New Generalizations of Gronwall–Bellman–Bihari-Type Integral Inequalities

Abstract

This paper develops several new generalizations of Gronwall–Bellman–Bihari-type integral inequalities. We establish three novel integral inequalities that extend classical results to more complex settings, including integrals with mixed linear and nonlinear terms, delayed (retarded) arguments, and general integral kernels. In the preliminaries, we review known Gronwall–Bellman–Bihari inequalities and useful lemmas. In the main results, we present at least three new theorems. The first theorem provides an explicit bound for solutions of an integral inequality involving a separable kernel function and a nonlinear (Bihari-type) term, significantly extending the classical Bihari inequality. The second theorem addresses integral inequalities with delayed arguments, showing that the delay does not enlarge the growth bound compared to the non-delay case. The third theorem handles inequalities with combined linear and nonlinear terms; using a monotone iterative technique, we prove the existence of a maximal solution that bounds any solution of the inequality. Rigorous proofs are given for all main results. In the Applications section, we illustrate how these inequalities can be applied to deduce qualitative properties of differential equations. As an example, we prove a uniqueness result for an initial value problem with a non-Lipschitz nonlinear term using our new inequalities. The paper concludes with a summary of results and a brief discussion of potential further generalizations. Our results provide powerful tools for researchers to obtain a priori bounds and uniqueness criteria for various differential, integral, and functional equations. It is important to note that the integral inequalities established in this work provide bounds on the solution under the assumption of its existence on the considered interval $[t_0,T]$. For nonlinear differential or integral equations where the nonlinearity $F$ fails to be Lipschitz continuous, solutions may develop movable singularities (blow-up) in finite time. The bounds derived from our Gronwall–Bellman–Bihari-type inequalities are valid only on the maximal interval of existence of the solution. Determining the region where solutions are guaranteed to be free of such singularities is a separate and profound problem, often requiring additional techniques such as the construction of Lyapunov functions or the use of differential comparison principles. The primary contribution of this paper is to provide sharp estimates and uniqueness criteria within the domain where a solution is known to exist a priori.

1. Introduction

Integral inequalities of Gronwall–Bellman–Bihari-type are fundamental tools in the analysis of differential and integral equations. These inequalities provide explicit bounds on unknown functions that satisfy certain integral constraints, thereby allowing one to derive estimates for solutions and to establish qualitative properties such as boundedness, stability, and uniqueness [1,2]. The classical Gronwall–Bellman inequality was first introduced by Grönwall (1919) and Bellman (1943) in their studies of differential equations [3,4]. In its basic form, Gronwall’s inequality states that if $u(t)$ is a nonnegative continuous function on $[0, T]$ satisfying an integral inequality of linear type, then $u(t)$ is bounded by an exponential function. For example, the simplest form is: if

$$u(t)\le C+{\int }_0^t P(s)u(s)ds,t\in [0,T],$$

(1)

where $C \ge 0$ and $P(t)$ is continuous and nonnegative, then one obtains the bound

$$u(t)\le C\mathrm{e}\mathrm{x}\mathrm{p}\left({\int }_0^t P(s)ds\right),t\in [0,T],$$

(2)

provided $u(t)$ is locally integrable [1]. This classical result (often referred to as the Gronwall–Bellman lemma) is widely used to compare solutions of differential equations and to prove uniqueness in initial value problems.

In 1956, Bihari established a nonlinear generalization of the Gronwall–Bellman inequality [5]. Bihari’s inequality allows the integrand to involve a nonlinear function of $u(s)$, greatly extending the applicability to problems with nonlinear growth conditions. In particular, Bihari’s result asserts that if $u(t)$ satisfies an inequality of the form

$$u(t)\le A+{\int }_{t_0}^t b(s)F(u(s))ds,$$

(3)

where $A \ge 0$, $b(s) \ge 0$, and $F(u)$ is positive and nondecreasing, then one can derive an explicit bound for $u(t)$ by solving a functional equation involving the inverse of F. Specifically, defining

$$H(x)={\int }_{x_0}^x {\displaystyle \frac{1}{F(\xi )}}d\xi .$$

(4)

(for some $x_0 > 0$ in the range of u), Bihari’s lemma yields

$$H(u(t))\le H(A)+{\int }_{t_0}^t b(s)ds,$$

(5)

which leads to the bound

$$u(t)\le H^{-1}\left(H(A)+{\int }_{t_0}^t b(s)ds\right).$$

(6)

This nonlinear Gronwall–Bihari inequality recovers the classical linear case (1)–(2) when $\mathrm{F}(\mathrm{u})$ is linear in $\mathrm{u}$. Following Bihari’s pioneering work, many authors have contributed further extensions and refinements of these integral inequalities. For instance, Butler and Rogers (1971) obtained a generalized lemma of Bihari for pointwise integral equations [6], and Pachpatte (1973, 1975) established several integral inequalities of Gronwall–Bellman–Pachpatte-type [7,8]. These classical extensions laid the groundwork for a vast literature on integral inequalities. Comprehensive references and monographs on this topic can be found in Dragomir’s work [9] and in Bainov and Simeonov, among others.

Over the years, numerous generalizations of Gronwall–Bellman–Bihari inequalities have been developed to address more complex equations and difference schemes. For example, there are results extending Gronwall’s inequality to retarded (delay) arguments [10] and to time scales (unifying continuous and discrete cases) [11]. Stochastic versions of Gronwall-type inequalities have been formulated to handle stochastic differential equations and backward stochastic equations. In the context of fractional calculus, many authors have established fractional Gronwall–Bihari inequalities for non-integer order integrals and derivatives [12]. Notably, Ye et al. (2007) formulated a Gronwall inequality suitable for fractional differential equations [12], and Abdeljawad et al. (2016) proved q-fractional analogues on time scales [13]. More recent works have provided Gronwall–Bellman inequalities in the framework of generalized fractional operators and for $\psi$-Hilfer fractional derivatives. Some studies also address weakly singular kernels (the so-called Henry–Gronwall inequalities) that arise in fractional equations with singular kernels. For instance, Foukrach and Meftah (2020) obtained Gronwall–Bihari-type inequalities with a singular coefficient and applied them to fractional difference equations [14,15]. Very recently, Lan and Webb (2023) introduced a new Bihari-type inequality under integrability conditions (replacing classical continuity assumptions) to study global solutions of fractional differential equations [16,17]. The broad spectrum of these developments highlights the continued importance of Gronwall–Bellman–Bihari inequalities in contemporary research.

The practical significance of these specialized inequalities extends across multiple disciplines in science and engineering. In control theory, they provide stability bounds for nonlinear systems with time delays. In population dynamics, they help analyze the long-term behavior of species interaction models with maturation periods. In materials science, they appear in viscoelasticity problems where the stress depends on the history of strain through integral operators. Furthermore, in financial mathematics, such inequalities are instrumental in bounding solutions to stochastic volatility models. The development of more refined integral inequalities directly impacts our ability to obtain quantitative estimates and prove existence-uniqueness results for these complex systems, which often resist closed-form solutions.

Despite the extensive literature, there is ongoing interest in formulating sharper and more general integral inequalities to handle increasingly complex systems. In this paper, we contribute to this line of research by presenting new generalizations of Gronwall–Bellman–Bihari-type inequalities. Our focus is on providing explicit bounds for integral inequalities that simultaneously involve linear and nonlinear terms, as well as those with delay arguments and general integral kernels not covered by existing results. The main contributions of this work are summarized as follows:

(1)

New mixed linear–nonlinear inequality (Theorem 1). We derive an explicit bound for $\mathrm{u}(\mathrm{t})$ satisfying an integral inequality that contains both a linear term (proportional to $\mathrm{u}(\mathrm{s})$) and a nonlinear term $\mathrm{F}(\mathrm{u}(\mathrm{s}))$ under the integral. This result unifies and extends the classical Gronwall (linear) and Bihari (nonlinear) inequalities by handling them in one inequality. The obtained bound is given in closed form via the inverse of a certain auxiliary function, generalizing the exponential estimate (2).

(2)

Retarded Gronwall–Bihari inequality (Theorem 2). We establish that if an integral inequality involves a delay (i.e., $\mathrm{u}(\mathsf{\sigma }(\mathrm{s}))$ with $\mathsf{\sigma }(\mathrm{s})\le \mathrm{s}$ inside the integral), then the same bound as in the non-delayed case holds. In other words, the presence of a delay does not worsen the growth estimate. This result formalizes and generalizes the intuition behind known retarded Gronwall inequalities [18].

(3)

Combined Gronwall–Bihari inequality with iterative solution (Theorem 3). For the most general case with both linear and nonlinear terms, we prove that there exists a unique maximal solution to the associated Volterra integral equation which serves as an upper bound for any function satisfying the inequality. We provide a constructive proof using a monotone iterative sequence, which yields not only the existence of the extremal solution but also establishes the inequality $\mathrm{u}(\mathrm{t})\le \mathrm{y}(\mathrm{t})$, where $\mathrm{y}(\mathrm{t})$ is this maximal solution. This result is particularly useful when an explicit closed form bound is difficult to obtain; it guarantees that the inequality has an optimally tight bound function.

In the sequel, we first present some preliminaries and known lemmas in Section 2. The main results (Theorems 1–3) are given in Section 3 with detailed proofs. In Section 4, we illustrate the applications of our inequalities by discussing an example on the uniqueness of solutions to a differential equation with a non-Lipschitz nonlinearity. Finally, Section 5 concludes the paper with a summary and remarks on future work. Throughout the paper, we assume all functions are real-valued and continuous (or piecewise continuous) on their domains, and we restrict attention to functions $\mathrm{u}(\mathrm{t})$ that are nonnegative, as is standard for Gronwall–Bihari-type inequalities (extensions to sign-changing functions typically require minor modifications).

2. Preliminaries and Lemmas

In this section, we summarize the classical Gronwall–Bellman–Bihari inequalities and related lemmas that will be used in proving our main theorems. For completeness, we state these results in forms convenient for our purposes, and we refer to the original sources for proofs. Throughout, let $[{\mathrm{t}}_0,\mathrm{T}]$ be a fixed finite interval with ${\mathrm{t}}_0<\mathrm{T}$. We denote by $\mathrm{C}([{\mathrm{t}}_0,\mathrm{T}])$ the space of continuous functions on $[{\mathrm{t}}_0,\mathrm{T}]$.

Lemma 1 

(Gronwall–Bellman Inequality [3,4]). Let $\mathrm{u}:[{\mathrm{t}}_0,\mathrm{T}]\to [0,\mathrm{\infty })$ be continuous. Suppose $\mathrm{a}\ge 0$ and $\mathrm{p}(\mathrm{t})$ is a continuous, nonnegative function on $[{\mathrm{t}}_0,\mathrm{T}]$ such that

$$u(t)\le a+{\int }_{t_0}^t p(s)u(s)ds,\forall t\in [t_0,T].$$

(7)

Then $u(t)$ is bounded by

$$u(t)\le a\mathrm{e}\mathrm{x}\mathrm{p}\left({\int }_{t_0}^t p(s)ds\right),t\in [t_0,T].$$

(8)

In particular, if $a=0$, then $u(t)=0$ for all $t\in [t_0,T]$.

Proof. 

See [9]. □

Lemma 2 

(Bihari’s Inequality [5]). Let $u:[t_0, T]\to [0, \infty )$ be continuous. Suppose $A \ge 0$, $b(t)$ is continuous and nonnegative on $[t_0, T]$, and $F:[0, \infty )\to [0, \infty )$ is a continuous, nondecreasing function. If

$$u(t)\le A+{\int }_{t_0}^t b(s)F(u(s))ds,\forall t\in [t_0,T],$$

(9)

then $u(t)$ satisfies

$$H(u(t))\le H(A)+{\int }_{t_0}^t b(s)ds,t\in [t_0,T],$$

(10)

where $H(x)$ is the function defined by

$$\mathrm{H}(\mathrm{x}):={\int }_{{\mathrm{x}}_0}^{\mathrm{x}} {\displaystyle \frac{1}{\mathrm{F}(\mathsf{\xi })}} \mathrm{d}\mathsf{\xi },$$

(11)

for some $x_0>0$ in the range of $u$ (one can take $x_0=A$ if $A>0$, or any small positive number if $A=0$). Since $F(\xi )>0$ for $\xi \ge 0$ and $F$ is nondecreasing, the function $H(x)$ is strictly increasing and continuous on $[0,\infty )$. Therefore, it is a bijection from $[0,\infty )$ onto its image $[H(0),\underset{x\to \infty }{lim} H(x))$, and the inverse function $H^{-1}$ exists and is continuous on this interval. Consequently,

$$\mathrm{u}(\mathrm{t})\le {\mathrm{H}}^{-1}\left(\mathrm{H}(\mathrm{A})+{\int }_{{\mathrm{t}}_0}^{\mathrm{t}} \mathrm{b}(\mathrm{s}) \mathrm{d}\mathrm{s}\right), \mathrm{t}\in [{\mathrm{t}}_0,\mathrm{T}].$$

(12)

In particular, if $A=0$ and

$${\int }_0^{ϵ} {\displaystyle \frac{1}{F(\xi )}}d\xi =\infty$$

(13)

for any $ϵ>0$ (an Osgood condition), then $u(t)\equiv 0$ on $[t_0,T]$.

Proof. 

See [5]. □

Remark 1. 

In Lemma 2, the function $H(x)$ is strictly increasing and continuous on its domain, hence invertible. In the special case $F(u)=u$ (linear dependence), choosing $x_0=A>0$, we have

$$H(x)={\int }_A^x {\displaystyle \frac{1}{\xi }}d\xi =\mathrm{l}\mathrm{n} x-\mathrm{l}\mathrm{n} A.$$

(14)

Then the bound (12) becomes

$$u(t)\le H^{-1}\left(H(A)+{\int }_{t_0}^t b(s)ds\right)=A\mathrm{e}\mathrm{x}\mathrm{p}\left({\int }_{t_0}^t b(s)ds\right),$$

(15)

which recovers the classical Gronwall–Bellman inequality (Lemma 1). In the case $F(u)=u^p$ for some $p>1$, one obtains the power-type estimate

$$u(t{)}^{1-p}\ge u(t_0{)}^{1-p}-(p-1){\int }_{t_0}^t b(s)ds,$$

(16)

which is a typical nonlinear Gronwall inequality used in reaction--diffusion and other contexts [17].

We will also make use of a simple comparison observation for nondecreasing functions under integration: if 0 $\le \sigma (s) \le s$ and $u(\cdot )$ is nondecreasing on $[t_0,T]$, then $u(\sigma (s)) \le u(s)$ for all $s$. Consequently,

$${\int }_{t_0}^t b(s)F(u(\sigma (s)))ds\le {\int }_{t_0}^t b(s)F(u(s))ds,\forall t\in [t_0,T],$$

(17)

whenever F is nondecreasing and b(s) ≥ 0. We will use this fact to compare delayed inequalities with non-delayed ones. If u is not known to be nondecreasing, one can consider the supremum sup_ {ξ∈[t0, s]}u(ξ) in place of u(s) to achieve a similar estimate. In most applications, u(t) arises as a solution of an equation or inequality that naturally preserves monotonicity, so this technical point is usually satisfied or can be assumed without loss of aeneralitv.

Having reviewed these fundamental inequalities, we proceed to our new results.

3. Main Results

In this section, we present three new theorems that generalize the Gronwall–Bellman–Bihari-type inequalities. Theorem 1 deals with an integral inequality involving a separable kernel and a nonlinear term, providing an explicit bound in terms of an inverse auxiliary function. Theorem 2 addresses inequalities with delay arguments, and Theorem 3 treats a combined linear and nonlinear integral inequality by means of a maximal solution. All proofs are given in detail for clarity.

Theorem 1. 

Let $f:[t_0, T]\to [0,\infty )$ and $g:[t_0, T]\to [0, \infty )$ be locally integrable functions. Let $F:[0, \infty )\to (0, \infty )$ be a continuous, nondecreasing function. Suppose $u:[t_0, T]\to [0, \infty )$ is a continuous function satisfying

$$u(t)\le a+{\int }_{t_0}^t f(t)g(s)F(u(s))ds,t\in [t_0,T],$$

(18)

for some constant $a\ge 0$. Define an auxiliary function $H(x)$ by

$$H(x):={\int }_0^x {\displaystyle \frac{1}{F(\xi )}}d\xi ,x\ge 0,$$

(19)

(which is well-defined on $[0,\infty )$ since $F(\xi )>0$ for $\xi \ge 0$). Then $H(u(t))$ is bounded by

$$H(u(t))\le H(a)+f(t){\int }_{t_0}^t g(\xi )d\xi ,t\in [t_0,T].$$

(20)

Equivalently, solving for $u(t)$ yields

$$u(t)\le H^{-1}\left(H(a)+f(t){\int }_{t_0}^t g(\xi )d\xi \right),t\in [t_0,T].$$

(21)

In particular, if $a=0$ and $H(x)$ diverges as $x\to 0^{+}$ (for example, ${\int }_0^{ϵ} {\displaystyle \frac{1}{F(\xi )}} d\xi =\infty$ for each $ϵ>0$), then $u(t)\equiv 0$ on $[t_0,T]$.

Proof. 

First, note that $\mathrm{H}(\mathrm{x})$ is continuous and strictly increasing on $[0,\infty )$ because $\mathrm{F}(\mathsf{\xi })>0$ and $\mathrm{F}$ is nondecreasing. Thus, $\mathrm{H}$ has an inverse function ${\mathrm{H}}^{-1}:[\mathrm{H}(0),\infty )\to [0,\infty )$. For convenience, denote

$$G(t):={\int }_{t_0}^t g(\xi )d\xi ,t\in [t_0,T].$$

(22)

Since $g$ is locally integrable, $\mathrm{G}(\mathrm{t})$ is absolutely continuous and hence differentiable almost everywhere, with ${\mathrm{G}}^{\prime }(\mathrm{t})=\mathrm{g}(\mathrm{t})$ for almost every $\mathrm{t}$. The subsequent steps involving differentiation thus hold almost everywhere, and the integral inequalities remain valid by standard properties of the Lebesgue integral. Since $\mathrm{g}(\mathsf{\xi })\ge 0$, $\mathrm{G}(\mathrm{t})$ is nondecreasing and $\mathrm{G}({\mathrm{t}}_0)=0$.

Our strategy is to transform inequality (18) by the substitution $\mathrm{H}(\mathrm{u}(\mathrm{t}))$. However, $\mathrm{u}(\mathrm{t})$ may not be differentiable. To justify the argument rigorously, one can apply a smoothing procedure or use an integral form of the argument. Here we present a heuristic proof that can be made rigorous by standard limit processes.

Assume for the moment that $\mathrm{u}(\mathrm{t})$ is differentiable on $({\mathrm{t}}_0,\mathrm{T})$ (the final result will hold by continuity even if it is not differentiable everywhere). Using (18) and the fact that $\mathrm{f}(\mathrm{t})$ does not depend on the integration variable $\mathrm{s}$, we differentiate both sides of (18) with respect to $\mathrm{t}$. By the Leibniz integral rule, we obtain

$$u^{\prime }(t)\le f^{\prime }(t){\int }_{t_0}^t g(s)F(u(s))ds+f(t)g(t)F(u(t)).$$

(23)

Using (16) again to bound the integral on the right-hand side, we have

$$\begin{array}{cc}\hfill u^{\prime }(t)& \le f^{\prime }(t)\left(u(t)-a\right)+f(t)g(t)F(u(t))\hfill \\ & \le f^{\prime }(t)u(t)-f^{\prime }(t)a+f(t)g(t)F(u(t)).\hfill \end{array}$$

(24)

Rearrange this inequality to isolate the terms involving $\mathrm{u}(\mathrm{t})$ and ${\mathrm{u}}^{\prime }(\mathrm{t})$:

$$u^{\prime }(t)-f^{\prime }(t)u(t)\le -f^{\prime }(t)a+f(t)g(t)F(u(t)).$$

(25)

Now consider the composite function $\mathrm{H}(\mathrm{u}(\mathrm{t}))$. Using the chain rule,

$${\displaystyle \frac{d}{dt}}H(u(t))=H^{\prime }(u(t))u^{\prime }(t)={\displaystyle \frac{u^{\prime }(t)}{F(u(t))}}.$$

(26)

Substitute the above inequality for ${\mathrm{u}}^{\prime }(\mathrm{t})$ into this derivative:

$${\displaystyle \frac{d}{dt}}H(u(t))\le {\displaystyle \frac{-f^{\prime }(t)a}{F(u(t))}}+{\displaystyle \frac{f(t)g(t)F(u(t))}{F(u(t))}}.$$

(27)

This simplifies to

$${\displaystyle \frac{d}{dt}}H(u(t))\le -f^{\prime }(t){\displaystyle \frac{a}{F(u(t))}}+f(t)g(t).$$

(28)

Observe that ${\displaystyle \frac{\mathrm{a}}{\mathrm{F}(\mathrm{u}(\mathrm{t}))}}\ge 0$ and $-{\mathrm{f}}^{\prime }(\mathrm{t})\le 0$ if ${\mathrm{f}}^{\prime }(\mathrm{t})$ is negative. Importantly, even if ${\mathrm{f}}^{\prime }(\mathrm{t})$ changes sign, we can still integrate this differential inequality from ${\mathrm{t}}_0$ to $\mathrm{t}$ and use integration by parts on the first term. Integrating, we get:

$$H(u(t))-H(u(t_0))\le -{\int }_{t_0}^t f^{\prime }(\xi ){\displaystyle \frac{a}{F(u(\xi ))}}d\xi +{\int }_{t_0}^t f(\xi )g(\xi )d\xi .$$

(29)

Since $\mathrm{u}({\mathrm{t}}_0)=\mathrm{u}({\mathrm{t}}_0)\le \mathrm{a}+{\int }_{{\mathrm{t}}_0}^{{\mathrm{t}}_0} \cdot =\mathrm{a}$, we have $\mathrm{u}({\mathrm{t}}_0)\le \mathrm{a}$ and hence $\mathrm{H}(\mathrm{u}({\mathrm{t}}_0))\le \mathrm{H}(\mathrm{a})$. Thus,

$$H(u(t))\le H(a)-a{\int }_{t_0}^t {\displaystyle \frac{f^{\prime }(\xi )}{F(u(\xi ))}}d\xi +{\int }_{t_0}^t f(\xi )g(\xi )d\xi .$$

(30)

For the second term on the right, we integrate by parts, treating ${\displaystyle \frac{1}{\mathrm{F}(\mathrm{u}(\mathsf{\xi }))}}$ as one factor and $\mathrm{a}{\mathrm{f}}^{\prime }(\mathsf{\xi })$ as the derivative of another factor

$$af(\xi ):=-a{\int }_{t_0}^t {\displaystyle \frac{f^{\prime }(\xi )}{F(u(\xi ))}}d\xi =-{\left[{\displaystyle \frac{af(\xi )}{F(u(\xi ))}}\right]}_{\xi =t_0}^{\xi =t}+{\int }_{t_0}^t af(\xi ){\displaystyle \frac{d}{d\xi }}\left({\displaystyle \frac{1}{F(u(\xi ))}}\right)d\xi .$$

(31)

At $\xi = t_0$, $f(t_0)$ is finite and $F(u(t_0)) \ge F(0) > 0$, so the boundary term is well-defined. Specifically,

$$-{\left[{\displaystyle \frac{af(\xi )}{F(u(\xi ))}}\right]}_{t_0}^t=-{\displaystyle \frac{af(t)}{F(u(t))}}+{\displaystyle \frac{af(t_0)}{F(u(t_0))}}.$$

(32)

The second term ${\displaystyle \frac{af(t_0)}{F(u(t_0))}}$ is nonnegative (since $a \ge 0$, $f(t_0) \ge 0$, and $F(u(t_0)) > 0$) and hence can be dropped to obtain an inequality. Meanwhile, the remaining integral becomes

$${\int }_{t_0}^t af(\xi )\left(-{\displaystyle \frac{F^{\prime }(u(\xi ))u^{\prime }(\xi )}{[F(u(\xi )){]}^2}}\right)d\xi .$$

(33)

This term is $\le 0$ because $a \ge 0$, $f(\xi ) \ge 0$, $F^{\prime }(u(\xi )) \ge 0$ (since $F$ is nondecreasing), and $u^{\prime }(\xi ) \ge 0$ for almost every $\xi$ (one can argue $u(t)$ is nondecreasing on $[t_0,T]$ under the integral inequality, or else take the nondecreasing upper bound in place of $u(\xi )$—either way the product is nonpositive). Therefore,

$$-a{\int }_{t_0}^t {\displaystyle \frac{f^{\prime }(\xi )}{F(u(\xi ))}}d\xi \le -{\displaystyle \frac{af(t)}{F(u(t))}}.$$

(34)

Combining everything, we obtain

$$H(u(t))\le H(a)-{\displaystyle \frac{af(t)}{F(u(t))}}+{\int }_{t_0}^t f(\xi )g(\xi )d\xi .$$

(35)

Finally, note that ${\displaystyle \frac{\mathrm{a}\mathrm{f}(\mathrm{t})}{\mathrm{F}(\mathrm{u}(\mathrm{t}))}}\ge 0$. Dropping this nonpositive term yields exactly the desired inequality (19):

$$H(u(t))\le H(a)+{\int }_{t_0}^t f(\xi )g(\xi )d\xi =H(a)+f(t)G(t).$$

(36)

This completes the proof of (20). Equivalently, since $\mathrm{H}$ is increasing, we can apply ${\mathrm{H}}^{-1}$ to both sides to obtain:

$$u(t)\le H^{-1}\left(H(a)+f(t)G(t)\right).$$

(37)

The final statement regarding the case $a = 0$ follows because if $H(x)$ diverges as $x\to 0^{+}$, then $H^{-1}(y) = 0$ for all finite $y$. In particular, $H^{-1}(f(t)G(t)) = 0$ for each fixed $t$, implying $u(t) = 0$. This is consistent with the uniqueness criterion mentioned in Lemma 2. □

Remark 2. 

The bound (20) in Theorem 1 is sharp. In fact, there are cases of equality: for example, if $\mathrm{u}(\mathrm{t})$ itself satisfies the integral equation $\mathrm{u}(\mathrm{t})=\mathrm{a}+{\int }_{{\mathrm{t}}_0}^{\mathrm{t}} \mathrm{f}(\mathrm{t})\mathrm{g}(\mathrm{s})\mathrm{F}(\mathrm{u}(\mathrm{s}))\mathrm{d}\mathrm{s}$ and the functions involved are sufficiently nice, then our derivation shows that (20) holds with equality. In general, for an inequality, (20) gives the least upper bound achievable under the given conditions. We also note that Theorem 1 recovers the classical Bihari inequality (Lemma 2) in the special case $\mathrm{f}(\mathrm{t})\equiv 1$ and $\mathrm{g}(\mathrm{s})=\mathrm{b}(\mathrm{s})$, since then $\mathrm{f}(\mathrm{t})\mathrm{G}(\mathrm{t})={\int }_{{\mathrm{t}}_0}^{\mathrm{t}} \mathrm{b}(\mathrm{s})\mathrm{d}\mathrm{s}$. Moreover, if we further specialize to $\mathrm{F}(\mathrm{u})=\mathrm{u}$, we recover Gronwall’s inequality (Lemma 1) as a special case of Bihari’s inequality. Thus, Theorem 1 indeed unifies and generalizes both Gronwall–Bellman and Bihari inequalities in a single statement.

Next, we address a retarded (delay) version of the Gronwall–Bihari inequality. Intuitively, if the integral inequality involves $\mathrm{u}$ evaluated at a previous time $\mathsf{\sigma }(\mathrm{s})\le \mathrm{s}$, one might expect that the solution grows no faster than in the absence of delay, since the feedback is delayed. Theorem 2 below confirms this intuition by showing that the same bound applies.

Theorem 2. 

Let $\mathrm{b}:[{\mathrm{t}}_0,\mathrm{T}]\to [0,\infty )$ be locally integrable and let $\mathsf{\sigma }:[{\mathrm{t}}_0,\mathrm{T}]\to [{\mathrm{t}}_0,\mathrm{T}]$ be a continuous function such that $\mathsf{\sigma }(\mathrm{s})\le \mathrm{s}$ for all $\mathrm{s}\in [{\mathrm{t}}_0,\mathrm{T}]$. Suppose $\mathrm{u}:[{\mathrm{t}}_0,\mathrm{T}]\to [0,\infty )$ is continuous and satisfies the retarded integral inequality

$$u(t)\le A+{\int }_{t_0}^t b(s)F(u(\sigma (s)))ds,t\in [t_0,T],$$

(38)

where $A\ge 0$ and $F:[0,\infty )\to [0,\infty )$ is continuous and nondecreasing. Then $u(t)$ obeys the same bound as in the non-delay case:

$$u(t)\le H^{-1}\left(H(A)+{\int }_{t_0}^t b(s)ds\right),t\in [t_0,T],$$

(39)

with $H(x)={\int }_0^x {\displaystyle \frac{1}{F(\xi )}} d\xi$ as in Lemma 2. In particular, if $A=0$ and ${\int }_0^{ϵ} {\displaystyle \frac{1}{F(\xi )}} d\xi =\infty$ for $ϵ>0$, then $u(t)\equiv 0$ on $[t_0,T]$.

Proof. 

The proof is straightforward by comparison with the non-delayed case. Since $\mathsf{\sigma }(\mathrm{s})\le \mathrm{s}$, for any $\mathrm{s}\le \mathrm{t}$ we have $\mathsf{\sigma }(\mathrm{s})\le \mathrm{t}$. If we assume (without loss of generality) that $\mathrm{u}(\mathrm{t})$ is nondecreasing in $\mathrm{t}$ (or else consider its nondecreasing upper envelope), then $\mathrm{u}(\mathsf{\sigma }(\mathrm{s}))\le \mathrm{u}(\mathrm{s})$ for $\mathrm{s}\le \mathrm{t}$. Because $\mathrm{F}$ is nondecreasing, it follows that $\mathrm{F}(\mathrm{u}(\mathsf{\sigma }(\mathrm{s})))\le \mathrm{F}(\mathrm{u}(\mathrm{s}))$. Therefore, the inequality (36) implies

$$u(t)\le A+{\int }_{t_0}^t b(s)F(u(s))ds.$$

(40)

This has exactly the form of the hypothesis in Lemma 2 (Bihari’s inequality). By Lemma 2, we immediately obtain

$$H(u(t))\le H(A)+{\int }_{t_0}^t b(s)ds,$$

(41)

which is equivalent to (39). The uniqueness part for $\mathrm{A}=0$ follows from the Osgood condition as before. □

Remark 3. 

Theorem 2 shows that delay does not affect the Gronwall–Bihari bound. This result generalizes known retarded Gronwall inequalities (which correspond to the linear case $\mathrm{F}(\mathrm{u})=\mathrm{u}$) [18]. In practice, it means that when proving stability or uniqueness for functional (delay) differential equations, one can use the same exponential or implicit bound as in the corresponding ordinary differential equation without delay. We emphasize that the condition $\mathsf{\sigma }(\mathrm{s})\le \mathrm{s}$ ensures that no advanced argument appears; it is essential that the inequality is retarded or at most neutral. If $\mathsf{\sigma }(\mathrm{s})$ were allowed to be greater than $\mathrm{s}$ (an advanced argument), then the inequality could potentially force faster growth than the non-delayed case, and different techniques would be needed.

Our final main result addresses an integral inequality that involves both a linear $\mathrm{u}(\mathrm{s})$ term and a nonlinear $\mathrm{F}(\mathrm{u}(\mathrm{s}))$ term, each with their own coefficient functions. Specifically, we consider inequalities of the form

$$\mathrm{u}(\mathrm{t})\le \mathsf{\varphi }(\mathrm{t})+{\int }_{{\mathrm{t}}_0}^{\mathrm{t}} \mathrm{p}(\mathrm{s})\mathrm{u}(\mathrm{s})\mathrm{d}\mathrm{s}+{\int }_{{\mathrm{t}}_0}^{\mathrm{t}} \mathrm{q}(\mathrm{s})\mathrm{F}(\mathrm{u}(\mathrm{s}))\mathrm{d}\mathrm{s}.$$

(42)

This type of inequality arises naturally in systems where one has a linear growth/decay rate $\mathrm{p}(\mathrm{s})$ and an additional nonlinear source term $\mathrm{q}(\mathrm{s})\mathrm{F}(\mathrm{u})$. While Theorem 1 provided a direct bound for a separable kernel case, a fully explicit bound for the combined case is difficult in general. Instead, Theorem 3 establishes the existence of a maximal solution to the associated integral equation and shows that $\mathrm{u}(\mathrm{t})$ is bounded above by this solution. This result guarantees that the inequality has an optimal (tightest possible) bound function, which is important for qualitative analysis (even if that bound function may not have a simple closed form).

Theorem 3. 

Let $\mathrm{p}:[{\mathrm{t}}_0,\mathrm{T}]\to [0,\infty )$ and $\mathrm{q}:[{\mathrm{t}}_0,\mathrm{T}]\to [0,\infty )$ be locally integrable functions. Let $\mathrm{F}:[0,\infty )\to [0,\infty )$ be a continuous, nondecreasing function. Suppose $\mathsf{\varphi }:[{\mathrm{t}}_0,\mathrm{T}]\to [0,\infty )$ is a continuous, nondecreasing function. If $\mathrm{u}:[{\mathrm{t}}_0,\mathrm{T}]\to [0,\infty )$ satisfies

$$u(t)\le \varphi (t)+{\int }_{t_0}^t p(s)u(s)ds+{\int }_{t_0}^t q(s)F(u(s))ds,t\in [t_0,T],$$

(43)

then there exists a unique continuous function $y:[t_0,T]\to [0,\infty )$ that solves the integral equation

$$y(t)=\varphi (t)+{\int }_{t_0}^t p(s)y(s)ds+{\int }_{t_0}^t q(s)F(y(s))ds,t\in [t_0,T],$$

(44)

and $u(t)$ is bounded above by this solution for all $t\in [t_0,T]$, i.e.,

$$u(t)\le y(t),\forall t\in [t_0,T].$$

(45)

Moreover, $y(t)$ is the minimal nonnegative solution of (43); any other continuous function $\tilde{y}(t)$ satisfying (43) must satisfy $\tilde{y}(t)\ge y(t)$ for all $t$. In particular, if $\varphi (t_0)=u(t_0)$, then $y(t_0)=u(t_0)$ and hence $u(t_0)=y(t_0)\le \tilde{y}(t_0)$ for any other solution $\tilde{y}$.

Proof. 

The proof is based on a monotone iteration (Picard iteration) method and a standard application of the Banach fixed-point theorem or the monotone convergence of successive approximations [17].

Define an integral operator $\mathcal{T}$ on $\mathrm{C}([{\mathrm{t}}_0,\mathrm{T}])$ by

$$(\mathcal{T}w)(t):=\varphi (t)+{\int }_{t_0}^t p(s)w(s)ds+{\int }_{t_0}^t q(s)F(w(s))ds.$$

(46)

We are seeking a function y such that y = Ty (a fixed point of T). By the standard theory of Volterra integral equations, such a solution exists and is unique provided T is a contraction or a monotone operator on an appropriate function space. In our case, $T$ may not be a contraction globally because $F$ could be only nondecreasing (not necessarily Lipschitz). However, we can construct the minimal solution by iteration as follows:

Define a sequence $\{y_n(t)\}$ on $[t_0,T]$ by starting with the initial function

$$y_0(t):=\varphi (t),t\in [t_0,T],$$

(47)

and then inductively setting

$$y_{n+1}(t):=\varphi (t)+{\int }_{t_0}^t p(s)y_n(s)ds+{\int }_{t_0}^t q(s)F(y_n(s))ds,t\in [t_0,T].$$

(48)

In other words, ${\mathrm{y}}_{\mathrm{n}+1}=\mathcal{T}{\mathrm{y}}_{\mathrm{n}}$ for each $\mathrm{n}\ge 0$. All ${\mathrm{y}}_{\mathrm{n}}(\mathrm{t})$ are well-defined continuous functions, since the integrals of continuous functions are continuous. Moreover, this sequence is monotonic increasing. We show this by induction: Clearly ${\mathrm{y}}_1(\mathrm{t})\ge \mathsf{\varphi }(\mathrm{t})={\mathrm{y}}_0(\mathrm{t})$ because $\mathrm{p},\mathrm{q},\mathrm{F}$ are nonnegative (note $\mathrm{F}({\mathrm{y}}_0(\mathrm{s}))=\mathrm{F}(\mathsf{\varphi }(\mathrm{s}))\ge 0$). Now assume ${\mathrm{y}}_{\mathrm{n}}(\mathrm{t})\ge {\mathrm{y}}_{\mathrm{n}-1}(\mathrm{t})$ for all $\mathrm{t}$. Then, using the monotonicity of $\mathrm{F}$, we have:

$$\begin{array}{cc}\hfill y_{n+1}(t)& =\varphi (t)+{\int }_{t_0}^t p(s)y_n(s)ds+{\int }_{t_0}^t q(s)F(y_n(s))ds,\hfill \\ \hfill y_n(t)& =\varphi (t)+{\int }_{t_0}^t p(s)y_{n-1}(s)ds+{\int }_{t_0}^t q(s)F(y_{n-1}(s))ds.\hfill \end{array}$$

(49)

Subtracting these expressions, we get

$$y_{n+1}(t)-y_n(t)={\int }_{t_0}^t p(s)\left[y_n(s)-y_{n-1}(s)\right]ds+{\int }_{t_0}^t q(s)\left[F(y_n(s))-F(y_{n-1}(s))\right]ds.$$

(50)

By the induction hypothesis, ${\mathrm{y}}_{\mathrm{n}}(\mathrm{s})-{\mathrm{y}}_{\mathrm{n}-1}(\mathrm{s})\ge 0$, and since $\mathrm{F}$ is nondecreasing, $\mathrm{F}({\mathrm{y}}_{\mathrm{n}}(\mathrm{s}))-\mathrm{F}({\mathrm{y}}_{\mathrm{n}-1}(\mathrm{s}))\ge 0$. Therefore, ${\mathrm{y}}_{\mathrm{n}+1}(\mathrm{t})-{\mathrm{y}}_{\mathrm{n}}(\mathrm{t})\ge 0$ for all $\mathrm{t}$. This proves that $\{{\mathrm{y}}_{\mathrm{n}}(\mathrm{t})\}$ is a nondecreasing sequence of continuous functions.

Next, we show that the sequence is bounded above on $[{\mathrm{t}}_0,\mathrm{T}]$. To see this, note that for each $\mathrm{n}$, ${\mathrm{y}}_{\mathrm{n}}(\mathrm{t})\le {\mathrm{y}}_{\mathrm{n}+1}(\mathrm{t})$, and by the integral inequality (42) and the definition of $\mathcal{T}$, we have

$$u(t)\le \varphi (t)+{\int }_{t_0}^t p(s)u(s)ds+{\int }_{t_0}^t q(s)F(u(s))ds=(\mathcal{T}u)(t).$$

(51)

In other words, $u(t) \le Tu(t)$. Now, comparing $y_0(t) = \mathsf{\varphi }(t)$ and $u(t)$, we have $y_0(t) = \mathsf{\varphi }(t)\le u(t)$ (since $u(t)$ satisfies (43) and $\mathsf{\varphi }(t)$ is the first term). Then, applying $T$ to both sides of $y_0(t) \le u(t)$ and using the monotonicity of $T$ (which follows from the nonnegativity of $p,q$ and the monotonicity of $F$), we get:

$$y_1(t)=\mathcal{T}{y}_0(t)\le \mathcal{T}u(t)\ge u(t).$$

(52)

In fact, we can show by induction that $y_n(t) \le u(t)$ for all $n$. The base case $y_0(t) \le u(t)$ is true. If $y_n(t) \le u(t)$, then

$$y_{n+1}(t)=\mathcal{T}{y}_n(t)\le \mathcal{T}u(t)\ge u(t),$$

(53)

where the last inequality is from (41). This shows that ${\mathrm{y}}_{\mathrm{n}}(\mathrm{t})\le \mathrm{u}(\mathrm{t})$ for all $\mathrm{n}$. Therefore, the sequence $\{{\mathrm{y}}_{\mathrm{n}}(\mathrm{t})\}$ is bounded above by $\mathrm{u}(\mathrm{t})$. Since it is also nondecreasing, it converges pointwise to a limit function $\mathrm{y}(\mathrm{t})$, i.e.,

$$y(t)=\underset{n\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}} y_n(t),t\in [t_0,T].$$

(54)

By the monotone convergence theorem and the continuity of the integrands, we can pass to the limit in the recurrence relation to obtain

$$y(t)=\varphi (t)+{\int }_{t_0}^t p(s)y(s)ds+{\int }_{t_0}^t q(s)F(y(s))ds.$$

(55)

Thus, $\mathrm{y}(\mathrm{t})$ is a solution of (44). Moreover, since ${\mathrm{y}}_{\mathrm{n}}(\mathrm{t})\le \mathrm{u}(\mathrm{t})$ for all $\mathrm{n}$, we have $\mathrm{y}(\mathrm{t})\le \mathrm{u}(\mathrm{t})$. But from (43) and the definition of $\mathcal{T}$, we also have $\mathrm{u}(\mathrm{t})\le \mathcal{T}\mathrm{u}(\mathrm{t})$. However, $\mathrm{y}(\mathrm{t})$ is the minimal solution of (44) because any other solution $\tilde{\mathrm{y}}(\mathrm{t})$ must satisfy ${\mathrm{y}}_0(\mathrm{t})\le \tilde{\mathrm{y}}(\mathrm{t})$, and then ${\mathrm{y}}_1(\mathrm{t})=\mathcal{T}{\mathrm{y}}_0(\mathrm{t})\le \mathcal{T}\tilde{\mathrm{y}}(\mathrm{t})=\tilde{\mathrm{y}}(\mathrm{t})$, and by induction ${\mathrm{y}}_{\mathrm{n}}(\mathrm{t})\le \tilde{\mathrm{y}}(\mathrm{t})$ for all $\mathrm{n}$, hence $\mathrm{y}(\mathrm{t})\le \tilde{\mathrm{y}}(\mathrm{t})$. This proves that $\mathrm{y}(\mathrm{t})$ is the minimal solution. In particular, $\mathrm{u}(\mathrm{t})\le \mathrm{y}(\mathrm{t})$ would imply $\mathrm{u}(\mathrm{t})=\mathrm{y}(\mathrm{t})$, so actually we must have $\mathrm{u}(\mathrm{t})\le \mathrm{y}(\mathrm{t})$. This establishes (46).

Finally, to show uniqueness of the solution $\mathrm{y}(\mathrm{t})$ of (44), suppose there is another solution $\tilde{\mathrm{y}}(\mathrm{t})$. Then $\tilde{\mathrm{y}}(\mathrm{t})\ge \mathrm{y}(\mathrm{t})$ by minimality. On the other hand, applying the same iteration starting from ${\tilde{\mathrm{y}}}_0(\mathrm{t})=\tilde{\mathrm{y}}(\mathrm{t})$, we would get ${\tilde{\mathrm{y}}}_{\mathrm{n}+1}(\mathrm{t})=\mathcal{T}{\tilde{\mathrm{y}}}_{\mathrm{n}}(\mathrm{t})$, and since ${\tilde{\mathrm{y}}}_0(\mathrm{t})$ is already a fixed point, the sequence is constant: ${\tilde{\mathrm{y}}}_{\mathrm{n}}(\mathrm{t})=\tilde{\mathrm{y}}(\mathrm{t})$ for all $\mathrm{n}$. But then the minimal solution $\mathrm{y}(\mathrm{t})$ is the limit of the sequence starting from $\mathsf{\varphi }(\mathrm{t})$, and since $\mathsf{\varphi }(\mathrm{t})\le \tilde{\mathrm{y}}(\mathrm{t})$, we have ${\mathrm{y}}_{\mathrm{n}}(\mathrm{t})\le {\tilde{\mathrm{y}}}_{\mathrm{n}}(\mathrm{t})=\tilde{\mathrm{y}}(\mathrm{t})$, so $\mathrm{y}(\mathrm{t})\le \tilde{\mathrm{y}}(\mathrm{t})$. To show the reverse inequality, note that $\tilde{\mathrm{y}}(\mathrm{t})$ also satisfies $\tilde{\mathrm{y}}(\mathrm{t})\ge \mathsf{\varphi }(\mathrm{t})$, so by minimality of $\mathrm{y}(\mathrm{t})$, we have $\tilde{\mathrm{y}}(\mathrm{t})\ge \mathrm{y}(\mathrm{t})$. Therefore, $\tilde{\mathrm{y}}(\mathrm{t})=\mathrm{y}(\mathrm{t})$, proving uniqueness. □

Remark 4. 

Theorem 3 provides a constructive method to obtain the bound function $\mathrm{y}(\mathrm{t})$ as the limit of a monotone sequence. In practice, one can compute ${\mathrm{y}}_{\mathrm{n}}(\mathrm{t})$ iteratively and then take the limit. This is particularly useful when the integral Equation (42) does not admit a closed-form solution. The theorem also guarantees that the inequality (41) is sharp in the sense that there exists a function (namely $\mathrm{y}(\mathrm{t})$) that attains equality and bounds all other solutions from above. This is a maximal solution result. We note that if $\mathrm{F}$ is Lipschitz continuous, then the operator $\mathcal{T}$ is a contraction on a sufficiently small interval, and the sequence $\{{\mathrm{y}}_{\mathrm{n}}(\mathrm{t})\}$ converges uniformly to the unique solution. In the non-Lipschitz case, the convergence may be only pointwise, but the existence of a minimal solution is still guaranteed by the monotonicity.

The existence of the maximal solution $y(t)$ in Theorem 3 provides a direct method to address the reviewer’s concern regarding the determination of a region without movable singularities. Specifically, for a given initial bound $\varphi (t)$, the maximal interval $\left[t_0,T^{*}\right]$ on which the solution $y(t)$ to Equation (44) remains finite precisely defines a domain in which any function $u(t)$ satisfying inequality (43) is guaranteed to be bounded, and hence free of movable singularities. Thus, the analysis of the integral Equation (44) yields a computable criterion for estimating the region of existence and boundedness for solutions of the original problem.

Corollary 1 

(Estimation of the Region without Movable Singularities). Consider the initial value problem (56) under the growth condition (57). Suppose there exist continuous, nonnegative functions $\varphi (t)$, $p(t)$, $q(t)$, and a continuous, nondecreasing function $F:[0,\infty )\to [0,\infty )$ such that for any solution $x(t)$, the function $u(t)=|x(t)|$ satisfies inequality (43) on its interval of existence.

Let $y^{*}(t)$ be the unique continuous solution to the integral equation

$$y(t)=\varphi (t)+{\int }_{t_0}^t p(s)y(s)ds+{\int }_{t_0}^t q(s)F(y(s))ds,t\in [t_0,T^{*}],$$

(56)

guaranteed by Theorem 3.

If $y^{*}(t)$ remains bounded on $[t_0,T^{*}]$, then any solution $x(t)$ of the IVP (56) satisfying (57) also remains bounded on $[t_0,T^{*}]$, and hence is free of movable singularities in this interval. Consequently, the maximal interval of existence of $x(t)$ extends at least up to $T^{*}$.

In practice, one may choose $\varphi (t)$ to be a constant $M>0$ bounding the initial data and the known part of the forcing term, and then determine $T^{*}$ as the largest value such that the solution $y^{*}(t)$ of (44) with this $\varphi (t)\equiv M$ exists and is finite for all $t\in [t_0,T^{*}]$. This provides a computable criterion to estimate a region without movable singularities.

4. Applications

4.1. Application to Uniqueness of Solutions for a Class of Nonlinear ODEs

In this subsection, we illustrate the utility of our new inequalities by applying them to prove a uniqueness result for an initial value problem (IVP) with a nonlinearity that may violate the standard Lipschitz condition. Consider the IVP

$$\left\{\begin{array}{ll}{\displaystyle \frac{\mathrm{d}\mathrm{x}}{\mathrm{d}\mathrm{t}}}=\mathrm{f}(\mathrm{t},\mathrm{x}(\mathrm{t})),& \mathrm{t}\in [{\mathrm{t}}_0,\mathrm{T}],\\ \mathrm{x}({\mathrm{t}}_0)={\mathrm{x}}_0,& \end{array}\right.$$

(57)

where $\mathrm{f}:[{\mathrm{t}}_0,\mathrm{T}]\times \mathbb{R}\to \mathbb{R}$ is continuous. Crucially, we assume that $\mathrm{f}$ satisfies a growth condition of the form

$$|\mathrm{f}(\mathrm{t},\mathrm{x})-\mathrm{f}(\mathrm{t},\mathrm{y})|\le \mathrm{p}(\mathrm{t})|\mathrm{x}-\mathrm{y}|+\mathrm{q}(\mathrm{t})|\mathrm{F}(\mathrm{x})-\mathrm{F}(\mathrm{y})|,\forall \mathrm{t}\in [{\mathrm{t}}_0,\mathrm{T}], \forall \mathrm{x},\mathrm{y}\in \mathbb{R},$$

(58)

where $\mathrm{p}(\mathrm{t}),\mathrm{q}(\mathrm{t})\ge 0$ are continuous, and $\mathrm{F}:\mathbb{R}\to \mathbb{R}$ is a continuous, nondecreasing function that is not necessarily Lipschitz continuous (a classic example being $\mathrm{F}(\mathrm{x})={\mathrm{x}}^{1/3}$).

A crucial remark on the domain of validity: The uniqueness result we are about to prove is of a local nature. Our analysis, and the application of Theorem 3, presupposes the existence of a continuous solution on the entire interval $[t_0,T]$. For non-Lipschitz nonlinearities, the solution of (56) may develop a movable singularity (blow-up) at some finite time $T_c\le T$. In such a case, the uniqueness guaranteed by our theorem holds on any subinterval $[t_0,\tau ]$ with $\tau <T_c$, which is the maximal interval of existence. The derivation of an a priori bound for the solution to ensure that $T_c$ can be extended to, or beyond, $T$—thus guaranteeing a solution free of singularities on $[t_0,T]$—is a separate problem. This typically requires additional arguments, such as the use of Lyapunov functions or differential comparison principles, to determine the region without movable singularities, as rightly pointed out by the reviewer.

Significance and Scope of the Result: The primary significance of this application is to demonstrate that Theorem 3 can be used to establish uniqueness of solutions for problems where the nonlinearity $\mathrm{F}$ is non-Lipschitz. This extends the classical uniqueness theory, which typically relies on the Lipschitz condition (a special case of (58) with $\mathrm{q}(\mathrm{t})\equiv 0$).

We emphasize the following points to address potential concerns:

Existence is Assumed: Our analysis here focuses exclusively on uniqueness. We assume that at least one continuous solution to the IVP (57) exists on $[{\mathrm{t}}_0,\mathrm{T}]$. The existence of such a solution under condition (58) is a separate question that may require additional hypotheses (e.g., compactness arguments or the Peano existence theorem for continuity, though the latter only guarantees existence, not uniqueness).

Continuity of Solutions: Under our assumption that $\mathrm{f}$ is continuous and a solution exists, the solution $\mathrm{x}(\mathrm{t})$ is automatically continuously differentiable (and hence continuous) by the fundamental theorem of calculus. The possibility of discontinuous solutions does not arise under the stated classical framework of (57).

Addressing Movable Singularities: The reviewer correctly notes that strong nonlinearities can lead to movable singularities, where solutions blow up in finite time. Our uniqueness result is local in nature and holds on any interval $[{\mathrm{t}}_0,\mathrm{T}]$ where a continuous solution is assumed to exist. If a solution develops a singularity at some time ${\mathrm{T}}_{\mathrm{c}}$, our result is valid on $[{\mathrm{t}}_0,\mathrm{T}]$ for any $\mathrm{T}<{\mathrm{T}}_{\mathrm{c}}$. Proving global existence (i.e., that ${\mathrm{T}}_{\mathrm{c}}$ can be extended arbitrarily) would require separate, additional analysis, such as deriving a priori bounds on the solution.

Proof of Uniqueness. 

Suppose $\mathrm{x}(\mathrm{t})$ and $\mathrm{y}(\mathrm{t})$ are two solutions of (58) with the same initial condition $\mathrm{x}({\mathrm{t}}_0)=\mathrm{y}({\mathrm{t}}_0)={\mathrm{x}}_0$. Then their difference satisfies

$$x(t)-y(t)={\int }_{t_0}^t [f(s,x(s))-f(s,y(s))]ds.$$

(59)

Taking absolute values and using (55), we obtain

$$|x(t)-y(t)|\le {\int }_{t_0}^t p(s)|x(s)-y(s)|ds+{\int }_{t_0}^t q(s)|F(x(s))-F(y(s))|ds.$$

(60)

Let $u(t)=|x(t)-y(t)|$. Since $F$ is nondecreasing, we have $|F(x(s))-F(y(s))|\le F(|x(s)-y(s)|)=F(u(s))$ (This holds if $F$ is also odd or if we define it on $[0,\infty )$; otherwise, we can work with $F(|x|)$ directly). Therefore,

$$u(t)\le {\int }_{t_0}^t p(s)u(s)ds+{\int }_{t_0}^t q(s)F(u(s))ds.$$

(61)

This is precisely the form of inequality (41) with $\varphi (t)\equiv 0$. By Theorem 3, there exists a unique continuous function $y^{*}(t)$ satisfying the corresponding integral equation

$$y^{*}(t)={\int }_{t_0}^t p(s)y^{*}(s)ds+{\int }_{t_0}^t q(s)F(y^{*}(s))ds,$$

(62)

and $u(t)\le y^{*}(t)$. Observe that $y^{*}(t)\equiv 0$ is a solution of this integral equation. By uniqueness of the solution guaranteed by Theorem 3, it must be the only solution. Therefore, $y^{*}(t)=0$ for all $t$, which implies $u(t)\le 0$. Hence, $u(t)=0$, proving that $x(t)=y(t)$ for all $t\in [t_0,T]$.

This application demonstrates that Theorem 3 provides a robust tool for proving uniqueness for differential equations with combined linear and nonlinear growth conditions, even when the nonlinear part is non-Lipschitz. The result is significant as it guarantees that if a solution exists, it is unique, which is a critical step in the analysis of such equations.

The previous application demonstrated the power of our inequalities in establishing sharp theoretical results, such as uniqueness. To further illustrate their practical utility in analyzing models derived from real-world processes, we now consider an application in population dynamics. □

4.2. Application to a Population Growth Model with Delay

To further demonstrate the practical utility of our results, we consider a classical model in population dynamics: the delayed logistic growth model, also known as Hutchinson’s equation [19]. The model describes the population density $N(t)$ under a self-regulation mechanism that incorporates a maturation period $\tau >0$:

$${\displaystyle \frac{dN}{dt}}=rN(t)\left(1-{\displaystyle \frac{N(t-\tau )}{K}}\right),t\ge t_0,$$

(63)

with an initial history condition $N(t)=\varphi (t)$ for $t\in [t_0-\tau ,t_0]$, where $\varphi$ is a positive continuous function. Here, $r>0$ is the intrinsic growth rate, and $K>0$ is the carrying capacity of the environment.

A fundamental question in the analysis of such models is whether the solutions remain bounded for all future time. We will use our Theorem 2 to establish the uniform boundedness of solutions to (60).

Let us assume a solution $N(t)$ exists on $[t_0,T]$. Integrating both sides of (60) from $t_0$ to $t$, we obtain:

$$N(t)=N(t_0)+{\int }_{t_0}^t rN(s)\left(1-{\displaystyle \frac{N(s-\tau )}{K}}\right)ds.$$

(64)

Taking absolute values is not necessary as $N(t)$ is positive. We aim to control the growth of $N(t)$. A direct application of the classical Gronwall inequality is not straightforward due to the presence of the delayed term $-N(s-\tau )$. However, we can rearrange the equation to isolate the growth term:

$$N(t)=N(t_0)+{\int }_{t_0}^t rN(s)ds-{\displaystyle \frac{r}{K}}{\int }_{t_0}^t N(s)N(s-\tau )ds.$$

(65)

Dropping the last (non-positive) term for an upper bound yields:

$$N(t)\le N(t_0)+{\int }_{t_0}^t rN(s)ds.$$

(66)

While this leads to an exponential bound via Lemma 1, it is often too coarse. A sharper bound can be derived using our nonlinear framework. Notice that for $t\ge t_0$, the solution satisfies

$${\displaystyle \frac{dN}{dt}}\le rN(t).$$

(67)

This suggests that the growth is at most exponential. Motivated by this, we define a function $u(t)$ that is designed to capture and bound the cumulative growth. Let us consider the function

$$u(t)=\underset{\theta \in [t_0-\tau ,t]}{\mathrm{s}\mathrm{u}\mathrm{p}} N(\theta ).$$

(68)

It follows that $N(s-\tau )\le u(s)$ for $s\in [t_0,t]$. From the differential Equation (64), we have:

$${\displaystyle \frac{dN}{ds}}\le rN(s)\le ru(s).$$

This implies that $N(s)$ cannot grow too rapidly compared to $u(s)$. More precisely, for any $s\in [t_0,t]$, it can be shown that $N(s)\le u(t_0)+{\int }_{t_0}^s ru(\xi )d\xi$. Taking the supremum over $s\in [t_0,t]$ on the left-hand side leads to an inequality of the form:

$$u(t)\le C+{\int }_{t_0}^t ru(s)ds,$$

(69)

for a suitable constant $C>0$ depending on the initial data. Applying the Gronwall–Bellman inequality (Lemma 1) to (66) immediately yields $u(t)\le C\mathrm{e}\mathrm{x}\mathrm{p}(r(t-t_0))$, and consequently, $N(t)$ is bounded on any finite interval $[t_0,T]$.

To showcase Theorem 2, consider a scenario with a nonlinear growth constraint. Suppose the growth rate is not constant but depends on the population size, such as $r(N)=r/(1+ϵN)$, leading to a modified equation:

$${\displaystyle \frac{dN}{dt}}={\displaystyle \frac{rN(t)}{1+ϵN(t)}}\left(1-{\displaystyle \frac{N(t-\tau )}{K}}\right).$$

(70)

For this model, one can derive an integral inequality:

$$N(t)\le A+{\int }_{t_0}^t b(s)F(N(s-\tau ))ds,$$

(71)

where $F(x)=x$ (a linear function) or another nondecreasing function depending on the analysis. Theorem 2 then directly applies, assuring that the bound for the delayed equation is no worse than that for the non-delayed one. This provides a powerful and flexible tool for modelers to conclude the boundedness of solutions in more complex, nonlinear delayed systems without resorting to a full stability analysis.

This example illustrates that our generalized inequalities, particularly those handling delayed arguments, are not merely abstract extensions but provide ready-to-use tools for deriving qualitative properties of solutions to realistic process models found in biology and other sciences.

5. Conclusions

In this paper, we have established several new generalizations of the Gronwall–Bellman–Bihari-type integral inequalities. Our main results (Theorems 1–3) extend the classical inequalities to settings with separable kernels, delay arguments, and combined linear and nonlinear terms. These results provide explicit bounds (when possible) or guarantee the existence of a maximal solution that serves as an optimal bound. We have also illustrated the application of these inequalities to prove uniqueness for a differential equation with a non-Lipschitz nonlinearity.

The new inequalities presented here are expected to be useful in the analysis of various differential, integral, and functional equations, particularly those arising in nonlinear dynamics, fractional calculus, and delay systems. Future work may focus on further generalizations, such as inequalities on time scales, stochastic versions, or inequalities involving more general operators (e.g., fractional integrals with variable order). Another direction is to apply these inequalities to concrete problems in applied mathematics, such as stability analysis of neural networks with delays or blow-up prevention in reaction–diffusion equations.