Existence of solutions for nonlinear mixed type integro-differential functional evolution equations with nonlocal conditions

Abstract

Using the Mönch fixed point theorem, this article proves the existence of mild solutions for nonlinear mixed type integro-differential functional evolution equations with nonlocal conditions in Banach spaces. Some restricted conditions on a priori estimation and measure of noncompactness estimation have been deleted, and compactness conditions of evolution operators or compactness conditions on a nonlinear term $f(t,X_r,X_r,X_r)$ have been weakened. Our results extend and improve many known results.

MSC:34G20, 34K30.

1 Introduction

Let $(X,\parallel \cdot \parallel )$ be a Banach space, $C[J,X]=\{x:J=[0,a]\to X,x(t)\text{ is continuous in }J\}$ with the norm ${\parallel x\parallel }_C={sup}_{t\in J}\parallel x(t)\parallel$. It is easy to verify that $C[J,X]$ is a Banach space. The space of X-valued Bochner integrable functions on J with the norm ${\parallel x\parallel }_1={\int }_0^a\parallel x(s)\parallel ds$ is denoted by $L[J,X]$. Consider the following nonlinear mixed type integro-differential functional evolution equations with nonlocal conditions in a Banach space $X(\mathit{IVP})$,

(1.1)

(1.2)

where

$$(K{x}_{{\sigma }_2})(t)={\int }_0^{t}k(t,s,x({\sigma }_2(s)))ds,(H{x}_{{\sigma }_3})(t)={\int }_0^{a}h(t,s,x({\sigma }_3(s)))ds,$$

(1.3)

A is the generator of a strongly continuous semigroup in the Banach space X, and $F(t)$ is a bounded operator for $t\in J$, $x_0\in X$, $f\in C[J\times X^3,X]$, $g:C[J,X]\to X$, $k\in C[\mathrm{△}\times X,X]$, $\mathrm{△}=\{(t,s)\in J\times J:s\le t\}$, $h\in C[J\times J\times X,X]$, ${\sigma }_i\in C[J,J]$ and ${\sigma }_i(t)\le t$ ($i=1,2,3$).

For the existence of mild solutions of integro-differential functional evolution equations in abstract spaces, there are many research results, see [1–16], and references therein. In order to obtain the existence and controllability of mild solutions in these study papers, usually, some restricted conditions on a priori estimation and compactness conditions of an evolution operator or compactness conditions on $f(t,X_r,X_r,X_r)$ are used.

Recently, using a fixed point theorem, Haribhau Laxman Tidkey and Machindra Baburao Dhakne [1] have studied the existence of mild solutions of IVP (1.1)-(1.2) when ${\sigma }_i(t)=t$ ($i=1,2,3$), the compactness of the resolvent operator and the restricted condition

$$M_1[\parallel x_0\parallel +G_1+Lrb+LKr{b}^2+L{K}_1{b}^2+LHr{b}^2+L{H}_1{b}^2+L_{1}b]\le r$$

with $M_1[Lb+LK{b}^2+LH{b}^2]<1$ is used. Malar [17] and Shi [18] studied the existence of mild solutions of semilinear mixed type integrodifferential evolution equations with the equicontinuous semigroup

$$\{\begin{array}{c}{x}^{\prime }(t)=Ax(t)+f(t,x(t),{\int }_0^{t}a(t,s)k(s,x(s))ds,\hfill \\ \phantom{x^{\prime }(t)=}{\int }_0^{a}b(t,s)h(s,x(s))ds),t\in [0,a],\hfill \\ x(0)=x_0+g(x).\hfill \end{array}$$

(1.4)

Solvability of the scalar equation

$$m(t)=K_1+K_2{\int }_0^{t}h(s,m(s),n(s),q(s))ds,t\in J$$

and the restricted condition on measure of noncompactness estimation

$${\int }_0^t[{\eta }_1(s)+k_1{\eta }_2(s)+k_2{\eta }_3(s)]ds\le K$$

are used in [17]. But estimations (3.15) and (3.21) in [18] seem to be incorrect, as they have no meaning.

In this paper, using the Mönch fixed point theorem, we investigate the existence of mild solutions of IVP (1.1)-(1.2). Some restricted conditions on a priori estimation and measure of noncompactness estimation have been deleted, and compactness conditions of a resolvent operator or compactness conditions on a nonlinear term $f(t,X_r,X_r,X_r)$ have been weakened. Our results extend and improve some corresponding results in papers [1–4, 6–21].

2 Preliminaries

We will make the following assumptions:

($H_1$) A generates a strongly continuous semigroup in the Banach space X.

($H_2$) $F(t)\in B(X)$, $0\le t\le a$. $F(t):Y\to Y$ and for $x(\cdot )$ continuous in Y, $AF(\cdot )x(\cdot )\in L^1[J,X]$. For $x\in X$, $F^{\prime }(t)x$ is continuous in $t\in J$, where $B(X)$ is the space of all linear and bounded operators on X, and Y is the Banach space formed from $D(A)$, the domain of A, endowed with the graph norm.

Definition 2.1 [5]

$R(t)$ is a resolvent operator of (1.1) with $f\equiv 0$ if $R(t)\in B(X)$ for $0\le t\le a$ and satisfies the following conditions:

• (1) $R(0)=I$, the identity operator on X,

• (2) for all $x\in X$, $R(t)x$ is continuous for $0\le t\le a$,

• (3) $R(t)\in B(Y)$, $0\le t\le a$; for $y\in Y$, $R(\cdot )y\in C^1[J,X]\cap C[J,Y]$ and

  $$\begin{array}{rcl}\frac{d}{dt}R(t)y& =& A[R(t)y+{\int }_0^{t}F(t-s)R(s)yds]\\ =& R(t)Ay+{\int }_0^{t}R(t-s)AF(s)yds,0\le t\le a.\end{array}$$

  (2.1)

The resolvent operator $R(t)$ is said to be equicontinuous if $\{t\to R(t)x:x\in B\}$ is equicontinuous for the entire bounded set $B\subset X$ and $t>0$. If $x\in C[J,X]$ satisfies the following integral equation:

$$x(t)=R(t)(x_0+g(x))+{\int }_0^{t}R(t-s)f(s,x({\sigma }_1(s)),(K{x}_{{\sigma }_2})(s),(H{x}_{{\sigma }_3})(s))ds,t\in J,$$

then x is said to be a mild solution IVP (1.1)-(1.2).

Lemma 2.2 [14]

Let the conditions ($H_1$), ($H_2$) be satisfied. Then (1.1) with $f\equiv 0$ has a unique resolvent operator.

The following lemma is obvious.

Lemma 2.3 Let the resolvent operator $R(t)$ be equicontinuous. If there is $\rho \in L[J,{\mathbb{R}}^{+}]$ such that $\parallel x(t)\parallel \le \rho (t)$ for a.e. $t\in J$, then the set $\{{\int }_0^{t}R(t-s)x(s)ds\}$ is equicontinuous.

Lemma 2.4 [22]

Let $V\in C[J,E]$ be an equicontinuous bounded subset. Then $\alpha (V(t))\in C[J,{\mathbb{R}}^{+}]$ (${\mathbb{R}}^{+}=[0,\mathrm{\infty })$), $\alpha (V)={max}_{t\in J}\alpha (V(t))$.

Lemma 2.5 [23]

Let $V=\{x_n\}\subset L[J,E]$ and there exists $\sigma \in L[J,{\mathbb{R}}^{+}]$ such that $\parallel x_n(t)\parallel \le \sigma (t)$ for any $x\in V$ and a.e. $t\in J$. Then $\alpha (V(t))\in L[J,{\mathbb{R}}^{+}]$ and

$$\alpha \left(\{{\int }_0^t{x}_n(s)ds:n\in \mathbb{N}\}\right)\le 2{\int }_0^t\alpha (V(s))ds,t\in J.$$

Lemma 2.6 [24] (Mönch)

Let E be a Banach space, Ω a closed convex subset in E and $y_0\in \mathrm{\Omega }$. Suppose that the continuous operator $F:\mathrm{\Omega }\to \mathrm{\Omega }$ has the following property:

$$V\subset \mathrm{\Omega }\mathit{\text{countable}},V\subset \overline{\mathit{co}}(\{y_0\}\cup F(V))\Rightarrow V\mathit{\text{is relatively compact}}.$$

Then F has a fixed point in Ω.

For $V\subset C[J,X]$, let $V(t)=\{x(t):x\in V\}$, $V_{{\sigma }_i}(t)=\{x({\sigma }_i(t)):x\in V\}$ ($i=1,2,3$), $(KV)(t)=\{(Kx)(t):x\in V\}$, $(HV)(t)=\{(Hx)(t):x\in V\}$ ($t\in J$), $X_r=\{x\in X:\parallel x\parallel \le r\}$ and $S_r=\{x\in C[J,X]:{\parallel x\parallel }_C\le r\}$ for any $r>0$. $\alpha (\cdot )$ and denote the Kuratowski measure of noncompactness in X and $C[J,X]$ respectively. For details on the properties of noncompact measure, we refer the reader to [22].

3 Existence of a mild solution

We make the following assumptions for convenience.

($H_3$) There exist constants $l_g>0$, $M>0$ and $4l_{g}M<1$ such that

$$\parallel g(x)-g(y)\parallel \le l_g{\parallel x-y\parallel }_C,x,y\in C[J,X],$$

and $g(0)=0$.

($H_3^{\prime }$) $g:C[J,X]\to E$ is continuous, compact and there exists a constant $N\ge 0$ such that $\parallel g(x)\parallel \le N$.

($H_4$) There exists $q\in C[J,{\mathbb{R}}^{+}]$ such that

$$\parallel f(t,x,y,z)\parallel \le q(t)(\parallel x\parallel +\parallel y\parallel +\parallel z\parallel ),t\in J,x,y,z\in X.$$

($H_5$) There exist $k_0\in C[\mathrm{△},{\mathbb{R}}^{+}]$, $h_0\in C[J\times J,{\mathbb{R}}^{+}]$ such that

$$\begin{array}{c}\parallel k(t,s,x)\parallel \le k_0(t,s)\parallel x\parallel ,(t,s)\in △,x\in X,\hfill \\ \parallel h(t,s,x)\parallel \le h_0(t,s)\parallel x\parallel ,t,s\in J,x\in X.\hfill \end{array}$$

($H_6$) For any $r>0$ and a bounded set $V_i\subset X_r$, there exist constants $l_i>0$ ($i=1,2,3$) such that

$$\alpha (f(t,V_1,V_2,V_3))\le l_1\alpha (V_1)+l_2\alpha (V_2)+l_3\alpha (V_3),t\in J.$$

($H_7$) For any $r>0$ and a bounded set $V\subset X_r$,

$$\begin{array}{c}\alpha (k(t,s,V))\le k_0(t,s)\alpha (V),(t,s)\in △,\hfill \\ \alpha (h(t,s,V))\le h_0(t,s)\alpha (V),t,s\in J.\hfill \end{array}$$

($H_8$) The resolvent operator $R(t)$ is equicontinuous and $\parallel R(t)\parallel \le M{e}^{-wt}$ for $t\in J$ and some positive number

$$w=max\{2M{q}_0(1+K_{0}a+H_{0}a),4M(l_1+2l_{2}a{K}_0+2l_{3}a{H}_0)\},$$

where $K_0={max}_{(t,s)\in \mathrm{△}}{k}_0(t,s)$, $H_0={max}_{t,s\in J}{h}_0(t,s)$, $q_0={max}_{t\in J}q(t)$.

Without loss of generality, we always suppose that $x_0=0$.

Theorem 3.1 Let conditions ($H_1$), ($H_2$), ($H_3$)-($H_8$) be satisfied. Then IVP (1.1)-(1.2) has at least one mild solution.

Proof Let

$$\begin{array}{rcl}(Fx)(t)& =& R(t)g(x)\\ +{\int }_0^{t}R(t-s)f(s,x({\sigma }_1(s)),(K{x}_{{\sigma }_2})(s),(H{x}_{{\sigma }_3})(s))ds,t\in J.\end{array}$$

(3.1)

We have by ($H_3$), ($H_4$) and ($H_5$),

(3.2)

Let

$$B_R=\{x\in C[J,X]:{\parallel x\parallel }_C\le R\}.$$

Then $B_R$ is a closed convex subset in $C[J,X]$, $0\in B_R$ and $F:B_R\to B_R$. Similar to the proof of [6] and [9], it is easy to verify that F is a continuous operator from $B_R$ into $B_R$. For $x\in B_R$, $s\in J$, ($H_4$) and ($H_5$) imply

$$\parallel f(s,x({\sigma }_1(s)),(K{x}_{{\sigma }_2})(s),(H{x}_{{\sigma }_3})(s))\parallel \le q(s)(1+{\int }_0^s{k}_0(s,r)dr+{\int }_0^a{h}_0(s,r)dr)R.$$

(3.3)

We can show from (3.3), ($H_8$) and Lemma 2.3 that $F(B_R)$ is an equicontinuous subset in $C[J,X]$.

Let $V\subset B_R$ be a countable set and $V\subset \overline{\mathit{co}}(\{0\}\cup F(V))$, then

$$V(t)\subset \overline{\mathit{co}}(\{0\}\cup (FV)(t)).$$

(3.4)

From equicontinuity of $F(B_R)$ and (3.4), we know that V is an equicontinuous subset in $C[J,X]$. By the properties of noncompact measure, the conditions ($H_3$), ($H_6$), ($H_7$), (3.4) and Lemma 2.5, we have

(3.5)

(3.5) together with Lemma 2.4 imply that , and so . Hence V is relatively compact in $C[J,X]$. Lemma 2.6 implies that F has a fixed point in $C[J,X]$. Then IVP (1.1)-(1.2) has at least one mild solution. The proof is completed. □

Theorem 3.2 Let the conditions ($H_1$), ($H_2$) and ($H_3^{\prime }$)-($H_8$) be satisfied. Then IVP (1.1)-(1.2) has at least one mild solution.

Proof Similar to (3.2) and (3.5), it is easy to verify

$$\parallel (Fx)(t)\parallel \le MN+M{q}_0(1+K_{0}a+H_{0}a)w^{-1}{\parallel x\parallel }_C=MN+\eta {\parallel x\parallel }_C,$$

where $\eta =M{q}_0(1+K_{0}a+H_{0}a)w^{-1}<1$. Taking $R>MN{(1-\eta )}^{-1}$, let $B_R=\{x\in C[J,X]:{\parallel x\parallel }_C\le R\}$. We have $F:B_R\to B_R$ and the inequality (3.5) is transformed into , $t\in J$.

The other proof is similar to the proof of Theorem 3.1, we omit it. □

4 An example

Let $X=L^2[0,\pi ]$. Consider the following partial functional integro-differential equation with a nonlocal condition,

$$\{\begin{array}{c}{u}_t(t,y)=u_y(t,y)+{\int }_0^{t}F(t-s)u_y(s,y)ds+{\gamma }_{1}sinu(t-r,y)\hfill \\ \phantom{u_t(t,y)=}+{\int }_0^t\frac{{\gamma }_{2}u(s-r,y)ds}{(1+t)}+{\int }_0^a\frac{{\gamma }_{3}u(s-r,y)ds}{(1+t){(1+s)}^2},0\le t\le a,\hfill \\ u(0,y)=u_0(y)+{\gamma }_{4}u(y),\hfill \end{array}$$

(4.1)

where $r,{\gamma }_i\in \mathbb{R}$ ($i=1,2,3,4$), ${\sigma }_1(t)={\sigma }_2(t)={\sigma }_3(t)=t-r$, $0\le r\le t\le a$, $F(t)$ satisfies the condition ($H_2$),

(4.2)

(4.3)

(4.4)

Let the operator A be defined by $Aw=w^{\prime }$, $w\in D(A)$ with the domain

$$D(A)=\{w\in E:w^{\prime }\in E,w^{\prime }\text{ is almost everywhere bounded}\}.$$

Then A generates a translation semigroup $R(t)$ and $R(t)$ is equicontinuous. The problem (4.1) can be regarded as a form of IVP (1.1)-(1.2). We have by (4.2), (4.3) and (4.4),

$$\begin{array}{c}\parallel f(t,u,v,z)\parallel \le |\gamma |(\parallel u\parallel +\parallel v\parallel +\parallel z\parallel ),|\gamma |=max\{|{\gamma }_1|,|{\gamma }_2|,|{\gamma }_3|\},u,v,z\in X,\hfill \\ \parallel k(t,s,u)\parallel \le \parallel u\parallel ,\parallel h(t,s,u)\parallel \le \parallel u\parallel ,u\in X,\hfill \end{array}$$

and

$$\parallel g(u)-g(v)\parallel \le |{\gamma }_4|{\parallel u-v\parallel }_C,g(0)=0.$$

${\gamma }_4$ and M can be chosen such that $4M|{\gamma }_4|<1$. In addition, for any $r>0$ and a bounded set $V_i\subset X_r$ ($i=1,2,3$), we can show that by the diagonal method,

$$\begin{array}{c}\alpha (f(t,V_1,V_2,V_3))\le |\gamma |(\alpha (V_1)+\alpha (V_2)+\alpha (V_3)),t\in J,\hfill \\ \alpha (k(t,s,V_1))\le \alpha (V_1),t,s\in \mathrm{△},\hfill \\ \alpha (h(t,s,V_1))\le \alpha (V_1),t,s\in [0,a].\hfill \end{array}$$

Hence all the conditions of Theorem 3.1 are satisfied, the problem (4.1) has at least one mild solution in $C[J,X]$.