frac {M_{3}}{(1-\gamma)^{2} \vert 2-m \vert }]<1\), then problem (2) admits an approximate solution.
Proof
Let \(H^{1}\) be equipped with \(d(x ,y )= \Vert x -y \Vert \), where \(\Vert x \Vert =\sup_{t\in I} \vert x(t) \vert \). Now, consider the selfmap \(\mathcal{F}:H^{1}\to H^{1}\) defined by
$$\begin{aligned} (\mathcal{F}x) (p)&=x(0)+(2-\mu)a_{\alpha} \biggl[ (\lambda k(p)^{\mathrm{CF}} \mathbb{C}_{\infty}^{(\delta,\delta)}x(p)+{ \rho}s(p)^{\mathrm{CF}} \mathbb{I}_{\infty}^{(\theta,\theta)}x(p) \\ &\quad {}+ \int_{0}^{p} f \bigl(w,x(w),(\varphi x) (w),q(w)^{\mathrm{CF}}\mathbb{C}_{(m,\infty)}^{ (\gamma, \gamma )}x(w) \bigr)\,dw \biggr] \\ &\quad{}+ b_{\alpha}(2-\mu) \int_{0}^{p} \biggl[ \lambda k(w)^{\mathrm{CF}} \mathbb{C}_{\infty }^{(\delta,\delta)}x(w)+{\rho}s(w)^{\mathrm{CF}} \mathbb{I}_{\infty }^{(\theta,\theta)}x(w)\,dw \\ &\quad {}+ \int_{0}^{w}f \bigl(r,x(r),(\varphi x) (r),q(r)^{\mathrm{CF}}\mathbb{C}_{(m,\infty)}^{ (\gamma, \gamma )}x(r) \bigr)\,dr \biggr] \,dw \end{aligned}$$
for all \(p\in I\) and \(x,y\in H^{1}\), where \(a_{\alpha}\) and \(b_{\alpha}\) are given in Lemma 3.10. As a result, utilizing Lemmas 3.5, 3.7, and 3.8, we get
$$\begin{aligned} & \biggl\vert \biggl[\lambda k(p)^{\mathrm{CF}}\mathbb{C}_{\infty }^{(\delta,\delta)}x(p)+{ \rho}s(p)^{\mathrm{CF}}\mathbb{I}_{\infty}^{(\theta,\theta)}x(p)+ \int_{0}^{p} f \bigl(w,x(w),(\varphi x) (w), q(w)^{\mathrm{CF}}\mathbb{C}_{(m,\infty)}^{(\gamma, \gamma)}x(w) \bigr)\,dw \biggr] \\ &\quad\quad{} - \biggl[\lambda k(p)^{\mathrm{CF}}\mathbb{C}_{\infty }^{(\delta,\delta)}y(p)+{ \rho} s(w)^{\mathrm{CF}}\mathbb{I}_{\infty}^{(\theta,\theta)}y(p) \\ &\quad\quad{} + \int _{0}^{p} f \bigl(w,y(w),(\varphi y) (w),q(w)^{\mathrm{CF}}\mathbb {C}_{(m,\infty)}^{(\gamma, \gamma)}y(w) \bigr)\,dw \biggr] \biggr\vert \\ &\quad \leq \biggl[\lambda\frac{M_{1}}{(1-\delta)^{2}}+\rho M_{2} \biggr] \Vert x-y \Vert + \xi_{1} \Vert x-y \Vert + \xi_{2} \gamma _{0} \Vert x-y \Vert +\xi_{3} \frac{M_{3}}{(1-\gamma)^{2} \vert 2-m \vert } \Vert x-y \Vert \\ &\quad\leq \biggl[\lambda\frac{M_{1}}{(1-\delta)^{2}}+\rho M_{2}+ \xi _{1} + \xi_{2} \gamma_{0} +\xi_{3} \frac{M_{3}}{(1-\gamma)^{2} \vert 2-m \vert } \biggr] \Vert x-y \Vert \end{aligned}$$
for all \(p\in I\) and \(x,y\in H^{1}\). As a result, we get
$$\begin{aligned} & \bigl\vert (\mathcal{F}x) (p)-(\mathcal{F}x) (p) \bigr\vert \\ &\quad \leq a_{\alpha} \vert 2-\mu \vert \biggl[\lambda\frac{M_{1}}{(1-\delta )^{2}}+\rho M_{2}+ \xi_{1} + \xi_{2} \gamma_{0} + \xi_{3} \frac{M_{3}}{(1-\gamma)^{2} \vert 2-m \vert } \biggr] \Vert x-y \Vert \\ &\quad \quad{}+b_{\alpha} \vert 2-\mu \vert \int _{0}^{p} \biggl[\lambda\frac{M_{1}}{(1-\delta)^{2}}+\rho M_{2}+ \xi_{1} + \xi_{2} \gamma_{0} + \xi_{3} \frac{M_{3}}{(1-\gamma)^{2} \vert 2-m \vert } \biggr] \Vert x-y \Vert \,ds \\ &\quad \leq \vert 2-\mu \vert \biggl[\lambda\frac{M_{1}}{(1-\delta )^{2}}+\rho M_{2}+ \xi_{1} + \xi_{2} \gamma_{0} +\xi_{3} \frac{M_{3}}{(1-\gamma)^{2} \vert 2-m \vert } \biggr] \Vert x-y \Vert \end{aligned}$$
for all \(p \in I\) and \(x,y \in H^{1}\). Now we consider the mappings \(j:[0,\infty)^{5} \to[0,\infty)\) and \(\alpha:H^{1}\times H^{1}\to[0,\infty)\) defined by \(\alpha(t,s) = 1\) and \(j(t_{1}, t_{2},t_{3},t_{4},t_{5})=\frac{\Delta}{3}(t_{1} +2t_{2})\). We can check that \(j \in\mathcal{R}\) and \(\mathcal{F}\) is a generalized α-contraction. With the help of Theorem 2.2, we conclude that \(\mathcal{F}\) possesses an approximate fixed point, which represents an approximate solution for the investigated problem (2). □
The next step is to study two applications to describe the reported results.
Example 1
Let us define \(\eta\in L^{\infty}([0,1])\) and \(\gamma ,\lambda:[0,1]\times[0,1]\to[0,\infty)\) by \(\eta(p)=\frac{\pi}{e^{(p+12)}}\), \(\gamma(p,s)=e^{ p-s}\) and \(\lambda(p,s)= \ln(5^{\sin(\pi p -s)})\). Then, we have \(\eta^{*}=\frac{\pi}{e^{12}}\), \(\gamma_{0}\leq e\), and \(\lambda_{0}\leq\ln5\). Let us consider \(\alpha=\frac{1}{5}\), \(\mu= \frac{1}{20}\), \(\beta =\frac{1}{4}\), \(\gamma=\frac{1}{2}\), \(\theta=\frac{3}{4}\), and \(\delta=\frac{3}{5}\). Consider the problem
$$ \begin{aligned}[b] ^{\mathrm{CF}}\mathbb{C}^{(\frac{1}{5},\frac {1}{5})}_{\infty}u_{1}^{\prime}(p)&= \frac{1}{20} { \bigl(}^{\mathrm{CF}}\mathbb{C}_{\infty} ^{(\frac{1}{4},\frac{1}{4})} u_{1}^{\prime}(p)+ ^{\mathrm{CF}} \mathbb{C}_{\infty} ^{(\frac{1}{2},\frac{1}{2})}u_{1}^{\prime}(p) { \bigr)} \\ &\quad{} +e^{-\pi(t+12)} \biggl[p+u_{1}^{\prime}(p)+ \int_{0}^{p} e^{ p-s}u_{1}^{\prime}(s) \,ds \\ &\quad{}+ \int_{0}^{p} \ln \bigl(5^{\sin(\pi p -s)} \bigr)u_{1}^{\prime}(s)\,ds+^{\mathrm{CF}}\mathbb{I}_{\infty}^{(\frac {3}{4},\frac{3}{4})} u_{1}^{\prime}(p)+^{\mathrm{CF}}\mathbb{C}_{\infty}^{(\frac{3}{5}, \frac{3}{5})}u_{1}^{\prime}(p) \biggr] \end{aligned} $$
(3)
with \(u_{1}^{\prime}(0)=0\). Considering \(f(p,x,y,w,u_{1},u_{2})= e^{-\pi(p+12)}(p+x+y+w+u_{1}+u_{2})\), we note that \(\Delta= [\eta^{*}(2+\gamma_{0} + \lambda_{0} + \frac{1}{(1-{\delta })^{2}})+{ \mu}( \frac{1}{(1-{\gamma})^{2}(1-{\beta})^{2}} )]<0/4447<1\). Now, by Theorem 3.11 problem (3) admits an approximate solution.
Example 2
Consider the function \(\lambda:[0,1] \times[0,1]\to[0,\infty)\) by \(\lambda(p,s )=\frac{e^{2p-s}}{e}\). Thus, \({\lambda_{0}\leq e}\). Let us consider \(\mu=3\), \(m=\frac{1}{2}\), \(\alpha=\frac{1}{4}\), \(\delta=\frac{1}{4}\), \(\theta=\frac{1}{2}\), \(\gamma=\frac{1}{2}\), \(\lambda= \frac{1}{200}\), \(\rho=\frac{1}{122}\), \(\xi_{1}=\frac{1}{320}\), \(\xi_{2}=\frac{1}{40}\), and \(\xi_{3}=\frac {1}{119}\). Let \(k(t)=\frac{2-p}{p+1}\), \(s(p)=\sin p\) and \(q(p)=\tan^{-1}(p)\). Then, \(M_{1}=\sup_{p\in[0,1]} \vert k(p) \vert =2\), \(M_{2}=\sup_{t\in[0,1]} \vert s(p) \vert =1\), and \(M_{3}=\sup_{t\in[0,1]} \vert q(p) \vert =\frac{\pi}{2} \). As a next step, we consider the problem
$$ \begin{aligned}[b] ^{\mathrm{CF}}\mathbb{C}_{(\mu,\infty)}^{(\frac {1}{4},\frac{1}{4})}x(p) &= \frac{1}{200}k(p)^{\mathrm{CF}}\mathbb{C}_{\infty}^{(\frac{1}{4}, \frac{1}{4})}x(p)+ \frac{1}{122}s(p)^{\mathrm{CF}}\mathbb{I}_{\infty}^{(\frac {1}{2},\frac{1}{2})}x(p) \\ &\quad{}+ \int_{0}^{p} \biggl[\frac{2}{56}s+ \frac{1}{320}x(s)+ \frac{1}{40} \int_{0}^{s}\frac{e^{2s-r}}{e}x(r) \,dr \\ &\quad{} + \frac{1}{119}\tan^{-1}(s) ^{\mathrm{CF}}\mathbb{C}_{(m,\infty )}^{(\frac{1}{2},\frac{1}{2})}x(s) \biggr]\,ds \end{aligned} $$
(4)
with \(x(0)=0\). Considering \(f(p,x_{1},y_{1},w_{1})= \frac{2}{56}p+\xi _{1}x_{1}+\xi_{2}y_{1}+\xi_{3}w_{1}\) for all \(p\in I\) and \(x_{1},y_{1},w_{1},v\in\mathbb{R}\), we note that
$$ \Delta= \vert 2-\mu \vert \biggl[\lambda\frac{M_{1}}{(1-\delta )^{2}}+\rho M_{2}+ \xi_{1} + \xi_{2} \gamma_{0} + \xi_{3} \frac{M_{3}}{(1-\gamma)^{2} \vert 2-m \vert } \biggr]< 0.111< 1. $$
Now, by Theorem 3.12, problem (4) admits an approximate solution.