Abstract
We establish the existence and uniqueness of solutions for a class of nonlinear Volterra integral and integro-differential equations using fixed-point theorems for a new variant of cyclic -contractive mappings. Nontrivial examples are given to support the usability of our results.
MSC:47H10, 54H25.
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1 Introduction and preliminaries
Integral equation methods are very useful for solving many problems in several applied fields like mathematical economics and optimal control theory, because these problems are often reduced to integral equations. Since these equations usually cannot be solved explicitly, so it is necessary to get different numerical techniques. There are numerous advanced and efficient methods, which have been focusing on the solution of integral equations.
Integral equations appear in many forms. Two distinct ways that depend on the limits of integration are used to characterize integral equations, namely:
1. If the limits of integration are fixed, the integral equation is called a Fredholm integral equation given in the form:
where a and b are constants.
2. If at least one limit is a variable, the equation is called a Volterra integral equation given in the form:
It is interesting to point out that any equation that includes both integrals and derivatives of the unknown function is called an integro-differential equation. The Volterra integro-differential equation is of the form:
Nonlinear analysis is a remarkable confluence of topology, analysis and applied mathematics. Indeed, the fixed-point theory is one of the most rapidly growing topic of nonlinear functional analysis. It is a vast and interdisciplinary subject whose study belongs to several mathematical domains such as: classical analysis, functional analysis, operator theory, topology and algebraic topology, etc. This topic has grown very rapidly perhaps due to its interesting applications in various fields within and out side the mathematics such as: integral equations, initial and boundary value problems for ordinary and partial differential equations, game theory, optimal control, nonlinear oscillations, variational inequalities, complementary problems, economics and others.
The celebrated Banach contraction principle is a fundamental piece both in several branches of functional analysis and in many applications. This important fixed-point theorem can be stated as follows.
Theorem 1.1 [1]
Let be a complete metric space and be a self-map of satisfying:
where k is a constant in . Then, has a unique fixed point .
Due to its relevance, generalizations of Banach’s fixed-point theorem have been studied by many authors (see, e.g., [2] and references cited therein). Many works have been done for getting the solution of linear and nonlinear Volterra integral and integro-differential equations using Banach’s fixed-point theorem (see Pachpatte [3, 4] and references cited therein).
A very important fact that condition (1.4) implies continuity of , suggests in a natural way the question of obtaining fixed-point results for metric spaces where the involved self-map is not necessarily continuous. This question is answered by Kirk et al. [5] and turned the area of investigation of fixed point by introducing cyclic representations and cyclic contractions, which can be stated as follows:
A mapping is called cyclic if and , where , ℬ are nonempty subsets of a metric space . Moreover, is called cyclic contraction if there exists such that for all and . Notice that although a contraction is continuous, cyclic contraction need not to be. This is one of the important gains of this theorem which motivates, in a natural way, the following notion.
Let be a complete metric space. Let p be a positive integer, be nonempty subsets of , and . Then is said to be a cyclic representation of with respect to if
-
(i)
, , are nonempty closed sets, and
-
(ii)
.
Following [5], a number of fixed-point theorems on cyclic representation of with respect to a self-mapping have appeared (see, e.g., [6–14]).
To continue the investigation specified in [5], a new variant of cyclic contractive mappings satisfying generalized altering distance function, which is followed by the existence and uniqueness of fixed points for such mappings is discussed here. The obtained result generalizes and improves many existing theorems in the literature. Some examples are given in the support of our results. Finally, applications to the solutions for a class of nonlinear Volterra integral and integro-differential equation using cyclic -contraction is presented.
2 Main results
In the sequel, we designate ℝ the set of all real numbers, the set of all real nonnegative numbers by and the set of all natural numbers by ℕ.
To introduce a new variant of cyclic contraction we need the notion of different type of altering distance function.
Definition 2.1 [15]
A function is called an altering distance function if the following properties are satisfied:
-
(a)
φ is continuous and nondecreasing, and
-
(b)
.
Let Φ denote the set of all functions satisfying
-
(a)
φ is continuous nondecreasing;
-
(b)
.
Let Ψ denote the set of all functions satisfying
-
(c)
(and finite) for all ;
-
(d)
.
Let Θ denote the set of all functions satisfying
-
(e)
θ is continuous;
-
(f)
if and only if .
The examples of function ψ are given in [14]; see also [5, 16].
Now, we give some examples of functions θ satisfying (e) and (f).
Example 2.1 The following functions belong to Θ:
-
(1)
;
-
(2)
;
-
(3)
;
-
(4)
.
Now we can state the notion of cyclic -contraction as the following.
Definition 2.2 Let be a metric space. Let p be a positive integer, be nonempty subsets of and . An operator is called cyclic -contractive, if
-
(I)
is a cyclic representation of with respect to ;
-
(II)
for any , (with ),
where , and .
Example 2.2 Let with the usual metric. Suppose and and . Define such that
Clearly, and are closed subsets of . Moreover for , so that is a cyclic representation of with respect to . Furthermore, let defined by
Now we show that satisfies cyclic -contractive.
For , (or , ).
-
When and , we deduce and equation (II) is trivially satisfied.
-
When and , we deduce , , and then equation (II) holds as it reduces to
Example 2.3 Let with the usual metric. Suppose and and . Define such that for all . It is clear that is a cyclic representation of with respect to .
Define by
It can be easily shown that the map is cyclic -contractive. Indeed, let and
-
When and , we deduce and .
-
When and , we deduce and .
-
When and , we deduce and .
Similarly other cases can be discussed. Hence, is a cyclic contractive -condition.
Our main result is the following.
Theorem 2.1 Let be a complete metric space, , nonempty closed subsets of and . Suppose the mapping is cyclic -contractive, for some , and . Then has a unique fixed point. Moreover, the fixed point of belongs to .
Proof Let (such a point exists since ). Define a sequence in by:
If there is such that , then for all , so is a fixed point of and .
Then we assume that
We shall prove that the sequence is nonincreasing with
Indeed, suppose that, for some ,
Using this together with the properties of functions ψ, φ, θ, we get
which, in view of the fact that , yields
which, in turn, by condition (a) we deduce that
Then, is a nonincreasing sequence of positive real numbers. This implies that there exists such that
Since
we deduce, passing to the limit as in (2.6) and using continuity of φ, that
From condition (c) and using (2.5), we have
We get
a contradiction, and thus . Hence, (2.3) is proved.
Now, we shall prove that is a Cauchy sequence in . Suppose to the contrary that is not a Cauchy sequence, then there exists for which we can find two sequences of positive integers and such that for all positive integers k,
Using (2.7) and the triangle inequality, we get
Passing to the limit as in the above inequality and using (2.3), we obtain
On the other hand, for all k, there exists such that . Then (for k large enough, ) and lie in different adjacently labeled sets and for certain .
Using the triangle inequality, we get
which, by (2.8), implies that
Using (2.3), we have
and
Again, using the triangle inequality, we get
Passing to the limit as in the above inequality, and using (2.11) and (2.9), we get
Therefore, from the inequality
we deduce, passing to the limit as , and using (2.3) and (2.12), that
Hence, by the continuity of φ and (2.13), we get
Using (II), we obtain
for all k. Now, it follows from (2.9), (2.10), (2.11) and the properties of Φ, Ψ, Θ we get
Now, combining (2.14) with the above inequality, we get
a contradiction since . Thus, we proved that is a Cauchy sequence in .
Since is complete, there exists such that
We shall prove that
From condition (I), and since , we have . Since is closed, from (2.18), we get that . Again, from the condition (I), we have . Since is closed, from (2.18), we get that . Continuing this process, we obtain (2.19).
Now, we shall prove that ξ is a fixed point of . Indeed, from (2.19), since for all n, there exists such that , applying (II) with and , we obtain
for all n. Passing to the limit as in (2.20), and using (2.18), we get
which holds unless , so
that is, ξ is a fixed point of .
Finally, we prove that ξ is the unique fixed point of . Assume that ζ is another fixed point of , that is, . By the condition (I), this implies that . Then we can apply (II) for and . We obtain
which, by the fact that , implies
a contradiction, and thus , that is, . 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 and in Theorem 2.1, then we get immediately the following fixed-point theorem.
Corollary 2.1 Let be a complete metric space and let satisfying the following condition: there exist , and in Theorem 2.1 such that
for all . Then has a unique fixed point.
Remark 2.1 Corollary 2.1 extends and generalizes many existing fixed-point theorems in the literature [1, 9, 10, 15–24].
Now, it is easy to state a corollary of Theorem 2.1 involving a contraction of integral type.
Corollary 2.2 Let satisfy the conditions of Theorem 2.1, except that condition (II) is replaced by the following: there exists a positive Lebesgue integrable function u on such that for each and that
Then has a unique fixed point. Moreover, the fixed point of belongs to .
A number of fixed-point results may be obtained by assuming different forms for the functions ψ, φ and θ. In particular, fixed-point results under various contractive conditions may be derived from the above theorems.
For example, if we consider , , we obtain the following results.
Corollary 2.3 Let be a complete metric space, , nonempty closed subsets of , and such that
-
(I)
is a cyclic representation of with respect to ;
-
(II)
for any , (with ),
(2.22)
where and .
Then has a unique fixed point. Moreover, the fixed point of belongs to .
Taking in (2.22), we obtain the following.
Corollary 2.4 Let be a complete metric space, , nonempty closed subsets of , and such that
-
(I)
is a cyclic representation of with respect to ;
-
(II)
for any , (with ),
where .
Then has a unique fixed point. Moreover, the fixed point of belongs to .
Next, we present some examples showing how our Theorem 2.1 can be used.
Example 2.4 Considering Example 2.3, satisfy all conditions of Theorem 2.1 (), and has a unique fixed point (which is ).
Example 2.5 Let endowed with the usual metric. Assume so that . Define such that and . It is clear that is a complete metric space and is a cyclic representation of with respect to . Define also by , and by for all .
It is easy to see that satisfy condition of cyclic -contraction and so all the hypotheses of Theorem 2.1 () are satisfied and has a unique fixed point (which is ).
3 An application to integro-differential equations
In this section we present an application of Corollary 2.3 to study the existence and uniqueness of solutions to certain Volterra and Fredholm type integro-differential equations. The examples are inspired by [3].
Consider the nonlinear Volterra type integro-differential equation of the form
for , where x, g, f are real functions. We shall denote . The functions g () and f (, ) are supposed to be continuous and continuously differentiable with respect to t.
For a real-valued function x, , continuous together with its first derivative for , we denote . Denote by ℰ the space of functions which fulfill the condition
where λ is a positive constant. Define the norm in the space ℰ as
It is easy to see that ℰ with the norm defined in (3.3) is a Banach space. We note that the condition (3.2) implies that there is a constant such that , . Using this fact in (3.3), we observe that
Define a mapping by
for . Note that, if is a fixed point of , then is a solution of the problem (3.1). We shall prove the existence of a fixed point of under the following conditions.
-
(I)
There exist , such that
and for all , we have
and
-
(II)
is continuous and satisfies for ,
-
(III)
The function f and its derivative satisfy the conditions
for , , where for .
-
(IV)
There exist nonnegative constants , such that and
for , where λ is given in (3.2).
-
(V)
There exist nonnegative constants , such that
for , where λ is given in (3.2).
We have the following result for the set
Theorem 3.1 Under the assumptions (I) to (V), the integro-differential problem (3.1) has a unique solution in the set .
Proof The proof of the theorem is divided into three parts.
(A): First, we show that maps ℰ into itself.
Differentiating both sides of (3.5) with respect to t, we get
Evidently, , are continuous on J. We verify that (3.2) is fulfilled. From (3.3), (3.6) and using conditions (IV), (V) and (3.4), we have
and
Combining (3.7) and (3.8), we get
It follows from (3.9) that . This proves that maps ℰ into itself.
(B): Define closed subsets of ℰ, and by
and
We shall prove that
Let , that is,
Using condition (II), we obtain that
The inequality (3.11) with condition (I) imply that
for all . The inequality (3.12) with condition (I) imply that
for all . Hence, we have .
Similarly, if , it can be proved that holds. Thus, (3.10) is fulfilled.
(C): We verify that the operator is a cyclic -contraction map.
Let , that is, for all ,
Using the properties (3.5) and (3.6) of and conditions (III), (IV) and (V), we conclude that
and
for . Combining (3.13) and (3.14), we get
From (3.15) we obtain (with )
Using the same technique, we can show that the above inequality also holds if we take . All other conditions of Corollary 2.3 are fulfilled for the complete metric space and restricted to (with ).
We conclude that the operator has a unique fixed point , and hence the integro-differential equation (3.1) has a unique solution in the set . □
4 An application to nonlinear Volterra integral equations in two variables
In this section, we present application of Theorem 2.1 to study the existence and uniqueness of solutions to certain nonlinear Volterra integral equations.
Consider the nonlinear Volterra integral equation in two variables of the form [4]:
where f, g, h are given functions and u is the unknown function to be found.
Let the class of continuous functions from the set to the set . We denote by , ; and .
Throughout, we assume that , , .
Denote by the space of functions which fulfill the condition
where λ is a positive constant. Define the norm in the space as
It is easy to see that with the norm defined in (4.3) is a Banach space. We note that the condition (4.2) implies that there is a constant such that . Using this fact in (4.3), we observe that
Define a mapping by
for . Note that, if is a fixed point of , then is a solution of the problem (4.1).
We shall prove the existence of a fixed point of under the following conditions.
-
(I)
There exist , such that
and for all , we have
and
-
(II)
The functions g, h in equation (4.1) satisfy the conditions
where , .
-
(III)
There exist nonnegative constants , such that
and
where λ is as given in (4.2).
-
(IV)
There exist such that for and that
We have the following result for the set
Theorem 4.1 Under the assumptions (I) to (IV), the integral problem (4.1) has a unique solution in the set .
Proof We proof of the theorem in three steps.
Step 1: First we show that maps into itself.
Evidently, is continuous on and . We verify that (4.2) is fulfilled. From (4.3), and using conditions (II), (III) and (4.4), we have
It follows from (4.6) that . This proves that maps into itself.
Step 2: Define closed subsets of , and by
and
We shall prove that
Let , that is,
Using condition (II), we obtain that
and
for , .
The inequalities (4.8) and (4.9) with conditions (I) and (IV) imply that
for all . Hence, we have .
Similarly, if , it can be proved that holds. Thus, (4.7) is fulfilled.
Step 3: We verify that the operator is a cyclic generalized ψ-weakly contractive mapping.
Let , that is, for all ,
Using the properties (4.5) of and conditions (II) and (III), we conclude that
From (4.10), we obtain (with )
Considering the functions defined by
we get
Using the same technique, we can show that the above inequality also holds if we take . All other conditions of Theorem 2.1 are fulfilled for the complete metric space and restricted to (with ).
We conclude that the operator has a unique fixed point and, hence, the integro-differential equation (4.1) has a unique solution in the set . □
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
The authors thank the referee for his valuable comments and suggestions for the improvement of the manuscript. P. Kumam was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-SCEC No. NRU56000508).
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Nashine, H.K., Pathak, R., Somvanshi, P.S. et al. Solutions for a class of nonlinear Volterra integral and integro-differential equation using cyclic -contraction. Adv Differ Equ 2013, 106 (2013). https://doi.org/10.1186/1687-1847-2013-106
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DOI: https://doi.org/10.1186/1687-1847-2013-106