Research Article | Open Access
A Nonlinear Weakly Singular Retarded Henry-Gronwall Type Integral Inequality and Its Application
We establish a class of new nonlinear retarded weakly singular integral inequality. Under several practical assumptions, the inequality is solved by adopting novel analysis techniques, and explicit bounds for the unknown functions are given clearly. An application of our result to the fractional differential equations with delay is shown at the end of the paper.
Integral inequalities play increasingly important roles in the study of existence, uniqueness, boundedness, oscillation, stability, invariant manifolds, and other qualitative properties of solutions of ordinary differential equations and integral equations. One of the best known and widely used inequalities in the study of nonlinear differential equations is Gronwall-Bellman inequality [1, 2], which can be stated as follows. If and are nonnegative continuous functions on an interval satisfying , , then , . Many papers are devoted to different generalizations of Bellman-Gronwall inequality. Very well-known generalization of Bellman-Gronwall inequality to the nonlinear case is the Bihari inequality . In 1956, Bihari  discussed the integral inequality where is a constant. In recent years, many researchers have devoted much effort to investigating weakly singular integral inequalities. For example, Henry  proposed a linear integral inequality with singular kernel to investigate some qualitative properties for a parabolic differential equation, and Sano and Kunimatsu  gave a modified version of Henry type inequality. However, such results are expressed by a complicated power series which are sometimes inconvenient for their applications. To avoid the shortcomings of these results, Medved’  presented a new method to discuss nonlinear singular integral inequalities of Henry type and their Bihari version is as follows: and the estimates of solutions are given. From then on, more attention has been paid to such inequalities with singular kernel; see [7–24] and the references cited therein. Ye and Gao  considered the integral inequality of Henry-Gronwall type with delay and Henry-Gronwall type retarded integral inequality with singular kernel In this paper, motivated by [6, 20], we discuss the nonlinear integral inequality of Henry-Gronwall type with delay and Henry-Gronwall type nonlinear retarded integral inequality with singular kernel
2. Main Results
Throughout this paper, denotes the set of real numbers, . For convenience, before giving our main results, we cite some useful lemmas and definitions in the discussion of our proof as follows.
Definition 1 (see ). Let be a real number and . We say that a function satisfies a condition (), if where is a continuous, nonnegative function.
Lemma 2 (discrete Jensen inequality ). Let be nonnegative real numbers, is real numbers, and is a natural number. Then
Lemma 3 (see ). (1) Let ; then
where is the gamma function.
(2) Let ; then
Proof. (1) Using a change of variables and successively, we have the estimate
Since , and .
(2) Using a change of variables and successively, we have the estimate Since , , , and .
Theorem 4. Suppose that are nonnegative continuous functions on , is a nonnegative continuous function on , , and , , are constants. Suppose that the function satisfies the following conditions:(1)) condition, that is, satisfies inequality (7);(2)subadditivity, that is, for all , .
If satisfies (5), then where
Proof. Define a function by the right side of (5), that is,
Then , , and is a nonnegative, nondecreasing, and continuous function with , .
For , by the subadditivity satisfied by , we conclude Letting in (18) and integrating both sides of inequality (18) from to , we obtain where is chosen arbitrarily.
Define a function by the right side of (19), that is, Then, the function is a nonnegative, nondecreasing, and continuous function with Differentiating , we have From (22), we obtain Using (21), from (23) we obtain where are defined by (14) and (15), respectively. From (24), we observe Let in (25); we have Since is chosen arbitrarily, from (26), we have the estimation
For , using the subadditivity of and monotony of , from (17) we have Letting in (28) and integrating both sides of inequality (28) from to and using (27) we obtain where , is seen as a constant, and is defined by (16).
Define a function by the right side of (29), that is, Obviously, is a nonnegative, nondecreasing, and continuous function with Differentiating , we have From (33), we have Using (31), from (34), we have It follows that In (36), let , and then we have Since is chosen arbitrarily, from (32) and (37), we obtain the estimation Noting that , from (27) and (38), we obtain our required estimations (13).
Theorem 6. Suppose that satisfy the corresponding conditions in Theorem 4; is a constant. If satisfies (6), then the following assertions hold.
(1) Suppose . Then where is defined by (14) in Theorem 4, and is defined in (7) in Definition 1.
(2) Suppose that . Then where and .
Proof. First we will prove assertion (1). Suppose that . Using Cauchy-Schwarz inequality, we obtain from (6) that
Since satisfies () condition, using (7) in Definition 1 and (9) in Lemma 3, from (53) we derive that
for all . Using discrete Jensen inequality (8) with , , from (54) we obtain
Let and . From (55) we have
We observe that
is defined by (45). By the definitions of , and in (42), (43), and (44), from (56) we see
We observe that (58) have the same form as (5) and satisfy the corresponding conditions in Theorem 4. Applying Theorem 4 to (58), we obtain our required estimations (39).
(2) Now let us prove assertion (2). Suppose . Let ; then . Using Hölder inequality, from (6) we obtain Since satisfies () condition, using (7) and (10), from (59) we derive for all . Using Jensen inequality (8), from (60) we have Let and . Then, we obtain from (61) that We observe that where is defined by (52). Using definitions of and in (49), (50), and (51), from (62) we have We observe that (64) have the same form as (5) and satisfy the corresponding conditions in Theorem 4. Applying Theorem 4 to (64), we obtain our required estimations (46).
3. Application to Fractional Differential Equations (FDEs) with Delay
In this section, we apply our result to the following fractional differential equations (FDEs) with delay (see ): where represents the Caputo fractional derivative of order , , and is as in Theorem 6.
Theorem 7. Suppose that
where are as in Theorem 6. Let . If is any solution of IVP (65), then the following estimates hold.
(1) Suppose . Then where is defined by (14) in Theorem 4, and are defined by (7) and (45), respectively.
(2) Suppose that . Then where and ; is defined by (52).
Conflict of Interests
The authors declare that they have no competing interests.
This research was supported by the National Natural Science Foundation of China (no. 11161018), the Guangxi Natural Science Foundation (no. 2012GXNSFAA053009), the Scientific Research Foundation of the Education Department of Guangxi Province (no. LX2014330), and the Foundation of Scientific Research Project of Fujian Province Education Department of China (no. JK2012049). The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.
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