Open Access

Cyclic generalized contractions and fixed point results with applications to an integral equation

  • Hemant Kumar Nashine1,
  • Wutiphol Sintunavarat2 and
  • Poom Kumam2Email author
Fixed Point Theory and Applications20122012:217

DOI: 10.1186/1687-1812-2012-217

Received: 13 June 2012

Accepted: 12 November 2012

Published: 28 November 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.

Keywords

fixed point cyclic generalized ( F , ψ , L ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq1_HTML.gif-contraction integral equation

1 Introduction and preliminaries

All the way through this paper, by R + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq2_HTML.gif, we designate the set of all real nonnegative numbers, while 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., [25]). 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq3_HTML.gif be a complete metric space and T be a mapping of X into itself satisfying
d ( T x , T y ) k d ( x , y ) , x , y X , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ1_HTML.gif
(1)

where k is a constant in ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq4_HTML.gif. Then T has a unique fixed point x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq5_HTML.gif.

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 B A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq6_HTML.gif is called cyclic if T ( A ) B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq7_HTML.gif and T ( B ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq8_HTML.gif, where A, B are nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq3_HTML.gif. Moreover, T is called a cyclic contraction if there exists k ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq9_HTML.gif such that d ( T x , T y ) k d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq10_HTML.gif for all x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq11_HTML.gif and y B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq12_HTML.gif. Notice that although a contraction is continuous, a cyclic contraction need not to be. This is one of the important gains of this theorem.

Definition 1.1 (See [6, 7])

Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq3_HTML.gif be a metric space. Let p be a positive integer, A 1 , A 2 , , A p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq13_HTML.gif be nonempty subsets of X, Y = i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq14_HTML.gif, and T : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq15_HTML.gif. Then Y is said to be a cyclic representation of Y with respect to T if
  1. (i)

    A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq16_HTML.gif, i = 1 , 2 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq17_HTML.gif are nonempty closed sets, and

     
  2. (ii)

    T ( A 1 ) A 2 , , T ( A p 1 ) A p , T ( A p ) A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq18_HTML.gif.

     

Following the paper in [6], a number of fixed point theorems on a cyclic representation of Y with respect to a self-mapping T have appeared (see, e.g., [3, 715]).

In this paper, we introduce a new class of cyclic generalized ( F , ψ , L ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq1_HTML.gif-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 F , ψ : [ 0 , + ) [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq19_HTML.gif such that
  1. (i)

    is nondecreasing, continuous, and F ( 0 ) = 0 < F ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq20_HTML.gif for every t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq21_HTML.gif;

     
  2. (ii)

    ψ is nondecreasing, right continuous, and ψ ( t ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq22_HTML.gif for every t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq23_HTML.gif.

     

Define F 1 = { F : F  satisfies (i) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq24_HTML.gif and Ψ 1 = { ψ : ψ  satisfies (ii) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq25_HTML.gif.

We state the notion of a cyclic generalized ( F , ψ , L ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq26_HTML.gif-contraction as follows.

Definition 2.1 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq3_HTML.gif be a metric space. Let p be a positive integer, A 1 , A 2 , , A p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq13_HTML.gif be nonempty subsets of X and Y = i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq14_HTML.gif. An operator T : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq27_HTML.gif is said to be a cyclic generalized ( F , ψ , L ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq1_HTML.gif-contraction for some ψ Ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq28_HTML.gif, F F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq29_HTML.gif, and L 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq30_HTML.gif if
  1. (a)

    Y = i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq31_HTML.gif is a cyclic representation of Y with respect to T;

     
  2. (b)
    for any ( x , y ) A i × A i + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq32_HTML.gif, i = 1 , 2 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq17_HTML.gif (with A p + 1 = A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq33_HTML.gif),
    F ( d ( T x , T y ) ) ψ ( F ( Θ ( x , y ) ) ) + L F ( Θ 1 ( x , y ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equa_HTML.gif
     
where
Θ ( x , y ) = max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) + d ( y , T x ) 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equb_HTML.gif
and
Θ 1 ( x , y ) = min { d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equc_HTML.gif

Our first main result is the following.

Theorem 2.1 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq34_HTML.gif be a complete metric space, p N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq35_HTML.gif, A 1 , A 2 , , A p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq36_HTML.gif be nonempty closed subsets of X, and Y = i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq37_HTML.gif. Suppose T : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq38_HTML.gif is a cyclic generalized ( F , ψ , L ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq1_HTML.gif-contraction mapping for some ψ Ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq39_HTML.gif and F F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq29_HTML.gif. Then T has a unique fixed point. Moreover, the fixed point of T belongs to i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq40_HTML.gif.

Proof Let x 0 A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq41_HTML.gif (such a point exists since A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq42_HTML.gif). Define the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq43_HTML.gif in X by
x n + 1 = T x n , n = 0 , 1 , 2 , . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equd_HTML.gif
We shall prove that
lim n d ( x n , x n + 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ2_HTML.gif
(2)
If, for some k, we have x k + 1 = x k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq44_HTML.gif, then (2) follows immediately. So, we can suppose that d ( x n , x n + 1 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq45_HTML.gif for all n. From the condition (a), we observe that for all n, there exists i = i ( n ) { 1 , 2 , , p } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq46_HTML.gif such that ( x n , x n + 1 ) A i × A i + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq47_HTML.gif. Then, from the condition (b), we have
F ( d ( x n , x n + 1 ) ) ψ ( F ( Θ ( x n 1 , x n ) ) ) + L F ( Θ 1 ( x n 1 , x n ) ) , n = 1 , 2 , . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ3_HTML.gif
(3)
On the other hand, we have
Θ ( x n 1 , x n ) = max { d ( x n 1 , x n ) , d ( x n + 1 , x n ) , 1 2 d ( x n 1 , x n + 1 ) } = max { d ( x n 1 , x n ) , d ( x n , x n + 1 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Eque_HTML.gif
and
Θ 1 ( x n 1 , x n ) = min { d ( x n 1 , x n ) , d ( x n , x n + 1 ) , d ( x n 1 , x n + 1 ) , d ( x n , x n ) } = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equf_HTML.gif
Suppose that max { d ( x k 1 , x k ) , d ( x k , x k + 1 ) } = d ( x k , x k + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq48_HTML.gif for some k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq49_HTML.gif. Then Θ ( x k 1 , x k ) = d ( x k , x k + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq50_HTML.gif, so
F ( d ( x k , x k + 1 ) ) ψ ( F ( d ( x k , x k + 1 ) ) ) < F ( d ( x k , x k + 1 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equg_HTML.gif
a contradiction. Hence,
Θ ( x n 1 , x n ) = d ( x n 1 , x n ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equh_HTML.gif
and thus
F ( d ( x n , x n + 1 ) ) ψ ( F ( d ( x n 1 , x n ) ) ) < F ( d ( x n 1 , x n ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ4_HTML.gif
(4)
Similarly, we have
F ( d ( x n 1 , x n ) ) < F ( d ( x n 2 , x n 1 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ5_HTML.gif
(5)
Thus, from (4) and (5), we get
F ( d ( x n + 1 , x n ) ) < F ( d ( x n , x n 1 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equi_HTML.gif
for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq51_HTML.gif. Now, from
F ( d ( x n + 1 , x n ) ) ψ ( F ( d ( x n , x n 1 ) ) ) < < ψ n ( F ( d ( x 1 , x 0 ) ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equj_HTML.gif

and the property of ψ, we obtain lim n F ( d ( x n + 1 , x n ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq52_HTML.gif, and consequently (2) holds.

Now, we shall prove that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq43_HTML.gif is a Cauchy sequence in ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq3_HTML.gif. Suppose, on the contrary, that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq43_HTML.gif is not a Cauchy sequence. Then there exists ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq53_HTML.gif for which we can find two sequences of positive integers { m ( k ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq54_HTML.gif and { n ( k ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq55_HTML.gif such that for all positive integers k,
m ( k ) > n ( k ) k , d ( x m ( k ) , x n ( k ) ) ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ6_HTML.gif
(6)
Further, corresponding to n ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq56_HTML.gif, we can choose m ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq57_HTML.gif in such a way that it is the smallest integer with m ( k ) > n ( k ) k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq58_HTML.gif satisfying (6). Then we have
d ( x m ( k ) 1 , x n ( k ) ) < ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ7_HTML.gif
(7)
Using (6), (7), and the triangular inequality, we get
ε d ( x n ( k ) , x m ( k ) ) d ( x n ( k ) , x m ( k ) 1 ) + d ( x m ( k ) 1 , x m ( k ) ) < ε + d ( x m ( k ) 1 , x m ( k ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equk_HTML.gif
Thus, we have
ε d ( x n ( k ) , x m ( k ) ) < ε + d ( x m ( k ) 1 , x m ( k ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equl_HTML.gif
Passing to the limit as k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq59_HTML.gif in the above inequality and using (2), we obtain
lim k d ( x n ( k ) , x m ( k ) ) = ε + . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ8_HTML.gif
(8)
On the other hand, for all k, there exists j ( k ) { 1 , , p } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq60_HTML.gif such that n ( k ) m ( k ) + j ( k ) 1 [ p ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq61_HTML.gif. Then x m ( k ) j ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq62_HTML.gif (for k large enough, m ( k ) > j ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq63_HTML.gif) and x n ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq64_HTML.gif lie in different adjacently labeled sets A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq65_HTML.gif and A i + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq66_HTML.gif for certain i { 1 , , p } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq67_HTML.gif. Using (b), we obtain
F ( d ( x m ( k ) j ( k ) + 1 , x n ( k ) + 1 ) ) ψ ( F ( Θ ( x m ( k ) j ( k ) , x n ( k ) ) ) ) + L F ( Θ 1 ( x m ( k ) j ( k ) , x n ( k ) ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ9_HTML.gif
(9)
for all k. Now, we have
Θ ( x m ( k ) j ( k ) , x n ( k ) ) = max { d ( x m ( k ) j ( k ) , x n ( k ) ) , d ( x m ( k ) j ( k ) + 1 , x m ( k ) j ( k ) ) , d ( x n ( k ) + 1 , x n ( k ) ) , d ( x m ( k ) j ( k ) , x n ( k ) + 1 ) + d ( x n ( k ) , x m ( k ) j ( k ) + 1 ) 2 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ10_HTML.gif
(10)
and
Θ 1 ( x m ( k ) j ( k ) , x n ( k ) ) = min { d ( x m ( k ) j ( k ) + 1 , x m ( k ) j ( k ) ) , d ( x n ( k ) + 1 , x n ( k ) ) , d ( x m ( k ) j ( k ) , x n ( k ) + 1 ) , d ( x n ( k ) , x m ( k ) j ( k ) + 1 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ11_HTML.gif
(11)
for all k. Using the triangular inequality, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equm_HTML.gif
which implies from (8) that
lim k d ( x m ( k ) j ( k ) , x n ( k ) ) = ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ12_HTML.gif
(12)
Using (2), we have
lim k d ( x m ( k ) j ( k ) + 1 , x m ( k ) j ( k ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ13_HTML.gif
(13)
and
lim k d ( x n ( k ) + 1 , x n ( k ) ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ14_HTML.gif
(14)
Again, using the triangular inequality, we get
| d ( x m ( k ) j ( k ) , x n ( k ) + 1 ) d ( x m ( k ) j ( k ) , x n ( k ) ) | d ( x n ( k ) , x n ( k ) + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equn_HTML.gif
Passing to the limit as k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq68_HTML.gif in the above inequality, using (14) and (12), we get
lim k d ( x m ( k ) j ( k ) , x n ( k ) + 1 ) = ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ15_HTML.gif
(15)
Similarly, we have
| d ( x n ( k ) , x m ( k ) j ( k ) + 1 ) d ( x m ( k ) j ( k ) , x n ( k ) ) | d ( x m ( k ) j ( k ) , x m ( k ) j ( k ) + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equo_HTML.gif
Passing to the limit as k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq68_HTML.gif, using (2) and (12), we obtain
lim k d ( x n ( k ) , x m ( k ) j ( k ) + 1 ) = ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ16_HTML.gif
(16)
Similarly, we have
lim k d ( x m ( k ) j ( k ) + 1 , x n ( k ) + 1 ) = ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ17_HTML.gif
(17)
Now, it follows from (12)-(16) and the continuity of φ that
lim k Θ ( x m ( k ) j ( k ) , x n ( k ) ) = max { ε , 0 } = ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ18_HTML.gif
(18)
and
lim k Θ 1 ( x m ( k ) j ( k ) , x n ( k ) ) = min { 0 , 0 , ε , ε } = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ19_HTML.gif
(19)
Passing to the limit as k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq68_HTML.gif in (9), using (17), (18), (19), and the condition (ii), we obtain
F ( ε ) ψ ( F ( ε ) ) + L 0 < F ( ε ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equp_HTML.gif

which is a contradiction. Thus, we proved that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq43_HTML.gif is a Cauchy sequence in ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq3_HTML.gif.

Since ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq3_HTML.gif is complete, there exists x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq5_HTML.gif such that
lim n x n = x . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ20_HTML.gif
(20)
We shall prove that
x i = 1 p A i . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ21_HTML.gif
(21)

From the condition (a), and since x 0 A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq41_HTML.gif, we have { x n p } n 0 A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq69_HTML.gif. Since A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq70_HTML.gif is closed, from (20), we get that x A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq71_HTML.gif. Again, from the condition (a), we have { x n p + 1 } n 0 A 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq72_HTML.gif. Since A 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq73_HTML.gif is closed, from (20), we get that x A 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq74_HTML.gif. Continuing this process, we obtain (21).

Now, we shall prove that x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq75_HTML.gif is a fixed point of T. Indeed, from (21), since for all n there exists i ( n ) { 1 , 2 , , p } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq76_HTML.gif such that x n A i ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq77_HTML.gif, applying (b) with x = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq78_HTML.gif and y = x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq79_HTML.gif, we obtain
F ( d ( T x , x n + 1 ) ) = F ( d ( T x , T x n ) ) ψ ( F ( Θ ( x , x n ) ) ) + L F ( Θ 1 ( x , x n ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ22_HTML.gif
(22)
for all n. On the other hand, we have
Θ ( x , x n ) = max { d ( x , x n ) , d ( x , T x ) , d ( x n , x n + 1 ) , d ( x , x n + 1 ) + d ( x n , T x ) 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equq_HTML.gif
and
Θ 1 ( x , x n ) = min { d ( x , T x ) , d ( x n , x n + 1 ) , d ( x , x n + 1 ) , d ( x n , T x ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equr_HTML.gif
Passing to the limit as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq80_HTML.gif in the above inequality and using (20), we obtain that
lim n Θ ( x , x n ) = max { d ( x , T x ) , 1 2 d ( x , T x ) } and lim n Θ 1 ( x , x n ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ23_HTML.gif
(23)
Passing to the limit as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq80_HTML.gif in (22), using (23) and (20), we get
F ( d ( x , T x ) ) ψ ( F ( max { d ( x , T x ) , 1 2 d ( x , T x ) } ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equs_HTML.gif
Suppose that d ( x , T x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq81_HTML.gif. In this case, we have
max { d ( x , T x ) , d ( x , T x ) 2 } = d ( x , T x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equt_HTML.gif
which implies that
F ( d ( x , T x ) ) ψ ( F ( d ( x , T x ) ) ) < F ( d ( x , T x ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equu_HTML.gif

a contradiction. Then we have d ( x , T x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq82_HTML.gif, that is, x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq75_HTML.gif is a fixed point of T.

Finally, we prove that x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq75_HTML.gif is the unique fixed point of T. Assume that y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq83_HTML.gif is another fixed point of T, that is, T y = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq84_HTML.gif. From the condition (a), this implies that y i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq85_HTML.gif. Then we can apply (b) for x = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq78_HTML.gif and y = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq86_HTML.gif. We obtain
F ( d ( x , y ) ) = F ( d ( T x , T y ) ) ψ ( F ( Θ ( x , y ) ) ) + L F ( Θ 1 ( x , y ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equv_HTML.gif
Since x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq75_HTML.gif and y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq83_HTML.gif are fixed points of T, we can show easily that Θ ( x , y ) = d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq87_HTML.gif and Θ 1 ( x , y ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq88_HTML.gif. If d ( x , y ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq89_HTML.gif, we get
F ( d ( x , y ) ) = F ( d ( T x , T y ) ) ψ ( F ( Θ ( x , y ) ) ) = ψ ( F ( d ( x , y ) ) ) < F ( d ( x , y ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equw_HTML.gif

a contradiction. Then we have d ( x , y ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq90_HTML.gif, that is, x = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq91_HTML.gif. 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq92_HTML.gif and A 1 = X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq93_HTML.gif in Theorem 2.1, then we get immediately the following fixed point theorem.

Corollary 2.1 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq3_HTML.gif be a complete metric space and T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq94_HTML.gif satisfy the following condition: there exist ψ Ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq28_HTML.gif, F F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq29_HTML.gif, and L 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq30_HTML.gif such that
F ( d ( T x , T y ) ) ψ ( F ( max { d ( x , y ) , d ( T x , x ) , d ( y , T y ) , d ( x , T y ) + d ( y , T x ) 2 } ) ) + L F ( min { d ( x , y ) , d ( T x , x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equx_HTML.gif

for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq95_HTML.gif. 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, 1621].

Corollary 2.2 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq34_HTML.gif be a complete metric space, p N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq35_HTML.gif, A 1 , A 2 , , A p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq36_HTML.gif be nonempty closed subsets of X, Y = i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq37_HTML.gif, and T : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq15_HTML.gif. Suppose that there exist ψ Ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq28_HTML.gif and F F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq29_HTML.gif such that

(a′) Y = i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq31_HTML.gif is a cyclic representation of Y with respect to T;

(b′) for any ( x , y ) A i × A i + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq32_HTML.gif, i = 1 , 2 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq17_HTML.gif (with A p + 1 = A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq33_HTML.gif),
F ( d ( T x , T y ) ) ψ ( F ( d ( x , y ) ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equy_HTML.gif

Then T has a unique fixed point. Moreover, the fixed point of T belongs to i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq40_HTML.gif.

Remark 2.2 Corollary 2.2 is similar to Theorem 2.1 in [7].

Remark 2.3 Taking in Corollary 2.2 ψ ( t ) = k t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq96_HTML.gif with k ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq9_HTML.gif, we obtain a generalized version of Theorem 1.3 in [6].

Corollary 2.3 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq34_HTML.gif be a complete metric space, p N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq35_HTML.gif, A 1 , A 2 , , A p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq36_HTML.gif be nonempty closed subsets of X, Y = i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq37_HTML.gif, and T : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq15_HTML.gif. Suppose that there exist ψ Ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq28_HTML.gif and F F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq29_HTML.gif such that

(a′) Y = i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq31_HTML.gif is a cyclic representation of Y with respect to T;

(b′) for any ( x , y ) A i × A i + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq32_HTML.gif, i = 1 , 2 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq17_HTML.gif (with A p + 1 = A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq33_HTML.gif),
F ( d ( T x , T y ) ) ψ ( F ( d ( x , T y ) + d ( y , T x ) 2 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equz_HTML.gif

Then T has a unique fixed point. Moreover, the fixed point of T belongs to i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq40_HTML.gif.

Remark 2.4 Taking in Corollary 2.3 ψ ( t ) = k t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq96_HTML.gif with k ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq9_HTML.gif, we obtain a generalized version of Theorem 3 in [13].

Corollary 2.4 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq34_HTML.gif be a complete metric space, p N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq35_HTML.gif, A 1 , A 2 , , A p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq36_HTML.gif be nonempty closed subsets of X, Y = i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq37_HTML.gif, and T : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq15_HTML.gif. Suppose that there exist ψ Ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq28_HTML.gif and F F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq29_HTML.gif such that

(a′) Y = i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq31_HTML.gif is a cyclic representation of Y with respect to T;

(b′) for any ( x , y ) A i × A i + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq32_HTML.gif, i = 1 , 2 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq17_HTML.gif (with A p + 1 = A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq33_HTML.gif),
F ( d ( T x , T y ) ) ψ ( F ( max { d ( x , T x ) , d ( y , T y ) } ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equaa_HTML.gif

Then T has a unique fixed point. Moreover, the fixed point of T belongs to i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq40_HTML.gif.

Remark 2.5 Taking in Corollary 2.4 ψ ( t ) = k t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq96_HTML.gif with k ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq9_HTML.gif, we obtain a generalized version of Theorem 5 in [13].

Corollary 2.5 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq34_HTML.gif be a complete metric space, p N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq35_HTML.gif, A 1 , A 2 , , A p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq36_HTML.gif be nonempty closed subsets of X, Y = i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq37_HTML.gif, and T : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq15_HTML.gif. Suppose that there exist ψ Ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq28_HTML.gif and F F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq29_HTML.gif such that
  1. (a)

    Y = i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq31_HTML.gif is a cyclic representation of Y with respect to T;

     
  2. (b)
    for any ( x , y ) A i × A i + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq32_HTML.gif, i = 1 , 2 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq17_HTML.gif (with A p + 1 = A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq33_HTML.gif),
    F ( d ( T x , T y ) ) ψ ( F ( max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) } ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equab_HTML.gif
     

Then T has a unique fixed point. Moreover, the fixed point of T belongs to i = 1 p A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq40_HTML.gif.

We provide some examples to illustrate our obtained Theorem 2.1.

Example 2.1 Let X = R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq97_HTML.gif with the usual metric. Suppose A 1 = [ 1 , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq98_HTML.gif and A 2 = [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq99_HTML.gif and Y = i = 1 2 A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq100_HTML.gif. Define T : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq101_HTML.gif such that T x = x 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq102_HTML.gif for all x Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq103_HTML.gif. It is clear that i = 1 2 A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq104_HTML.gif is a cyclic representation of Y with respect to T. Let ψ Ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq28_HTML.gif be defined by ψ ( t ) = t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq105_HTML.gif and F F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq29_HTML.gif of the form F ( t ) = k t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq106_HTML.gif, k > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq107_HTML.gif. For all x , y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq108_HTML.gif and L 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq30_HTML.gif, we have
F ( d ( T x , T y ) ) = k d ( T x , T y ) = k | x y | 3 k | x y | 2 k ( Θ ( x , y ) ) 2 k ( Θ ( x , y ) ) 2 + L k Θ 1 ( x , y ) = ψ ( F ( Θ ( x , y ) ) ) + L F ( Θ 1 ( x , y ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equac_HTML.gif

So, T is a cyclic generalized ( F , ψ , L ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq1_HTML.gif-contraction for any L 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq30_HTML.gif. Therefore, all conditions of Theorem 2.1 are satisfied ( p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq109_HTML.gif), and so T has a unique fixed point (which is x = 0 i = 1 2 A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq110_HTML.gif).

Example 2.2 Let X = R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq97_HTML.gif with the usual metric. Suppose A 1 = [ π / 2 , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq111_HTML.gif and A 2 = [ 0 , π / 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq112_HTML.gif and Y = i = 1 2 A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq100_HTML.gif. Define the mapping T : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq113_HTML.gif by
T x = { 1 3 x | cos ( 1 / x ) | if  x [ π / 2 , 0 ) ( 0 , π / 2 ] , 0 if  x = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equad_HTML.gif

Clearly, we have T ( A 1 ) A 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq114_HTML.gif and T ( A 2 ) A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq115_HTML.gif. Moreover, A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq70_HTML.gif and A 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq73_HTML.gif are nonempty closed subsets of X. Therefore, i = 1 2 A i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq104_HTML.gif is a cyclic representation of Y with respect to T.

Now, let ( x , y ) A 1 × A 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq116_HTML.gif with x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq117_HTML.gif and y 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq118_HTML.gif, we have
d ( T x , T y ) = | T x T y | = | 1 3 x | cos ( 1 / x ) | + 1 3 y | cos ( 1 / y ) | | = 1 3 | | x | | cos ( 1 / x ) | + | y | | cos ( 1 / y ) | | 1 3 ( | x | + | y | ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equae_HTML.gif
On the other hand, we have
| x | = x x + 1 3 | x cos ( 1 / x ) | = x 1 3 x | cos ( 1 / x ) | | x + 1 3 x | cos ( 1 / x ) | | = d ( x , T x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equaf_HTML.gif
and
| y | = y y + 1 3 | y cos ( 1 / y ) | = | y + 1 3 y | cos ( 1 / y ) | | = d ( y , T y ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equag_HTML.gif
Then we have
d ( T x , T y ) 2 3 max { d ( x , T x ) , d ( y , T y ) } 2 3 max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , ( d ( x , T y ) + d ( y , T x ) 2 ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equah_HTML.gif
Define the function ψ : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq119_HTML.gif by ψ ( t ) = 2 t 3 , for all  t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq120_HTML.gif and F F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq29_HTML.gif of the form F ( t ) = k t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq106_HTML.gif, k > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq107_HTML.gif and L 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq30_HTML.gif. Then we have
F ( d ( T x , T y ) ) ψ ( F ( Θ ( x , y ) ) ) + L F ( Θ ( x , y ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ24_HTML.gif
(24)

Moreover, we can show that (24) holds if x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq121_HTML.gif or y = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq122_HTML.gif. Similarly, we also get (24) holds for ( x , y ) A 2 × A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq123_HTML.gif.

Now, all the conditions of Theorem 2.1 are satisfied (with p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq124_HTML.gif), we deduce that T has a unique fixed point x A 1 A 2 = { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq125_HTML.gif.

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
u ( t ) = 0 T G ( t , s ) f ( s , u ( s ) ) d s for all  t [ 0 , T ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ25_HTML.gif
(25)

where T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq126_HTML.gif, f : [ 0 , T ] × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq127_HTML.gif and G : [ 0 , T ] × [ 0 , T ] [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq128_HTML.gif are continuous functions.

Let X = C ( [ 0 , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq129_HTML.gif be the set of real continuous functions on [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq130_HTML.gif. We endow X with the standard metric
d ( u , v ) = max t [ 0 , T ] | u ( t ) v ( t ) | for all  u , v X . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equai_HTML.gif

It is well known that ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq131_HTML.gif is a complete metric space.

Let ( α , β ) X 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq132_HTML.gif, ( α 0 , β 0 ) R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq133_HTML.gif such that
α 0 α ( t ) β ( t ) β 0 for all  t [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ26_HTML.gif
(26)
We suppose that for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq134_HTML.gif, we have
α ( t ) 0 T G ( t , s ) f ( s , β ( s ) ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ27_HTML.gif
(27)
and
β ( t ) 0 T G ( t , s ) f ( s , α ( s ) ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ28_HTML.gif
(28)
We suppose that for all s [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq135_HTML.gif, f ( s , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq136_HTML.gif is a decreasing function, that is,
x , y R , x y f ( s , x ) f ( s , y ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ29_HTML.gif
(29)
We suppose that
sup t [ 0 , T ] 0 T G ( t , s ) d s 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ30_HTML.gif
(30)
Finally, we suppose that, for all s [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq135_HTML.gif, for all x , y R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq137_HTML.gif with ( x β 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq138_HTML.gif and y α 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq139_HTML.gif) or ( x α 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq140_HTML.gif and y β 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq141_HTML.gif),
| f ( s , x ) f ( s , y ) | ψ ( max { | x y | , | x T x | , | y T y | , | x T y | + | y T x | 2 } ) + L min { | x T x | , | y T y | , | x T y | , | y T x | } , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ31_HTML.gif
(31)

where ψ : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq119_HTML.gif is a nondecreasing function that belongs to Ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq142_HTML.gif and L 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq30_HTML.gif.

Now, define the set
C = { u C ( [ 0 , T ] ) : α u ( t ) β  for all  t [ 0 , T ] } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equaj_HTML.gif

We have the following result.

Theorem 3.1 Under the assumptions (26)-(31), problem (25) has one and only one solution u C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq143_HTML.gif.

Proof Define the closed subsets of X, A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq70_HTML.gif and A 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq73_HTML.gif, by
A 1 = { u X : u β } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equak_HTML.gif
and
A 2 = { u X : u α } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equal_HTML.gif
Define the mapping T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq144_HTML.gif by
T u ( t ) = 0 T G ( t , s ) f ( s , u ( s ) ) d s  for all  t [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equam_HTML.gif
We shall prove that
T ( A 1 ) A 2 and T ( A 2 ) A 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equ32_HTML.gif
(32)
Let u A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq145_HTML.gif, that is,
u ( s ) β ( s ) for all  s [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equan_HTML.gif
Using condition (29), since G ( t , s ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq146_HTML.gif for all t , s [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq147_HTML.gif, we obtain that
G ( t , s ) f ( s , u ( s ) ) G ( t , s ) f ( s , β ( s ) ) for all  t , s [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equao_HTML.gif
The above inequality with condition (27) implies that
0 T G ( t , s ) f ( s , u ( s ) ) d s 0 T G ( t , s ) f ( s , β ( s ) ) d s α ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equap_HTML.gif

for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq134_HTML.gif. Then we have T u A 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq148_HTML.gif.

Similarly, let u A 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq149_HTML.gif, that is,
u ( s ) α ( s ) for all  s [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equaq_HTML.gif
Using condition (29), since G ( t , s ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq146_HTML.gif for all t , s [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq147_HTML.gif, we obtain that
G ( t , s ) f ( s , u ( s ) ) G ( t , s ) f ( s , α ( s ) ) for all  t , s [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equar_HTML.gif
The above inequality with condition (28) implies that
0 T G ( t , s ) f ( s , u ( s ) ) d s 0 T G ( t , s ) f ( s , α ( s ) ) d s β ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equas_HTML.gif

for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq134_HTML.gif. Then we have T u A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq150_HTML.gif. Finally, we deduce that (32) holds.

Now, let ( u , v ) A 1 × A 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq151_HTML.gif, that is, for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq134_HTML.gif,
u ( t ) β ( t ) , v ( t ) α ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equat_HTML.gif
This implies, from condition (26), that for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq134_HTML.gif,
u ( t ) β 0 , v ( t ) α 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equau_HTML.gif
Now, using conditions (30) and (31), we can write that for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq134_HTML.gif, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equav_HTML.gif
This implies that
F ( d ( T u , T v ) ) ψ ( F ( Θ ( u , v ) ) ) + L F ( Θ 1 ( u , v ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_Equaw_HTML.gif

where F F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq29_HTML.gif of the form F ( t ) = t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq152_HTML.gif. Using the same technique, we can show that the above inequality holds also if we take ( u , v ) A 2 × A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq153_HTML.gif.

Now, all the conditions of Theorem 2.1 are satisfied (with p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq124_HTML.gif), we deduce that T has a unique fixed point u A 1 A 2 = C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq154_HTML.gif, that is, u C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_Article_325_IEq143_HTML.gif is the unique solution to (25). □

Declarations

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).

Authors’ Affiliations

(1)
Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Mandir Hasaud
(2)
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT)

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