 Research
 Open Access
 Published:
Cyclic generalized contractions and fixed point results with applications to an integral equation
Fixed Point Theory and Applications volume 2012, Article number: 217 (2012)
Abstract
We set up a new variant of cyclic generalized contractive mappings for a map in a metric space and present existence and uniqueness results of fixed points for such mappings. Our results generalize or improve many existing fixed point theorems in the literature. To illustrate our results, we give some examples. At the same time as applications of the presented theorems, we prove an existence theorem for solutions of a class of nonlinear integral equations.
MSC:47H10, 54H25.
1 Introduction and preliminaries
All the way through this paper, by {\mathbb{R}}^{+}, we designate the set of all real nonnegative numbers, while ℕ is the set of all natural numbers.
The celebrated Banach’s [1] contraction mapping principle is one of the cornerstones in the development of nonlinear analysis. This principle has been extended and improved in many ways over the years (see, e.g., [2–5]). Fixed point theorems have applications not only in various branches of mathematics but also in economics, chemistry, biology, computer science, engineering, and other fields. In particular, such theorems are used to demonstrate the existence and uniqueness of a solution of differential equations, integral equations, functional equations, partial differential equations, and others. Owing to the magnitude, generalizations of the Banach fixed point theorem have been explored heavily by many authors. This celebrated theorem can be stated as follows.
Theorem 1.1 ([1])
Let (X,d) be a complete metric space and T be a mapping of X into itself satisfying
where k is a constant in (0,1). Then T has a unique fixed point {x}^{\ast}\in X.
Inequality (1) implies the continuity of T. A natural question is whether we can find contractive conditions which will imply the existence of a fixed point in a complete metric space but will not imply continuity.
On the other hand, cyclic representations and cyclic contractions were introduced by Kirk et al. [6]. A mapping T:A\cup B\to A\cup B is called cyclic if T(A)\subseteq B and T(B)\subseteq A, where A, B are nonempty subsets of a metric space (X,d). Moreover, T is called a cyclic contraction if there exists k\in (0,1) such that d(Tx,Ty)\le kd(x,y) for all x\in A and y\in B. Notice that although a contraction is continuous, a cyclic contraction need not to be. This is one of the important gains of this theorem.
Let (X,d) be a metric space. Let p be a positive integer, {A}_{1},{A}_{2},\dots ,{A}_{p} be nonempty subsets of X, Y={\bigcup}_{i=1}^{p}{A}_{i}, and T:Y\to Y. Then Y is said to be a cyclic representation of Y with respect to T if

(i)
{A}_{i}, i=1,2,\dots ,p are nonempty closed sets, and

(ii)
T({A}_{1})\subseteq {A}_{2},\dots ,T({A}_{p1})\subseteq {A}_{p},T({A}_{p})\subseteq {A}_{1}.
Following the paper in [6], a number of fixed point theorems on a cyclic representation of Y with respect to a selfmapping T have appeared (see, e.g., [3, 7–15]).
In this paper, we introduce a new class of cyclic generalized (\mathcal{F},\psi ,L)contractive mappings, and then investigate the existence and uniqueness of fixed points for such mappings. Our main result generalizes and improves many existing theorems in the literature. We supply appropriate examples to make obvious the validity of the propositions of our results. To end with, as applications of the presented theorems, we achieve fixed point results for a generalized contraction of integral type and we prove an existence theorem for solutions of a system of integral equations.
2 Main results
In this section, we introduce two new notions of a cyclic contraction and establish new results for such mappings.
In the sequel, we fixed the set of functions by \mathcal{F},\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) such that

(i)
ℱ is nondecreasing, continuous, and \mathcal{F}(0)=0<\mathcal{F}(t) for every t>0;

(ii)
ψ is nondecreasing, right continuous, and \psi (t)<t for every t>0.
Define {\mathbf{F}}_{1}=\{\mathcal{F}:\mathcal{F}\text{satisfies (i)}\} and {\mathrm{\Psi}}_{1}=\{\psi :\psi \text{satisfies (ii)}\}.
We state the notion of a cyclic generalized (\mathcal{F},\psi ,L)contraction as follows.
Definition 2.1 Let (X,d) be a metric space. Let p be a positive integer, {A}_{1},{A}_{2},\dots ,{A}_{p} be nonempty subsets of X and Y={\bigcup}_{i=1}^{p}{A}_{i}. An operator T:Y\to Y is said to be a cyclic generalized (\mathcal{F},\psi ,L)contraction for some \psi \in {\mathrm{\Psi}}_{1}, \mathcal{F}\in {\mathbf{F}}_{1}, and L\ge 0 if

(a)
Y={\bigcup}_{i=1}^{p}{A}_{i} is a cyclic representation of Y with respect to T;

(b)
for any (x,y)\in {A}_{i}\times {A}_{i+1}, i=1,2,\dots ,p (with {A}_{p+1}={A}_{1}),
\mathcal{F}(d(Tx,Ty))\le \psi \left(\mathcal{F}(\mathrm{\Theta}(x,y))\right)+L\mathcal{F}({\mathrm{\Theta}}_{1}(x,y)),
where
and
Our first main result is the following.
Theorem 2.1 Let (X,d) be a complete metric space, p\in \mathbb{N}, {A}_{1},{A}_{2},\dots ,{A}_{p} be nonempty closed subsets of X, and Y={\bigcup}_{i=1}^{p}{A}_{i}. Suppose T:Y\to Y is a cyclic generalized (\mathcal{F},\psi ,L)contraction mapping for some \psi \in {\mathrm{\Psi}}_{1} and \mathcal{F}\in {\mathbf{F}}_{1}. Then T has a unique fixed point. Moreover, the fixed point of T belongs to {\bigcap}_{i=1}^{p}{A}_{i}.
Proof Let {x}_{0}\in {A}_{1} (such a point exists since {A}_{1}\ne \mathrm{\varnothing}). Define the sequence \{{x}_{n}\} in X by
We shall prove that
If, for some k, we have {x}_{k+1}={x}_{k}, then (2) follows immediately. So, we can suppose that d({x}_{n},{x}_{n+1})>0 for all n. From the condition (a), we observe that for all n, there exists i=i(n)\in \{1,2,\dots ,p\} such that ({x}_{n},{x}_{n+1})\in {A}_{i}\times {A}_{i+1}. Then, from the condition (b), we have
On the other hand, we have
and
Suppose that max\{d({x}_{k1},{x}_{k}),d({x}_{k},{x}_{k+1})\}=d({x}_{k},{x}_{k+1}) for some k\in \mathbb{N}. Then \mathrm{\Theta}({x}_{k1},{x}_{k})=d({x}_{k},{x}_{k+1}), so
a contradiction. Hence,
and thus
Similarly, we have
Thus, from (4) and (5), we get
for all n\in \mathbb{N}. Now, from
and the property of ψ, we obtain {lim}_{n\to \mathrm{\infty}}\mathcal{F}(d({x}_{n+1},{x}_{n}))=0, and consequently (2) holds.
Now, we shall prove that \{{x}_{n}\} is a Cauchy sequence in (X,d). Suppose, on the contrary, that \{{x}_{n}\} is not a Cauchy sequence. Then there exists \epsilon >0 for which we can find two sequences of positive integers \{m(k)\} and \{n(k)\} such that for all positive integers k,
Further, corresponding to n(k), we can choose m(k) in such a way that it is the smallest integer with m(k)>n(k)\ge k satisfying (6). Then we have
Using (6), (7), and the triangular inequality, we get
Thus, we have
Passing to the limit as k\to \mathrm{\infty} in the above inequality and using (2), we obtain
On the other hand, for all k, there exists j(k)\in \{1,\dots ,p\} such that n(k)m(k)+j(k)\equiv 1[p]. Then {x}_{m(k)j(k)} (for k large enough, m(k)>j(k)) and {x}_{n(k)} lie in different adjacently labeled sets {A}_{i} and {A}_{i+1} for certain i\in \{1,\dots ,p\}. Using (b), we obtain
for all k. Now, we have
and
for all k. Using the triangular inequality, we get
which implies from (8) that
Using (2), we have
and
Again, using the triangular inequality, we get
Passing to the limit as k\to \mathrm{\infty} in the above inequality, using (14) and (12), we get
Similarly, we have
Passing to the limit as k\to \mathrm{\infty}, using (2) and (12), we obtain
Similarly, we have
Now, it follows from (12)(16) and the continuity of φ that
and
Passing to the limit as k\to \mathrm{\infty} in (9), using (17), (18), (19), and the condition (ii), we obtain
which is a contradiction. Thus, we proved that \{{x}_{n}\} is a Cauchy sequence in (X,d).
Since (X,d) is complete, there exists {x}^{\ast}\in X such that
We shall prove that
From the condition (a), and since {x}_{0}\in {A}_{1}, we have {\{{x}_{np}\}}_{n\ge 0}\subseteq {A}_{1}. Since {A}_{1} is closed, from (20), we get that {x}^{\ast}\in {A}_{1}. Again, from the condition (a), we have {\{{x}_{np+1}\}}_{n\ge 0}\subseteq {A}_{2}. Since {A}_{2} is closed, from (20), we get that {x}^{\ast}\in {A}_{2}. Continuing this process, we obtain (21).
Now, we shall prove that {x}^{\ast} is a fixed point of T. Indeed, from (21), since for all n there exists i(n)\in \{1,2,\dots ,p\} such that {x}_{n}\in {A}_{i(n)}, applying (b) with x={x}^{\ast} and y={x}_{n}, we obtain
for all n. On the other hand, we have
and
Passing to the limit as n\to \mathrm{\infty} in the above inequality and using (20), we obtain that
Passing to the limit as n\to \mathrm{\infty} in (22), using (23) and (20), we get
Suppose that d({x}^{\ast},T{x}^{\ast})>0. In this case, we have
which implies that
a contradiction. Then we have d({x}^{\ast},T{x}^{\ast})=0, that is, {x}^{\ast} is a fixed point of T.
Finally, we prove that {x}^{\ast} is the unique fixed point of T. Assume that {y}^{\ast} is another fixed point of T, that is, T{y}^{\ast}={y}^{\ast}. From the condition (a), this implies that {y}^{\ast}\in {\bigcap}_{i=1}^{p}{A}_{i}. Then we can apply (b) for x={x}^{\ast} and y={y}^{\ast}. We obtain
Since {x}^{\ast} and {y}^{\ast} are fixed points of T, we can show easily that \mathrm{\Theta}({x}^{\ast},{y}^{\ast})=d({x}^{\ast},{y}^{\ast}) and {\mathrm{\Theta}}_{1}({x}^{\ast},{y}^{\ast})=0. If d({x}^{\ast},{y}^{\ast})>0, we get
a contradiction. Then we have d({x}^{\ast},{y}^{\ast})=0, that is, {x}^{\ast}={y}^{\ast}. Thus, we proved the uniqueness of the fixed point. □
In the following, we deduce some fixed point theorems from our main result given by Theorem 2.1.
If we take p=1 and {A}_{1}=X in Theorem 2.1, then we get immediately the following fixed point theorem.
Corollary 2.1 Let (X,d) be a complete metric space and T:X\to X satisfy the following condition: there exist \psi \in {\mathrm{\Psi}}_{1}, \mathcal{F}\in {\mathbf{F}}_{1}, and L\ge 0 such that
for all x,y\in X. Then T has a unique fixed point.
Remark 2.1 Corollary 2.1 extends and generalizes many existing fixed point theorems in the literature [1, 16–21].
Corollary 2.2 Let (X,d) be a complete metric space, p\in \mathbb{N}, {A}_{1},{A}_{2},\dots ,{A}_{p} be nonempty closed subsets of X, Y={\bigcup}_{i=1}^{p}{A}_{i}, and T:Y\to Y. Suppose that there exist \psi \in {\mathrm{\Psi}}_{1} and \mathcal{F}\in {\mathbf{F}}_{1} such that
(a′) Y={\bigcup}_{i=1}^{p}{A}_{i} is a cyclic representation of Y with respect to T;
(b′) for any (x,y)\in {A}_{i}\times {A}_{i+1}, i=1,2,\dots ,p (with {A}_{p+1}={A}_{1}),
Then T has a unique fixed point. Moreover, the fixed point of T belongs to {\bigcap}_{i=1}^{p}{A}_{i}.
Remark 2.2 Corollary 2.2 is similar to Theorem 2.1 in [7].
Remark 2.3 Taking in Corollary 2.2 \psi (t)=kt with k\in (0,1), we obtain a generalized version of Theorem 1.3 in [6].
Corollary 2.3 Let (X,d) be a complete metric space, p\in \mathbb{N}, {A}_{1},{A}_{2},\dots ,{A}_{p} be nonempty closed subsets of X, Y={\bigcup}_{i=1}^{p}{A}_{i}, and T:Y\to Y. Suppose that there exist \psi \in {\mathrm{\Psi}}_{1} and \mathcal{F}\in {\mathbf{F}}_{1} such that
(a′) Y={\bigcup}_{i=1}^{p}{A}_{i} is a cyclic representation of Y with respect to T;
(b′) for any (x,y)\in {A}_{i}\times {A}_{i+1}, i=1,2,\dots ,p (with {A}_{p+1}={A}_{1}),
Then T has a unique fixed point. Moreover, the fixed point of T belongs to {\bigcap}_{i=1}^{p}{A}_{i}.
Remark 2.4 Taking in Corollary 2.3 \psi (t)=kt with k\in (0,1), we obtain a generalized version of Theorem 3 in [13].
Corollary 2.4 Let (X,d) be a complete metric space, p\in \mathbb{N}, {A}_{1},{A}_{2},\dots ,{A}_{p} be nonempty closed subsets of X, Y={\bigcup}_{i=1}^{p}{A}_{i}, and T:Y\to Y. Suppose that there exist \psi \in {\mathrm{\Psi}}_{1} and \mathcal{F}\in {\mathbf{F}}_{1} such that
(a′) Y={\bigcup}_{i=1}^{p}{A}_{i} is a cyclic representation of Y with respect to T;
(b′) for any (x,y)\in {A}_{i}\times {A}_{i+1}, i=1,2,\dots ,p (with {A}_{p+1}={A}_{1}),
Then T has a unique fixed point. Moreover, the fixed point of T belongs to {\bigcap}_{i=1}^{p}{A}_{i}.
Remark 2.5 Taking in Corollary 2.4 \psi (t)=kt with k\in (0,1), we obtain a generalized version of Theorem 5 in [13].
Corollary 2.5 Let (X,d) be a complete metric space, p\in \mathbb{N}, {A}_{1},{A}_{2},\dots ,{A}_{p} be nonempty closed subsets of X, Y={\bigcup}_{i=1}^{p}{A}_{i}, and T:Y\to Y. Suppose that there exist \psi \in {\mathrm{\Psi}}_{1} and \mathcal{F}\in {\mathbf{F}}_{1} such that

(a)
Y={\bigcup}_{i=1}^{p}{A}_{i} is a cyclic representation of Y with respect to T;

(b)
for any (x,y)\in {A}_{i}\times {A}_{i+1}, i=1,2,\dots ,p (with {A}_{p+1}={A}_{1}),
\mathcal{F}(d(Tx,Ty))\le \psi \left(\mathcal{F}(max\{d(x,y),d(x,Tx),d(y,Ty)\})\right).
Then T has a unique fixed point. Moreover, the fixed point of T belongs to {\bigcap}_{i=1}^{p}{A}_{i}.
We provide some examples to illustrate our obtained Theorem 2.1.
Example 2.1 Let X=\mathbb{R} with the usual metric. Suppose {A}_{1}=[1,0] and {A}_{2}=[0,1] and Y={\bigcup}_{i=1}^{2}{A}_{i}. Define T:Y\to Y such that Tx=\frac{x}{3} for all x\in Y. It is clear that {\bigcup}_{i=1}^{2}{A}_{i} is a cyclic representation of Y with respect to T. Let \psi \in {\mathrm{\Psi}}_{1} be defined by \psi (t)=\frac{t}{2} and \mathcal{F}\in {\mathbf{F}}_{1} of the form \mathcal{F}(t)=kt, k>0. For all x,y\in Y and L\ge 0, we have
So, T is a cyclic generalized (\mathcal{F},\psi ,L)contraction for any L\ge 0. Therefore, all conditions of Theorem 2.1 are satisfied (p=2), and so T has a unique fixed point (which is {x}^{\ast}=0\in {\bigcap}_{i=1}^{2}{A}_{i}).
Example 2.2 Let X=\mathbb{R} with the usual metric. Suppose {A}_{1}=[\pi /2,0] and {A}_{2}=[0,\pi /2] and Y={\bigcup}_{i=1}^{2}{A}_{i}. Define the mapping T:Y\to Y by
Clearly, we have T({A}_{1})\subseteq {A}_{2} and T({A}_{2})\subseteq {A}_{1}. Moreover, {A}_{1} and {A}_{2} are nonempty closed subsets of X. Therefore, {\bigcup}_{i=1}^{2}{A}_{i} is a cyclic representation of Y with respect to T.
Now, let (x,y)\in {A}_{1}\times {A}_{2} with x\ne 0 and y\ne 0, we have
On the other hand, we have
and
Then we have
Define the function \psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) by \psi (t)=\frac{2t}{3},\text{for all}t\ge 0 and \mathcal{F}\in {\mathbf{F}}_{1} of the form \mathcal{F}(t)=kt, k>0 and L\ge 0. Then we have
Moreover, we can show that (24) holds if x=0 or y=0. Similarly, we also get (24) holds for (x,y)\in {A}_{2}\times {A}_{1}.
Now, all the conditions of Theorem 2.1 are satisfied (with p=2), we deduce that T has a unique fixed point {x}^{\ast}\in {A}_{1}\cap {A}_{2}=\{0\}.
3 An application to an integral equation
In this section, we apply the result given by Theorem 2.1 to study the existence and uniqueness of solutions to a class of nonlinear integral equations.
We consider the nonlinear integral equation
where T>0, f:[0,T]\times \mathbb{R}\to \mathbb{R} and G:[0,T]\times [0,T]\to [0,\mathrm{\infty}) are continuous functions.
Let X=C([0,T]) be the set of real continuous functions on [0,T]. We endow X with the standard metric
It is well known that (X,{d}_{\mathrm{\infty}}) is a complete metric space.
Let (\alpha ,\beta )\in {X}^{2}, ({\alpha}_{0},{\beta}_{0})\in {\mathbb{R}}^{2} such that
We suppose that for all t\in [0,T], we have
and
We suppose that for all s\in [0,T], f(s,\cdot ) is a decreasing function, that is,
We suppose that
Finally, we suppose that, for all s\in [0,T], for all x,y\in \mathbb{R} with (x\le {\beta}_{0} and y\ge {\alpha}_{0}) or (x\ge {\alpha}_{0} and y\le {\beta}_{0}),
where \psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) is a nondecreasing function that belongs to {\mathrm{\Psi}}_{1} and L\ge 0.
Now, define the set
We have the following result.
Theorem 3.1 Under the assumptions (26)(31), problem (25) has one and only one solution {u}^{\ast}\in \mathcal{C}.
Proof Define the closed subsets of X, {A}_{1} and {A}_{2}, by
and
Define the mapping T:X\to X by
We shall prove that
Let u\in {A}_{1}, that is,
Using condition (29), since G(t,s)\ge 0 for all t,s\in [0,T], we obtain that
The above inequality with condition (27) implies that
for all t\in [0,T]. Then we have Tu\in {A}_{2}.
Similarly, let u\in {A}_{2}, that is,
Using condition (29), since G(t,s)\ge 0 for all t,s\in [0,T], we obtain that
The above inequality with condition (28) implies that
for all t\in [0,T]. Then we have Tu\in {A}_{1}. Finally, we deduce that (32) holds.
Now, let (u,v)\in {A}_{1}\times {A}_{2}, that is, for all t\in [0,T],
This implies, from condition (26), that for all t\in [0,T],
Now, using conditions (30) and (31), we can write that for all t\in [0,T], we have
This implies that
where \mathcal{F}\in {\mathbf{F}}_{1} of the form \mathcal{F}(t)=t. Using the same technique, we can show that the above inequality holds also if we take (u,v)\in {A}_{2}\times {A}_{1}.
Now, all the conditions of Theorem 2.1 are satisfied (with p=2), we deduce that T has a unique fixed point {u}^{\ast}\in {A}_{1}\cap {A}_{2}=\mathcal{C}, that is, {u}^{\ast}\in \mathcal{C} is the unique solution to (25). □
References
Banach S: Sur les opérations dans les ensembles abstraits et leur application aux equations itegrales. Fundam. Math. 1922, 3: 133–181.
Chauhan S, Sintunavarat W, Kumam P: Common fixed point theorems for weakly compatible mappings in fuzzy metric spaces using (JCLR) property. Appl. Math. 2012, 3(9):976–982. 10.4236/am.2012.39145
Pacurar M: Fixed point theory for cyclic Berinde operators. Fixed Point Theory 2011, 12(2):419–428.
Sintunavarat W, Cho YJ, Kumam P: Common fixed point theorems for c distance in ordered cone metric spaces. Comput. Math. Appl. 2011, 62: 1969–1978. 10.1016/j.camwa.2011.06.040
Sintunavarat W, Kumam P: Generalized common fixed point theorems in complex valued metric spaces and applications. J. Inequal. Appl. 2012., 2012: Article ID 84
Kirk WA, Srinivasan PS, Veeramani P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 2003, 4(1):79–89.
Pacurar M, Rus IA: Fixed point theory for cyclic φ contractions. Nonlinear Anal. 2010, 72: 1181–1187. 10.1016/j.na.2009.08.002
Agarwal RP, Alghamdi MA, Shahzad N: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 40
Aydi H, Vetro C, Sintunavarat W, Kumam P: Coincidence and fixed points for contractions and cyclical contractions in partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 124
Karapınar E: Fixed point theory for cyclic weak ϕ contraction. Appl. Math. Lett. 2011, 24(6):822–825. 10.1016/j.aml.2010.12.016
Karapınar E, Sadaranagni K: Fixed point theory for cyclic (\varphi \text{}\psi )contractions. Fixed Point Theory Appl. 2011., 2011: Article ID 69
Mongkolkeha C, Kumam P: Best proximity point theorems for generalized cyclic contractions in ordered metric spaces. J. Optim. Theory Appl. 2012, 155: 215–226. 10.1007/s109570129991y
Petric MA: Some results concerning cyclical contractive mappings. Gen. Math. 2010, 18(4):213–226.
Rus IA: Cyclic representations and fixed points. Ann. T. Popoviciu Semin. Funct. Equ. Approx. Convexity 2005, 3: 171–178.
Sintunavarat W, Kumam P: Common fixed point theorem for cyclic generalized multivalued contraction mappings. Appl. Math. Lett. 2012, 25(11):1849–1855. 10.1016/j.aml.2012.02.045
Bianchini RMT: Su un problema di S. Reich riguardante la teoria dei punti fissi. Boll. Unione Mat. Ital., Ser. IV 1972, 5: 103–108.
Boyd DW, Wong JS: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20: 458–469. 10.1090/S00029939196902395599
Chatterjea SK: Fixed point theorems. C. R. Acad. Bulgare Sci. 1972, 25: 727–730.
Hardy GE, Rogers TD: A generalization of a fixed point theorem of Reich. Can. Math. Bull. 1973, 16(2):201–206.
Kannan R: Some results on fixed points. Bull. Calcutta Math. Soc. 1968, 10: 71–76.
Reich S: Some remarks concerning contraction mappings. Can. Math. Bull. 1971, 14: 121–124. 10.4153/CMB19710249
Acknowledgements
The second author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST). Moreover, the third author was supported by the Commission on Higher Education (CHE), the Thailand Research Fund (TRF) and the King Mongkut’s University of Technology Thonburi (KMUTT) (Grant No. MRG5580213).
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Nashine, H.K., Sintunavarat, W. & Kumam, P. Cyclic generalized contractions and fixed point results with applications to an integral equation. Fixed Point Theory Appl 2012, 217 (2012). https://doi.org/10.1186/168718122012217
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/168718122012217
Keywords
 fixed point
 cyclic generalized (\mathcal{F},\psi ,L)contraction
 integral equation